Properties

Label 55.4.a.a.1.1
Level $55$
Weight $4$
Character 55.1
Self dual yes
Analytic conductor $3.245$
Analytic rank $1$
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] = [55,4,Mod(1,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 55.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: \(3.24510505032\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 55.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -9.00000 q^{7} -15.0000 q^{8} -18.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -9.00000 q^{7} -15.0000 q^{8} -18.0000 q^{9} -5.00000 q^{10} +11.0000 q^{11} +21.0000 q^{12} +2.00000 q^{13} -9.00000 q^{14} +15.0000 q^{15} +41.0000 q^{16} +21.0000 q^{17} -18.0000 q^{18} -85.0000 q^{19} +35.0000 q^{20} +27.0000 q^{21} +11.0000 q^{22} +22.0000 q^{23} +45.0000 q^{24} +25.0000 q^{25} +2.00000 q^{26} +135.000 q^{27} +63.0000 q^{28} -165.000 q^{29} +15.0000 q^{30} -83.0000 q^{31} +161.000 q^{32} -33.0000 q^{33} +21.0000 q^{34} +45.0000 q^{35} +126.000 q^{36} +1.00000 q^{37} -85.0000 q^{38} -6.00000 q^{39} +75.0000 q^{40} -478.000 q^{41} +27.0000 q^{42} -8.00000 q^{43} -77.0000 q^{44} +90.0000 q^{45} +22.0000 q^{46} +126.000 q^{47} -123.000 q^{48} -262.000 q^{49} +25.0000 q^{50} -63.0000 q^{51} -14.0000 q^{52} -683.000 q^{53} +135.000 q^{54} -55.0000 q^{55} +135.000 q^{56} +255.000 q^{57} -165.000 q^{58} -290.000 q^{59} -105.000 q^{60} +257.000 q^{61} -83.0000 q^{62} +162.000 q^{63} -167.000 q^{64} -10.0000 q^{65} -33.0000 q^{66} +776.000 q^{67} -147.000 q^{68} -66.0000 q^{69} +45.0000 q^{70} -313.000 q^{71} +270.000 q^{72} +902.000 q^{73} +1.00000 q^{74} -75.0000 q^{75} +595.000 q^{76} -99.0000 q^{77} -6.00000 q^{78} +830.000 q^{79} -205.000 q^{80} +81.0000 q^{81} -478.000 q^{82} +842.000 q^{83} -189.000 q^{84} -105.000 q^{85} -8.00000 q^{86} +495.000 q^{87} -165.000 q^{88} +25.0000 q^{89} +90.0000 q^{90} -18.0000 q^{91} -154.000 q^{92} +249.000 q^{93} +126.000 q^{94} +425.000 q^{95} -483.000 q^{96} -1784.00 q^{97} -262.000 q^{98} -198.000 q^{99} +O(q^{100})\)

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 0.353553 0.176777 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) −3.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −7.00000 −0.875000
\(5\) −5.00000 −0.447214
\(6\) −3.00000 −0.204124
\(7\) −9.00000 −0.485954 −0.242977 0.970032i \(-0.578124\pi\)
−0.242977 + 0.970032i \(0.578124\pi\)
\(8\) −15.0000 −0.662913
\(9\) −18.0000 −0.666667
\(10\) −5.00000 −0.158114
\(11\) 11.0000 0.301511
\(12\) 21.0000 0.505181
\(13\) 2.00000 0.0426692 0.0213346 0.999772i \(-0.493208\pi\)
0.0213346 + 0.999772i \(0.493208\pi\)
\(14\) −9.00000 −0.171811
\(15\) 15.0000 0.258199
\(16\) 41.0000 0.640625
\(17\) 21.0000 0.299603 0.149801 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) −18.0000 −0.235702
\(19\) −85.0000 −1.02633 −0.513167 0.858289i \(-0.671528\pi\)
−0.513167 + 0.858289i \(0.671528\pi\)
\(20\) 35.0000 0.391312
\(21\) 27.0000 0.280566
\(22\) 11.0000 0.106600
\(23\) 22.0000 0.199449 0.0997243 0.995015i \(-0.468204\pi\)
0.0997243 + 0.995015i \(0.468204\pi\)
\(24\) 45.0000 0.382733
\(25\) 25.0000 0.200000
\(26\) 2.00000 0.0150859
\(27\) 135.000 0.962250
\(28\) 63.0000 0.425210
\(29\) −165.000 −1.05654 −0.528271 0.849076i \(-0.677160\pi\)
−0.528271 + 0.849076i \(0.677160\pi\)
\(30\) 15.0000 0.0912871
\(31\) −83.0000 −0.480879 −0.240439 0.970664i \(-0.577292\pi\)
−0.240439 + 0.970664i \(0.577292\pi\)
\(32\) 161.000 0.889408
\(33\) −33.0000 −0.174078
\(34\) 21.0000 0.105926
\(35\) 45.0000 0.217325
\(36\) 126.000 0.583333
\(37\) 1.00000 0.00444322 0.00222161 0.999998i \(-0.499293\pi\)
0.00222161 + 0.999998i \(0.499293\pi\)
\(38\) −85.0000 −0.362864
\(39\) −6.00000 −0.0246351
\(40\) 75.0000 0.296464
\(41\) −478.000 −1.82076 −0.910379 0.413776i \(-0.864210\pi\)
−0.910379 + 0.413776i \(0.864210\pi\)
\(42\) 27.0000 0.0991950
\(43\) −8.00000 −0.0283718 −0.0141859 0.999899i \(-0.504516\pi\)
−0.0141859 + 0.999899i \(0.504516\pi\)
\(44\) −77.0000 −0.263822
\(45\) 90.0000 0.298142
\(46\) 22.0000 0.0705157
\(47\) 126.000 0.391042 0.195521 0.980699i \(-0.437360\pi\)
0.195521 + 0.980699i \(0.437360\pi\)
\(48\) −123.000 −0.369865
\(49\) −262.000 −0.763848
\(50\) 25.0000 0.0707107
\(51\) −63.0000 −0.172976
\(52\) −14.0000 −0.0373356
\(53\) −683.000 −1.77014 −0.885069 0.465461i \(-0.845889\pi\)
−0.885069 + 0.465461i \(0.845889\pi\)
\(54\) 135.000 0.340207
\(55\) −55.0000 −0.134840
\(56\) 135.000 0.322145
\(57\) 255.000 0.592554
\(58\) −165.000 −0.373544
\(59\) −290.000 −0.639912 −0.319956 0.947432i \(-0.603668\pi\)
−0.319956 + 0.947432i \(0.603668\pi\)
\(60\) −105.000 −0.225924
\(61\) 257.000 0.539434 0.269717 0.962940i \(-0.413070\pi\)
0.269717 + 0.962940i \(0.413070\pi\)
\(62\) −83.0000 −0.170016
\(63\) 162.000 0.323970
\(64\) −167.000 −0.326172
\(65\) −10.0000 −0.0190823
\(66\) −33.0000 −0.0615457
\(67\) 776.000 1.41498 0.707489 0.706725i \(-0.249828\pi\)
0.707489 + 0.706725i \(0.249828\pi\)
\(68\) −147.000 −0.262152
\(69\) −66.0000 −0.115152
\(70\) 45.0000 0.0768361
\(71\) −313.000 −0.523187 −0.261593 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) 270.000 0.441942
\(73\) 902.000 1.44618 0.723090 0.690754i \(-0.242721\pi\)
0.723090 + 0.690754i \(0.242721\pi\)
\(74\) 1.00000 0.00157091
\(75\) −75.0000 −0.115470
\(76\) 595.000 0.898042
\(77\) −99.0000 −0.146521
\(78\) −6.00000 −0.00870982
\(79\) 830.000 1.18205 0.591027 0.806652i \(-0.298723\pi\)
0.591027 + 0.806652i \(0.298723\pi\)
\(80\) −205.000 −0.286496
\(81\) 81.0000 0.111111
\(82\) −478.000 −0.643735
\(83\) 842.000 1.11351 0.556756 0.830676i \(-0.312046\pi\)
0.556756 + 0.830676i \(0.312046\pi\)
\(84\) −189.000 −0.245495
\(85\) −105.000 −0.133986
\(86\) −8.00000 −0.0100310
\(87\) 495.000 0.609995
\(88\) −165.000 −0.199876
\(89\) 25.0000 0.0297752 0.0148876 0.999889i \(-0.495261\pi\)
0.0148876 + 0.999889i \(0.495261\pi\)
\(90\) 90.0000 0.105409
\(91\) −18.0000 −0.0207353
\(92\) −154.000 −0.174517
\(93\) 249.000 0.277635
\(94\) 126.000 0.138254
\(95\) 425.000 0.458990
\(96\) −483.000 −0.513500
\(97\) −1784.00 −1.86740 −0.933700 0.358057i \(-0.883439\pi\)
−0.933700 + 0.358057i \(0.883439\pi\)
\(98\) −262.000 −0.270061
\(99\) −198.000 −0.201008
\(100\) −175.000 −0.175000
\(101\) −298.000 −0.293585 −0.146793 0.989167i \(-0.546895\pi\)
−0.146793 + 0.989167i \(0.546895\pi\)
\(102\) −63.0000 −0.0611562
\(103\) 1832.00 1.75255 0.876273 0.481814i \(-0.160022\pi\)
0.876273 + 0.481814i \(0.160022\pi\)
\(104\) −30.0000 −0.0282860
\(105\) −135.000 −0.125473
\(106\) −683.000 −0.625838
\(107\) −1404.00 −1.26850 −0.634251 0.773127i \(-0.718692\pi\)
−0.634251 + 0.773127i \(0.718692\pi\)
\(108\) −945.000 −0.841969
\(109\) 1050.00 0.922677 0.461338 0.887224i \(-0.347369\pi\)
0.461338 + 0.887224i \(0.347369\pi\)
\(110\) −55.0000 −0.0476731
\(111\) −3.00000 −0.00256529
\(112\) −369.000 −0.311314
\(113\) −1668.00 −1.38860 −0.694302 0.719684i \(-0.744287\pi\)
−0.694302 + 0.719684i \(0.744287\pi\)
\(114\) 255.000 0.209499
\(115\) −110.000 −0.0891961
\(116\) 1155.00 0.924475
\(117\) −36.0000 −0.0284462
\(118\) −290.000 −0.226243
\(119\) −189.000 −0.145593
\(120\) −225.000 −0.171163
\(121\) 121.000 0.0909091
\(122\) 257.000 0.190719
\(123\) 1434.00 1.05121
\(124\) 581.000 0.420769
\(125\) −125.000 −0.0894427
\(126\) 162.000 0.114541
\(127\) −2384.00 −1.66571 −0.832857 0.553488i \(-0.813297\pi\)
−0.832857 + 0.553488i \(0.813297\pi\)
\(128\) −1455.00 −1.00473
\(129\) 24.0000 0.0163805
\(130\) −10.0000 −0.00674660
\(131\) 1167.00 0.778330 0.389165 0.921168i \(-0.372764\pi\)
0.389165 + 0.921168i \(0.372764\pi\)
\(132\) 231.000 0.152318
\(133\) 765.000 0.498751
\(134\) 776.000 0.500270
\(135\) −675.000 −0.430331
\(136\) −315.000 −0.198610
\(137\) −1164.00 −0.725892 −0.362946 0.931810i \(-0.618229\pi\)
−0.362946 + 0.931810i \(0.618229\pi\)
\(138\) −66.0000 −0.0407123
\(139\) 1260.00 0.768862 0.384431 0.923154i \(-0.374398\pi\)
0.384431 + 0.923154i \(0.374398\pi\)
\(140\) −315.000 −0.190160
\(141\) −378.000 −0.225768
\(142\) −313.000 −0.184974
\(143\) 22.0000 0.0128653
\(144\) −738.000 −0.427083
\(145\) 825.000 0.472500
\(146\) 902.000 0.511302
\(147\) 786.000 0.441008
\(148\) −7.00000 −0.00388781
\(149\) −55.0000 −0.0302401 −0.0151201 0.999886i \(-0.504813\pi\)
−0.0151201 + 0.999886i \(0.504813\pi\)
\(150\) −75.0000 −0.0408248
\(151\) −598.000 −0.322282 −0.161141 0.986931i \(-0.551517\pi\)
−0.161141 + 0.986931i \(0.551517\pi\)
\(152\) 1275.00 0.680369
\(153\) −378.000 −0.199735
\(154\) −99.0000 −0.0518029
\(155\) 415.000 0.215055
\(156\) 42.0000 0.0215557
\(157\) 1321.00 0.671511 0.335756 0.941949i \(-0.391008\pi\)
0.335756 + 0.941949i \(0.391008\pi\)
\(158\) 830.000 0.417919
\(159\) 2049.00 1.02199
\(160\) −805.000 −0.397755
\(161\) −198.000 −0.0969229
\(162\) 81.0000 0.0392837
\(163\) 577.000 0.277265 0.138632 0.990344i \(-0.455729\pi\)
0.138632 + 0.990344i \(0.455729\pi\)
\(164\) 3346.00 1.59316
\(165\) 165.000 0.0778499
\(166\) 842.000 0.393686
\(167\) −1169.00 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) −405.000 −0.185991
\(169\) −2193.00 −0.998179
\(170\) −105.000 −0.0473714
\(171\) 1530.00 0.684222
\(172\) 56.0000 0.0248253
\(173\) 1542.00 0.677665 0.338833 0.940847i \(-0.389968\pi\)
0.338833 + 0.940847i \(0.389968\pi\)
\(174\) 495.000 0.215666
\(175\) −225.000 −0.0971909
\(176\) 451.000 0.193156
\(177\) 870.000 0.369453
\(178\) 25.0000 0.0105271
\(179\) −560.000 −0.233834 −0.116917 0.993142i \(-0.537301\pi\)
−0.116917 + 0.993142i \(0.537301\pi\)
\(180\) −630.000 −0.260875
\(181\) −3058.00 −1.25580 −0.627899 0.778295i \(-0.716085\pi\)
−0.627899 + 0.778295i \(0.716085\pi\)
\(182\) −18.0000 −0.00733104
\(183\) −771.000 −0.311442
\(184\) −330.000 −0.132217
\(185\) −5.00000 −0.00198707
\(186\) 249.000 0.0981590
\(187\) 231.000 0.0903337
\(188\) −882.000 −0.342162
\(189\) −1215.00 −0.467610
\(190\) 425.000 0.162278
\(191\) −1828.00 −0.692510 −0.346255 0.938140i \(-0.612547\pi\)
−0.346255 + 0.938140i \(0.612547\pi\)
\(192\) 501.000 0.188315
\(193\) 577.000 0.215199 0.107599 0.994194i \(-0.465684\pi\)
0.107599 + 0.994194i \(0.465684\pi\)
\(194\) −1784.00 −0.660225
\(195\) 30.0000 0.0110172
\(196\) 1834.00 0.668367
\(197\) −2164.00 −0.782633 −0.391316 0.920256i \(-0.627980\pi\)
−0.391316 + 0.920256i \(0.627980\pi\)
\(198\) −198.000 −0.0710669
\(199\) −4425.00 −1.57628 −0.788141 0.615495i \(-0.788956\pi\)
−0.788141 + 0.615495i \(0.788956\pi\)
\(200\) −375.000 −0.132583
\(201\) −2328.00 −0.816938
\(202\) −298.000 −0.103798
\(203\) 1485.00 0.513431
\(204\) 441.000 0.151354
\(205\) 2390.00 0.814268
\(206\) 1832.00 0.619619
\(207\) −396.000 −0.132966
\(208\) 82.0000 0.0273350
\(209\) −935.000 −0.309451
\(210\) −135.000 −0.0443614
\(211\) −1793.00 −0.585001 −0.292500 0.956265i \(-0.594487\pi\)
−0.292500 + 0.956265i \(0.594487\pi\)
\(212\) 4781.00 1.54887
\(213\) 939.000 0.302062
\(214\) −1404.00 −0.448483
\(215\) 40.0000 0.0126883
\(216\) −2025.00 −0.637888
\(217\) 747.000 0.233685
\(218\) 1050.00 0.326215
\(219\) −2706.00 −0.834952
\(220\) 385.000 0.117985
\(221\) 42.0000 0.0127838
\(222\) −3.00000 −0.000906968 0
\(223\) 1142.00 0.342933 0.171466 0.985190i \(-0.445150\pi\)
0.171466 + 0.985190i \(0.445150\pi\)
\(224\) −1449.00 −0.432212
\(225\) −450.000 −0.133333
\(226\) −1668.00 −0.490946
\(227\) 4906.00 1.43446 0.717231 0.696836i \(-0.245409\pi\)
0.717231 + 0.696836i \(0.245409\pi\)
\(228\) −1785.00 −0.518485
\(229\) 1130.00 0.326081 0.163040 0.986619i \(-0.447870\pi\)
0.163040 + 0.986619i \(0.447870\pi\)
\(230\) −110.000 −0.0315356
\(231\) 297.000 0.0845938
\(232\) 2475.00 0.700395
\(233\) −5403.00 −1.51915 −0.759576 0.650419i \(-0.774593\pi\)
−0.759576 + 0.650419i \(0.774593\pi\)
\(234\) −36.0000 −0.0100572
\(235\) −630.000 −0.174879
\(236\) 2030.00 0.559923
\(237\) −2490.00 −0.682459
\(238\) −189.000 −0.0514750
\(239\) 5090.00 1.37759 0.688797 0.724955i \(-0.258139\pi\)
0.688797 + 0.724955i \(0.258139\pi\)
\(240\) 615.000 0.165409
\(241\) −6578.00 −1.75820 −0.879100 0.476637i \(-0.841856\pi\)
−0.879100 + 0.476637i \(0.841856\pi\)
\(242\) 121.000 0.0321412
\(243\) −3888.00 −1.02640
\(244\) −1799.00 −0.472005
\(245\) 1310.00 0.341603
\(246\) 1434.00 0.371661
\(247\) −170.000 −0.0437929
\(248\) 1245.00 0.318781
\(249\) −2526.00 −0.642887
\(250\) −125.000 −0.0316228
\(251\) 3102.00 0.780066 0.390033 0.920801i \(-0.372464\pi\)
0.390033 + 0.920801i \(0.372464\pi\)
\(252\) −1134.00 −0.283473
\(253\) 242.000 0.0601360
\(254\) −2384.00 −0.588919
\(255\) 315.000 0.0773571
\(256\) −119.000 −0.0290527
\(257\) 4866.00 1.18106 0.590531 0.807015i \(-0.298918\pi\)
0.590531 + 0.807015i \(0.298918\pi\)
\(258\) 24.0000 0.00579137
\(259\) −9.00000 −0.00215920
\(260\) 70.0000 0.0166970
\(261\) 2970.00 0.704362
\(262\) 1167.00 0.275181
\(263\) −2163.00 −0.507134 −0.253567 0.967318i \(-0.581604\pi\)
−0.253567 + 0.967318i \(0.581604\pi\)
\(264\) 495.000 0.115398
\(265\) 3415.00 0.791629
\(266\) 765.000 0.176335
\(267\) −75.0000 −0.0171907
\(268\) −5432.00 −1.23811
\(269\) 7020.00 1.59114 0.795571 0.605861i \(-0.207171\pi\)
0.795571 + 0.605861i \(0.207171\pi\)
\(270\) −675.000 −0.152145
\(271\) 4812.00 1.07863 0.539314 0.842105i \(-0.318684\pi\)
0.539314 + 0.842105i \(0.318684\pi\)
\(272\) 861.000 0.191933
\(273\) 54.0000 0.0119715
\(274\) −1164.00 −0.256642
\(275\) 275.000 0.0603023
\(276\) 462.000 0.100758
\(277\) 5176.00 1.12273 0.561364 0.827569i \(-0.310277\pi\)
0.561364 + 0.827569i \(0.310277\pi\)
\(278\) 1260.00 0.271834
\(279\) 1494.00 0.320586
\(280\) −675.000 −0.144068
\(281\) 1242.00 0.263671 0.131835 0.991272i \(-0.457913\pi\)
0.131835 + 0.991272i \(0.457913\pi\)
\(282\) −378.000 −0.0798212
\(283\) 7402.00 1.55478 0.777391 0.629018i \(-0.216543\pi\)
0.777391 + 0.629018i \(0.216543\pi\)
\(284\) 2191.00 0.457788
\(285\) −1275.00 −0.264998
\(286\) 22.0000 0.00454856
\(287\) 4302.00 0.884805
\(288\) −2898.00 −0.592938
\(289\) −4472.00 −0.910238
\(290\) 825.000 0.167054
\(291\) 5352.00 1.07814
\(292\) −6314.00 −1.26541
\(293\) −3578.00 −0.713410 −0.356705 0.934217i \(-0.616100\pi\)
−0.356705 + 0.934217i \(0.616100\pi\)
\(294\) 786.000 0.155920
\(295\) 1450.00 0.286177
\(296\) −15.0000 −0.00294546
\(297\) 1485.00 0.290129
\(298\) −55.0000 −0.0106915
\(299\) 44.0000 0.00851032
\(300\) 525.000 0.101036
\(301\) 72.0000 0.0137874
\(302\) −598.000 −0.113944
\(303\) 894.000 0.169502
\(304\) −3485.00 −0.657495
\(305\) −1285.00 −0.241242
\(306\) −378.000 −0.0706171
\(307\) −9094.00 −1.69063 −0.845313 0.534272i \(-0.820586\pi\)
−0.845313 + 0.534272i \(0.820586\pi\)
\(308\) 693.000 0.128206
\(309\) −5496.00 −1.01183
\(310\) 415.000 0.0760336
\(311\) −7443.00 −1.35709 −0.678543 0.734561i \(-0.737388\pi\)
−0.678543 + 0.734561i \(0.737388\pi\)
\(312\) 90.0000 0.0163309
\(313\) 2822.00 0.509613 0.254807 0.966992i \(-0.417988\pi\)
0.254807 + 0.966992i \(0.417988\pi\)
\(314\) 1321.00 0.237415
\(315\) −810.000 −0.144884
\(316\) −5810.00 −1.03430
\(317\) −1869.00 −0.331147 −0.165573 0.986197i \(-0.552947\pi\)
−0.165573 + 0.986197i \(0.552947\pi\)
\(318\) 2049.00 0.361328
\(319\) −1815.00 −0.318560
\(320\) 835.000 0.145868
\(321\) 4212.00 0.732370
\(322\) −198.000 −0.0342674
\(323\) −1785.00 −0.307492
\(324\) −567.000 −0.0972222
\(325\) 50.0000 0.00853385
\(326\) 577.000 0.0980278
\(327\) −3150.00 −0.532708
\(328\) 7170.00 1.20700
\(329\) −1134.00 −0.190029
\(330\) 165.000 0.0275241
\(331\) −11408.0 −1.89438 −0.947191 0.320670i \(-0.896092\pi\)
−0.947191 + 0.320670i \(0.896092\pi\)
\(332\) −5894.00 −0.974323
\(333\) −18.0000 −0.00296214
\(334\) −1169.00 −0.191511
\(335\) −3880.00 −0.632797
\(336\) 1107.00 0.179738
\(337\) 10251.0 1.65700 0.828498 0.559992i \(-0.189196\pi\)
0.828498 + 0.559992i \(0.189196\pi\)
\(338\) −2193.00 −0.352910
\(339\) 5004.00 0.801711
\(340\) 735.000 0.117238
\(341\) −913.000 −0.144990
\(342\) 1530.00 0.241909
\(343\) 5445.00 0.857150
\(344\) 120.000 0.0188080
\(345\) 330.000 0.0514974
\(346\) 1542.00 0.239591
\(347\) −11494.0 −1.77819 −0.889093 0.457727i \(-0.848664\pi\)
−0.889093 + 0.457727i \(0.848664\pi\)
\(348\) −3465.00 −0.533746
\(349\) 5690.00 0.872718 0.436359 0.899773i \(-0.356268\pi\)
0.436359 + 0.899773i \(0.356268\pi\)
\(350\) −225.000 −0.0343622
\(351\) 270.000 0.0410585
\(352\) 1771.00 0.268167
\(353\) −4398.00 −0.663122 −0.331561 0.943434i \(-0.607575\pi\)
−0.331561 + 0.943434i \(0.607575\pi\)
\(354\) 870.000 0.130621
\(355\) 1565.00 0.233976
\(356\) −175.000 −0.0260533
\(357\) 567.000 0.0840583
\(358\) −560.000 −0.0826730
\(359\) −8840.00 −1.29960 −0.649801 0.760104i \(-0.725148\pi\)
−0.649801 + 0.760104i \(0.725148\pi\)
\(360\) −1350.00 −0.197642
\(361\) 366.000 0.0533605
\(362\) −3058.00 −0.443991
\(363\) −363.000 −0.0524864
\(364\) 126.000 0.0181434
\(365\) −4510.00 −0.646751
\(366\) −771.000 −0.110112
\(367\) −5564.00 −0.791385 −0.395693 0.918383i \(-0.629495\pi\)
−0.395693 + 0.918383i \(0.629495\pi\)
\(368\) 902.000 0.127772
\(369\) 8604.00 1.21384
\(370\) −5.00000 −0.000702534 0
\(371\) 6147.00 0.860206
\(372\) −1743.00 −0.242931
\(373\) 9222.00 1.28015 0.640076 0.768311i \(-0.278903\pi\)
0.640076 + 0.768311i \(0.278903\pi\)
\(374\) 231.000 0.0319378
\(375\) 375.000 0.0516398
\(376\) −1890.00 −0.259227
\(377\) −330.000 −0.0450819
\(378\) −1215.00 −0.165325
\(379\) 4670.00 0.632933 0.316467 0.948604i \(-0.397503\pi\)
0.316467 + 0.948604i \(0.397503\pi\)
\(380\) −2975.00 −0.401617
\(381\) 7152.00 0.961701
\(382\) −1828.00 −0.244839
\(383\) −378.000 −0.0504305 −0.0252153 0.999682i \(-0.508027\pi\)
−0.0252153 + 0.999682i \(0.508027\pi\)
\(384\) 4365.00 0.580079
\(385\) 495.000 0.0655261
\(386\) 577.000 0.0760843
\(387\) 144.000 0.0189146
\(388\) 12488.0 1.63397
\(389\) 10110.0 1.31773 0.658865 0.752261i \(-0.271037\pi\)
0.658865 + 0.752261i \(0.271037\pi\)
\(390\) 30.0000 0.00389515
\(391\) 462.000 0.0597554
\(392\) 3930.00 0.506365
\(393\) −3501.00 −0.449369
\(394\) −2164.00 −0.276702
\(395\) −4150.00 −0.528631
\(396\) 1386.00 0.175882
\(397\) 12186.0 1.54055 0.770274 0.637713i \(-0.220119\pi\)
0.770274 + 0.637713i \(0.220119\pi\)
\(398\) −4425.00 −0.557300
\(399\) −2295.00 −0.287954
\(400\) 1025.00 0.128125
\(401\) −4573.00 −0.569488 −0.284744 0.958604i \(-0.591909\pi\)
−0.284744 + 0.958604i \(0.591909\pi\)
\(402\) −2328.00 −0.288831
\(403\) −166.000 −0.0205187
\(404\) 2086.00 0.256887
\(405\) −405.000 −0.0496904
\(406\) 1485.00 0.181525
\(407\) 11.0000 0.00133968
\(408\) 945.000 0.114668
\(409\) 2280.00 0.275645 0.137822 0.990457i \(-0.455990\pi\)
0.137822 + 0.990457i \(0.455990\pi\)
\(410\) 2390.00 0.287887
\(411\) 3492.00 0.419094
\(412\) −12824.0 −1.53348
\(413\) 2610.00 0.310968
\(414\) −396.000 −0.0470105
\(415\) −4210.00 −0.497978
\(416\) 322.000 0.0379504
\(417\) −3780.00 −0.443903
\(418\) −935.000 −0.109408
\(419\) −3700.00 −0.431401 −0.215700 0.976460i \(-0.569203\pi\)
−0.215700 + 0.976460i \(0.569203\pi\)
\(420\) 945.000 0.109789
\(421\) 3612.00 0.418143 0.209071 0.977900i \(-0.432956\pi\)
0.209071 + 0.977900i \(0.432956\pi\)
\(422\) −1793.00 −0.206829
\(423\) −2268.00 −0.260695
\(424\) 10245.0 1.17345
\(425\) 525.000 0.0599206
\(426\) 939.000 0.106795
\(427\) −2313.00 −0.262140
\(428\) 9828.00 1.10994
\(429\) −66.0000 −0.00742776
\(430\) 40.0000 0.00448598
\(431\) 792.000 0.0885135 0.0442567 0.999020i \(-0.485908\pi\)
0.0442567 + 0.999020i \(0.485908\pi\)
\(432\) 5535.00 0.616442
\(433\) −4888.00 −0.542500 −0.271250 0.962509i \(-0.587437\pi\)
−0.271250 + 0.962509i \(0.587437\pi\)
\(434\) 747.000 0.0826202
\(435\) −2475.00 −0.272798
\(436\) −7350.00 −0.807342
\(437\) −1870.00 −0.204701
\(438\) −2706.00 −0.295200
\(439\) 15100.0 1.64165 0.820824 0.571181i \(-0.193515\pi\)
0.820824 + 0.571181i \(0.193515\pi\)
\(440\) 825.000 0.0893871
\(441\) 4716.00 0.509232
\(442\) 42.0000 0.00451977
\(443\) −11188.0 −1.19991 −0.599953 0.800036i \(-0.704814\pi\)
−0.599953 + 0.800036i \(0.704814\pi\)
\(444\) 21.0000 0.00224463
\(445\) −125.000 −0.0133159
\(446\) 1142.00 0.121245
\(447\) 165.000 0.0174591
\(448\) 1503.00 0.158505
\(449\) −12070.0 −1.26864 −0.634319 0.773071i \(-0.718719\pi\)
−0.634319 + 0.773071i \(0.718719\pi\)
\(450\) −450.000 −0.0471405
\(451\) −5258.00 −0.548979
\(452\) 11676.0 1.21503
\(453\) 1794.00 0.186069
\(454\) 4906.00 0.507159
\(455\) 90.0000 0.00927311
\(456\) −3825.00 −0.392811
\(457\) −12449.0 −1.27427 −0.637133 0.770754i \(-0.719880\pi\)
−0.637133 + 0.770754i \(0.719880\pi\)
\(458\) 1130.00 0.115287
\(459\) 2835.00 0.288293
\(460\) 770.000 0.0780466
\(461\) 2957.00 0.298745 0.149372 0.988781i \(-0.452275\pi\)
0.149372 + 0.988781i \(0.452275\pi\)
\(462\) 297.000 0.0299084
\(463\) −9738.00 −0.977458 −0.488729 0.872436i \(-0.662539\pi\)
−0.488729 + 0.872436i \(0.662539\pi\)
\(464\) −6765.00 −0.676848
\(465\) −1245.00 −0.124162
\(466\) −5403.00 −0.537101
\(467\) −13779.0 −1.36534 −0.682672 0.730725i \(-0.739182\pi\)
−0.682672 + 0.730725i \(0.739182\pi\)
\(468\) 252.000 0.0248904
\(469\) −6984.00 −0.687614
\(470\) −630.000 −0.0618292
\(471\) −3963.00 −0.387697
\(472\) 4350.00 0.424205
\(473\) −88.0000 −0.00855443
\(474\) −2490.00 −0.241286
\(475\) −2125.00 −0.205267
\(476\) 1323.00 0.127394
\(477\) 12294.0 1.18009
\(478\) 5090.00 0.487053
\(479\) 16320.0 1.55674 0.778371 0.627804i \(-0.216046\pi\)
0.778371 + 0.627804i \(0.216046\pi\)
\(480\) 2415.00 0.229644
\(481\) 2.00000 0.000189589 0
\(482\) −6578.00 −0.621618
\(483\) 594.000 0.0559585
\(484\) −847.000 −0.0795455
\(485\) 8920.00 0.835126
\(486\) −3888.00 −0.362887
\(487\) −2744.00 −0.255323 −0.127662 0.991818i \(-0.540747\pi\)
−0.127662 + 0.991818i \(0.540747\pi\)
\(488\) −3855.00 −0.357598
\(489\) −1731.00 −0.160079
\(490\) 1310.00 0.120775
\(491\) −853.000 −0.0784019 −0.0392010 0.999231i \(-0.512481\pi\)
−0.0392010 + 0.999231i \(0.512481\pi\)
\(492\) −10038.0 −0.919813
\(493\) −3465.00 −0.316543
\(494\) −170.000 −0.0154831
\(495\) 990.000 0.0898933
\(496\) −3403.00 −0.308063
\(497\) 2817.00 0.254245
\(498\) −2526.00 −0.227295
\(499\) −6840.00 −0.613628 −0.306814 0.951769i \(-0.599263\pi\)
−0.306814 + 0.951769i \(0.599263\pi\)
\(500\) 875.000 0.0782624
\(501\) 3507.00 0.312737
\(502\) 3102.00 0.275795
\(503\) −5128.00 −0.454565 −0.227283 0.973829i \(-0.572984\pi\)
−0.227283 + 0.973829i \(0.572984\pi\)
\(504\) −2430.00 −0.214763
\(505\) 1490.00 0.131295
\(506\) 242.000 0.0212613
\(507\) 6579.00 0.576299
\(508\) 16688.0 1.45750
\(509\) −18160.0 −1.58139 −0.790695 0.612210i \(-0.790281\pi\)
−0.790695 + 0.612210i \(0.790281\pi\)
\(510\) 315.000 0.0273499
\(511\) −8118.00 −0.702777
\(512\) 11521.0 0.994455
\(513\) −11475.0 −0.987590
\(514\) 4866.00 0.417568
\(515\) −9160.00 −0.783763
\(516\) −168.000 −0.0143329
\(517\) 1386.00 0.117904
\(518\) −9.00000 −0.000763392 0
\(519\) −4626.00 −0.391250
\(520\) 150.000 0.0126499
\(521\) 6462.00 0.543388 0.271694 0.962384i \(-0.412416\pi\)
0.271694 + 0.962384i \(0.412416\pi\)
\(522\) 2970.00 0.249029
\(523\) 5932.00 0.495962 0.247981 0.968765i \(-0.420233\pi\)
0.247981 + 0.968765i \(0.420233\pi\)
\(524\) −8169.00 −0.681039
\(525\) 675.000 0.0561132
\(526\) −2163.00 −0.179299
\(527\) −1743.00 −0.144073
\(528\) −1353.00 −0.111518
\(529\) −11683.0 −0.960220
\(530\) 3415.00 0.279883
\(531\) 5220.00 0.426608
\(532\) −5355.00 −0.436407
\(533\) −956.000 −0.0776904
\(534\) −75.0000 −0.00607784
\(535\) 7020.00 0.567292
\(536\) −11640.0 −0.938006
\(537\) 1680.00 0.135004
\(538\) 7020.00 0.562553
\(539\) −2882.00 −0.230309
\(540\) 4725.00 0.376540
\(541\) 12647.0 1.00506 0.502530 0.864560i \(-0.332403\pi\)
0.502530 + 0.864560i \(0.332403\pi\)
\(542\) 4812.00 0.381353
\(543\) 9174.00 0.725035
\(544\) 3381.00 0.266469
\(545\) −5250.00 −0.412634
\(546\) 54.0000 0.00423258
\(547\) −13524.0 −1.05712 −0.528560 0.848896i \(-0.677268\pi\)
−0.528560 + 0.848896i \(0.677268\pi\)
\(548\) 8148.00 0.635156
\(549\) −4626.00 −0.359623
\(550\) 275.000 0.0213201
\(551\) 14025.0 1.08436
\(552\) 990.000 0.0763355
\(553\) −7470.00 −0.574424
\(554\) 5176.00 0.396944
\(555\) 15.0000 0.00114723
\(556\) −8820.00 −0.672754
\(557\) 25066.0 1.90679 0.953394 0.301729i \(-0.0975639\pi\)
0.953394 + 0.301729i \(0.0975639\pi\)
\(558\) 1494.00 0.113344
\(559\) −16.0000 −0.00121060
\(560\) 1845.00 0.139224
\(561\) −693.000 −0.0521542
\(562\) 1242.00 0.0932217
\(563\) 12102.0 0.905930 0.452965 0.891528i \(-0.350366\pi\)
0.452965 + 0.891528i \(0.350366\pi\)
\(564\) 2646.00 0.197547
\(565\) 8340.00 0.621003
\(566\) 7402.00 0.549698
\(567\) −729.000 −0.0539949
\(568\) 4695.00 0.346827
\(569\) −20860.0 −1.53690 −0.768451 0.639909i \(-0.778972\pi\)
−0.768451 + 0.639909i \(0.778972\pi\)
\(570\) −1275.00 −0.0936910
\(571\) 20637.0 1.51249 0.756245 0.654289i \(-0.227032\pi\)
0.756245 + 0.654289i \(0.227032\pi\)
\(572\) −154.000 −0.0112571
\(573\) 5484.00 0.399821
\(574\) 4302.00 0.312826
\(575\) 550.000 0.0398897
\(576\) 3006.00 0.217448
\(577\) 3266.00 0.235642 0.117821 0.993035i \(-0.462409\pi\)
0.117821 + 0.993035i \(0.462409\pi\)
\(578\) −4472.00 −0.321818
\(579\) −1731.00 −0.124245
\(580\) −5775.00 −0.413438
\(581\) −7578.00 −0.541116
\(582\) 5352.00 0.381181
\(583\) −7513.00 −0.533716
\(584\) −13530.0 −0.958691
\(585\) 180.000 0.0127215
\(586\) −3578.00 −0.252228
\(587\) 3351.00 0.235623 0.117811 0.993036i \(-0.462412\pi\)
0.117811 + 0.993036i \(0.462412\pi\)
\(588\) −5502.00 −0.385882
\(589\) 7055.00 0.493542
\(590\) 1450.00 0.101179
\(591\) 6492.00 0.451853
\(592\) 41.0000 0.00284644
\(593\) −20258.0 −1.40286 −0.701430 0.712738i \(-0.747455\pi\)
−0.701430 + 0.712738i \(0.747455\pi\)
\(594\) 1485.00 0.102576
\(595\) 945.000 0.0651113
\(596\) 385.000 0.0264601
\(597\) 13275.0 0.910066
\(598\) 44.0000 0.00300885
\(599\) 20445.0 1.39459 0.697296 0.716784i \(-0.254387\pi\)
0.697296 + 0.716784i \(0.254387\pi\)
\(600\) 1125.00 0.0765466
\(601\) −1498.00 −0.101672 −0.0508359 0.998707i \(-0.516189\pi\)
−0.0508359 + 0.998707i \(0.516189\pi\)
\(602\) 72.0000 0.00487459
\(603\) −13968.0 −0.943318
\(604\) 4186.00 0.281997
\(605\) −605.000 −0.0406558
\(606\) 894.000 0.0599278
\(607\) 23461.0 1.56879 0.784393 0.620265i \(-0.212975\pi\)
0.784393 + 0.620265i \(0.212975\pi\)
\(608\) −13685.0 −0.912829
\(609\) −4455.00 −0.296430
\(610\) −1285.00 −0.0852920
\(611\) 252.000 0.0166855
\(612\) 2646.00 0.174768
\(613\) −1718.00 −0.113196 −0.0565982 0.998397i \(-0.518025\pi\)
−0.0565982 + 0.998397i \(0.518025\pi\)
\(614\) −9094.00 −0.597726
\(615\) −7170.00 −0.470118
\(616\) 1485.00 0.0971304
\(617\) 26276.0 1.71448 0.857238 0.514920i \(-0.172178\pi\)
0.857238 + 0.514920i \(0.172178\pi\)
\(618\) −5496.00 −0.357737
\(619\) −15560.0 −1.01035 −0.505177 0.863016i \(-0.668573\pi\)
−0.505177 + 0.863016i \(0.668573\pi\)
\(620\) −2905.00 −0.188174
\(621\) 2970.00 0.191919
\(622\) −7443.00 −0.479802
\(623\) −225.000 −0.0144694
\(624\) −246.000 −0.0157819
\(625\) 625.000 0.0400000
\(626\) 2822.00 0.180175
\(627\) 2805.00 0.178662
\(628\) −9247.00 −0.587572
\(629\) 21.0000 0.00133120
\(630\) −810.000 −0.0512241
\(631\) −28433.0 −1.79382 −0.896910 0.442214i \(-0.854193\pi\)
−0.896910 + 0.442214i \(0.854193\pi\)
\(632\) −12450.0 −0.783599
\(633\) 5379.00 0.337750
\(634\) −1869.00 −0.117078
\(635\) 11920.0 0.744930
\(636\) −14343.0 −0.894240
\(637\) −524.000 −0.0325928
\(638\) −1815.00 −0.112628
\(639\) 5634.00 0.348791
\(640\) 7275.00 0.449328
\(641\) 9847.00 0.606760 0.303380 0.952870i \(-0.401885\pi\)
0.303380 + 0.952870i \(0.401885\pi\)
\(642\) 4212.00 0.258932
\(643\) 2177.00 0.133519 0.0667593 0.997769i \(-0.478734\pi\)
0.0667593 + 0.997769i \(0.478734\pi\)
\(644\) 1386.00 0.0848075
\(645\) −120.000 −0.00732557
\(646\) −1785.00 −0.108715
\(647\) 4466.00 0.271370 0.135685 0.990752i \(-0.456676\pi\)
0.135685 + 0.990752i \(0.456676\pi\)
\(648\) −1215.00 −0.0736570
\(649\) −3190.00 −0.192941
\(650\) 50.0000 0.00301717
\(651\) −2241.00 −0.134918
\(652\) −4039.00 −0.242607
\(653\) 11507.0 0.689592 0.344796 0.938678i \(-0.387948\pi\)
0.344796 + 0.938678i \(0.387948\pi\)
\(654\) −3150.00 −0.188341
\(655\) −5835.00 −0.348080
\(656\) −19598.0 −1.16642
\(657\) −16236.0 −0.964120
\(658\) −1134.00 −0.0671853
\(659\) 12825.0 0.758105 0.379052 0.925375i \(-0.376250\pi\)
0.379052 + 0.925375i \(0.376250\pi\)
\(660\) −1155.00 −0.0681187
\(661\) −8818.00 −0.518881 −0.259441 0.965759i \(-0.583538\pi\)
−0.259441 + 0.965759i \(0.583538\pi\)
\(662\) −11408.0 −0.669765
\(663\) −126.000 −0.00738075
\(664\) −12630.0 −0.738161
\(665\) −3825.00 −0.223048
\(666\) −18.0000 −0.00104728
\(667\) −3630.00 −0.210726
\(668\) 8183.00 0.473967
\(669\) −3426.00 −0.197992
\(670\) −3880.00 −0.223728
\(671\) 2827.00 0.162645
\(672\) 4347.00 0.249537
\(673\) −9263.00 −0.530553 −0.265277 0.964172i \(-0.585463\pi\)
−0.265277 + 0.964172i \(0.585463\pi\)
\(674\) 10251.0 0.585836
\(675\) 3375.00 0.192450
\(676\) 15351.0 0.873407
\(677\) −1184.00 −0.0672154 −0.0336077 0.999435i \(-0.510700\pi\)
−0.0336077 + 0.999435i \(0.510700\pi\)
\(678\) 5004.00 0.283448
\(679\) 16056.0 0.907471
\(680\) 1575.00 0.0888213
\(681\) −14718.0 −0.828186
\(682\) −913.000 −0.0512618
\(683\) −1693.00 −0.0948475 −0.0474238 0.998875i \(-0.515101\pi\)
−0.0474238 + 0.998875i \(0.515101\pi\)
\(684\) −10710.0 −0.598695
\(685\) 5820.00 0.324629
\(686\) 5445.00 0.303048
\(687\) −3390.00 −0.188263
\(688\) −328.000 −0.0181757
\(689\) −1366.00 −0.0755304
\(690\) 330.000 0.0182071
\(691\) 13022.0 0.716903 0.358452 0.933548i \(-0.383305\pi\)
0.358452 + 0.933548i \(0.383305\pi\)
\(692\) −10794.0 −0.592957
\(693\) 1782.00 0.0976805
\(694\) −11494.0 −0.628683
\(695\) −6300.00 −0.343845
\(696\) −7425.00 −0.404373
\(697\) −10038.0 −0.545504
\(698\) 5690.00 0.308553
\(699\) 16209.0 0.877083
\(700\) 1575.00 0.0850420
\(701\) 1177.00 0.0634161 0.0317080 0.999497i \(-0.489905\pi\)
0.0317080 + 0.999497i \(0.489905\pi\)
\(702\) 270.000 0.0145164
\(703\) −85.0000 −0.00456022
\(704\) −1837.00 −0.0983445
\(705\) 1890.00 0.100967
\(706\) −4398.00 −0.234449
\(707\) 2682.00 0.142669
\(708\) −6090.00 −0.323271
\(709\) −24130.0 −1.27817 −0.639084 0.769137i \(-0.720686\pi\)
−0.639084 + 0.769137i \(0.720686\pi\)
\(710\) 1565.00 0.0827231
\(711\) −14940.0 −0.788036
\(712\) −375.000 −0.0197384
\(713\) −1826.00 −0.0959106
\(714\) 567.000 0.0297191
\(715\) −110.000 −0.00575352
\(716\) 3920.00 0.204605
\(717\) −15270.0 −0.795354
\(718\) −8840.00 −0.459479
\(719\) 13785.0 0.715012 0.357506 0.933911i \(-0.383627\pi\)
0.357506 + 0.933911i \(0.383627\pi\)
\(720\) 3690.00 0.190997
\(721\) −16488.0 −0.851658
\(722\) 366.000 0.0188658
\(723\) 19734.0 1.01510
\(724\) 21406.0 1.09882
\(725\) −4125.00 −0.211308
\(726\) −363.000 −0.0185567
\(727\) −17654.0 −0.900620 −0.450310 0.892872i \(-0.648686\pi\)
−0.450310 + 0.892872i \(0.648686\pi\)
\(728\) 270.000 0.0137457
\(729\) 9477.00 0.481481
\(730\) −4510.00 −0.228661
\(731\) −168.000 −0.00850028
\(732\) 5397.00 0.272512
\(733\) 14412.0 0.726220 0.363110 0.931746i \(-0.381715\pi\)
0.363110 + 0.931746i \(0.381715\pi\)
\(734\) −5564.00 −0.279797
\(735\) −3930.00 −0.197225
\(736\) 3542.00 0.177391
\(737\) 8536.00 0.426632
\(738\) 8604.00 0.429157
\(739\) 16480.0 0.820334 0.410167 0.912011i \(-0.365470\pi\)
0.410167 + 0.912011i \(0.365470\pi\)
\(740\) 35.0000 0.00173868
\(741\) 510.000 0.0252838
\(742\) 6147.00 0.304129
\(743\) 30127.0 1.48755 0.743777 0.668428i \(-0.233033\pi\)
0.743777 + 0.668428i \(0.233033\pi\)
\(744\) −3735.00 −0.184048
\(745\) 275.000 0.0135238
\(746\) 9222.00 0.452602
\(747\) −15156.0 −0.742341
\(748\) −1617.00 −0.0790419
\(749\) 12636.0 0.616434
\(750\) 375.000 0.0182574
\(751\) 36577.0 1.77725 0.888624 0.458636i \(-0.151662\pi\)
0.888624 + 0.458636i \(0.151662\pi\)
\(752\) 5166.00 0.250511
\(753\) −9306.00 −0.450371
\(754\) −330.000 −0.0159388
\(755\) 2990.00 0.144129
\(756\) 8505.00 0.409159
\(757\) 19386.0 0.930774 0.465387 0.885107i \(-0.345915\pi\)
0.465387 + 0.885107i \(0.345915\pi\)
\(758\) 4670.00 0.223776
\(759\) −726.000 −0.0347195
\(760\) −6375.00 −0.304270
\(761\) −18218.0 −0.867808 −0.433904 0.900959i \(-0.642864\pi\)
−0.433904 + 0.900959i \(0.642864\pi\)
\(762\) 7152.00 0.340013
\(763\) −9450.00 −0.448379
\(764\) 12796.0 0.605946
\(765\) 1890.00 0.0893243
\(766\) −378.000 −0.0178299
\(767\) −580.000 −0.0273045
\(768\) 357.000 0.0167736
\(769\) 9650.00 0.452520 0.226260 0.974067i \(-0.427350\pi\)
0.226260 + 0.974067i \(0.427350\pi\)
\(770\) 495.000 0.0231670
\(771\) −14598.0 −0.681886
\(772\) −4039.00 −0.188299
\(773\) 12617.0 0.587066 0.293533 0.955949i \(-0.405169\pi\)
0.293533 + 0.955949i \(0.405169\pi\)
\(774\) 144.000 0.00668730
\(775\) −2075.00 −0.0961757
\(776\) 26760.0 1.23792
\(777\) 27.0000 0.00124661
\(778\) 10110.0 0.465888
\(779\) 40630.0 1.86870
\(780\) −210.000 −0.00964001
\(781\) −3443.00 −0.157747
\(782\) 462.000 0.0211267
\(783\) −22275.0 −1.01666
\(784\) −10742.0 −0.489340
\(785\) −6605.00 −0.300309
\(786\) −3501.00 −0.158876
\(787\) −22364.0 −1.01295 −0.506474 0.862255i \(-0.669051\pi\)
−0.506474 + 0.862255i \(0.669051\pi\)
\(788\) 15148.0 0.684803
\(789\) 6489.00 0.292794
\(790\) −4150.00 −0.186899
\(791\) 15012.0 0.674798
\(792\) 2970.00 0.133250
\(793\) 514.000 0.0230172
\(794\) 12186.0 0.544666
\(795\) −10245.0 −0.457047
\(796\) 30975.0 1.37925
\(797\) −17594.0 −0.781947 −0.390973 0.920402i \(-0.627862\pi\)
−0.390973 + 0.920402i \(0.627862\pi\)
\(798\) −2295.00 −0.101807
\(799\) 2646.00 0.117157
\(800\) 4025.00 0.177882
\(801\) −450.000 −0.0198501
\(802\) −4573.00 −0.201344
\(803\) 9922.00 0.436040
\(804\) 16296.0 0.714820
\(805\) 990.000 0.0433452
\(806\) −166.000 −0.00725447
\(807\) −21060.0 −0.918646
\(808\) 4470.00 0.194621
\(809\) −45030.0 −1.95695 −0.978474 0.206371i \(-0.933834\pi\)
−0.978474 + 0.206371i \(0.933834\pi\)
\(810\) −405.000 −0.0175682
\(811\) −2943.00 −0.127426 −0.0637131 0.997968i \(-0.520294\pi\)
−0.0637131 + 0.997968i \(0.520294\pi\)
\(812\) −10395.0 −0.449252
\(813\) −14436.0 −0.622746
\(814\) 11.0000 0.000473648 0
\(815\) −2885.00 −0.123996
\(816\) −2583.00 −0.110813
\(817\) 680.000 0.0291190
\(818\) 2280.00 0.0974552
\(819\) 324.000 0.0138235
\(820\) −16730.0 −0.712484
\(821\) −4038.00 −0.171653 −0.0858265 0.996310i \(-0.527353\pi\)
−0.0858265 + 0.996310i \(0.527353\pi\)
\(822\) 3492.00 0.148172
\(823\) −5688.00 −0.240913 −0.120456 0.992719i \(-0.538436\pi\)
−0.120456 + 0.992719i \(0.538436\pi\)
\(824\) −27480.0 −1.16179
\(825\) −825.000 −0.0348155
\(826\) 2610.00 0.109944
\(827\) −16824.0 −0.707410 −0.353705 0.935357i \(-0.615078\pi\)
−0.353705 + 0.935357i \(0.615078\pi\)
\(828\) 2772.00 0.116345
\(829\) −22940.0 −0.961085 −0.480542 0.876972i \(-0.659560\pi\)
−0.480542 + 0.876972i \(0.659560\pi\)
\(830\) −4210.00 −0.176062
\(831\) −15528.0 −0.648207
\(832\) −334.000 −0.0139175
\(833\) −5502.00 −0.228851
\(834\) −3780.00 −0.156943
\(835\) 5845.00 0.242245
\(836\) 6545.00 0.270770
\(837\) −11205.0 −0.462726
\(838\) −3700.00 −0.152523
\(839\) 12040.0 0.495431 0.247716 0.968833i \(-0.420320\pi\)
0.247716 + 0.968833i \(0.420320\pi\)
\(840\) 2025.00 0.0831775
\(841\) 2836.00 0.116282
\(842\) 3612.00 0.147836
\(843\) −3726.00 −0.152230
\(844\) 12551.0 0.511876
\(845\) 10965.0 0.446399
\(846\) −2268.00 −0.0921696
\(847\) −1089.00 −0.0441777
\(848\) −28003.0 −1.13399
\(849\) −22206.0 −0.897654
\(850\) 525.000 0.0211851
\(851\) 22.0000 0.000886193 0
\(852\) −6573.00 −0.264304
\(853\) −14338.0 −0.575526 −0.287763 0.957702i \(-0.592912\pi\)
−0.287763 + 0.957702i \(0.592912\pi\)
\(854\) −2313.00 −0.0926806
\(855\) −7650.00 −0.305994
\(856\) 21060.0 0.840907
\(857\) −17619.0 −0.702280 −0.351140 0.936323i \(-0.614206\pi\)
−0.351140 + 0.936323i \(0.614206\pi\)
\(858\) −66.0000 −0.00262611
\(859\) −25550.0 −1.01485 −0.507424 0.861696i \(-0.669402\pi\)
−0.507424 + 0.861696i \(0.669402\pi\)
\(860\) −280.000 −0.0111022
\(861\) −12906.0 −0.510842
\(862\) 792.000 0.0312942
\(863\) 36922.0 1.45636 0.728180 0.685385i \(-0.240366\pi\)
0.728180 + 0.685385i \(0.240366\pi\)
\(864\) 21735.0 0.855833
\(865\) −7710.00 −0.303061
\(866\) −4888.00 −0.191803
\(867\) 13416.0 0.525526
\(868\) −5229.00 −0.204474
\(869\) 9130.00 0.356403
\(870\) −2475.00 −0.0964487
\(871\) 1552.00 0.0603760
\(872\) −15750.0 −0.611654
\(873\) 32112.0 1.24493
\(874\) −1870.00 −0.0723726
\(875\) 1125.00 0.0434651
\(876\) 18942.0 0.730583
\(877\) 11486.0 0.442252 0.221126 0.975245i \(-0.429027\pi\)
0.221126 + 0.975245i \(0.429027\pi\)
\(878\) 15100.0 0.580410
\(879\) 10734.0 0.411887
\(880\) −2255.00 −0.0863819
\(881\) −2958.00 −0.113119 −0.0565593 0.998399i \(-0.518013\pi\)
−0.0565593 + 0.998399i \(0.518013\pi\)
\(882\) 4716.00 0.180041
\(883\) −5173.00 −0.197152 −0.0985761 0.995130i \(-0.531429\pi\)
−0.0985761 + 0.995130i \(0.531429\pi\)
\(884\) −294.000 −0.0111858
\(885\) −4350.00 −0.165224
\(886\) −11188.0 −0.424230
\(887\) −33424.0 −1.26524 −0.632620 0.774462i \(-0.718021\pi\)
−0.632620 + 0.774462i \(0.718021\pi\)
\(888\) 45.0000 0.00170056
\(889\) 21456.0 0.809461
\(890\) −125.000 −0.00470788
\(891\) 891.000 0.0335013
\(892\) −7994.00 −0.300066
\(893\) −10710.0 −0.401340
\(894\) 165.000 0.00617274
\(895\) 2800.00 0.104574
\(896\) 13095.0 0.488251
\(897\) −132.000 −0.00491344
\(898\) −12070.0 −0.448531
\(899\) 13695.0 0.508069
\(900\) 3150.00 0.116667
\(901\) −14343.0 −0.530338
\(902\) −5258.00 −0.194093
\(903\) −216.000 −0.00796017
\(904\) 25020.0 0.920523
\(905\) 15290.0 0.561610
\(906\) 1794.00 0.0657855
\(907\) 27101.0 0.992143 0.496072 0.868282i \(-0.334775\pi\)
0.496072 + 0.868282i \(0.334775\pi\)
\(908\) −34342.0 −1.25515
\(909\) 5364.00 0.195723
\(910\) 90.0000 0.00327854
\(911\) −23893.0 −0.868947 −0.434473 0.900685i \(-0.643065\pi\)
−0.434473 + 0.900685i \(0.643065\pi\)
\(912\) 10455.0 0.379605
\(913\) 9262.00 0.335737
\(914\) −12449.0 −0.450521
\(915\) 3855.00 0.139281
\(916\) −7910.00 −0.285321
\(917\) −10503.0 −0.378233
\(918\) 2835.00 0.101927
\(919\) −34980.0 −1.25559 −0.627793 0.778380i \(-0.716042\pi\)
−0.627793 + 0.778380i \(0.716042\pi\)
\(920\) 1650.00 0.0591292
\(921\) 27282.0 0.976083
\(922\) 2957.00 0.105622
\(923\) −626.000 −0.0223240
\(924\) −2079.00 −0.0740196
\(925\) 25.0000 0.000888643 0
\(926\) −9738.00 −0.345584
\(927\) −32976.0 −1.16836
\(928\) −26565.0 −0.939697
\(929\) 21595.0 0.762658 0.381329 0.924439i \(-0.375467\pi\)
0.381329 + 0.924439i \(0.375467\pi\)
\(930\) −1245.00 −0.0438980
\(931\) 22270.0 0.783963
\(932\) 37821.0 1.32926
\(933\) 22329.0 0.783514
\(934\) −13779.0 −0.482722
\(935\) −1155.00 −0.0403984
\(936\) 540.000 0.0188573
\(937\) 31446.0 1.09637 0.548184 0.836358i \(-0.315320\pi\)
0.548184 + 0.836358i \(0.315320\pi\)
\(938\) −6984.00 −0.243108
\(939\) −8466.00 −0.294225
\(940\) 4410.00 0.153020
\(941\) −24353.0 −0.843661 −0.421831 0.906675i \(-0.638612\pi\)
−0.421831 + 0.906675i \(0.638612\pi\)
\(942\) −3963.00 −0.137072
\(943\) −10516.0 −0.363147
\(944\) −11890.0 −0.409943
\(945\) 6075.00 0.209121
\(946\) −88.0000 −0.00302445
\(947\) −22089.0 −0.757968 −0.378984 0.925403i \(-0.623727\pi\)
−0.378984 + 0.925403i \(0.623727\pi\)
\(948\) 17430.0 0.597152
\(949\) 1804.00 0.0617074
\(950\) −2125.00 −0.0725727
\(951\) 5607.00 0.191188
\(952\) 2835.00 0.0965156
\(953\) −37893.0 −1.28801 −0.644006 0.765021i \(-0.722729\pi\)
−0.644006 + 0.765021i \(0.722729\pi\)
\(954\) 12294.0 0.417225
\(955\) 9140.00 0.309700
\(956\) −35630.0 −1.20539
\(957\) 5445.00 0.183920
\(958\) 16320.0 0.550392
\(959\) 10476.0 0.352750
\(960\) −2505.00 −0.0842172
\(961\) −22902.0 −0.768756
\(962\) 2.00000 6.70297e−5 0
\(963\) 25272.0 0.845669
\(964\) 46046.0 1.53843
\(965\) −2885.00 −0.0962398
\(966\) 594.000 0.0197843
\(967\) 40601.0 1.35020 0.675098 0.737728i \(-0.264101\pi\)
0.675098 + 0.737728i \(0.264101\pi\)
\(968\) −1815.00 −0.0602648
\(969\) 5355.00 0.177531
\(970\) 8920.00 0.295262
\(971\) −1188.00 −0.0392634 −0.0196317 0.999807i \(-0.506249\pi\)
−0.0196317 + 0.999807i \(0.506249\pi\)
\(972\) 27216.0 0.898100
\(973\) −11340.0 −0.373632
\(974\) −2744.00 −0.0902705
\(975\) −150.000 −0.00492702
\(976\) 10537.0 0.345575
\(977\) −17024.0 −0.557468 −0.278734 0.960368i \(-0.589915\pi\)
−0.278734 + 0.960368i \(0.589915\pi\)
\(978\) −1731.00 −0.0565964
\(979\) 275.000 0.00897757
\(980\) −9170.00 −0.298903
\(981\) −18900.0 −0.615118
\(982\) −853.000 −0.0277193
\(983\) 43262.0 1.40371 0.701853 0.712322i \(-0.252356\pi\)
0.701853 + 0.712322i \(0.252356\pi\)
\(984\) −21510.0 −0.696864
\(985\) 10820.0 0.350004
\(986\) −3465.00 −0.111915
\(987\) 3402.00 0.109713
\(988\) 1190.00 0.0383188
\(989\) −176.000 −0.00565872
\(990\) 990.000 0.0317821
\(991\) −18328.0 −0.587496 −0.293748 0.955883i \(-0.594903\pi\)
−0.293748 + 0.955883i \(0.594903\pi\)
\(992\) −13363.0 −0.427697
\(993\) 34224.0 1.09372
\(994\) 2817.00 0.0898891
\(995\) 22125.0 0.704934
\(996\) 17682.0 0.562526
\(997\) 34196.0 1.08626 0.543128 0.839650i \(-0.317240\pi\)
0.543128 + 0.839650i \(0.317240\pi\)
\(998\) −6840.00 −0.216950
\(999\) 135.000 0.00427549
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 55.4.a.a.1.1 1
3.2 odd 2 495.4.a.a.1.1 1
4.3 odd 2 880.4.a.j.1.1 1
5.2 odd 4 275.4.b.a.199.2 2
5.3 odd 4 275.4.b.a.199.1 2
5.4 even 2 275.4.a.a.1.1 1
11.10 odd 2 605.4.a.b.1.1 1
15.14 odd 2 2475.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.a.a.1.1 1 1.1 even 1 trivial
275.4.a.a.1.1 1 5.4 even 2
275.4.b.a.199.1 2 5.3 odd 4
275.4.b.a.199.2 2 5.2 odd 4
495.4.a.a.1.1 1 3.2 odd 2
605.4.a.b.1.1 1 11.10 odd 2
880.4.a.j.1.1 1 4.3 odd 2
2475.4.a.h.1.1 1 15.14 odd 2