Properties

Label 4010.2.a.a.1.1
Level $4010$
Weight $2$
Character 4010.1
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, 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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +8.00000 q^{21} -3.00000 q^{22} +2.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} +2.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -4.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -8.00000 q^{38} -10.0000 q^{39} -1.00000 q^{40} -9.00000 q^{41} -8.00000 q^{42} +5.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -2.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +5.00000 q^{52} -9.00000 q^{53} -4.00000 q^{54} +3.00000 q^{55} +4.00000 q^{56} -16.0000 q^{57} -2.00000 q^{60} -1.00000 q^{61} -5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} +6.00000 q^{66} -4.00000 q^{67} +4.00000 q^{70} -3.00000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +10.0000 q^{74} -2.00000 q^{75} +8.00000 q^{76} -12.0000 q^{77} +10.0000 q^{78} -1.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +9.00000 q^{82} +3.00000 q^{83} +8.00000 q^{84} -5.00000 q^{86} -3.00000 q^{88} +15.0000 q^{89} -1.00000 q^{90} -20.0000 q^{91} -10.0000 q^{93} +8.00000 q^{95} +2.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −2.00000 −0.577350
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 4.00000 1.06904
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 8.00000 1.74574
\(22\) −3.00000 −0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −5.00000 −0.980581
\(27\) 4.00000 0.769800
\(28\) −4.00000 −0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 2.00000 0.365148
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −8.00000 −1.29777
\(39\) −10.0000 −1.60128
\(40\) −1.00000 −0.158114
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) −8.00000 −1.23443
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 3.00000 0.452267
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −4.00000 −0.544331
\(55\) 3.00000 0.404520
\(56\) 4.00000 0.534522
\(57\) −16.0000 −2.11925
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −5.00000 −0.635001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 10.0000 1.16248
\(75\) −2.00000 −0.230940
\(76\) 8.00000 0.917663
\(77\) −12.0000 −1.36753
\(78\) 10.0000 1.13228
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 9.00000 0.993884
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 8.00000 0.872872
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) −1.00000 −0.105409
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 2.00000 0.204124
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −9.00000 −0.909137
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −5.00000 −0.490290
\(105\) 8.00000 0.780720
\(106\) 9.00000 0.874157
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 4.00000 0.384900
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −3.00000 −0.286039
\(111\) 20.0000 1.89832
\(112\) −4.00000 −0.377964
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 16.0000 1.49854
\(115\) 0 0
\(116\) 0 0
\(117\) 5.00000 0.462250
\(118\) 0 0
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −2.00000 −0.181818
\(122\) 1.00000 0.0905357
\(123\) 18.0000 1.62301
\(124\) 5.00000 0.449013
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) −5.00000 −0.438529
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) −32.0000 −2.77475
\(134\) 4.00000 0.345547
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) 15.0000 1.25436
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) −18.0000 −1.48461
\(148\) −10.0000 −0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 2.00000 0.163299
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 5.00000 0.401610
\(156\) −10.0000 −0.800641
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 1.00000 0.0795557
\(159\) 18.0000 1.42749
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −9.00000 −0.702782
\(165\) −6.00000 −0.467099
\(166\) −3.00000 −0.232845
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) −8.00000 −0.617213
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 5.00000 0.381246
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −15.0000 −1.12430
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 1.00000 0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 20.0000 1.48250
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) −8.00000 −0.580381
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) −2.00000 −0.144338
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −14.0000 −1.00514
\(195\) −10.0000 −0.716115
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −3.00000 −0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 24.0000 1.66011
\(210\) −8.00000 −0.552052
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −9.00000 −0.618123
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 5.00000 0.340997
\(216\) −4.00000 −0.272166
\(217\) −20.0000 −1.35769
\(218\) 10.0000 0.677285
\(219\) 14.0000 0.946032
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) −20.0000 −1.34231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) −15.0000 −0.997785
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −16.0000 −1.05963
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 24.0000 1.57908
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −5.00000 −0.326860
\(235\) 0 0
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −2.00000 −0.129099
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 2.00000 0.128565
\(243\) 10.0000 0.641500
\(244\) −1.00000 −0.0640184
\(245\) 9.00000 0.574989
\(246\) −18.0000 −1.14764
\(247\) 40.0000 2.54514
\(248\) −5.00000 −0.317500
\(249\) −6.00000 −0.380235
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 10.0000 0.622573
\(259\) 40.0000 2.48548
\(260\) 5.00000 0.310087
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 6.00000 0.369274
\(265\) −9.00000 −0.552866
\(266\) 32.0000 1.96205
\(267\) −30.0000 −1.83597
\(268\) −4.00000 −0.244339
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) −4.00000 −0.243432
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 40.0000 2.42091
\(274\) 18.0000 1.08742
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) −8.00000 −0.479808
\(279\) 5.00000 0.299342
\(280\) 4.00000 0.239046
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −3.00000 −0.178017
\(285\) −16.0000 −0.947758
\(286\) −15.0000 −0.886969
\(287\) 36.0000 2.12501
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) −7.00000 −0.409644
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 18.0000 1.04978
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 12.0000 0.696311
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −20.0000 −1.15278
\(302\) 4.00000 0.230174
\(303\) −36.0000 −2.06815
\(304\) 8.00000 0.458831
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −1.00000 −0.0570730 −0.0285365 0.999593i \(-0.509085\pi\)
−0.0285365 + 0.999593i \(0.509085\pi\)
\(308\) −12.0000 −0.683763
\(309\) −28.0000 −1.59286
\(310\) −5.00000 −0.283981
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 10.0000 0.566139
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 10.0000 0.564333
\(315\) −4.00000 −0.225374
\(316\) −1.00000 −0.0562544
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −18.0000 −1.00939
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 5.00000 0.277350
\(326\) −2.00000 −0.110770
\(327\) 20.0000 1.10600
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 3.00000 0.164646
\(333\) −10.0000 −0.547997
\(334\) −15.0000 −0.820763
\(335\) −4.00000 −0.218543
\(336\) 8.00000 0.436436
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) −12.0000 −0.652714
\(339\) −30.0000 −1.62938
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) −8.00000 −0.432590
\(343\) −8.00000 −0.431959
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 4.00000 0.213809
\(351\) 20.0000 1.06752
\(352\) −3.00000 −0.159901
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) 16.0000 0.840941
\(363\) 4.00000 0.209946
\(364\) −20.0000 −1.04828
\(365\) −7.00000 −0.366397
\(366\) −2.00000 −0.104542
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 10.0000 0.519875
\(371\) 36.0000 1.86903
\(372\) −10.0000 −0.518476
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 0 0
\(378\) 16.0000 0.822951
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 8.00000 0.410391
\(381\) 14.0000 0.717242
\(382\) −15.0000 −0.767467
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 2.00000 0.102062
\(385\) −12.0000 −0.611577
\(386\) −20.0000 −1.01797
\(387\) 5.00000 0.254164
\(388\) 14.0000 0.710742
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 10.0000 0.506370
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −1.00000 −0.0503155
\(396\) 3.00000 0.150756
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −20.0000 −1.00251
\(399\) 64.0000 3.20401
\(400\) 1.00000 0.0500000
\(401\) −1.00000 −0.0499376
\(402\) −8.00000 −0.399004
\(403\) 25.0000 1.24534
\(404\) 18.0000 0.895533
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 9.00000 0.444478
\(411\) 36.0000 1.77575
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) −5.00000 −0.245145
\(417\) −16.0000 −0.783523
\(418\) −24.0000 −1.17388
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 8.00000 0.390360
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 4.00000 0.193574
\(428\) 0 0
\(429\) −30.0000 −1.44841
\(430\) −5.00000 −0.241121
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 4.00000 0.192450
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) −14.0000 −0.668946
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −3.00000 −0.143019
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 20.0000 0.949158
\(445\) 15.0000 0.711068
\(446\) 4.00000 0.189405
\(447\) 12.0000 0.567581
\(448\) −4.00000 −0.188982
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −27.0000 −1.27138
\(452\) 15.0000 0.705541
\(453\) 8.00000 0.375873
\(454\) −24.0000 −1.12638
\(455\) −20.0000 −0.937614
\(456\) 16.0000 0.749269
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) −24.0000 −1.11658
\(463\) 41.0000 1.90543 0.952716 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 24.0000 1.11178
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 5.00000 0.231125
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) 15.0000 0.689701
\(474\) −2.00000 −0.0918630
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) −6.00000 −0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) −50.0000 −2.27980
\(482\) −17.0000 −0.774329
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 14.0000 0.635707
\(486\) −10.0000 −0.453609
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 1.00000 0.0452679
\(489\) −4.00000 −0.180886
\(490\) −9.00000 −0.406579
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 18.0000 0.811503
\(493\) 0 0
\(494\) −40.0000 −1.79969
\(495\) 3.00000 0.134840
\(496\) 5.00000 0.224507
\(497\) 12.0000 0.538274
\(498\) 6.00000 0.268866
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) −30.0000 −1.34030
\(502\) 12.0000 0.535586
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 4.00000 0.178174
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) −24.0000 −1.06588
\(508\) −7.00000 −0.310575
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) −1.00000 −0.0441942
\(513\) 32.0000 1.41283
\(514\) −15.0000 −0.661622
\(515\) 14.0000 0.616914
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) −40.0000 −1.75750
\(519\) 24.0000 1.05348
\(520\) −5.00000 −0.219265
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 0 0
\(525\) 8.00000 0.349149
\(526\) 18.0000 0.784837
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) −23.0000 −1.00000
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) −32.0000 −1.38738
\(533\) −45.0000 −1.94917
\(534\) 30.0000 1.29823
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −18.0000 −0.776757
\(538\) 15.0000 0.646696
\(539\) 27.0000 1.16297
\(540\) 4.00000 0.172133
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) −8.00000 −0.343629
\(543\) 32.0000 1.37325
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) −40.0000 −1.71184
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −18.0000 −0.768922
\(549\) −1.00000 −0.0426790
\(550\) −3.00000 −0.127920
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −5.00000 −0.212430
\(555\) 20.0000 0.848953
\(556\) 8.00000 0.339276
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −5.00000 −0.211667
\(559\) 25.0000 1.05739
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 15.0000 0.631055
\(566\) −8.00000 −0.336265
\(567\) 44.0000 1.84783
\(568\) 3.00000 0.125877
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 16.0000 0.670166
\(571\) −46.0000 −1.92504 −0.962520 0.271211i \(-0.912576\pi\)
−0.962520 + 0.271211i \(0.912576\pi\)
\(572\) 15.0000 0.627182
\(573\) −30.0000 −1.25327
\(574\) −36.0000 −1.50261
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −1.00000 −0.0416305 −0.0208153 0.999783i \(-0.506626\pi\)
−0.0208153 + 0.999783i \(0.506626\pi\)
\(578\) 17.0000 0.707107
\(579\) −40.0000 −1.66234
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 28.0000 1.16064
\(583\) −27.0000 −1.11823
\(584\) 7.00000 0.289662
\(585\) 5.00000 0.206725
\(586\) 21.0000 0.867502
\(587\) 27.0000 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) −18.0000 −0.742307
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) −10.0000 −0.410997
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −40.0000 −1.63709
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 2.00000 0.0816497
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 20.0000 0.815139
\(603\) −4.00000 −0.162893
\(604\) −4.00000 −0.162758
\(605\) −2.00000 −0.0813116
\(606\) 36.0000 1.46240
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 1.00000 0.0404888
\(611\) 0 0
\(612\) 0 0
\(613\) 35.0000 1.41364 0.706818 0.707395i \(-0.250130\pi\)
0.706818 + 0.707395i \(0.250130\pi\)
\(614\) 1.00000 0.0403567
\(615\) 18.0000 0.725830
\(616\) 12.0000 0.483494
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 28.0000 1.12633
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) 5.00000 0.200805
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −60.0000 −2.40385
\(624\) −10.0000 −0.400320
\(625\) 1.00000 0.0400000
\(626\) −17.0000 −0.679457
\(627\) −48.0000 −1.91694
\(628\) −10.0000 −0.399043
\(629\) 0 0
\(630\) 4.00000 0.159364
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 1.00000 0.0397779
\(633\) 8.00000 0.317971
\(634\) −3.00000 −0.119145
\(635\) −7.00000 −0.277787
\(636\) 18.0000 0.713746
\(637\) 45.0000 1.78296
\(638\) 0 0
\(639\) −3.00000 −0.118678
\(640\) −1.00000 −0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −10.0000 −0.393750
\(646\) 0 0
\(647\) −39.0000 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) −5.00000 −0.196116
\(651\) 40.0000 1.56772
\(652\) 2.00000 0.0783260
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −6.00000 −0.233550
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −3.00000 −0.116423
\(665\) −32.0000 −1.24091
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) 15.0000 0.580367
\(669\) 8.00000 0.309298
\(670\) 4.00000 0.154533
\(671\) −3.00000 −0.115814
\(672\) −8.00000 −0.308607
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −5.00000 −0.192593
\(675\) 4.00000 0.153960
\(676\) 12.0000 0.461538
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 30.0000 1.15214
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) −15.0000 −0.574380
\(683\) 42.0000 1.60709 0.803543 0.595247i \(-0.202946\pi\)
0.803543 + 0.595247i \(0.202946\pi\)
\(684\) 8.00000 0.305888
\(685\) −18.0000 −0.687745
\(686\) 8.00000 0.305441
\(687\) −28.0000 −1.06827
\(688\) 5.00000 0.190623
\(689\) −45.0000 −1.71436
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) −12.0000 −0.456172
\(693\) −12.0000 −0.455842
\(694\) −18.0000 −0.683271
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 48.0000 1.81553
\(700\) −4.00000 −0.151186
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) −20.0000 −0.754851
\(703\) −80.0000 −3.01726
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −72.0000 −2.70784
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 3.00000 0.112588
\(711\) −1.00000 −0.0375029
\(712\) −15.0000 −0.562149
\(713\) 0 0
\(714\) 0 0
\(715\) 15.0000 0.560968
\(716\) 9.00000 0.336346
\(717\) −12.0000 −0.448148
\(718\) 9.00000 0.335877
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) −56.0000 −2.08555
\(722\) −45.0000 −1.67473
\(723\) −34.0000 −1.26447
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 20.0000 0.741249
\(729\) 13.0000 0.481481
\(730\) 7.00000 0.259082
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −17.0000 −0.627481
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 9.00000 0.331295
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −10.0000 −0.367607
\(741\) −80.0000 −2.93887
\(742\) −36.0000 −1.32160
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 10.0000 0.366618
\(745\) −6.00000 −0.219823
\(746\) −32.0000 −1.17160
\(747\) 3.00000 0.109764
\(748\) 0 0
\(749\) 0 0
\(750\) 2.00000 0.0730297
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) −16.0000 −0.581914
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) −11.0000 −0.399538
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) −14.0000 −0.507166
\(763\) 40.0000 1.44810
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 12.0000 0.432450
\(771\) −30.0000 −1.08042
\(772\) 20.0000 0.719816
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −5.00000 −0.179721
\(775\) 5.00000 0.179605
\(776\) −14.0000 −0.502571
\(777\) −80.0000 −2.86998
\(778\) −9.00000 −0.322666
\(779\) −72.0000 −2.57967
\(780\) −10.0000 −0.358057
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 6.00000 0.213741
\(789\) 36.0000 1.28163
\(790\) 1.00000 0.0355784
\(791\) −60.0000 −2.13335
\(792\) −3.00000 −0.106600
\(793\) −5.00000 −0.177555
\(794\) −38.0000 −1.34857
\(795\) 18.0000 0.638394
\(796\) 20.0000 0.708881
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) −64.0000 −2.26558
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 15.0000 0.529999
\(802\) 1.00000 0.0353112
\(803\) −21.0000 −0.741074
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −25.0000 −0.880587
\(807\) 30.0000 1.05605
\(808\) −18.0000 −0.633238
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 11.0000 0.386501
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 30.0000 1.05150
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) −14.0000 −0.489499
\(819\) −20.0000 −0.698857
\(820\) −9.00000 −0.314294
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) −36.0000 −1.25564
\(823\) −49.0000 −1.70803 −0.854016 0.520246i \(-0.825840\pi\)
−0.854016 + 0.520246i \(0.825840\pi\)
\(824\) −14.0000 −0.487713
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) −3.00000 −0.104132
\(831\) −10.0000 −0.346896
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) 15.0000 0.519096
\(836\) 24.0000 0.830057
\(837\) 20.0000 0.691301
\(838\) −12.0000 −0.414533
\(839\) 15.0000 0.517858 0.258929 0.965896i \(-0.416631\pi\)
0.258929 + 0.965896i \(0.416631\pi\)
\(840\) −8.00000 −0.276026
\(841\) −29.0000 −1.00000
\(842\) −2.00000 −0.0689246
\(843\) −12.0000 −0.413302
\(844\) −4.00000 −0.137686
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 8.00000 0.274883
\(848\) −9.00000 −0.309061
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 6.00000 0.205557
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −4.00000 −0.136877
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −15.0000 −0.512390 −0.256195 0.966625i \(-0.582469\pi\)
−0.256195 + 0.966625i \(0.582469\pi\)
\(858\) 30.0000 1.02418
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 5.00000 0.170499
\(861\) −72.0000 −2.45375
\(862\) −21.0000 −0.715263
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) −4.00000 −0.136083
\(865\) −12.0000 −0.408012
\(866\) −2.00000 −0.0679628
\(867\) 34.0000 1.15470
\(868\) −20.0000 −0.678844
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 10.0000 0.338643
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) 14.0000 0.473016
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −20.0000 −0.674967
\(879\) 42.0000 1.41662
\(880\) 3.00000 0.101130
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −9.00000 −0.303046
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −20.0000 −0.671156
\(889\) 28.0000 0.939090
\(890\) −15.0000 −0.502801
\(891\) −33.0000 −1.10554
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 9.00000 0.300837
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 27.0000 0.899002
\(903\) 40.0000 1.33112
\(904\) −15.0000 −0.498893
\(905\) −16.0000 −0.531858
\(906\) −8.00000 −0.265782
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 24.0000 0.796468
\(909\) 18.0000 0.597022
\(910\) 20.0000 0.662994
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) −16.0000 −0.529813
\(913\) 9.00000 0.297857
\(914\) −17.0000 −0.562310
\(915\) 2.00000 0.0661180
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) 47.0000 1.55039 0.775193 0.631724i \(-0.217652\pi\)
0.775193 + 0.631724i \(0.217652\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 30.0000 0.987997
\(923\) −15.0000 −0.493731
\(924\) 24.0000 0.789542
\(925\) −10.0000 −0.328798
\(926\) −41.0000 −1.34734
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 10.0000 0.327913
\(931\) 72.0000 2.35970
\(932\) −24.0000 −0.786146
\(933\) 24.0000 0.785725
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) −5.00000 −0.163430
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) −16.0000 −0.522419
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −51.0000 −1.66255 −0.831276 0.555860i \(-0.812389\pi\)
−0.831276 + 0.555860i \(0.812389\pi\)
\(942\) −20.0000 −0.651635
\(943\) 0 0
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) −15.0000 −0.487692
\(947\) −15.0000 −0.487435 −0.243717 0.969846i \(-0.578367\pi\)
−0.243717 + 0.969846i \(0.578367\pi\)
\(948\) 2.00000 0.0649570
\(949\) −35.0000 −1.13615
\(950\) −8.00000 −0.259554
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 9.00000 0.291386
\(955\) 15.0000 0.485389
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 0 0
\(959\) 72.0000 2.32500
\(960\) −2.00000 −0.0645497
\(961\) −6.00000 −0.193548
\(962\) 50.0000 1.61206
\(963\) 0 0
\(964\) 17.0000 0.547533
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 10.0000 0.320750
\(973\) −32.0000 −1.02587
\(974\) 34.0000 1.08943
\(975\) −10.0000 −0.320256
\(976\) −1.00000 −0.0320092
\(977\) −51.0000 −1.63163 −0.815817 0.578310i \(-0.803713\pi\)
−0.815817 + 0.578310i \(0.803713\pi\)
\(978\) 4.00000 0.127906
\(979\) 45.0000 1.43821
\(980\) 9.00000 0.287494
\(981\) −10.0000 −0.319275
\(982\) −33.0000 −1.05307
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) −18.0000 −0.573819
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 40.0000 1.27257
\(989\) 0 0
\(990\) −3.00000 −0.0953463
\(991\) −55.0000 −1.74713 −0.873566 0.486705i \(-0.838199\pi\)
−0.873566 + 0.486705i \(0.838199\pi\)
\(992\) −5.00000 −0.158750
\(993\) 26.0000 0.825085
\(994\) −12.0000 −0.380617
\(995\) 20.0000 0.634043
\(996\) −6.00000 −0.190117
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −20.0000 −0.633089
\(999\) −40.0000 −1.26554
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 4010.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.a.1.1 1 1.1 even 1 trivial