Properties

Label 6762.2.a.bn.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} +4.00000 q^{20} -5.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} +4.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} +4.00000 q^{40} -6.00000 q^{41} -6.00000 q^{43} -5.00000 q^{44} +4.00000 q^{45} +1.00000 q^{46} -9.00000 q^{47} +1.00000 q^{48} +11.0000 q^{50} +3.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -20.0000 q^{55} +6.00000 q^{57} +1.00000 q^{58} +8.00000 q^{59} +4.00000 q^{60} +2.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -5.00000 q^{66} -2.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -7.00000 q^{71} +1.00000 q^{72} +11.0000 q^{73} -6.00000 q^{74} +11.0000 q^{75} +6.00000 q^{76} +2.00000 q^{78} -1.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -16.0000 q^{83} +12.0000 q^{85} -6.00000 q^{86} +1.00000 q^{87} -5.00000 q^{88} -18.0000 q^{89} +4.00000 q^{90} +1.00000 q^{92} +10.0000 q^{93} -9.00000 q^{94} +24.0000 q^{95} +1.00000 q^{96} +4.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 4.00000 0.730297
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 6.00000 0.973329
\(39\) 2.00000 0.320256
\(40\) 4.00000 0.632456
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −5.00000 −0.753778
\(45\) 4.00000 0.596285
\(46\) 1.00000 0.147442
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) 3.00000 0.420084
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −20.0000 −2.69680
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 1.00000 0.131306
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 4.00000 0.516398
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) −5.00000 −0.615457
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 3.00000 0.363803
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −6.00000 −0.697486
\(75\) 11.0000 1.27017
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −6.00000 −0.646997
\(87\) 1.00000 0.107211
\(88\) −5.00000 −0.533002
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 10.0000 1.03695
\(94\) −9.00000 −0.928279
\(95\) 24.0000 2.46235
\(96\) 1.00000 0.102062
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 11.0000 1.10000
\(101\) −17.0000 −1.69156 −0.845782 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(102\) 3.00000 0.297044
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −20.0000 −1.90693
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 6.00000 0.561951
\(115\) 4.00000 0.373002
\(116\) 1.00000 0.0928477
\(117\) 2.00000 0.184900
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 4.00000 0.365148
\(121\) 14.0000 1.27273
\(122\) 2.00000 0.181071
\(123\) −6.00000 −0.541002
\(124\) 10.0000 0.898027
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) 8.00000 0.701646
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) −5.00000 −0.435194
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 4.00000 0.344265
\(136\) 3.00000 0.257248
\(137\) 13.0000 1.11066 0.555332 0.831628i \(-0.312591\pi\)
0.555332 + 0.831628i \(0.312591\pi\)
\(138\) 1.00000 0.0851257
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) −7.00000 −0.587427
\(143\) −10.0000 −0.836242
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 24.0000 1.96616 0.983078 0.183186i \(-0.0586410\pi\)
0.983078 + 0.183186i \(0.0586410\pi\)
\(150\) 11.0000 0.898146
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 6.00000 0.486664
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 40.0000 3.21288
\(156\) 2.00000 0.160128
\(157\) 3.00000 0.239426 0.119713 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −6.00000 −0.475831
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −6.00000 −0.468521
\(165\) −20.0000 −1.55700
\(166\) −16.0000 −1.24184
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 6.00000 0.458831
\(172\) −6.00000 −0.457496
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 8.00000 0.601317
\(178\) −18.0000 −1.34916
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 4.00000 0.298142
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) −24.0000 −1.76452
\(186\) 10.0000 0.733236
\(187\) −15.0000 −1.09691
\(188\) −9.00000 −0.656392
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 4.00000 0.287183
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) −5.00000 −0.355335
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) 11.0000 0.777817
\(201\) −2.00000 −0.141069
\(202\) −17.0000 −1.19612
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) −24.0000 −1.67623
\(206\) 9.00000 0.627060
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) −6.00000 −0.412082
\(213\) −7.00000 −0.479632
\(214\) −12.0000 −0.820303
\(215\) −24.0000 −1.63679
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −5.00000 −0.338643
\(219\) 11.0000 0.743311
\(220\) −20.0000 −1.34840
\(221\) 6.00000 0.403604
\(222\) −6.00000 −0.402694
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 6.00000 0.399114
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 6.00000 0.397360
\(229\) 3.00000 0.198246 0.0991228 0.995075i \(-0.468396\pi\)
0.0991228 + 0.995075i \(0.468396\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 2.00000 0.130744
\(235\) −36.0000 −2.34838
\(236\) 8.00000 0.520756
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 4.00000 0.258199
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 12.0000 0.763542
\(248\) 10.0000 0.635001
\(249\) −16.0000 −1.01396
\(250\) 24.0000 1.51789
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 10.0000 0.627456
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) −6.00000 −0.373544
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) 1.00000 0.0618984
\(262\) 16.0000 0.988483
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −5.00000 −0.307729
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) −2.00000 −0.122169
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 4.00000 0.243432
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 13.0000 0.785359
\(275\) −55.0000 −3.31662
\(276\) 1.00000 0.0601929
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −5.00000 −0.299880
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) −9.00000 −0.535942
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −7.00000 −0.415374
\(285\) 24.0000 1.42164
\(286\) −10.0000 −0.591312
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 4.00000 0.234888
\(291\) 4.00000 0.234484
\(292\) 11.0000 0.643726
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 32.0000 1.86311
\(296\) −6.00000 −0.348743
\(297\) −5.00000 −0.290129
\(298\) 24.0000 1.39028
\(299\) 2.00000 0.115663
\(300\) 11.0000 0.635085
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) −17.0000 −0.976624
\(304\) 6.00000 0.344124
\(305\) 8.00000 0.458079
\(306\) 3.00000 0.171499
\(307\) −29.0000 −1.65512 −0.827559 0.561379i \(-0.810271\pi\)
−0.827559 + 0.561379i \(0.810271\pi\)
\(308\) 0 0
\(309\) 9.00000 0.511992
\(310\) 40.0000 2.27185
\(311\) −33.0000 −1.87126 −0.935629 0.352985i \(-0.885167\pi\)
−0.935629 + 0.352985i \(0.885167\pi\)
\(312\) 2.00000 0.113228
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 3.00000 0.169300
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) −5.00000 −0.279946
\(320\) 4.00000 0.223607
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 1.00000 0.0555556
\(325\) 22.0000 1.22034
\(326\) −1.00000 −0.0553849
\(327\) −5.00000 −0.276501
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −20.0000 −1.10096
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −16.0000 −0.878114
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) 12.0000 0.650791
\(341\) −50.0000 −2.70765
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 4.00000 0.215353
\(346\) 3.00000 0.161281
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 1.00000 0.0536056
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −5.00000 −0.266501
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 8.00000 0.425195
\(355\) −28.0000 −1.48609
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 4.00000 0.210819
\(361\) 17.0000 0.894737
\(362\) 7.00000 0.367912
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 44.0000 2.30307
\(366\) 2.00000 0.104542
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) −24.0000 −1.24770
\(371\) 0 0
\(372\) 10.0000 0.518476
\(373\) −17.0000 −0.880227 −0.440113 0.897942i \(-0.645062\pi\)
−0.440113 + 0.897942i \(0.645062\pi\)
\(374\) −15.0000 −0.775632
\(375\) 24.0000 1.23935
\(376\) −9.00000 −0.464140
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 24.0000 1.23117
\(381\) 10.0000 0.512316
\(382\) −4.00000 −0.204658
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −6.00000 −0.304997
\(388\) 4.00000 0.203069
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 8.00000 0.405096
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 21.0000 1.05796
\(395\) −4.00000 −0.201262
\(396\) −5.00000 −0.251259
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 21.0000 1.05263
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 20.0000 0.996271
\(404\) −17.0000 −0.845782
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) 30.0000 1.48704
\(408\) 3.00000 0.148522
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −24.0000 −1.18528
\(411\) 13.0000 0.641243
\(412\) 9.00000 0.443398
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −64.0000 −3.14164
\(416\) 2.00000 0.0980581
\(417\) −5.00000 −0.244851
\(418\) −30.0000 −1.46735
\(419\) −35.0000 −1.70986 −0.854931 0.518742i \(-0.826401\pi\)
−0.854931 + 0.518742i \(0.826401\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) −11.0000 −0.535472
\(423\) −9.00000 −0.437595
\(424\) −6.00000 −0.291386
\(425\) 33.0000 1.60074
\(426\) −7.00000 −0.339151
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −10.0000 −0.482805
\(430\) −24.0000 −1.15738
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 1.00000 0.0481125
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −5.00000 −0.239457
\(437\) 6.00000 0.287019
\(438\) 11.0000 0.525600
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) −20.0000 −0.953463
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −40.0000 −1.90046 −0.950229 0.311553i \(-0.899151\pi\)
−0.950229 + 0.311553i \(0.899151\pi\)
\(444\) −6.00000 −0.284747
\(445\) −72.0000 −3.41313
\(446\) −22.0000 −1.04173
\(447\) 24.0000 1.13516
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 11.0000 0.518545
\(451\) 30.0000 1.41264
\(452\) 6.00000 0.282216
\(453\) 12.0000 0.563809
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) 3.00000 0.140181
\(459\) 3.00000 0.140028
\(460\) 4.00000 0.186501
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 1.00000 0.0464238
\(465\) 40.0000 1.85496
\(466\) 16.0000 0.741186
\(467\) 5.00000 0.231372 0.115686 0.993286i \(-0.463093\pi\)
0.115686 + 0.993286i \(0.463093\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −36.0000 −1.66056
\(471\) 3.00000 0.138233
\(472\) 8.00000 0.368230
\(473\) 30.0000 1.37940
\(474\) −1.00000 −0.0459315
\(475\) 66.0000 3.02829
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 5.00000 0.228695
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 4.00000 0.182574
\(481\) −12.0000 −0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 16.0000 0.726523
\(486\) 1.00000 0.0453609
\(487\) 14.0000 0.634401 0.317200 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) 2.00000 0.0905357
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 3.00000 0.135113
\(494\) 12.0000 0.539906
\(495\) −20.0000 −0.898933
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 27.0000 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(500\) 24.0000 1.07331
\(501\) 8.00000 0.357414
\(502\) −27.0000 −1.20507
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) −68.0000 −3.02596
\(506\) −5.00000 −0.222277
\(507\) −9.00000 −0.399704
\(508\) 10.0000 0.443678
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.00000 0.264906
\(514\) 30.0000 1.32324
\(515\) 36.0000 1.58635
\(516\) −6.00000 −0.264135
\(517\) 45.0000 1.97910
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 8.00000 0.350823
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 1.00000 0.0437688
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 30.0000 1.30682
\(528\) −5.00000 −0.217597
\(529\) 1.00000 0.0434783
\(530\) −24.0000 −1.04249
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −18.0000 −0.778936
\(535\) −48.0000 −2.07522
\(536\) −2.00000 −0.0863868
\(537\) −2.00000 −0.0863064
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) −10.0000 −0.429537
\(543\) 7.00000 0.300399
\(544\) 3.00000 0.128624
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 13.0000 0.555332
\(549\) 2.00000 0.0853579
\(550\) −55.0000 −2.34521
\(551\) 6.00000 0.255609
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) −24.0000 −1.01874
\(556\) −5.00000 −0.212047
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 10.0000 0.423334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) −13.0000 −0.548372
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) −9.00000 −0.378968
\(565\) 24.0000 1.00969
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −7.00000 −0.293713
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 24.0000 1.00525
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) −10.0000 −0.418121
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) 1.00000 0.0416667
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) −8.00000 −0.332756
\(579\) 6.00000 0.249351
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 30.0000 1.24247
\(584\) 11.0000 0.455183
\(585\) 8.00000 0.330759
\(586\) −4.00000 −0.165238
\(587\) −48.0000 −1.98117 −0.990586 0.136892i \(-0.956289\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) 32.0000 1.31742
\(591\) 21.0000 0.863825
\(592\) −6.00000 −0.246598
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) 21.0000 0.859473
\(598\) 2.00000 0.0817861
\(599\) −35.0000 −1.43006 −0.715031 0.699093i \(-0.753587\pi\)
−0.715031 + 0.699093i \(0.753587\pi\)
\(600\) 11.0000 0.449073
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 12.0000 0.488273
\(605\) 56.0000 2.27672
\(606\) −17.0000 −0.690578
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −18.0000 −0.728202
\(612\) 3.00000 0.121268
\(613\) −7.00000 −0.282727 −0.141364 0.989958i \(-0.545149\pi\)
−0.141364 + 0.989958i \(0.545149\pi\)
\(614\) −29.0000 −1.17034
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −9.00000 −0.362326 −0.181163 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 9.00000 0.362033
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 40.0000 1.60644
\(621\) 1.00000 0.0401286
\(622\) −33.0000 −1.32318
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 41.0000 1.64000
\(626\) 20.0000 0.799361
\(627\) −30.0000 −1.19808
\(628\) 3.00000 0.119713
\(629\) −18.0000 −0.717707
\(630\) 0 0
\(631\) −11.0000 −0.437903 −0.218952 0.975736i \(-0.570264\pi\)
−0.218952 + 0.975736i \(0.570264\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −11.0000 −0.437211
\(634\) 2.00000 0.0794301
\(635\) 40.0000 1.58735
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −5.00000 −0.197952
\(639\) −7.00000 −0.276916
\(640\) 4.00000 0.158114
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) −12.0000 −0.473602
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 18.0000 0.708201
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 1.00000 0.0392837
\(649\) −40.0000 −1.57014
\(650\) 22.0000 0.862911
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −7.00000 −0.273931 −0.136966 0.990576i \(-0.543735\pi\)
−0.136966 + 0.990576i \(0.543735\pi\)
\(654\) −5.00000 −0.195515
\(655\) 64.0000 2.50069
\(656\) −6.00000 −0.234261
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) −27.0000 −1.05177 −0.525885 0.850555i \(-0.676266\pi\)
−0.525885 + 0.850555i \(0.676266\pi\)
\(660\) −20.0000 −0.778499
\(661\) −15.0000 −0.583432 −0.291716 0.956505i \(-0.594226\pi\)
−0.291716 + 0.956505i \(0.594226\pi\)
\(662\) −8.00000 −0.310929
\(663\) 6.00000 0.233021
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 1.00000 0.0387202
\(668\) 8.00000 0.309529
\(669\) −22.0000 −0.850569
\(670\) −8.00000 −0.309067
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 14.0000 0.539260
\(675\) 11.0000 0.423390
\(676\) −9.00000 −0.346154
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 12.0000 0.460179
\(681\) 3.00000 0.114960
\(682\) −50.0000 −1.91460
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 6.00000 0.229416
\(685\) 52.0000 1.98682
\(686\) 0 0
\(687\) 3.00000 0.114457
\(688\) −6.00000 −0.228748
\(689\) −12.0000 −0.457164
\(690\) 4.00000 0.152277
\(691\) 49.0000 1.86405 0.932024 0.362397i \(-0.118041\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) −20.0000 −0.758643
\(696\) 1.00000 0.0379049
\(697\) −18.0000 −0.681799
\(698\) −18.0000 −0.681310
\(699\) 16.0000 0.605176
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 2.00000 0.0754851
\(703\) −36.0000 −1.35777
\(704\) −5.00000 −0.188445
\(705\) −36.0000 −1.35584
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 21.0000 0.788672 0.394336 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) −28.0000 −1.05082
\(711\) −1.00000 −0.0375029
\(712\) −18.0000 −0.674579
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) −40.0000 −1.49592
\(716\) −2.00000 −0.0747435
\(717\) 5.00000 0.186728
\(718\) −4.00000 −0.149279
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −10.0000 −0.371904
\(724\) 7.00000 0.260153
\(725\) 11.0000 0.408530
\(726\) 14.0000 0.519589
\(727\) 43.0000 1.59478 0.797391 0.603463i \(-0.206213\pi\)
0.797391 + 0.603463i \(0.206213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 44.0000 1.62851
\(731\) −18.0000 −0.665754
\(732\) 2.00000 0.0739221
\(733\) −43.0000 −1.58824 −0.794121 0.607760i \(-0.792068\pi\)
−0.794121 + 0.607760i \(0.792068\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 10.0000 0.368355
\(738\) −6.00000 −0.220863
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) −24.0000 −0.882258
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 10.0000 0.366618
\(745\) 96.0000 3.51717
\(746\) −17.0000 −0.622414
\(747\) −16.0000 −0.585409
\(748\) −15.0000 −0.548454
\(749\) 0 0
\(750\) 24.0000 0.876356
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −9.00000 −0.328196
\(753\) −27.0000 −0.983935
\(754\) 2.00000 0.0728357
\(755\) 48.0000 1.74690
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −16.0000 −0.581146
\(759\) −5.00000 −0.181489
\(760\) 24.0000 0.870572
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 10.0000 0.362262
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 12.0000 0.433861
\(766\) 6.00000 0.216789
\(767\) 16.0000 0.577727
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 6.00000 0.215945
\(773\) −32.0000 −1.15096 −0.575480 0.817816i \(-0.695185\pi\)
−0.575480 + 0.817816i \(0.695185\pi\)
\(774\) −6.00000 −0.215666
\(775\) 110.000 3.95132
\(776\) 4.00000 0.143592
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) −36.0000 −1.28983
\(780\) 8.00000 0.286446
\(781\) 35.0000 1.25240
\(782\) 3.00000 0.107280
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 16.0000 0.570701
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 21.0000 0.748094
\(789\) −24.0000 −0.854423
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) 4.00000 0.142044
\(794\) −26.0000 −0.922705
\(795\) −24.0000 −0.851192
\(796\) 21.0000 0.744325
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) 11.0000 0.388909
\(801\) −18.0000 −0.635999
\(802\) 15.0000 0.529668
\(803\) −55.0000 −1.94091
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 21.0000 0.739235
\(808\) −17.0000 −0.598058
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 4.00000 0.140546
\(811\) 25.0000 0.877869 0.438934 0.898519i \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 30.0000 1.05150
\(815\) −4.00000 −0.140114
\(816\) 3.00000 0.105021
\(817\) −36.0000 −1.25948
\(818\) 5.00000 0.174821
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 13.0000 0.453427
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 9.00000 0.313530
\(825\) −55.0000 −1.91485
\(826\) 0 0
\(827\) −43.0000 −1.49526 −0.747628 0.664117i \(-0.768807\pi\)
−0.747628 + 0.664117i \(0.768807\pi\)
\(828\) 1.00000 0.0347524
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −64.0000 −2.22147
\(831\) −4.00000 −0.138758
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −5.00000 −0.173136
\(835\) 32.0000 1.10741
\(836\) −30.0000 −1.03757
\(837\) 10.0000 0.345651
\(838\) −35.0000 −1.20905
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −35.0000 −1.20618
\(843\) −13.0000 −0.447744
\(844\) −11.0000 −0.378636
\(845\) −36.0000 −1.23844
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) 33.0000 1.13189
\(851\) −6.00000 −0.205677
\(852\) −7.00000 −0.239816
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) −12.0000 −0.410152
\(857\) 2.00000 0.0683187 0.0341593 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) −10.0000 −0.341394
\(859\) 51.0000 1.74010 0.870049 0.492966i \(-0.164087\pi\)
0.870049 + 0.492966i \(0.164087\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) 1.00000 0.0340404 0.0170202 0.999855i \(-0.494582\pi\)
0.0170202 + 0.999855i \(0.494582\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0000 0.408012
\(866\) 22.0000 0.747590
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 5.00000 0.169613
\(870\) 4.00000 0.135613
\(871\) −4.00000 −0.135535
\(872\) −5.00000 −0.169321
\(873\) 4.00000 0.135379
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −40.0000 −1.34993
\(879\) −4.00000 −0.134917
\(880\) −20.0000 −0.674200
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 6.00000 0.201802
\(885\) 32.0000 1.07567
\(886\) −40.0000 −1.34383
\(887\) 49.0000 1.64526 0.822629 0.568578i \(-0.192506\pi\)
0.822629 + 0.568578i \(0.192506\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −72.0000 −2.41345
\(891\) −5.00000 −0.167506
\(892\) −22.0000 −0.736614
\(893\) −54.0000 −1.80704
\(894\) 24.0000 0.802680
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 12.0000 0.400445
\(899\) 10.0000 0.333519
\(900\) 11.0000 0.366667
\(901\) −18.0000 −0.599667
\(902\) 30.0000 0.998891
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 28.0000 0.930751
\(906\) 12.0000 0.398673
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 3.00000 0.0995585
\(909\) −17.0000 −0.563854
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 6.00000 0.198680
\(913\) 80.0000 2.64761
\(914\) 12.0000 0.396925
\(915\) 8.00000 0.264472
\(916\) 3.00000 0.0991228
\(917\) 0 0
\(918\) 3.00000 0.0990148
\(919\) 17.0000 0.560778 0.280389 0.959886i \(-0.409536\pi\)
0.280389 + 0.959886i \(0.409536\pi\)
\(920\) 4.00000 0.131876
\(921\) −29.0000 −0.955582
\(922\) 14.0000 0.461065
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) −66.0000 −2.17007
\(926\) −20.0000 −0.657241
\(927\) 9.00000 0.295599
\(928\) 1.00000 0.0328266
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 40.0000 1.31165
\(931\) 0 0
\(932\) 16.0000 0.524097
\(933\) −33.0000 −1.08037
\(934\) 5.00000 0.163605
\(935\) −60.0000 −1.96221
\(936\) 2.00000 0.0653720
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) −36.0000 −1.17419
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 3.00000 0.0977453
\(943\) −6.00000 −0.195387
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 22.0000 0.714150
\(950\) 66.0000 2.14132
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 7.00000 0.226752 0.113376 0.993552i \(-0.463833\pi\)
0.113376 + 0.993552i \(0.463833\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) 5.00000 0.161712
\(957\) −5.00000 −0.161627
\(958\) −26.0000 −0.840022
\(959\) 0 0
\(960\) 4.00000 0.129099
\(961\) 69.0000 2.22581
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) 14.0000 0.449977
\(969\) 18.0000 0.578243
\(970\) 16.0000 0.513729
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 14.0000 0.448589
\(975\) 22.0000 0.704564
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 90.0000 2.87641
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) −12.0000 −0.382935
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −6.00000 −0.191273
\(985\) 84.0000 2.67646
\(986\) 3.00000 0.0955395
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −6.00000 −0.190789
\(990\) −20.0000 −0.635642
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 10.0000 0.317500
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 84.0000 2.66298
\(996\) −16.0000 −0.506979
\(997\) −48.0000 −1.52018 −0.760088 0.649821i \(-0.774844\pi\)
−0.760088 + 0.649821i \(0.774844\pi\)
\(998\) 27.0000 0.854670
\(999\) −6.00000 −0.189832
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 6762.2.a.bn.1.1 1
7.2 even 3 966.2.i.a.277.1 2
7.4 even 3 966.2.i.a.415.1 yes 2
7.6 odd 2 6762.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.a.277.1 2 7.2 even 3
966.2.i.a.415.1 yes 2 7.4 even 3
6762.2.a.w.1.1 1 7.6 odd 2
6762.2.a.bn.1.1 1 1.1 even 1 trivial