Properties

Label 6762.2.a.y.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 $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -3.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} +3.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -9.00000 q^{37} -3.00000 q^{39} -3.00000 q^{40} -9.00000 q^{41} -3.00000 q^{43} +4.00000 q^{44} -3.00000 q^{45} +1.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -4.00000 q^{51} +3.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -12.0000 q^{55} +3.00000 q^{58} -6.00000 q^{59} +3.00000 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -9.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -9.00000 q^{74} -4.00000 q^{75} -3.00000 q^{78} +8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -4.00000 q^{83} -12.0000 q^{85} -3.00000 q^{86} -3.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -6.00000 q^{93} +7.00000 q^{94} -1.00000 q^{96} +7.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 3.00000 0.588348
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 3.00000 0.547723
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) −3.00000 −0.474342
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −3.00000 −0.457496 −0.228748 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) 4.00000 0.603023
\(45\) −3.00000 −0.447214
\(46\) 1.00000 0.147442
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −4.00000 −0.560112
\(52\) 3.00000 0.416025
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 3.00000 0.387298
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.00000 −1.11631
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −9.00000 −1.04623
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −3.00000 −0.323498
\(87\) −3.00000 −0.321634
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −6.00000 −0.622171
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 4.00000 0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −4.00000 −0.396059
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) −12.0000 −1.14416
\(111\) 9.00000 0.854242
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 3.00000 0.278543
\(117\) 3.00000 0.277350
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 9.00000 0.811503
\(124\) 6.00000 0.538816
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.00000 0.264135
\(130\) −9.00000 −0.789352
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 3.00000 0.258199
\(136\) 4.00000 0.342997
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −9.00000 −0.763370 −0.381685 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) −7.00000 −0.589506
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) −9.00000 −0.739795
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) −4.00000 −0.326599
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −18.0000 −1.44579
\(156\) −3.00000 −0.240192
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 8.00000 0.636446
\(159\) 4.00000 0.317221
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −9.00000 −0.702782
\(165\) 12.0000 0.934199
\(166\) −4.00000 −0.310460
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) −3.00000 −0.228748
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 6.00000 0.450988
\(178\) 14.0000 1.04934
\(179\) 19.0000 1.42013 0.710063 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) −3.00000 −0.223607
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) 27.0000 1.98508
\(186\) −6.00000 −0.439941
\(187\) 16.0000 1.17004
\(188\) 7.00000 0.510527
\(189\) 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
\(193\) −17.0000 −1.22369 −0.611843 0.790979i \(-0.709572\pi\)
−0.611843 + 0.790979i \(0.709572\pi\)
\(194\) 7.00000 0.502571
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 4.00000 0.284268
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 4.00000 0.282843
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 27.0000 1.88576
\(206\) −5.00000 −0.348367
\(207\) 1.00000 0.0695048
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −4.00000 −0.274721
\(213\) 6.00000 0.411113
\(214\) 8.00000 0.546869
\(215\) 9.00000 0.613795
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 3.00000 0.203186
\(219\) −8.00000 −0.540590
\(220\) −12.0000 −0.809040
\(221\) 12.0000 0.807207
\(222\) 9.00000 0.604040
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 9.00000 0.598671
\(227\) −11.0000 −0.730096 −0.365048 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 3.00000 0.196116
\(235\) −21.0000 −1.36989
\(236\) −6.00000 −0.390567
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 3.00000 0.193649
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) 4.00000 0.253490
\(250\) 3.00000 0.189737
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 7.00000 0.439219
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 3.00000 0.186772
\(259\) 0 0
\(260\) −9.00000 −0.558156
\(261\) 3.00000 0.185695
\(262\) 6.00000 0.370681
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) −4.00000 −0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 3.00000 0.182574
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) 16.0000 0.964836
\(276\) −1.00000 −0.0601929
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) −9.00000 −0.539784
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 23.0000 1.37206 0.686032 0.727571i \(-0.259351\pi\)
0.686032 + 0.727571i \(0.259351\pi\)
\(282\) −7.00000 −0.416844
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −9.00000 −0.528498
\(291\) −7.00000 −0.410347
\(292\) 8.00000 0.468165
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) −9.00000 −0.523114
\(297\) −4.00000 −0.232104
\(298\) 16.0000 0.926855
\(299\) 3.00000 0.173494
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 15.0000 0.863153
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 4.00000 0.228665
\(307\) −15.0000 −0.856095 −0.428048 0.903756i \(-0.640798\pi\)
−0.428048 + 0.903756i \(0.640798\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) −18.0000 −1.02233
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) −3.00000 −0.169842
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) 4.00000 0.224309
\(319\) 12.0000 0.671871
\(320\) −3.00000 −0.167705
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 12.0000 0.665640
\(326\) 16.0000 0.886158
\(327\) −3.00000 −0.165900
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 12.0000 0.660578
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −4.00000 −0.219529
\(333\) −9.00000 −0.493197
\(334\) 20.0000 1.09435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) −4.00000 −0.217571
\(339\) −9.00000 −0.488813
\(340\) −12.0000 −0.650791
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 0 0
\(344\) −3.00000 −0.161749
\(345\) 3.00000 0.161515
\(346\) 6.00000 0.322562
\(347\) 23.0000 1.23470 0.617352 0.786687i \(-0.288205\pi\)
0.617352 + 0.786687i \(0.288205\pi\)
\(348\) −3.00000 −0.160817
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 4.00000 0.213201
\(353\) 29.0000 1.54351 0.771757 0.635917i \(-0.219378\pi\)
0.771757 + 0.635917i \(0.219378\pi\)
\(354\) 6.00000 0.318896
\(355\) 18.0000 0.955341
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 19.0000 1.00418
\(359\) −1.00000 −0.0527780 −0.0263890 0.999652i \(-0.508401\pi\)
−0.0263890 + 0.999652i \(0.508401\pi\)
\(360\) −3.00000 −0.158114
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 10.0000 0.522708
\(367\) −31.0000 −1.61819 −0.809093 0.587680i \(-0.800041\pi\)
−0.809093 + 0.587680i \(0.800041\pi\)
\(368\) 1.00000 0.0521286
\(369\) −9.00000 −0.468521
\(370\) 27.0000 1.40366
\(371\) 0 0
\(372\) −6.00000 −0.311086
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 16.0000 0.827340
\(375\) −3.00000 −0.154919
\(376\) 7.00000 0.360997
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) −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
\(386\) −17.0000 −0.865277
\(387\) −3.00000 −0.152499
\(388\) 7.00000 0.355371
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 9.00000 0.455733
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) −23.0000 −1.15872
\(395\) −24.0000 −1.20757
\(396\) 4.00000 0.201008
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 5.00000 0.250627
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) −4.00000 −0.199502
\(403\) 18.0000 0.896644
\(404\) 14.0000 0.696526
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −36.0000 −1.78445
\(408\) −4.00000 −0.198030
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 27.0000 1.33343
\(411\) 15.0000 0.739895
\(412\) −5.00000 −0.246332
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 12.0000 0.589057
\(416\) 3.00000 0.147087
\(417\) 9.00000 0.440732
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −21.0000 −1.02348 −0.511739 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) −12.0000 −0.584151
\(423\) 7.00000 0.340352
\(424\) −4.00000 −0.194257
\(425\) 16.0000 0.776114
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) −12.0000 −0.579365
\(430\) 9.00000 0.434019
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −23.0000 −1.10531 −0.552655 0.833410i \(-0.686385\pi\)
−0.552655 + 0.833410i \(0.686385\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) 3.00000 0.143674
\(437\) 0 0
\(438\) −8.00000 −0.382255
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) 9.00000 0.427121
\(445\) −42.0000 −1.99099
\(446\) 14.0000 0.662919
\(447\) −16.0000 −0.756774
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 4.00000 0.188562
\(451\) −36.0000 −1.69517
\(452\) 9.00000 0.423324
\(453\) −15.0000 −0.704761
\(454\) −11.0000 −0.516256
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) −12.0000 −0.560723
\(459\) −4.00000 −0.186704
\(460\) −3.00000 −0.139876
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 3.00000 0.139272
\(465\) 18.0000 0.834730
\(466\) −10.0000 −0.463241
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) −21.0000 −0.968658
\(471\) −8.00000 −0.368621
\(472\) −6.00000 −0.276172
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) −12.0000 −0.548867
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 3.00000 0.136931
\(481\) −27.0000 −1.23109
\(482\) 5.00000 0.227744
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −21.0000 −0.953561
\(486\) −1.00000 −0.0453609
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) −10.0000 −0.452679
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 9.00000 0.405751
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 6.00000 0.269408
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 3.00000 0.134164
\(501\) −20.0000 −0.893534
\(502\) −19.0000 −0.848012
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) 4.00000 0.177822
\(507\) 4.00000 0.177646
\(508\) 7.00000 0.310575
\(509\) −44.0000 −1.95027 −0.975133 0.221621i \(-0.928865\pi\)
−0.975133 + 0.221621i \(0.928865\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) 15.0000 0.660979
\(516\) 3.00000 0.132068
\(517\) 28.0000 1.23144
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −9.00000 −0.394676
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 3.00000 0.131306
\(523\) −42.0000 −1.83653 −0.918266 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) 24.0000 1.04546
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) −27.0000 −1.16950
\(534\) −14.0000 −0.605839
\(535\) −24.0000 −1.03761
\(536\) 4.00000 0.172774
\(537\) −19.0000 −0.819911
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) −20.0000 −0.859074
\(543\) −2.00000 −0.0858282
\(544\) 4.00000 0.171499
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −15.0000 −0.640768
\(549\) −10.0000 −0.426790
\(550\) 16.0000 0.682242
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 20.0000 0.849719
\(555\) −27.0000 −1.14609
\(556\) −9.00000 −0.381685
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 6.00000 0.254000
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 23.0000 0.970196
\(563\) −15.0000 −0.632175 −0.316087 0.948730i \(-0.602369\pi\)
−0.316087 + 0.948730i \(0.602369\pi\)
\(564\) −7.00000 −0.294753
\(565\) −27.0000 −1.13590
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 12.0000 0.501745
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 17.0000 0.706496
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) −7.00000 −0.290159
\(583\) −16.0000 −0.662652
\(584\) 8.00000 0.331042
\(585\) −9.00000 −0.372104
\(586\) −6.00000 −0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 18.0000 0.741048
\(591\) 23.0000 0.946094
\(592\) −9.00000 −0.369898
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 16.0000 0.655386
\(597\) −5.00000 −0.204636
\(598\) 3.00000 0.122679
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) −4.00000 −0.163299
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 15.0000 0.610341
\(605\) −15.0000 −0.609837
\(606\) −14.0000 −0.568711
\(607\) −30.0000 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 30.0000 1.21466
\(611\) 21.0000 0.849569
\(612\) 4.00000 0.161690
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) −15.0000 −0.605351
\(615\) −27.0000 −1.08875
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 5.00000 0.201129
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) −18.0000 −0.722897
\(621\) −1.00000 −0.0401286
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) −3.00000 −0.120096
\(625\) −29.0000 −1.16000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) 12.0000 0.476957
\(634\) 5.00000 0.198575
\(635\) −21.0000 −0.833360
\(636\) 4.00000 0.158610
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) −6.00000 −0.237356
\(640\) −3.00000 −0.118585
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) −8.00000 −0.315735
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) −9.00000 −0.354375
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 31.0000 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) −3.00000 −0.117309
\(655\) −18.0000 −0.703318
\(656\) −9.00000 −0.351391
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 12.0000 0.467099
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 8.00000 0.310929
\(663\) −12.0000 −0.466041
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −9.00000 −0.348743
\(667\) 3.00000 0.116160
\(668\) 20.0000 0.773823
\(669\) −14.0000 −0.541271
\(670\) −12.0000 −0.463600
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −12.0000 −0.462223
\(675\) −4.00000 −0.153960
\(676\) −4.00000 −0.153846
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) −9.00000 −0.345643
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) 11.0000 0.421521
\(682\) 24.0000 0.919007
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 45.0000 1.71936
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) −3.00000 −0.114374
\(689\) −12.0000 −0.457164
\(690\) 3.00000 0.114208
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 23.0000 0.873068
\(695\) 27.0000 1.02417
\(696\) −3.00000 −0.113715
\(697\) −36.0000 −1.36360
\(698\) 26.0000 0.984115
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) −3.00000 −0.113228
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 21.0000 0.790906
\(706\) 29.0000 1.09143
\(707\) 0 0
\(708\) 6.00000 0.225494
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 18.0000 0.675528
\(711\) 8.00000 0.300023
\(712\) 14.0000 0.524672
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) 19.0000 0.710063
\(717\) 12.0000 0.448148
\(718\) −1.00000 −0.0373197
\(719\) 7.00000 0.261056 0.130528 0.991445i \(-0.458333\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −5.00000 −0.185952
\(724\) 2.00000 0.0743294
\(725\) 12.0000 0.445669
\(726\) −5.00000 −0.185567
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −24.0000 −0.888280
\(731\) −12.0000 −0.443836
\(732\) 10.0000 0.369611
\(733\) 16.0000 0.590973 0.295487 0.955347i \(-0.404518\pi\)
0.295487 + 0.955347i \(0.404518\pi\)
\(734\) −31.0000 −1.14423
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 16.0000 0.589368
\(738\) −9.00000 −0.331295
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 27.0000 0.992540
\(741\) 0 0
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) −6.00000 −0.219971
\(745\) −48.0000 −1.75858
\(746\) 22.0000 0.805477
\(747\) −4.00000 −0.146352
\(748\) 16.0000 0.585018
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 7.00000 0.255264
\(753\) 19.0000 0.692398
\(754\) 9.00000 0.327761
\(755\) −45.0000 −1.63772
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −25.0000 −0.908041
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −7.00000 −0.253583
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) −1.00000 −0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) −26.0000 −0.936367
\(772\) −17.0000 −0.611843
\(773\) −19.0000 −0.683383 −0.341691 0.939812i \(-0.611000\pi\)
−0.341691 + 0.939812i \(0.611000\pi\)
\(774\) −3.00000 −0.107833
\(775\) 24.0000 0.862105
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 9.00000 0.322252
\(781\) −24.0000 −0.858788
\(782\) 4.00000 0.143040
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) −6.00000 −0.214013
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −23.0000 −0.819341
\(789\) −21.0000 −0.747620
\(790\) −24.0000 −0.853882
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −30.0000 −1.06533
\(794\) −34.0000 −1.20661
\(795\) −12.0000 −0.425596
\(796\) 5.00000 0.177220
\(797\) 49.0000 1.73567 0.867835 0.496853i \(-0.165511\pi\)
0.867835 + 0.496853i \(0.165511\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 4.00000 0.141421
\(801\) 14.0000 0.494666
\(802\) 38.0000 1.34183
\(803\) 32.0000 1.12926
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 18.0000 0.634023
\(807\) −6.00000 −0.211210
\(808\) 14.0000 0.492518
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) −3.00000 −0.105409
\(811\) −33.0000 −1.15879 −0.579393 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) −36.0000 −1.26180
\(815\) −48.0000 −1.68137
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) −16.0000 −0.559427
\(819\) 0 0
\(820\) 27.0000 0.942881
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 15.0000 0.523185
\(823\) 7.00000 0.244005 0.122002 0.992530i \(-0.461068\pi\)
0.122002 + 0.992530i \(0.461068\pi\)
\(824\) −5.00000 −0.174183
\(825\) −16.0000 −0.557048
\(826\) 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 \(0.729400\pi\)
\(830\) 12.0000 0.416526
\(831\) −20.0000 −0.693792
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 9.00000 0.311645
\(835\) −60.0000 −2.07639
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 4.00000 0.138178
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −21.0000 −0.723708
\(843\) −23.0000 −0.792162
\(844\) −12.0000 −0.413057
\(845\) 12.0000 0.412813
\(846\) 7.00000 0.240665
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) −6.00000 −0.205919
\(850\) 16.0000 0.548795
\(851\) −9.00000 −0.308516
\(852\) 6.00000 0.205557
\(853\) −7.00000 −0.239675 −0.119838 0.992793i \(-0.538237\pi\)
−0.119838 + 0.992793i \(0.538237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) −12.0000 −0.409673
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 9.00000 0.306897
\(861\) 0 0
\(862\) 21.0000 0.715263
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −23.0000 −0.781572
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 9.00000 0.305129
\(871\) 12.0000 0.406604
\(872\) 3.00000 0.101593
\(873\) 7.00000 0.236914
\(874\) 0 0
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) −36.0000 −1.21563 −0.607817 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) 10.0000 0.337484
\(879\) 6.00000 0.202375
\(880\) −12.0000 −0.404520
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 54.0000 1.81724 0.908622 0.417619i \(-0.137135\pi\)
0.908622 + 0.417619i \(0.137135\pi\)
\(884\) 12.0000 0.403604
\(885\) −18.0000 −0.605063
\(886\) 25.0000 0.839891
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 9.00000 0.302020
\(889\) 0 0
\(890\) −42.0000 −1.40784
\(891\) 4.00000 0.134005
\(892\) 14.0000 0.468755
\(893\) 0 0
\(894\) −16.0000 −0.535120
\(895\) −57.0000 −1.90530
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 30.0000 1.00111
\(899\) 18.0000 0.600334
\(900\) 4.00000 0.133333
\(901\) −16.0000 −0.533037
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) −6.00000 −0.199447
\(906\) −15.0000 −0.498342
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) −11.0000 −0.365048
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −49.0000 −1.62344 −0.811721 0.584045i \(-0.801469\pi\)
−0.811721 + 0.584045i \(0.801469\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 24.0000 0.793849
\(915\) −30.0000 −0.991769
\(916\) −12.0000 −0.396491
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 15.0000 0.494267
\(922\) 22.0000 0.724531
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) −36.0000 −1.18367
\(926\) 13.0000 0.427207
\(927\) −5.00000 −0.164222
\(928\) 3.00000 0.0984798
\(929\) −9.00000 −0.295280 −0.147640 0.989041i \(-0.547168\pi\)
−0.147640 + 0.989041i \(0.547168\pi\)
\(930\) 18.0000 0.590243
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) −28.0000 −0.916679
\(934\) −13.0000 −0.425373
\(935\) −48.0000 −1.56977
\(936\) 3.00000 0.0980581
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 0 0
\(939\) 34.0000 1.10955
\(940\) −21.0000 −0.684944
\(941\) −3.00000 −0.0977972 −0.0488986 0.998804i \(-0.515571\pi\)
−0.0488986 + 0.998804i \(0.515571\pi\)
\(942\) −8.00000 −0.260654
\(943\) −9.00000 −0.293080
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −31.0000 −1.00736 −0.503682 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(948\) −8.00000 −0.259828
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) −5.00000 −0.162136
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −4.00000 −0.129505
\(955\) 36.0000 1.16493
\(956\) −12.0000 −0.388108
\(957\) −12.0000 −0.387905
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) 5.00000 0.161290
\(962\) −27.0000 −0.870515
\(963\) 8.00000 0.257796
\(964\) 5.00000 0.161039
\(965\) 51.0000 1.64175
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −21.0000 −0.674269
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 13.0000 0.416547
\(975\) −12.0000 −0.384308
\(976\) −10.0000 −0.320092
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) −16.0000 −0.511624
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) 3.00000 0.0957826
\(982\) −12.0000 −0.382935
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 9.00000 0.286910
\(985\) 69.0000 2.19852
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) −12.0000 −0.381385
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 6.00000 0.190500
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) −15.0000 −0.475532
\(996\) 4.00000 0.126745
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 34.0000 1.07625
\(999\) 9.00000 0.284747
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.y.1.1 1
7.6 odd 2 966.2.a.k.1.1 1
21.20 even 2 2898.2.a.a.1.1 1
28.27 even 2 7728.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.k.1.1 1 7.6 odd 2
2898.2.a.a.1.1 1 21.20 even 2
6762.2.a.y.1.1 1 1.1 even 1 trivial
7728.2.a.j.1.1 1 28.27 even 2