↑

Properties

Label 6762.2.a.h.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -6.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{46} -1.00000 q^{48} +5.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +2.00000 q^{57} +6.00000 q^{58} -8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -10.0000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -1.00000 q^{72} -14.0000 q^{73} -8.00000 q^{74} +5.00000 q^{75} -2.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -6.00000 q^{83} -2.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} -6.00000 q^{89} +1.00000 q^{92} +8.00000 q^{93} +1.00000 q^{96} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −8.00000 −0.929981
\(75\) 5.00000 0.577350
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) −5.00000 −0.500000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 1.00000 0.0851257
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) −5.00000 −0.408248
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −8.00000 −0.636446
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 2.00000 0.152499
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 − 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 36.0000 2.63258
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 5.00000 0.353553
\(201\) 10.0000 0.705346
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 8.00000 0.536925
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −6.00000 −0.399114
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 2.00000 0.132453
\(229\) −8.00000 −0.528655 −0.264327 − 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −10.0000 −0.610847
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −30.0000 −1.80907
\(276\) −1.00000 −0.0601929
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −16.0000 −0.959616
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −14.0000 −0.819288
\(293\) −24.0000 −1.40209 −0.701047 − 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −6.00000 −0.348155
\(298\) −12.0000 −0.695141
\(299\) −2.00000 −0.115663
\(300\) 5.00000 0.288675
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −2.00000 −0.113228
\(313\) 34.0000 1.92179 0.960897 − 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −12.0000 −0.672927
\(319\) −36.0000 −2.01561
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 10.0000 0.554700
\(326\) 16.0000 0.886158
\(327\) 4.00000 0.221201
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −6.00000 −0.329293
\(333\) 8.00000 0.438397
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 6.00000 0.321634
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 36.0000 1.90001 0.950004 − 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 8.00000 0.420471
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 − 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 12.0000 0.613973
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 2.00000 0.101666
\(388\) 10.0000 0.507673
\(389\) 24.0000 1.21685 0.608424 − 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −10.0000 −0.498755
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 48.0000 2.37927
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −16.0000 −0.783523
\(418\) 12.0000 0.586939
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 − 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −2.00000 −0.0956730
\(438\) −14.0000 −0.668946
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 5.00000 0.235702
\(451\) −36.0000 −1.69517
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −34.0000 −1.59045 −0.795226 − 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 8.00000 0.373815
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 8.00000 0.367452
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 24.0000 1.09773
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 8.00000 0.362143
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −36.0000 −1.62136
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 6.00000 0.267793
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 42.0000 1.86162 0.930809 − 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 6.00000 0.262613
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) −6.00000 −0.261116
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) 24.0000 1.03568
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 − 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −16.0000 −0.687259
\(543\) 8.00000 0.343313
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) −8.00000 −0.341432
\(550\) 30.0000 1.27920
\(551\) 12.0000 0.511217
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 8.00000 0.338667
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) −6.00000 −0.253095
\(563\) −18.0000 −0.758610 −0.379305 − 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −10.0000 −0.420331
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 − 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) −12.0000 −0.501745
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −5.00000 −0.208514
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −19.0000 −0.790296
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −72.0000 −2.98194
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 8.00000 0.328798
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) −16.0000 −0.654836
\(598\) 2.00000 0.0817861
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −5.00000 −0.204124
\(601\) −14.0000 −0.571072 −0.285536 − 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 20.0000 0.807792 0.403896 − 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 4.00000 0.160904
\(619\) −38.0000 −1.52735 −0.763674 − 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) −34.0000 −1.35891
\(627\) 12.0000 0.479234
\(628\) 4.00000 0.159617
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −6.00000 −0.236801
\(643\) 34.0000 1.34083 0.670415 − 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −10.0000 −0.392232
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −20.0000 −0.777322
\(663\) 12.0000 0.466041
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −6.00000 −0.232321
\(668\) −24.0000 −0.928588
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 22.0000 0.847408
\(675\) 5.00000 0.192450
\(676\) −9.00000 −0.346154
\(677\) −24.0000 −0.922395 −0.461197 − 0.887298i \(-0.652580\pi\)
−0.461197 + 0.887298i \(0.652580\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 48.0000 1.83801
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 2.00000 0.0762493
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −36.0000 −1.36360
\(698\) −10.0000 −0.378506
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −16.0000 −0.603451
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 − 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 24.0000 0.896296
\(718\) −36.0000 −1.34351
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −10.0000 −0.371904
\(724\) −8.00000 −0.297318
\(725\) 30.0000 1.11417
\(726\) 25.0000 0.927837
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 8.00000 0.295689
\(733\) −20.0000 −0.738717 −0.369358 − 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −60.0000 −2.21013
\(738\) 6.00000 0.220863
\(739\) 8.00000 0.294285 0.147142 − 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 16.0000 0.585802
\(747\) −6.00000 −0.219529
\(748\) 36.0000 1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −52.0000 −1.88997 −0.944986 − 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 10.0000 0.363216
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −22.0000 −0.791797
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 40.0000 1.43684
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) −6.00000 −0.214560
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) −6.00000 −0.213741
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) 16.0000 0.568177
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) −84.0000 −2.96430
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) −48.0000 −1.68240
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −4.00000 −0.139942
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −6.00000 −0.209274
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −4.00000 −0.139347
\(825\) 30.0000 1.04447
\(826\) 0 0
\(827\) −30.0000 −1.04320 −0.521601 − 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 8.00000 0.276520
\(838\) −18.0000 −0.621800
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) −10.0000 −0.343199
\(850\) 30.0000 1.02899
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) −12.0000 −0.409673
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 38.0000 1.29129
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 4.00000 0.135457
\(873\) 10.0000 0.338449
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 8.00000 0.269987
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 − 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) −6.00000 −0.200223
\(899\) 48.0000 1.60089
\(900\) −5.00000 −0.166667
\(901\) −72.0000 −2.39867
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 2.00000 0.0664089 0.0332045 − 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −18.0000 −0.597351
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 2.00000 0.0662266
\(913\) −36.0000 −1.19143
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) 16.0000 0.525793
\(927\) 4.00000 0.131377
\(928\) 6.00000 0.196960
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 46.0000 1.50275 0.751377 − 0.659873i \(-0.229390\pi\)
0.751377 + 0.659873i \(0.229390\pi\)
\(938\) 0 0
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 − 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 4.00000 0.130327
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) −8.00000 −0.259828
\(949\) 28.0000 0.908918
\(950\) −10.0000 −0.324443
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 36.0000 1.16371
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 16.0000 0.515861
\(963\) −6.00000 −0.193347
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −25.0000 −0.803530
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −10.0000 −0.320256
\(976\) −8.00000 −0.256074
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −16.0000 −0.511624
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −60.0000 −1.91370 −0.956851 − 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −14.0000 −0.443384 −0.221692 − 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 16.0000 0.506471
\(999\) −8.00000 −0.253109
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.h.1.1 1
7.6 odd 2 966.2.a.f.1.1 ✓ 1
21.20 even 2 2898.2.a.o.1.1 1
28.27 even 2 7728.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.f.1.1 ✓ 1 7.6 odd 2
2898.2.a.o.1.1 1 21.20 even 2
6762.2.a.h.1.1 1 1.1 even 1 trivial
7728.2.a.c.1.1 1 28.27 even 2