Properties

Label 4014.2.a.b.1.1
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $2$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, 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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 4014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -2.00000 q^{23} -4.00000 q^{25} +6.00000 q^{26} -4.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} +8.00000 q^{34} +4.00000 q^{35} -1.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +6.00000 q^{43} -2.00000 q^{44} +2.00000 q^{46} +3.00000 q^{47} +9.00000 q^{49} +4.00000 q^{50} -6.00000 q^{52} +14.0000 q^{53} +2.00000 q^{55} +4.00000 q^{56} +10.0000 q^{59} -6.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -13.0000 q^{67} -8.00000 q^{68} -4.00000 q^{70} +6.00000 q^{71} -11.0000 q^{73} +1.00000 q^{74} -6.00000 q^{76} +8.00000 q^{77} -5.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} -9.00000 q^{83} +8.00000 q^{85} -6.00000 q^{86} +2.00000 q^{88} -12.0000 q^{89} +24.0000 q^{91} -2.00000 q^{92} -3.00000 q^{94} +6.00000 q^{95} +18.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 32.0000 2.93344
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) 13.0000 1.12303
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −23.0000 −1.87171 −0.935857 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 5.00000 0.397779
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −13.0000 −0.988372 −0.494186 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) 0 0
\(175\) 16.0000 1.20949
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −24.0000 −1.77900
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 7.00000 0.487713
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −1.00000 −0.0677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 48.0000 3.22883
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) −32.0000 −2.07425
\(239\) −27.0000 −1.74648 −0.873242 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 36.0000 2.29063
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −7.00000 −0.432461
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) −24.0000 −1.47153
\(267\) 0 0
\(268\) −13.0000 −0.794101
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −18.0000 −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) −11.0000 −0.643726
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 23.0000 1.32350
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −9.00000 −0.493939
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 13.0000 0.710266
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) −16.0000 −0.855236
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) −15.0000 −0.792775
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) 24.0000 1.25794
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −56.0000 −2.90738
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 22.0000 1.12562
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 18.0000 0.913812
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 5.00000 0.251577
\(396\) 0 0
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −7.00000 −0.344865
\(413\) −40.0000 −1.96827
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −14.0000 −0.679900
\(425\) 32.0000 1.55223
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 0 0
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −48.0000 −2.28313
\(443\) −3.00000 −0.142534 −0.0712672 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 15.0000 0.703985
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −30.0000 −1.39422 −0.697109 0.716965i \(-0.745531\pi\)
−0.697109 + 0.716965i \(0.745531\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 52.0000 2.40114
\(470\) 3.00000 0.138380
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 32.0000 1.46672
\(477\) 0 0
\(478\) 27.0000 1.23495
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) −23.0000 −1.04762
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) −5.00000 −0.223161
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) 44.0000 1.94645
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 7.00000 0.308457
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 7.00000 0.305796
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 14.0000 0.608121
\(531\) 0 0
\(532\) 24.0000 1.04053
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 13.0000 0.561514
\(537\) 0 0
\(538\) 15.0000 0.646696
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) −1.00000 −0.0428353
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −17.0000 −0.726204
\(549\) 0 0
\(550\) −8.00000 −0.341121
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 18.0000 0.756596
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) 37.0000 1.54840 0.774201 0.632940i \(-0.218152\pi\)
0.774201 + 0.632940i \(0.218152\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 33.0000 1.37381 0.686904 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) −28.0000 −1.15964
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −38.0000 −1.56843 −0.784214 0.620491i \(-0.786934\pi\)
−0.784214 + 0.620491i \(0.786934\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) −32.0000 −1.31187
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) −23.0000 −0.935857
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −25.0000 −1.01472 −0.507359 0.861735i \(-0.669378\pi\)
−0.507359 + 0.861735i \(0.669378\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 23.0000 0.928204
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 43.0000 1.72832 0.864158 0.503221i \(-0.167852\pi\)
0.864158 + 0.503221i \(0.167852\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 0 0
\(623\) 48.0000 1.92308
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 3.00000 0.119428 0.0597141 0.998216i \(-0.480981\pi\)
0.0597141 + 0.998216i \(0.480981\pi\)
\(632\) 5.00000 0.198889
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −54.0000 −2.13956
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) −24.0000 −0.941357
\(651\) 0 0
\(652\) −11.0000 −0.430793
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) −7.00000 −0.273513
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) −13.0000 −0.502234
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) −72.0000 −2.76311
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −23.0000 −0.880071 −0.440035 0.897980i \(-0.645034\pi\)
−0.440035 + 0.897980i \(0.645034\pi\)
\(684\) 0 0
\(685\) 17.0000 0.649537
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) −84.0000 −3.20015
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) −13.0000 −0.494186
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) −5.00000 −0.189253
\(699\) 0 0
\(700\) 16.0000 0.604743
\(701\) −4.00000 −0.151078 −0.0755390 0.997143i \(-0.524068\pi\)
−0.0755390 + 0.997143i \(0.524068\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 15.0000 0.560576
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) 3.00000 0.111881 0.0559406 0.998434i \(-0.482184\pi\)
0.0559406 + 0.998434i \(0.482184\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) −7.00000 −0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) −24.0000 −0.889499
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 26.0000 0.957722
\(738\) 0 0
\(739\) 3.00000 0.110357 0.0551784 0.998477i \(-0.482427\pi\)
0.0551784 + 0.998477i \(0.482427\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 56.0000 2.05582
\(743\) −41.0000 −1.50414 −0.752072 0.659081i \(-0.770945\pi\)
−0.752072 + 0.659081i \(0.770945\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) −18.0000 −0.659027
\(747\) 0 0
\(748\) 16.0000 0.585018
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) 0 0
\(755\) 23.0000 0.837056
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −22.0000 −0.795932
\(765\) 0 0
\(766\) 0 0
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 23.0000 0.819861 0.409931 0.912117i \(-0.365553\pi\)
0.409931 + 0.912117i \(0.365553\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) −5.00000 −0.177892
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) 22.0000 0.776363
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) 49.0000 1.72275 0.861374 0.507971i \(-0.169604\pi\)
0.861374 + 0.507971i \(0.169604\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 13.0000 0.453152 0.226576 0.973994i \(-0.427247\pi\)
0.226576 + 0.973994i \(0.427247\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 40.0000 1.39178
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 32.0000 1.11141 0.555703 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) −9.00000 −0.312395
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) −72.0000 −2.49465
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −8.00000 −0.275698
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) −32.0000 −1.09759
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −4.00000 −0.136241
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) −31.0000 −1.05342
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) 78.0000 2.64293
\(872\) −1.00000 −0.0338643
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 1.00000 0.0337484
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) 47.0000 1.58168 0.790838 0.612026i \(-0.209645\pi\)
0.790838 + 0.612026i \(0.209645\pi\)
\(884\) 48.0000 1.61441
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) −9.00000 −0.302190 −0.151095 0.988519i \(-0.548280\pi\)
−0.151095 + 0.988519i \(0.548280\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) −112.000 −3.73126
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 18.0000 0.597680 0.298840 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) −15.0000 −0.497792
\(909\) 0 0
\(910\) 24.0000 0.795592
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −12.0000 −0.396925
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) −28.0000 −0.924641
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 0 0
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 30.0000 0.985861
\(927\) 0 0
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) −52.0000 −1.69786
\(939\) 0 0
\(940\) −3.00000 −0.0978492
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 66.0000 2.14245
\(950\) −24.0000 −0.778663
\(951\) 0 0
\(952\) −32.0000 −1.03713
\(953\) 47.0000 1.52248 0.761240 0.648471i \(-0.224591\pi\)
0.761240 + 0.648471i \(0.224591\pi\)
\(954\) 0 0
\(955\) 22.0000 0.711903
\(956\) −27.0000 −0.873242
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 68.0000 2.19583
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) 23.0000 0.740780
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 18.0000 0.577945
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −17.0000 −0.543878 −0.271939 0.962314i \(-0.587665\pi\)
−0.271939 + 0.962314i \(0.587665\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) 16.0000 0.506471
\(999\) 0 0
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 4014.2.a.b.1.1 1
3.2 odd 2 4014.2.a.f.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.a.b.1.1 1 1.1 even 1 trivial
4014.2.a.f.1.1 yes 1 3.2 odd 2