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Properties

Label 4998.2.a.b.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -6.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +14.0000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{57} +10.0000 q^{58} -14.0000 q^{59} +2.00000 q^{60} +14.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} -6.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -2.00000 q^{85} +8.00000 q^{86} +10.0000 q^{87} +4.00000 q^{88} +2.00000 q^{89} +2.00000 q^{90} +6.00000 q^{92} -8.00000 q^{93} -4.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −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
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) −6.00000 −0.884652
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 10.0000 1.31306
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 2.00000 0.258199
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −2.00000 −0.223607
\(81\) 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
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) 10.0000 1.07211
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −8.00000 −0.829561
\(94\) −4.00000 −0.412568
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 4.00000 0.398015 0.199007 − 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 1.00000 0.0990148
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.00000 −0.766261 −0.383131 − 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 − 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 4.00000 0.374634
\(115\) −12.0000 −1.11901
\(116\) −10.0000 −0.928477
\(117\) −2.00000 −0.184900
\(118\) 14.0000 1.28880
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 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
\(129\) 8.00000 0.704361
\(130\) −4.00000 −0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) −1.00000 −0.0857493
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 6.00000 0.510754
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 10.0000 0.839181
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 20.0000 1.66091
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −4.00000 −0.318223
\(159\) −14.0000 −1.11027
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 18.0000 1.40987 0.704934 − 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) 6.00000 0.465690
\(167\) 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
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) −8.00000 −0.609994
\(173\) −22.0000 −1.67263 −0.836315 − 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 14.0000 1.05230
\(178\) −2.00000 −0.149906
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) −4.00000 −0.292509
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 4.00000 0.284268
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −4.00000 −0.281439
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) −12.0000 −0.838116
\(206\) 14.0000 0.975426
\(207\) 6.00000 0.417029
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 14.0000 0.961524
\(213\) 10.0000 0.685189
\(214\) 8.00000 0.546869
\(215\) 16.0000 1.09119
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 8.00000 0.541828
\(219\) −14.0000 −0.946032
\(220\) 8.00000 0.539360
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 20.0000 1.33038
\(227\) −28.0000 −1.85843 −0.929213 − 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) −14.0000 −0.911322
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 2.00000 0.129099
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) −8.00000 −0.508001
\(249\) 6.00000 0.380235
\(250\) −12.0000 −0.758947
\(251\) 10.0000 0.631194 0.315597 − 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) −8.00000 −0.501965
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) −28.0000 −1.72003
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −2.00000 −0.121716
\(271\) 10.0000 0.607457 0.303728 − 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) −6.00000 −0.361158
\(277\) −32.0000 −1.92269 −0.961347 − 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) −12.0000 −0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 4.00000 0.238197
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −10.0000 −0.593391
\(285\) 8.00000 0.473879
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −20.0000 −1.17444
\(291\) −2.00000 −0.117242
\(292\) 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
\(295\) 28.0000 1.63022
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) −2.00000 −0.115857
\(299\) −12.0000 −0.693978
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −4.00000 −0.229794
\(304\) 4.00000 0.229416
\(305\) −28.0000 −1.60328
\(306\) −1.00000 −0.0571662
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 16.0000 0.908739
\(311\) −16.0000 −0.907277 −0.453638 − 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −2.00000 −0.113228
\(313\) 18.0000 1.01742 0.508710 − 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 14.0000 0.785081
\(319\) 40.0000 2.23957
\(320\) −2.00000 −0.111803
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −18.0000 −0.996928
\(327\) 8.00000 0.442401
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) 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
\(339\) 20.0000 1.08625
\(340\) −2.00000 −0.108465
\(341\) −32.0000 −1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 12.0000 0.646058
\(346\) 22.0000 1.18273
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 10.0000 0.536056
\(349\) −6.00000 −0.321173 −0.160586 − 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −14.0000 −0.744092
\(355\) 20.0000 1.06149
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 14.0000 0.731792
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 6.00000 0.312772
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 4.00000 0.206835
\(375\) −12.0000 −0.619677
\(376\) −4.00000 −0.206284
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 − 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −8.00000 −0.410391
\(381\) −8.00000 −0.409852
\(382\) −24.0000 −1.22795
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 4.00000 0.202548
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −8.00000 −0.402524
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 4.00000 0.199502
\(403\) −16.0000 −0.797017
\(404\) 4.00000 0.199007
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) −24.0000 −1.18672 −0.593362 − 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 12.0000 0.592638
\(411\) −2.00000 −0.0986527
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) −12.0000 −0.587643
\(418\) 16.0000 0.782586
\(419\) 16.0000 0.781651 0.390826 − 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 − 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 4.00000 0.194487
\(424\) −14.0000 −0.679900
\(425\) −1.00000 −0.0485071
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −8.00000 −0.386244
\(430\) −16.0000 −0.771589
\(431\) 22.0000 1.05970 0.529851 − 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) −20.0000 −0.958927
\(436\) −8.00000 −0.383131
\(437\) 24.0000 1.14808
\(438\) 14.0000 0.668946
\(439\) 12.0000 0.572729 0.286364 − 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) −2.00000 −0.0947027
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 1.00000 0.0471405
\(451\) −24.0000 −1.13012
\(452\) −20.0000 −0.940721
\(453\) 8.00000 0.375873
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 2.00000 0.0934539
\(459\) −1.00000 −0.0466760
\(460\) −12.0000 −0.559503
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −10.0000 −0.464238
\(465\) 16.0000 0.741982
\(466\) −8.00000 −0.370593
\(467\) 26.0000 1.20314 0.601568 − 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) 14.0000 0.645086
\(472\) 14.0000 0.644402
\(473\) 32.0000 1.47136
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 14.0000 0.641016
\(478\) −12.0000 −0.548867
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 − 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −14.0000 −0.633750
\(489\) −18.0000 −0.813988
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) −10.0000 −0.450377
\(494\) 8.00000 0.359937
\(495\) 8.00000 0.359573
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 6.00000 0.268597 0.134298 − 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 12.0000 0.536656
\(501\) −8.00000 −0.357414
\(502\) −10.0000 −0.446322
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 24.0000 1.06693
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) −16.0000 −0.709188 −0.354594 − 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 22.0000 0.970378
\(515\) 28.0000 1.23383
\(516\) 8.00000 0.352180
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) −4.00000 −0.175412
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 10.0000 0.437688
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) 4.00000 0.174078
\(529\) 13.0000 0.565217
\(530\) 28.0000 1.21624
\(531\) −14.0000 −0.607548
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 2.00000 0.0865485
\(535\) 16.0000 0.691740
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −10.0000 −0.431131
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −20.0000 −0.859867 −0.429934 − 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) −10.0000 −0.429537
\(543\) −2.00000 −0.0858282
\(544\) −1.00000 −0.0428746
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 − 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 2.00000 0.0854358
\(549\) 14.0000 0.597505
\(550\) −4.00000 −0.170561
\(551\) −40.0000 −1.70406
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 32.0000 1.35955
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −8.00000 −0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 30.0000 1.26547
\(563\) 14.0000 0.590030 0.295015 − 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) −4.00000 −0.168430
\(565\) 40.0000 1.68281
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) −8.00000 −0.335083
\(571\) −30.0000 −1.25546 −0.627730 − 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 8.00000 0.334497
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −28.0000 −1.16566 −0.582828 − 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 10.0000 0.415586
\(580\) 20.0000 0.830455
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −56.0000 −2.31928
\(584\) −14.0000 −0.579324
\(585\) 4.00000 0.165380
\(586\) 24.0000 0.991431
\(587\) 46.0000 1.89862 0.949312 − 0.314337i \(-0.101782\pi\)
0.949312 + 0.314337i \(0.101782\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) −28.0000 −1.15274
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 4.00000 0.163709
\(598\) 12.0000 0.490716
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 22.0000 0.897399 0.448699 − 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −10.0000 −0.406558
\(606\) 4.00000 0.162489
\(607\) −12.0000 −0.487065 −0.243532 − 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) −8.00000 −0.323645
\(612\) 1.00000 0.0404226
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 16.0000 0.645707
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 − 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) −14.0000 −0.563163
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) −16.0000 −0.642575
\(621\) −6.00000 −0.240772
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −18.0000 −0.719425
\(627\) 16.0000 0.638978
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −4.00000 −0.159111
\(633\) −2.00000 −0.0794929
\(634\) 30.0000 1.19145
\(635\) −16.0000 −0.634941
\(636\) −14.0000 −0.555136
\(637\) 0 0
\(638\) −40.0000 −1.58362
\(639\) −10.0000 −0.395594
\(640\) 2.00000 0.0790569
\(641\) −32.0000 −1.26392 −0.631962 − 0.774999i \(-0.717750\pi\)
−0.631962 + 0.774999i \(0.717750\pi\)
\(642\) −8.00000 −0.315735
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −4.00000 −0.157378
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 56.0000 2.19819
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 18.0000 0.704934
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −8.00000 −0.311400
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 4.00000 0.155464
\(663\) 2.00000 0.0776736
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) −60.0000 −2.32321
\(668\) 8.00000 0.309529
\(669\) −2.00000 −0.0773245
\(670\) 8.00000 0.309067
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 22.0000 0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 − 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −20.0000 −0.768095
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 28.0000 1.07296
\(682\) 32.0000 1.22534
\(683\) −32.0000 −1.22445 −0.612223 − 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) 4.00000 0.152944
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) −8.00000 −0.304997
\(689\) −28.0000 −1.06672
\(690\) −12.0000 −0.456832
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −24.0000 −0.910372
\(696\) −10.0000 −0.379049
\(697\) 6.00000 0.227266
\(698\) 6.00000 0.227103
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 14.0000 0.526152
\(709\) 40.0000 1.50223 0.751116 − 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) −20.0000 −0.750587
\(711\) 4.00000 0.150012
\(712\) −2.00000 −0.0749532
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) −12.0000 −0.448148
\(718\) 20.0000 0.746393
\(719\) −48.0000 −1.79010 −0.895049 − 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −26.0000 −0.966950
\(724\) 2.00000 0.0743294
\(725\) 10.0000 0.371391
\(726\) 5.00000 0.185567
\(727\) −26.0000 −0.964287 −0.482143 − 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) −8.00000 −0.295891
\(732\) −14.0000 −0.517455
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 8.00000 0.293294
\(745\) −4.00000 −0.146549
\(746\) −10.0000 −0.366126
\(747\) −6.00000 −0.219529
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 4.00000 0.145865
\(753\) −10.0000 −0.364420
\(754\) −20.0000 −0.728357
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 − 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 18.0000 0.653789
\(759\) 24.0000 0.871145
\(760\) 8.00000 0.290191
\(761\) −26.0000 −0.942499 −0.471250 − 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) −2.00000 −0.0723102
\(766\) 24.0000 0.867155
\(767\) 28.0000 1.01102
\(768\) −1.00000 −0.0360844
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −10.0000 −0.359908
\(773\) 24.0000 0.863220 0.431610 − 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 8.00000 0.287554
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) 40.0000 1.43131
\(782\) −6.00000 −0.214560
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 18.0000 0.641223
\(789\) −8.00000 −0.284808
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −28.0000 −0.994309
\(794\) 2.00000 0.0709773
\(795\) 28.0000 0.993058
\(796\) −4.00000 −0.141776
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 1.00000 0.0353553
\(801\) 2.00000 0.0706665
\(802\) 12.0000 0.423735
\(803\) −56.0000 −1.97620
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −10.0000 −0.352017
\(808\) −4.00000 −0.140720
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 2.00000 0.0702728
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) −36.0000 −1.26102
\(816\) −1.00000 −0.0350070
\(817\) −32.0000 −1.11954
\(818\) 24.0000 0.839140
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 2.00000 0.0697580
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 14.0000 0.487713
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 6.00000 0.208514
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) −12.0000 −0.416526
\(831\) 32.0000 1.11007
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) −16.0000 −0.553372
\(837\) −8.00000 −0.276520
\(838\) −16.0000 −0.552711
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 30.0000 1.03387
\(843\) 30.0000 1.03325
\(844\) 2.00000 0.0688428
\(845\) 18.0000 0.619219
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) 14.0000 0.480762
\(849\) 4.00000 0.137280
\(850\) 1.00000 0.0342997
\(851\) 0 0
\(852\) 10.0000 0.342594
\(853\) −2.00000 −0.0684787 −0.0342393 − 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 8.00000 0.273434
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 8.00000 0.273115
\(859\) 12.0000 0.409435 0.204717 − 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −22.0000 −0.749323
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 1.00000 0.0340207
\(865\) 44.0000 1.49604
\(866\) −4.00000 −0.135926
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 20.0000 0.678064
\(871\) −8.00000 −0.271070
\(872\) 8.00000 0.270914
\(873\) 2.00000 0.0676897
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −12.0000 −0.405211 −0.202606 − 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) −12.0000 −0.404980
\(879\) 24.0000 0.809500
\(880\) 8.00000 0.269680
\(881\) −2.00000 −0.0673817 −0.0336909 − 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −28.0000 −0.941210
\(886\) 12.0000 0.403148
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) −4.00000 −0.134005
\(892\) 2.00000 0.0669650
\(893\) 16.0000 0.535420
\(894\) 2.00000 0.0668900
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) −12.0000 −0.400445
\(899\) −80.0000 −2.66815
\(900\) −1.00000 −0.0333333
\(901\) 14.0000 0.466408
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) −4.00000 −0.132964
\(906\) −8.00000 −0.265782
\(907\) 46.0000 1.52740 0.763702 − 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) −28.0000 −0.929213
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 − 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) −4.00000 −0.132453
\(913\) 24.0000 0.794284
\(914\) 22.0000 0.727695
\(915\) 28.0000 0.925651
\(916\) −2.00000 −0.0660819
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −48.0000 −1.58337 −0.791687 − 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 12.0000 0.395628
\(921\) 16.0000 0.527218
\(922\) 24.0000 0.790398
\(923\) 20.0000 0.658308
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) −14.0000 −0.459820
\(928\) 10.0000 0.328266
\(929\) −54.0000 −1.77168 −0.885841 − 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) −16.0000 −0.524661
\(931\) 0 0
\(932\) 8.00000 0.262049
\(933\) 16.0000 0.523816
\(934\) −26.0000 −0.850746
\(935\) 8.00000 0.261628
\(936\) 2.00000 0.0653720
\(937\) −36.0000 −1.17607 −0.588034 − 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) −8.00000 −0.260931
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −14.0000 −0.456145
\(943\) 36.0000 1.17232
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −4.00000 −0.129914
\(949\) −28.0000 −0.908918
\(950\) 4.00000 0.129777
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −14.0000 −0.453267
\(955\) −48.0000 −1.55324
\(956\) 12.0000 0.388108
\(957\) −40.0000 −1.29302
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 26.0000 0.837404
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −5.00000 −0.160706
\(969\) −4.00000 −0.128499
\(970\) 4.00000 0.128432
\(971\) 22.0000 0.706014 0.353007 − 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −28.0000 −0.897178
\(975\) −2.00000 −0.0640513
\(976\) 14.0000 0.448129
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 18.0000 0.575577
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) −12.0000 −0.382935
\(983\) 16.0000 0.510321 0.255160 − 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 6.00000 0.191273
\(985\) −36.0000 −1.14706
\(986\) 10.0000 0.318465
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −48.0000 −1.52631
\(990\) −8.00000 −0.254257
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −8.00000 −0.254000
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 6.00000 0.190117
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −6.00000 −0.189927
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4998.2.a.b.1.1 ✓ 1
7.6 odd 2 4998.2.a.t.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4998.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
4998.2.a.t.1.1 yes 1 7.6 odd 2