Properties

Label 4002.2.a.j.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
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 \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -5.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +2.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -5.00000 q^{41} +4.00000 q^{42} +10.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} +2.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -4.00000 q^{56} -2.00000 q^{57} -1.00000 q^{58} +3.00000 q^{59} -1.00000 q^{60} +5.00000 q^{61} -5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{66} +5.00000 q^{67} -2.00000 q^{68} -1.00000 q^{69} -4.00000 q^{70} -9.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -7.00000 q^{74} +4.00000 q^{75} +2.00000 q^{76} -4.00000 q^{77} -1.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +6.00000 q^{83} +4.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} +1.00000 q^{87} +1.00000 q^{88} +1.00000 q^{90} -4.00000 q^{91} +1.00000 q^{92} +5.00000 q^{93} +2.00000 q^{95} -1.00000 q^{96} -18.0000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.00000 −0.408248
\(7\) −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
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.00000 0.872872
\(22\) 1.00000 0.213201
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −1.00000 −0.185695
\(30\) −1.00000 −0.182574
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 2.00000 0.324443
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 4.00000 0.617213
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −4.00000 −0.565685
\(51\) 2.00000 0.280056
\(52\) 1.00000 0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) −4.00000 −0.534522
\(57\) −2.00000 −0.264906
\(58\) −1.00000 −0.131306
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −5.00000 −0.635001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −1.00000 −0.123091
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −2.00000 −0.242536
\(69\) −1.00000 −0.120386
\(70\) −4.00000 −0.478091
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −7.00000 −0.813733
\(75\) 4.00000 0.461880
\(76\) 2.00000 0.229416
\(77\) −4.00000 −0.455842
\(78\) −1.00000 −0.113228
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 4.00000 0.436436
\(85\) −2.00000 −0.216930
\(86\) 10.0000 1.07833
\(87\) 1.00000 0.107211
\(88\) 1.00000 0.106600
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.00000 −0.419314
\(92\) 1.00000 0.104257
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) −1.00000 −0.102062
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 9.00000 0.909137
\(99\) 1.00000 0.100504
\(100\) −4.00000 −0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 2.00000 0.198030
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 1.00000 0.0980581
\(105\) 4.00000 0.390360
\(106\) −10.0000 −0.971286
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 1.00000 0.0953463
\(111\) 7.00000 0.664411
\(112\) −4.00000 −0.377964
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −2.00000 −0.187317
\(115\) 1.00000 0.0932505
\(116\) −1.00000 −0.0928477
\(117\) 1.00000 0.0924500
\(118\) 3.00000 0.276172
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 5.00000 0.452679
\(123\) 5.00000 0.450835
\(124\) −5.00000 −0.449013
\(125\) −9.00000 −0.804984
\(126\) −4.00000 −0.356348
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 1.00000 0.0877058
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −8.00000 −0.693688
\(134\) 5.00000 0.431934
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −9.00000 −0.755263
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) −1.00000 −0.0830455
\(146\) −10.0000 −0.827606
\(147\) −9.00000 −0.742307
\(148\) −7.00000 −0.575396
\(149\) 11.0000 0.901155 0.450578 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) 4.00000 0.326599
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 2.00000 0.162221
\(153\) −2.00000 −0.161690
\(154\) −4.00000 −0.322329
\(155\) −5.00000 −0.401610
\(156\) −1.00000 −0.0800641
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 4.00000 0.318223
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) −5.00000 −0.390434
\(165\) −1.00000 −0.0778499
\(166\) 6.00000 0.465690
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 4.00000 0.308607
\(169\) −12.0000 −0.923077
\(170\) −2.00000 −0.153393
\(171\) 2.00000 0.152944
\(172\) 10.0000 0.762493
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 1.00000 0.0758098
\(175\) 16.0000 1.20949
\(176\) 1.00000 0.0753778
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) −4.00000 −0.296500
\(183\) −5.00000 −0.369611
\(184\) 1.00000 0.0737210
\(185\) −7.00000 −0.514650
\(186\) 5.00000 0.366618
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 2.00000 0.145095
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −18.0000 −1.29232
\(195\) −1.00000 −0.0716115
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 13.0000 0.921546 0.460773 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) −4.00000 −0.282843
\(201\) −5.00000 −0.352673
\(202\) 3.00000 0.211079
\(203\) 4.00000 0.280745
\(204\) 2.00000 0.140028
\(205\) −5.00000 −0.349215
\(206\) −11.0000 −0.766406
\(207\) 1.00000 0.0695048
\(208\) 1.00000 0.0693375
\(209\) 2.00000 0.138343
\(210\) 4.00000 0.276026
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) −10.0000 −0.686803
\(213\) 9.00000 0.616670
\(214\) −18.0000 −1.23045
\(215\) 10.0000 0.681994
\(216\) −1.00000 −0.0680414
\(217\) 20.0000 1.35769
\(218\) −10.0000 −0.677285
\(219\) 10.0000 0.675737
\(220\) 1.00000 0.0674200
\(221\) −2.00000 −0.134535
\(222\) 7.00000 0.469809
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −4.00000 −0.267261
\(225\) −4.00000 −0.266667
\(226\) −14.0000 −0.931266
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −2.00000 −0.132453
\(229\) −23.0000 −1.51988 −0.759941 0.649992i \(-0.774772\pi\)
−0.759941 + 0.649992i \(0.774772\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) −1.00000 −0.0656532
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) −4.00000 −0.259828
\(238\) 8.00000 0.518563
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 5.00000 0.320092
\(245\) 9.00000 0.574989
\(246\) 5.00000 0.318788
\(247\) 2.00000 0.127257
\(248\) −5.00000 −0.317500
\(249\) −6.00000 −0.380235
\(250\) −9.00000 −0.569210
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) −4.00000 −0.251976
\(253\) 1.00000 0.0628695
\(254\) −1.00000 −0.0627456
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −10.0000 −0.622573
\(259\) 28.0000 1.73984
\(260\) 1.00000 0.0620174
\(261\) −1.00000 −0.0618984
\(262\) −10.0000 −0.617802
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −10.0000 −0.614295
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) 5.00000 0.305424
\(269\) 29.0000 1.76816 0.884081 0.467334i \(-0.154786\pi\)
0.884081 + 0.467334i \(0.154786\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −27.0000 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(272\) −2.00000 −0.121268
\(273\) 4.00000 0.242091
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) −1.00000 −0.0601929
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) −4.00000 −0.239046
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 21.0000 1.24832 0.624160 0.781296i \(-0.285441\pi\)
0.624160 + 0.781296i \(0.285441\pi\)
\(284\) −9.00000 −0.534052
\(285\) −2.00000 −0.118470
\(286\) 1.00000 0.0591312
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −1.00000 −0.0587220
\(291\) 18.0000 1.05518
\(292\) −10.0000 −0.585206
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) −9.00000 −0.524891
\(295\) 3.00000 0.174667
\(296\) −7.00000 −0.406867
\(297\) −1.00000 −0.0580259
\(298\) 11.0000 0.637213
\(299\) 1.00000 0.0578315
\(300\) 4.00000 0.230940
\(301\) −40.0000 −2.30556
\(302\) 16.0000 0.920697
\(303\) −3.00000 −0.172345
\(304\) 2.00000 0.114708
\(305\) 5.00000 0.286299
\(306\) −2.00000 −0.114332
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) −4.00000 −0.227921
\(309\) 11.0000 0.625768
\(310\) −5.00000 −0.283981
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 22.0000 1.24153
\(315\) −4.00000 −0.225374
\(316\) 4.00000 0.225018
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 10.0000 0.560772
\(319\) −1.00000 −0.0559893
\(320\) 1.00000 0.0559017
\(321\) 18.0000 1.00466
\(322\) −4.00000 −0.222911
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 1.00000 0.0553849
\(327\) 10.0000 0.553001
\(328\) −5.00000 −0.276079
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 6.00000 0.329293
\(333\) −7.00000 −0.383598
\(334\) −3.00000 −0.164153
\(335\) 5.00000 0.273179
\(336\) 4.00000 0.218218
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −12.0000 −0.652714
\(339\) 14.0000 0.760376
\(340\) −2.00000 −0.108465
\(341\) −5.00000 −0.270765
\(342\) 2.00000 0.108148
\(343\) −8.00000 −0.431959
\(344\) 10.0000 0.539164
\(345\) −1.00000 −0.0538382
\(346\) −4.00000 −0.215041
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 1.00000 0.0536056
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 16.0000 0.855236
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) −3.00000 −0.159448
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) −8.00000 −0.423405
\(358\) 4.00000 0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 6.00000 0.315353
\(363\) 10.0000 0.524864
\(364\) −4.00000 −0.209657
\(365\) −10.0000 −0.523424
\(366\) −5.00000 −0.261354
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 1.00000 0.0521286
\(369\) −5.00000 −0.260290
\(370\) −7.00000 −0.363913
\(371\) 40.0000 2.07670
\(372\) 5.00000 0.259238
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) −2.00000 −0.103418
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 4.00000 0.205738
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 2.00000 0.102598
\(381\) 1.00000 0.0512316
\(382\) 3.00000 0.153493
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) 14.0000 0.712581
\(387\) 10.0000 0.508329
\(388\) −18.0000 −0.913812
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −2.00000 −0.101144
\(392\) 9.00000 0.454569
\(393\) 10.0000 0.504433
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) 1.00000 0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 13.0000 0.651631
\(399\) 8.00000 0.400501
\(400\) −4.00000 −0.200000
\(401\) 29.0000 1.44819 0.724095 0.689700i \(-0.242257\pi\)
0.724095 + 0.689700i \(0.242257\pi\)
\(402\) −5.00000 −0.249377
\(403\) −5.00000 −0.249068
\(404\) 3.00000 0.149256
\(405\) 1.00000 0.0496904
\(406\) 4.00000 0.198517
\(407\) −7.00000 −0.346977
\(408\) 2.00000 0.0990148
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) −5.00000 −0.246932
\(411\) 18.0000 0.887875
\(412\) −11.0000 −0.541931
\(413\) −12.0000 −0.590481
\(414\) 1.00000 0.0491473
\(415\) 6.00000 0.294528
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 4.00000 0.195180
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) −9.00000 −0.438113
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 8.00000 0.388057
\(426\) 9.00000 0.436051
\(427\) −20.0000 −0.967868
\(428\) −18.0000 −0.870063
\(429\) −1.00000 −0.0482805
\(430\) 10.0000 0.482243
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 20.0000 0.960031
\(435\) 1.00000 0.0479463
\(436\) −10.0000 −0.478913
\(437\) 2.00000 0.0956730
\(438\) 10.0000 0.477818
\(439\) 18.0000 0.859093 0.429547 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 1.00000 0.0476731
\(441\) 9.00000 0.428571
\(442\) −2.00000 −0.0951303
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 7.00000 0.332205
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) −11.0000 −0.520282
\(448\) −4.00000 −0.188982
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) −4.00000 −0.188562
\(451\) −5.00000 −0.235441
\(452\) −14.0000 −0.658505
\(453\) −16.0000 −0.751746
\(454\) 18.0000 0.844782
\(455\) −4.00000 −0.187523
\(456\) −2.00000 −0.0936586
\(457\) −40.0000 −1.87112 −0.935561 0.353166i \(-0.885105\pi\)
−0.935561 + 0.353166i \(0.885105\pi\)
\(458\) −23.0000 −1.07472
\(459\) 2.00000 0.0933520
\(460\) 1.00000 0.0466252
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 4.00000 0.186097
\(463\) −30.0000 −1.39422 −0.697109 0.716965i \(-0.745531\pi\)
−0.697109 + 0.716965i \(0.745531\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 5.00000 0.231869
\(466\) −24.0000 −1.11178
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 1.00000 0.0462250
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 3.00000 0.138086
\(473\) 10.0000 0.459800
\(474\) −4.00000 −0.183726
\(475\) −8.00000 −0.367065
\(476\) 8.00000 0.366679
\(477\) −10.0000 −0.457869
\(478\) −5.00000 −0.228695
\(479\) −3.00000 −0.137073 −0.0685367 0.997649i \(-0.521833\pi\)
−0.0685367 + 0.997649i \(0.521833\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −7.00000 −0.319173
\(482\) −26.0000 −1.18427
\(483\) 4.00000 0.182006
\(484\) −10.0000 −0.454545
\(485\) −18.0000 −0.817338
\(486\) −1.00000 −0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 5.00000 0.226339
\(489\) −1.00000 −0.0452216
\(490\) 9.00000 0.406579
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 5.00000 0.225417
\(493\) 2.00000 0.0900755
\(494\) 2.00000 0.0899843
\(495\) 1.00000 0.0449467
\(496\) −5.00000 −0.224507
\(497\) 36.0000 1.61482
\(498\) −6.00000 −0.268866
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) −9.00000 −0.402492
\(501\) 3.00000 0.134030
\(502\) −3.00000 −0.133897
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −4.00000 −0.178174
\(505\) 3.00000 0.133498
\(506\) 1.00000 0.0444554
\(507\) 12.0000 0.532939
\(508\) −1.00000 −0.0443678
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 2.00000 0.0885615
\(511\) 40.0000 1.76950
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −12.0000 −0.529297
\(515\) −11.0000 −0.484718
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) 28.0000 1.23025
\(519\) 4.00000 0.175581
\(520\) 1.00000 0.0438529
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 9.00000 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) −10.0000 −0.436852
\(525\) −16.0000 −0.698297
\(526\) −12.0000 −0.523225
\(527\) 10.0000 0.435607
\(528\) −1.00000 −0.0435194
\(529\) 1.00000 0.0434783
\(530\) −10.0000 −0.434372
\(531\) 3.00000 0.130189
\(532\) −8.00000 −0.346844
\(533\) −5.00000 −0.216574
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 5.00000 0.215967
\(537\) −4.00000 −0.172613
\(538\) 29.0000 1.25028
\(539\) 9.00000 0.387657
\(540\) −1.00000 −0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −27.0000 −1.15975
\(543\) −6.00000 −0.257485
\(544\) −2.00000 −0.0857493
\(545\) −10.0000 −0.428353
\(546\) 4.00000 0.171184
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −18.0000 −0.768922
\(549\) 5.00000 0.213395
\(550\) −4.00000 −0.170561
\(551\) −2.00000 −0.0852029
\(552\) −1.00000 −0.0425628
\(553\) −16.0000 −0.680389
\(554\) −3.00000 −0.127458
\(555\) 7.00000 0.297133
\(556\) 0 0
\(557\) 35.0000 1.48300 0.741499 0.670954i \(-0.234115\pi\)
0.741499 + 0.670954i \(0.234115\pi\)
\(558\) −5.00000 −0.211667
\(559\) 10.0000 0.422955
\(560\) −4.00000 −0.169031
\(561\) 2.00000 0.0844401
\(562\) 26.0000 1.09674
\(563\) 7.00000 0.295015 0.147507 0.989061i \(-0.452875\pi\)
0.147507 + 0.989061i \(0.452875\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 21.0000 0.882696
\(567\) −4.00000 −0.167984
\(568\) −9.00000 −0.377632
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 39.0000 1.63210 0.816050 0.577982i \(-0.196160\pi\)
0.816050 + 0.577982i \(0.196160\pi\)
\(572\) 1.00000 0.0418121
\(573\) −3.00000 −0.125327
\(574\) 20.0000 0.834784
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) −1.00000 −0.0415227
\(581\) −24.0000 −0.995688
\(582\) 18.0000 0.746124
\(583\) −10.0000 −0.414158
\(584\) −10.0000 −0.413803
\(585\) 1.00000 0.0413449
\(586\) 4.00000 0.165238
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −9.00000 −0.371154
\(589\) −10.0000 −0.412043
\(590\) 3.00000 0.123508
\(591\) −6.00000 −0.246807
\(592\) −7.00000 −0.287698
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 8.00000 0.327968
\(596\) 11.0000 0.450578
\(597\) −13.0000 −0.532055
\(598\) 1.00000 0.0408930
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 4.00000 0.163299
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) −40.0000 −1.63028
\(603\) 5.00000 0.203616
\(604\) 16.0000 0.651031
\(605\) −10.0000 −0.406558
\(606\) −3.00000 −0.121867
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 2.00000 0.0811107
\(609\) −4.00000 −0.162088
\(610\) 5.00000 0.202444
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 7.00000 0.282497
\(615\) 5.00000 0.201619
\(616\) −4.00000 −0.161165
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 11.0000 0.442485
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −5.00000 −0.200805
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) −16.0000 −0.639489
\(627\) −2.00000 −0.0798723
\(628\) 22.0000 0.877896
\(629\) 14.0000 0.558217
\(630\) −4.00000 −0.159364
\(631\) −33.0000 −1.31371 −0.656855 0.754017i \(-0.728113\pi\)
−0.656855 + 0.754017i \(0.728113\pi\)
\(632\) 4.00000 0.159111
\(633\) 9.00000 0.357718
\(634\) 3.00000 0.119145
\(635\) −1.00000 −0.0396838
\(636\) 10.0000 0.396526
\(637\) 9.00000 0.356593
\(638\) −1.00000 −0.0395904
\(639\) −9.00000 −0.356034
\(640\) 1.00000 0.0395285
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 18.0000 0.710403
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −4.00000 −0.157622
\(645\) −10.0000 −0.393750
\(646\) −4.00000 −0.157378
\(647\) 15.0000 0.589711 0.294855 0.955542i \(-0.404729\pi\)
0.294855 + 0.955542i \(0.404729\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.00000 0.117760
\(650\) −4.00000 −0.156893
\(651\) −20.0000 −0.783862
\(652\) 1.00000 0.0391630
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) 10.0000 0.391031
\(655\) −10.0000 −0.390732
\(656\) −5.00000 −0.195217
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) −16.0000 −0.621858
\(663\) 2.00000 0.0776736
\(664\) 6.00000 0.232845
\(665\) −8.00000 −0.310227
\(666\) −7.00000 −0.271244
\(667\) −1.00000 −0.0387202
\(668\) −3.00000 −0.116073
\(669\) −2.00000 −0.0773245
\(670\) 5.00000 0.193167
\(671\) 5.00000 0.193023
\(672\) 4.00000 0.154303
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −9.00000 −0.346667
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) 16.0000 0.614930 0.307465 0.951559i \(-0.400519\pi\)
0.307465 + 0.951559i \(0.400519\pi\)
\(678\) 14.0000 0.537667
\(679\) 72.0000 2.76311
\(680\) −2.00000 −0.0766965
\(681\) −18.0000 −0.689761
\(682\) −5.00000 −0.191460
\(683\) −49.0000 −1.87493 −0.937466 0.348076i \(-0.886835\pi\)
−0.937466 + 0.348076i \(0.886835\pi\)
\(684\) 2.00000 0.0764719
\(685\) −18.0000 −0.687745
\(686\) −8.00000 −0.305441
\(687\) 23.0000 0.877505
\(688\) 10.0000 0.381246
\(689\) −10.0000 −0.380970
\(690\) −1.00000 −0.0380693
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) −4.00000 −0.152057
\(693\) −4.00000 −0.151947
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 10.0000 0.378777
\(698\) 15.0000 0.567758
\(699\) 24.0000 0.907763
\(700\) 16.0000 0.604743
\(701\) 51.0000 1.92624 0.963122 0.269066i \(-0.0867150\pi\)
0.963122 + 0.269066i \(0.0867150\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −14.0000 −0.528020
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) −12.0000 −0.451306
\(708\) −3.00000 −0.112747
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) −9.00000 −0.337764
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) −8.00000 −0.299392
\(715\) 1.00000 0.0373979
\(716\) 4.00000 0.149487
\(717\) 5.00000 0.186728
\(718\) 24.0000 0.895672
\(719\) 21.0000 0.783168 0.391584 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 1.00000 0.0372678
\(721\) 44.0000 1.63865
\(722\) −15.0000 −0.558242
\(723\) 26.0000 0.966950
\(724\) 6.00000 0.222988
\(725\) 4.00000 0.148556
\(726\) 10.0000 0.371135
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) −20.0000 −0.739727
\(732\) −5.00000 −0.184805
\(733\) −39.0000 −1.44050 −0.720249 0.693716i \(-0.755972\pi\)
−0.720249 + 0.693716i \(0.755972\pi\)
\(734\) −26.0000 −0.959678
\(735\) −9.00000 −0.331970
\(736\) 1.00000 0.0368605
\(737\) 5.00000 0.184177
\(738\) −5.00000 −0.184053
\(739\) 31.0000 1.14035 0.570177 0.821522i \(-0.306875\pi\)
0.570177 + 0.821522i \(0.306875\pi\)
\(740\) −7.00000 −0.257325
\(741\) −2.00000 −0.0734718
\(742\) 40.0000 1.46845
\(743\) −41.0000 −1.50414 −0.752072 0.659081i \(-0.770945\pi\)
−0.752072 + 0.659081i \(0.770945\pi\)
\(744\) 5.00000 0.183309
\(745\) 11.0000 0.403009
\(746\) 24.0000 0.878702
\(747\) 6.00000 0.219529
\(748\) −2.00000 −0.0731272
\(749\) 72.0000 2.63082
\(750\) 9.00000 0.328634
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) 3.00000 0.109326
\(754\) −1.00000 −0.0364179
\(755\) 16.0000 0.582300
\(756\) 4.00000 0.145479
\(757\) −35.0000 −1.27210 −0.636048 0.771649i \(-0.719432\pi\)
−0.636048 + 0.771649i \(0.719432\pi\)
\(758\) 32.0000 1.16229
\(759\) −1.00000 −0.0362977
\(760\) 2.00000 0.0725476
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 1.00000 0.0362262
\(763\) 40.0000 1.44810
\(764\) 3.00000 0.108536
\(765\) −2.00000 −0.0723102
\(766\) 16.0000 0.578103
\(767\) 3.00000 0.108324
\(768\) −1.00000 −0.0360844
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) −4.00000 −0.144150
\(771\) 12.0000 0.432169
\(772\) 14.0000 0.503871
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 10.0000 0.359443
\(775\) 20.0000 0.718421
\(776\) −18.0000 −0.646162
\(777\) −28.0000 −1.00449
\(778\) 10.0000 0.358517
\(779\) −10.0000 −0.358287
\(780\) −1.00000 −0.0358057
\(781\) −9.00000 −0.322045
\(782\) −2.00000 −0.0715199
\(783\) 1.00000 0.0357371
\(784\) 9.00000 0.321429
\(785\) 22.0000 0.785214
\(786\) 10.0000 0.356688
\(787\) −13.0000 −0.463400 −0.231700 0.972787i \(-0.574429\pi\)
−0.231700 + 0.972787i \(0.574429\pi\)
\(788\) 6.00000 0.213741
\(789\) 12.0000 0.427211
\(790\) 4.00000 0.142314
\(791\) 56.0000 1.99113
\(792\) 1.00000 0.0355335
\(793\) 5.00000 0.177555
\(794\) 14.0000 0.496841
\(795\) 10.0000 0.354663
\(796\) 13.0000 0.460773
\(797\) 36.0000 1.27519 0.637593 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 29.0000 1.02403
\(803\) −10.0000 −0.352892
\(804\) −5.00000 −0.176336
\(805\) −4.00000 −0.140981
\(806\) −5.00000 −0.176117
\(807\) −29.0000 −1.02085
\(808\) 3.00000 0.105540
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 1.00000 0.0351364
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 4.00000 0.140372
\(813\) 27.0000 0.946931
\(814\) −7.00000 −0.245350
\(815\) 1.00000 0.0350285
\(816\) 2.00000 0.0700140
\(817\) 20.0000 0.699711
\(818\) 12.0000 0.419570
\(819\) −4.00000 −0.139771
\(820\) −5.00000 −0.174608
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 18.0000 0.627822
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) −11.0000 −0.383203
\(825\) 4.00000 0.139262
\(826\) −12.0000 −0.417533
\(827\) −9.00000 −0.312961 −0.156480 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 1.00000 0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 6.00000 0.208263
\(831\) 3.00000 0.104069
\(832\) 1.00000 0.0346688
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 2.00000 0.0691714
\(837\) 5.00000 0.172825
\(838\) −26.0000 −0.898155
\(839\) 53.0000 1.82976 0.914882 0.403722i \(-0.132284\pi\)
0.914882 + 0.403722i \(0.132284\pi\)
\(840\) 4.00000 0.138013
\(841\) 1.00000 0.0344828
\(842\) −17.0000 −0.585859
\(843\) −26.0000 −0.895488
\(844\) −9.00000 −0.309793
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 40.0000 1.37442
\(848\) −10.0000 −0.343401
\(849\) −21.0000 −0.720718
\(850\) 8.00000 0.274398
\(851\) −7.00000 −0.239957
\(852\) 9.00000 0.308335
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) −20.0000 −0.684386
\(855\) 2.00000 0.0683986
\(856\) −18.0000 −0.615227
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) −1.00000 −0.0341394
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 10.0000 0.340997
\(861\) −20.0000 −0.681598
\(862\) 16.0000 0.544962
\(863\) −43.0000 −1.46374 −0.731869 0.681446i \(-0.761351\pi\)
−0.731869 + 0.681446i \(0.761351\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −4.00000 −0.136004
\(866\) 18.0000 0.611665
\(867\) 13.0000 0.441503
\(868\) 20.0000 0.678844
\(869\) 4.00000 0.135691
\(870\) 1.00000 0.0339032
\(871\) 5.00000 0.169419
\(872\) −10.0000 −0.338643
\(873\) −18.0000 −0.609208
\(874\) 2.00000 0.0676510
\(875\) 36.0000 1.21702
\(876\) 10.0000 0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 18.0000 0.607471
\(879\) −4.00000 −0.134917
\(880\) 1.00000 0.0337100
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 9.00000 0.303046
\(883\) −54.0000 −1.81724 −0.908622 0.417619i \(-0.862865\pi\)
−0.908622 + 0.417619i \(0.862865\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −3.00000 −0.100844
\(886\) 12.0000 0.403148
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 7.00000 0.234905
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) −11.0000 −0.367895
\(895\) 4.00000 0.133705
\(896\) −4.00000 −0.133631
\(897\) −1.00000 −0.0333890
\(898\) 15.0000 0.500556
\(899\) 5.00000 0.166759
\(900\) −4.00000 −0.133333
\(901\) 20.0000 0.666297
\(902\) −5.00000 −0.166482
\(903\) 40.0000 1.33112
\(904\) −14.0000 −0.465633
\(905\) 6.00000 0.199447
\(906\) −16.0000 −0.531564
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 18.0000 0.597351
\(909\) 3.00000 0.0995037
\(910\) −4.00000 −0.132599
\(911\) 39.0000 1.29213 0.646064 0.763283i \(-0.276414\pi\)
0.646064 + 0.763283i \(0.276414\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 6.00000 0.198571
\(914\) −40.0000 −1.32308
\(915\) −5.00000 −0.165295
\(916\) −23.0000 −0.759941
\(917\) 40.0000 1.32092
\(918\) 2.00000 0.0660098
\(919\) −35.0000 −1.15454 −0.577272 0.816552i \(-0.695883\pi\)
−0.577272 + 0.816552i \(0.695883\pi\)
\(920\) 1.00000 0.0329690
\(921\) −7.00000 −0.230658
\(922\) −9.00000 −0.296399
\(923\) −9.00000 −0.296239
\(924\) 4.00000 0.131590
\(925\) 28.0000 0.920634
\(926\) −30.0000 −0.985861
\(927\) −11.0000 −0.361287
\(928\) −1.00000 −0.0328266
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 5.00000 0.163956
\(931\) 18.0000 0.589926
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 33.0000 1.07979
\(935\) −2.00000 −0.0654070
\(936\) 1.00000 0.0326860
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −20.0000 −0.653023
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −39.0000 −1.27136 −0.635682 0.771951i \(-0.719281\pi\)
−0.635682 + 0.771951i \(0.719281\pi\)
\(942\) −22.0000 −0.716799
\(943\) −5.00000 −0.162822
\(944\) 3.00000 0.0976417
\(945\) 4.00000 0.130120
\(946\) 10.0000 0.325128
\(947\) −14.0000 −0.454939 −0.227469 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) −4.00000 −0.129914
\(949\) −10.0000 −0.324614
\(950\) −8.00000 −0.259554
\(951\) −3.00000 −0.0972817
\(952\) 8.00000 0.259281
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) −10.0000 −0.323762
\(955\) 3.00000 0.0970777
\(956\) −5.00000 −0.161712
\(957\) 1.00000 0.0323254
\(958\) −3.00000 −0.0969256
\(959\) 72.0000 2.32500
\(960\) −1.00000 −0.0322749
\(961\) −6.00000 −0.193548
\(962\) −7.00000 −0.225689
\(963\) −18.0000 −0.580042
\(964\) −26.0000 −0.837404
\(965\) 14.0000 0.450676
\(966\) 4.00000 0.128698
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) −10.0000 −0.321412
\(969\) 4.00000 0.128499
\(970\) −18.0000 −0.577945
\(971\) 39.0000 1.25157 0.625785 0.779996i \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −18.0000 −0.576757
\(975\) 4.00000 0.128103
\(976\) 5.00000 0.160046
\(977\) −11.0000 −0.351921 −0.175961 0.984397i \(-0.556303\pi\)
−0.175961 + 0.984397i \(0.556303\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) −10.0000 −0.319275
\(982\) −12.0000 −0.382935
\(983\) −33.0000 −1.05254 −0.526268 0.850319i \(-0.676409\pi\)
−0.526268 + 0.850319i \(0.676409\pi\)
\(984\) 5.00000 0.159394
\(985\) 6.00000 0.191176
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 10.0000 0.317982
\(990\) 1.00000 0.0317821
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −5.00000 −0.158750
\(993\) 16.0000 0.507745
\(994\) 36.0000 1.14185
\(995\) 13.0000 0.412128
\(996\) −6.00000 −0.190117
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 30.0000 0.949633
\(999\) 7.00000 0.221470
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 4002.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.j.1.1 1 1.1 even 1 trivial