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Properties

Label 6762.2.a.s.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: yes
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{57} -2.00000 q^{58} +8.00000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +12.0000 q^{85} -10.0000 q^{86} +2.00000 q^{87} -2.00000 q^{88} -2.00000 q^{89} -2.00000 q^{90} +1.00000 q^{92} -2.00000 q^{93} -2.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 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
\(15\) 2.00000 0.516398
\(16\) 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
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −6.00000 −1.02899
\(35\) 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
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 2.00000 0.301511
\(45\) 2.00000 0.298142
\(46\) −1.00000 −0.147442
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −2.00000 −0.246183
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 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
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −10.0000 −1.07833
\(87\) 2.00000 0.214423
\(88\) −2.00000 −0.213201
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −2.00000 −0.207390
\(94\) −2.00000 −0.206284
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) 2.00000 0.186501
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −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
\(129\) 10.0000 0.880451
\(130\) −4.00000 −0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 2.00000 0.172133
\(136\) −6.00000 −0.514496
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) −8.00000 −0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) −4.00000 −0.317221
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 12.0000 0.931381
\(167\) −2.00000 −0.154765 −0.0773823 − 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) −4.00000 −0.305888
\(172\) 10.0000 0.762493
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 8.00000 0.601317
\(178\) 2.00000 0.149906
\(179\) 24.0000 1.79384 0.896922 − 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) 12.0000 0.877527
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −2.00000 −0.142134
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.00000 −0.141069
\(202\) −14.0000 −0.985037
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −4.00000 −0.274721
\(213\) 8.00000 0.548151
\(214\) 2.00000 0.136717
\(215\) 20.0000 1.36399
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 4.00000 0.270295
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 − 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) 28.0000 1.85843 0.929213 − 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.00000 0.260931
\(236\) 8.00000 0.520756
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 2.00000 0.127000
\(249\) −12.0000 −0.760469
\(250\) 12.0000 0.758947
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −8.00000 −0.501965
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 4.00000 0.249513 0.124757 − 0.992187i \(-0.460185\pi\)
0.124757 + 0.992187i \(0.460185\pi\)
\(258\) −10.0000 −0.622573
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −2.00000 −0.123091
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) −2.00000 −0.122169
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −2.00000 −0.121716
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −2.00000 −0.120605
\(276\) 1.00000 0.0601929
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 12.0000 0.719712
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −2.00000 −0.119098
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 8.00000 0.474713
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) −2.00000 −0.117242
\(292\) 4.00000 0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 4.00000 0.231714
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 14.0000 0.804279
\(304\) −4.00000 −0.229416
\(305\) −4.00000 −0.229039
\(306\) −6.00000 −0.342997
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) −2.00000 −0.113410 −0.0567048 − 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) −2.00000 −0.113228
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 4.00000 0.224309
\(319\) 4.00000 0.223957
\(320\) 2.00000 0.111803
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 2.00000 0.109435
\(335\) −4.00000 −0.218543
\(336\) 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
\(339\) −14.0000 −0.760376
\(340\) 12.0000 0.650791
\(341\) −4.00000 −0.216612
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −10.0000 −0.539164
\(345\) 2.00000 0.107676
\(346\) 14.0000 0.752645
\(347\) −36.0000 −1.93258 −0.966291 − 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 2.00000 0.107211
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −8.00000 −0.425195
\(355\) 16.0000 0.849192
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 2.00000 0.104542
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −32.0000 −1.65690 −0.828449 − 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) −12.0000 −0.620505
\(375\) −12.0000 −0.619677
\(376\) −2.00000 −0.103142
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 − 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −8.00000 −0.410391
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 10.0000 0.508329
\(388\) −2.00000 −0.101535
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −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
\(396\) 2.00000 0.100504
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 2.00000 0.0997509
\(403\) −4.00000 −0.199254
\(404\) 14.0000 0.696526
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −24.0000 −1.17811
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 8.00000 0.391293
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −8.00000 −0.389434
\(423\) 2.00000 0.0972433
\(424\) 4.00000 0.194257
\(425\) −6.00000 −0.291043
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −2.00000 −0.0966736
\(429\) 4.00000 0.193122
\(430\) −20.0000 −0.964486
\(431\) −20.0000 −0.963366 −0.481683 − 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 4.00000 0.191565
\(437\) −4.00000 −0.191346
\(438\) −4.00000 −0.191127
\(439\) 6.00000 0.286364 0.143182 − 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −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
\(446\) 14.0000 0.662919
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 16.0000 0.751746
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 14.0000 0.654892 0.327446 − 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −14.0000 −0.654177
\(459\) 6.00000 0.280056
\(460\) 2.00000 0.0932505
\(461\) 42.0000 1.95614 0.978068 − 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 2.00000 0.0928477
\(465\) −4.00000 −0.185496
\(466\) 22.0000 1.01913
\(467\) 4.00000 0.185098 0.0925490 − 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 2.00000 0.0921551
\(472\) −8.00000 −0.368230
\(473\) 20.0000 0.919601
\(474\) −4.00000 −0.183726
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 − 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 2.00000 0.0905357
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) 4.00000 0.179787
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −12.0000 −0.536656
\(501\) −2.00000 −0.0893534
\(502\) 4.00000 0.178529
\(503\) −20.0000 −0.891756 −0.445878 − 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) −2.00000 −0.0889108
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −4.00000 −0.176432
\(515\) 16.0000 0.705044
\(516\) 10.0000 0.440225
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −4.00000 −0.175412
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −12.0000 −0.522728
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 8.00000 0.347498
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 2.00000 0.0865485
\(535\) −4.00000 −0.172935
\(536\) 2.00000 0.0863868
\(537\) 24.0000 1.03568
\(538\) −2.00000 −0.0862261
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −38.0000 −1.63375 −0.816874 − 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −2.00000 −0.0858282
\(544\) −6.00000 −0.257248
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 2.00000 0.0852803
\(551\) −8.00000 −0.340811
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 2.00000 0.0846668
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 18.0000 0.759284
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 2.00000 0.0842152
\(565\) −28.0000 −1.17797
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 8.00000 0.335083
\(571\) −18.0000 −0.753277 −0.376638 − 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 − 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) −19.0000 −0.790296
\(579\) 14.0000 0.581820
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −8.00000 −0.331326
\(584\) −4.00000 −0.165521
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) −4.00000 −0.165098 −0.0825488 − 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −16.0000 −0.658710
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 4.00000 0.163709
\(598\) −2.00000 −0.0817861
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000 0.0408248
\(601\) −4.00000 −0.163163 −0.0815817 − 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 16.0000 0.651031
\(605\) −14.0000 −0.569181
\(606\) −14.0000 −0.568711
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 4.00000 0.161823
\(612\) 6.00000 0.242536
\(613\) 44.0000 1.77714 0.888572 − 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 − 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −8.00000 −0.321807
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −4.00000 −0.160644
\(621\) 1.00000 0.0401286
\(622\) 2.00000 0.0801927
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) −8.00000 −0.319489
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 − 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) −4.00000 −0.159111
\(633\) 8.00000 0.317971
\(634\) 18.0000 0.714871
\(635\) 16.0000 0.634941
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 2.00000 0.0789337
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 20.0000 0.787499
\(646\) 24.0000 0.944267
\(647\) 42.0000 1.65119 0.825595 − 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −14.0000 −0.545363 −0.272681 − 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 4.00000 0.155700
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) −2.00000 −0.0773823
\(669\) −14.0000 −0.541271
\(670\) 4.00000 0.154533
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) 28.0000 1.07296
\(682\) 4.00000 0.153168
\(683\) 16.0000 0.612223 0.306111 − 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −4.00000 −0.152944
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 10.0000 0.381246
\(689\) −8.00000 −0.304776
\(690\) −2.00000 −0.0761387
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) −24.0000 −0.910372
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 30.0000 1.13552
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 − 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 4.00000 0.150649
\(706\) 0 0
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 16.0000 0.600893 0.300446 − 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) −16.0000 −0.600469
\(711\) 4.00000 0.150012
\(712\) 2.00000 0.0749532
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 18.0000 0.671287 0.335643 − 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 10.0000 0.371904
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 7.00000 0.259794
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) 60.0000 2.21918
\(732\) −2.00000 −0.0739221
\(733\) 46.0000 1.69905 0.849524 − 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 2.00000 0.0733236
\(745\) −8.00000 −0.293097
\(746\) 32.0000 1.17160
\(747\) −12.0000 −0.439057
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 2.00000 0.0729325
\(753\) −4.00000 −0.145768
\(754\) −4.00000 −0.145671
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) 4.00000 0.145382 0.0726912 − 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 30.0000 1.08965
\(759\) 2.00000 0.0725954
\(760\) 8.00000 0.290191
\(761\) −4.00000 −0.145000 −0.0724999 − 0.997368i \(-0.523098\pi\)
−0.0724999 + 0.997368i \(0.523098\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 14.0000 0.503871
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −10.0000 −0.359443
\(775\) 2.00000 0.0718421
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 16.0000 0.572525
\(782\) −6.00000 −0.214560
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −18.0000 −0.641223
\(789\) −16.0000 −0.569615
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) −4.00000 −0.142044
\(794\) −2.00000 −0.0709773
\(795\) −8.00000 −0.283731
\(796\) 4.00000 0.141776
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −18.0000 −0.635602
\(803\) 8.00000 0.282314
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 2.00000 0.0704033
\(808\) −14.0000 −0.492518
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 6.00000 0.210042
\(817\) −40.0000 −1.39942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 − 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −42.0000 −1.46048 −0.730242 − 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 1.00000 0.0347524
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 24.0000 0.833052
\(831\) −14.0000 −0.485655
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −4.00000 −0.138426
\(836\) −8.00000 −0.276686
\(837\) −2.00000 −0.0691301
\(838\) −20.0000 −0.690889
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 8.00000 0.275371
\(845\) −18.0000 −0.619219
\(846\) −2.00000 −0.0687614
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 28.0000 0.960958
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) 50.0000 1.71197 0.855984 − 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 2.00000 0.0683586
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −4.00000 −0.136558
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 20.0000 0.681994
\(861\) 0 0
\(862\) 20.0000 0.681203
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.0000 −0.952029
\(866\) −14.0000 −0.475739
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) −4.00000 −0.135613
\(871\) −4.00000 −0.135535
\(872\) −4.00000 −0.135457
\(873\) −2.00000 −0.0676897
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −6.00000 −0.202490
\(879\) 6.00000 0.202375
\(880\) 4.00000 0.134840
\(881\) −46.0000 −1.54978 −0.774890 − 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 12.0000 0.403604
\(885\) 16.0000 0.537834
\(886\) 12.0000 0.403148
\(887\) 30.0000 1.00730 0.503651 − 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) −8.00000 −0.267710
\(894\) 4.00000 0.133780
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 6.00000 0.200223
\(899\) −4.00000 −0.133407
\(900\) −1.00000 −0.0333333
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −4.00000 −0.132964
\(906\) −16.0000 −0.531564
\(907\) −2.00000 −0.0664089 −0.0332045 − 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) 28.0000 0.929213
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 − 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) −4.00000 −0.132453
\(913\) −24.0000 −0.794284
\(914\) −14.0000 −0.463079
\(915\) −4.00000 −0.132236
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −12.0000 −0.395843 −0.197922 − 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 16.0000 0.527218
\(922\) −42.0000 −1.38320
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) −48.0000 −1.57483 −0.787414 − 0.616424i \(-0.788581\pi\)
−0.787414 + 0.616424i \(0.788581\pi\)
\(930\) 4.00000 0.131165
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) −2.00000 −0.0654771
\(934\) −4.00000 −0.130884
\(935\) 24.0000 0.784884
\(936\) −2.00000 −0.0653720
\(937\) 6.00000 0.196011 0.0980057 − 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 4.00000 0.130466
\(941\) −54.0000 −1.76035 −0.880175 − 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 44.0000 1.42981 0.714904 − 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 4.00000 0.129914
\(949\) 8.00000 0.259691
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) 0 0
\(957\) 4.00000 0.129302
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) 10.0000 0.322078
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 7.00000 0.224989
\(969\) −24.0000 −0.770991
\(970\) 4.00000 0.128432
\(971\) −4.00000 −0.128366 −0.0641831 − 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 26.0000 0.831814 0.415907 − 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) −12.0000 −0.383718
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 12.0000 0.382935
\(983\) −4.00000 −0.127580 −0.0637901 − 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 10.0000 0.317982
\(990\) −4.00000 −0.127128
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 2.00000 0.0635001
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) −12.0000 −0.380235
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −12.0000 −0.379853
\(999\) 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 6762.2.a.s.1.1 yes 1
7.6 odd 2 6762.2.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.d.1.1 ✓ 1 7.6 odd 2
6762.2.a.s.1.1 yes 1 1.1 even 1 trivial