Properties

Label 550.2.a.a.1.1
Level $550$
Weight $2$
Character 550.1
Self dual yes
Analytic conductor $4.392$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,2,Mod(1,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} -7.00000 q^{19} +8.00000 q^{21} +1.00000 q^{22} +3.00000 q^{23} +2.00000 q^{24} -5.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} -10.0000 q^{39} +12.0000 q^{41} -8.00000 q^{42} +5.00000 q^{43} -1.00000 q^{44} -3.00000 q^{46} -2.00000 q^{48} +9.00000 q^{49} +5.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} +4.00000 q^{56} +14.0000 q^{57} -3.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +14.0000 q^{67} -6.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} +4.00000 q^{74} -7.00000 q^{76} +4.00000 q^{77} +10.0000 q^{78} -4.00000 q^{79} -11.0000 q^{81} -12.0000 q^{82} -15.0000 q^{83} +8.00000 q^{84} -5.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +3.00000 q^{89} -20.0000 q^{91} +3.00000 q^{92} -10.0000 q^{93} +2.00000 q^{96} -13.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 4.00000 0.769800
\(28\) −4.00000 −0.755929
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 7.00000 1.13555
\(39\) −10.0000 −1.60128
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −8.00000 −1.23443
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 14.0000 1.85435
\(58\) −3.00000 −0.393919
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −5.00000 −0.635001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 4.00000 0.455842
\(78\) 10.0000 1.13228
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −12.0000 −1.32518
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 8.00000 0.872872
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 3.00000 0.312772
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −9.00000 −0.909137
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 4.00000 0.384900
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −14.0000 −1.31122
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 5.00000 0.462250
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −24.0000 −2.16401
\(124\) 5.00000 0.449013
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 2.00000 0.174078
\(133\) 28.0000 2.42791
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 6.00000 0.510754
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.00000 −0.251754
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) −18.0000 −1.48461
\(148\) −4.00000 −0.328798
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 7.00000 0.567775
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −10.0000 −0.800641
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 4.00000 0.318223
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 11.0000 0.864242
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 15.0000 1.16423
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −8.00000 −0.617213
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 5.00000 0.381246
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −24.0000 −1.80395
\(178\) −3.00000 −0.224860
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 20.0000 1.48250
\(183\) 20.0000 1.47844
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) −2.00000 −0.144338
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 1.00000 0.0710669
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) −28.0000 −1.97497
\(202\) 9.00000 0.633238
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) 3.00000 0.208514
\(208\) 5.00000 0.346688
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −6.00000 −0.411113
\(214\) −15.0000 −1.02538
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −20.0000 −1.35769
\(218\) −5.00000 −0.338643
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 14.0000 0.927173
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) −3.00000 −0.196960
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −5.00000 −0.326860
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 10.0000 0.641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 24.0000 1.53018
\(247\) −35.0000 −2.22700
\(248\) −5.00000 −0.317500
\(249\) 30.0000 1.90117
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) −4.00000 −0.251976
\(253\) −3.00000 −0.188608
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 10.0000 0.622573
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −21.0000 −1.29738
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −28.0000 −1.71679
\(267\) −6.00000 −0.367194
\(268\) 14.0000 0.855186
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 40.0000 2.42091
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 13.0000 0.779688
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) −48.0000 −2.83335
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 26.0000 1.52415
\(292\) 8.00000 0.468165
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 18.0000 1.04978
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) 15.0000 0.867472
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 16.0000 0.920697
\(303\) 18.0000 1.03407
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 4.00000 0.227921
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 10.0000 0.566139
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 12.0000 0.672927
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) −30.0000 −1.67444
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −10.0000 −0.553001
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −15.0000 −0.823232
\(333\) −4.00000 −0.219199
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 8.00000 0.436436
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −12.0000 −0.652714
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 7.00000 0.378517
\(343\) −8.00000 −0.431959
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) 1.00000 0.0533002
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) 3.00000 0.159000
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −20.0000 −1.05118
\(363\) −2.00000 −0.104973
\(364\) −20.0000 −1.04828
\(365\) 0 0
\(366\) −20.0000 −1.04542
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 3.00000 0.156386
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) −10.0000 −0.518476
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 16.0000 0.822951
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) −15.0000 −0.767467
\(383\) −3.00000 −0.153293 −0.0766464 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 5.00000 0.254164
\(388\) −13.0000 −0.659975
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) −42.0000 −2.11862
\(394\) −3.00000 −0.151138
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −11.0000 −0.551380
\(399\) −56.0000 −2.80351
\(400\) 0 0
\(401\) −9.00000 −0.449439 −0.224719 0.974424i \(-0.572147\pi\)
−0.224719 + 0.974424i \(0.572147\pi\)
\(402\) 28.0000 1.39651
\(403\) 25.0000 1.24534
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −7.00000 −0.344865
\(413\) −48.0000 −2.36193
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 26.0000 1.27323
\(418\) −7.00000 −0.342381
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 40.0000 1.93574
\(428\) 15.0000 0.725052
\(429\) 10.0000 0.482805
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000 0.192450
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) −21.0000 −1.00457
\(438\) 16.0000 0.764510
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 12.0000 0.567581
\(448\) −4.00000 −0.188982
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −6.00000 −0.282216
\(453\) 32.0000 1.50349
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) −14.0000 −0.655610
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 8.00000 0.372194
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 5.00000 0.231125
\(469\) −56.0000 −2.58584
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) −12.0000 −0.552345
\(473\) −5.00000 −0.229900
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 6.00000 0.274434
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 10.0000 0.455488
\(483\) 24.0000 1.09204
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 29.0000 1.31412 0.657058 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 10.0000 0.452679
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) −24.0000 −1.08200
\(493\) 0 0
\(494\) 35.0000 1.57472
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) −12.0000 −0.538274
\(498\) −30.0000 −1.34433
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) −48.0000 −2.14448
\(502\) −24.0000 −1.07117
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 3.00000 0.133366
\(507\) −24.0000 −1.06588
\(508\) −16.0000 −0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) −1.00000 −0.0441942
\(513\) −28.0000 −1.23623
\(514\) 9.00000 0.396973
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) −42.0000 −1.84360
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) −3.00000 −0.131306
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 21.0000 0.917389
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 28.0000 1.21395
\(533\) 60.0000 2.59889
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) −24.0000 −1.03568
\(538\) −30.0000 −1.29339
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −40.0000 −1.71656
\(544\) 0 0
\(545\) 0 0
\(546\) −40.0000 −1.71184
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 3.00000 0.128154
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −21.0000 −0.894630
\(552\) 6.00000 0.255377
\(553\) 16.0000 0.680389
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) 27.0000 1.14403 0.572013 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) −5.00000 −0.211667
\(559\) 25.0000 1.05739
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 44.0000 1.84783
\(568\) −3.00000 −0.125877
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) −5.00000 −0.209061
\(573\) −30.0000 −1.25327
\(574\) 48.0000 2.00348
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 60.0000 2.48922
\(582\) −26.0000 −1.07773
\(583\) −6.00000 −0.248495
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −18.0000 −0.742307
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −22.0000 −0.900400
\(598\) −15.0000 −0.613396
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 20.0000 0.815139
\(603\) 14.0000 0.570124
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 7.00000 0.283887
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) −14.0000 −0.563163
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 12.0000 0.481543
\(622\) −3.00000 −0.120289
\(623\) −12.0000 −0.480770
\(624\) −10.0000 −0.400320
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) −14.0000 −0.559106
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 4.00000 0.159111
\(633\) 8.00000 0.317971
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 45.0000 1.78296
\(638\) 3.00000 0.118771
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 30.0000 1.18401
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 11.0000 0.432121
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 40.0000 1.56772
\(652\) −16.0000 −0.626608
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 3.00000 0.116863 0.0584317 0.998291i \(-0.481390\pi\)
0.0584317 + 0.998291i \(0.481390\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 15.0000 0.582113
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 9.00000 0.348481
\(668\) 24.0000 0.928588
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) −8.00000 −0.308607
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) −12.0000 −0.460857
\(679\) 52.0000 1.99558
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 5.00000 0.191460
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −7.00000 −0.267652
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −4.00000 −0.152610
\(688\) 5.00000 0.190623
\(689\) 30.0000 1.14291
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 21.0000 0.798300
\(693\) 4.00000 0.151947
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −11.0000 −0.416356
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) −20.0000 −0.754851
\(703\) 28.0000 1.05604
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) 36.0000 1.35392
\(708\) −24.0000 −0.901975
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) −3.00000 −0.112430
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 12.0000 0.448148
\(718\) 24.0000 0.895672
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) −30.0000 −1.11648
\(723\) 20.0000 0.743808
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 5.00000 0.185440 0.0927199 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 20.0000 0.741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 20.0000 0.739221
\(733\) 5.00000 0.184679 0.0923396 0.995728i \(-0.470565\pi\)
0.0923396 + 0.995728i \(0.470565\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −14.0000 −0.515697
\(738\) −12.0000 −0.441726
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 70.0000 2.57151
\(742\) 24.0000 0.881068
\(743\) 54.0000 1.98107 0.990534 0.137268i \(-0.0438322\pi\)
0.990534 + 0.137268i \(0.0438322\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) −15.0000 −0.548821
\(748\) 0 0
\(749\) −60.0000 −2.19235
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) −48.0000 −1.74922
\(754\) −15.0000 −0.546268
\(755\) 0 0
\(756\) −16.0000 −0.581914
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −32.0000 −1.15924
\(763\) −20.0000 −0.724049
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 3.00000 0.108394
\(767\) 60.0000 2.16647
\(768\) −2.00000 −0.0721688
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −4.00000 −0.143963
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −5.00000 −0.179721
\(775\) 0 0
\(776\) 13.0000 0.466673
\(777\) −32.0000 −1.14799
\(778\) 24.0000 0.860442
\(779\) −84.0000 −3.00961
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) 12.0000 0.428845
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 42.0000 1.49809
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 3.00000 0.106871
\(789\) 0 0
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 1.00000 0.0355335
\(793\) −50.0000 −1.77555
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 56.0000 1.98238
\(799\) 0 0
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) 9.00000 0.317801
\(803\) −8.00000 −0.282314
\(804\) −28.0000 −0.987484
\(805\) 0 0
\(806\) −25.0000 −0.880587
\(807\) −60.0000 −2.11210
\(808\) 9.00000 0.316619
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −12.0000 −0.421117
\(813\) −4.00000 −0.140286
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) −35.0000 −1.22449
\(818\) 4.00000 0.139857
\(819\) −20.0000 −0.698857
\(820\) 0 0
\(821\) −9.00000 −0.314102 −0.157051 0.987590i \(-0.550199\pi\)
−0.157051 + 0.987590i \(0.550199\pi\)
\(822\) 6.00000 0.209274
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) −27.0000 −0.938882 −0.469441 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) 3.00000 0.104257
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) −26.0000 −0.900306
\(835\) 0 0
\(836\) 7.00000 0.242100
\(837\) 20.0000 0.691301
\(838\) −30.0000 −1.03633
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) 6.00000 0.206041
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) −6.00000 −0.205557
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) −15.0000 −0.512689
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) −10.0000 −0.341394
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 0 0
\(861\) 96.0000 3.27167
\(862\) −12.0000 −0.408722
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −11.0000 −0.373795
\(867\) 34.0000 1.15470
\(868\) −20.0000 −0.678844
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 70.0000 2.37186
\(872\) −5.00000 −0.169321
\(873\) −13.0000 −0.439983
\(874\) 21.0000 0.710336
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) −20.0000 −0.674967
\(879\) 36.0000 1.21425
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) −9.00000 −0.303046
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) −8.00000 −0.268462
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) −30.0000 −1.00167
\(898\) 27.0000 0.901002
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 0 0
\(902\) 12.0000 0.399556
\(903\) 40.0000 1.33112
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) −3.00000 −0.0995585
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 14.0000 0.463586
\(913\) 15.0000 0.496428
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −84.0000 −2.77392
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 6.00000 0.197599
\(923\) 15.0000 0.493731
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 13.0000 0.427207
\(927\) −7.00000 −0.229910
\(928\) −3.00000 −0.0984798
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) −63.0000 −2.06474
\(932\) −18.0000 −0.589610
\(933\) −6.00000 −0.196431
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) −5.00000 −0.163430
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 56.0000 1.82846
\(939\) −52.0000 −1.69696
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 28.0000 0.912289
\(943\) 36.0000 1.17232
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 5.00000 0.162564
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 8.00000 0.259828
\(949\) 40.0000 1.29845
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 6.00000 0.193952
\(958\) −6.00000 −0.193851
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 20.0000 0.644826
\(963\) 15.0000 0.483368
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 10.0000 0.320750
\(973\) 52.0000 1.66704
\(974\) −29.0000 −0.929220
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −32.0000 −1.02325
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) −9.00000 −0.287202
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 24.0000 0.765092
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −35.0000 −1.11350
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −5.00000 −0.158750
\(993\) −64.0000 −2.03098
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 30.0000 0.950586
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 10.0000 0.316544
\(999\) −16.0000 −0.506218
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 550.2.a.a.1.1 1
3.2 odd 2 4950.2.a.y.1.1 1
4.3 odd 2 4400.2.a.bc.1.1 1
5.2 odd 4 550.2.b.d.199.1 2
5.3 odd 4 550.2.b.d.199.2 2
5.4 even 2 550.2.a.m.1.1 yes 1
11.10 odd 2 6050.2.a.bb.1.1 1
15.2 even 4 4950.2.c.ba.199.2 2
15.8 even 4 4950.2.c.ba.199.1 2
15.14 odd 2 4950.2.a.u.1.1 1
20.3 even 4 4400.2.b.e.4049.2 2
20.7 even 4 4400.2.b.e.4049.1 2
20.19 odd 2 4400.2.a.d.1.1 1
55.54 odd 2 6050.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
550.2.a.a.1.1 1 1.1 even 1 trivial
550.2.a.m.1.1 yes 1 5.4 even 2
550.2.b.d.199.1 2 5.2 odd 4
550.2.b.d.199.2 2 5.3 odd 4
4400.2.a.d.1.1 1 20.19 odd 2
4400.2.a.bc.1.1 1 4.3 odd 2
4400.2.b.e.4049.1 2 20.7 even 4
4400.2.b.e.4049.2 2 20.3 even 4
4950.2.a.u.1.1 1 15.14 odd 2
4950.2.a.y.1.1 1 3.2 odd 2
4950.2.c.ba.199.1 2 15.8 even 4
4950.2.c.ba.199.2 2 15.2 even 4
6050.2.a.n.1.1 1 55.54 odd 2
6050.2.a.bb.1.1 1 11.10 odd 2