Properties

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

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

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -3.00000 q^{20} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{36} -5.00000 q^{37} -3.00000 q^{39} +3.00000 q^{40} -5.00000 q^{41} -7.00000 q^{43} +4.00000 q^{44} -3.00000 q^{45} +1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -3.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} -12.0000 q^{55} -1.00000 q^{58} +2.00000 q^{59} -3.00000 q^{60} +6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +9.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} -1.00000 q^{69} +10.0000 q^{71} -1.00000 q^{72} +5.00000 q^{74} +4.00000 q^{75} +3.00000 q^{78} +4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} -4.00000 q^{83} +7.00000 q^{86} +1.00000 q^{87} -4.00000 q^{88} -10.0000 q^{89} +3.00000 q^{90} -1.00000 q^{92} +2.00000 q^{93} -3.00000 q^{94} -1.00000 q^{96} -19.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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 \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −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
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 3.00000 0.547723
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 3.00000 0.474342
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 4.00000 0.603023
\(45\) −3.00000 −0.447214
\(46\) 1.00000 0.147442
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) −3.00000 −0.387298
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 9.00000 1.11631
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 5.00000 0.581238
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) 1.00000 0.107211
\(88\) −4.00000 −0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 2.00000 0.207390
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −19.0000 −1.92916 −0.964579 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 4.00000 0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 12.0000 1.14416
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 1.00000 0.0928477
\(117\) −3.00000 −0.277350
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −5.00000 −0.450835
\(124\) 2.00000 0.179605
\(125\) 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
\(129\) −7.00000 −0.616316
\(130\) −9.00000 −0.789352
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 1.00000 0.0851257
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −10.0000 −0.839181
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) −5.00000 −0.410997
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) −4.00000 −0.326599
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) −3.00000 −0.240192
\(157\) 24.0000 1.91541 0.957704 0.287754i \(-0.0929087\pi\)
0.957704 + 0.287754i \(0.0929087\pi\)
\(158\) −4.00000 −0.318223
\(159\) 12.0000 0.951662
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −5.00000 −0.390434
\(165\) −12.0000 −0.934199
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) −7.00000 −0.533745
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 2.00000 0.150329
\(178\) 10.0000 0.749532
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) −3.00000 −0.223607
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 1.00000 0.0737210
\(185\) 15.0000 1.10282
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 19.0000 1.36412
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) −21.0000 −1.49619 −0.748094 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) −4.00000 −0.284268
\(199\) 23.0000 1.63043 0.815213 0.579161i \(-0.196620\pi\)
0.815213 + 0.579161i \(0.196620\pi\)
\(200\) −4.00000 −0.282843
\(201\) −12.0000 −0.846415
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) −1.00000 −0.0696733
\(207\) −1.00000 −0.0695048
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 12.0000 0.824163
\(213\) 10.0000 0.685189
\(214\) −4.00000 −0.273434
\(215\) 21.0000 1.43219
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) −12.0000 −0.809040
\(221\) 0 0
\(222\) 5.00000 0.335578
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −7.00000 −0.465633
\(227\) 5.00000 0.331862 0.165931 0.986137i \(-0.446937\pi\)
0.165931 + 0.986137i \(0.446937\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 3.00000 0.196116
\(235\) −9.00000 −0.587095
\(236\) 2.00000 0.130189
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −3.00000 −0.193649
\(241\) −9.00000 −0.579741 −0.289870 0.957066i \(-0.593612\pi\)
−0.289870 + 0.957066i \(0.593612\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 5.00000 0.318788
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) −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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 7.00000 0.435801
\(259\) 0 0
\(260\) 9.00000 0.558156
\(261\) 1.00000 0.0618984
\(262\) −18.0000 −1.11204
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) −4.00000 −0.246183
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 3.00000 0.182574
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) 16.0000 0.964836
\(276\) −1.00000 −0.0601929
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 7.00000 0.419832
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) −3.00000 −0.178647
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 3.00000 0.176166
\(291\) −19.0000 −1.11380
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 5.00000 0.290619
\(297\) 4.00000 0.232104
\(298\) −12.0000 −0.695141
\(299\) 3.00000 0.173494
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 17.0000 0.978240
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 6.00000 0.340777
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 3.00000 0.169842
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) −12.0000 −0.672927
\(319\) 4.00000 0.223957
\(320\) −3.00000 −0.167705
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −12.0000 −0.665640
\(326\) 12.0000 0.664619
\(327\) −1.00000 −0.0553001
\(328\) 5.00000 0.276079
\(329\) 0 0
\(330\) 12.0000 0.660578
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) −5.00000 −0.273998
\(334\) −12.0000 −0.656611
\(335\) 36.0000 1.96689
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 4.00000 0.217571
\(339\) 7.00000 0.380188
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 7.00000 0.377415
\(345\) 3.00000 0.161515
\(346\) 2.00000 0.107521
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 1.00000 0.0536056
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) −4.00000 −0.213201
\(353\) 17.0000 0.904819 0.452409 0.891810i \(-0.350565\pi\)
0.452409 + 0.891810i \(0.350565\pi\)
\(354\) −2.00000 −0.106299
\(355\) −30.0000 −1.59223
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 3.00000 0.158114
\(361\) −19.0000 −1.00000
\(362\) 18.0000 0.946059
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −5.00000 −0.260290
\(370\) −15.0000 −0.779813
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) −3.00000 −0.154713
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 20.0000 1.02329
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) −7.00000 −0.355830
\(388\) −19.0000 −0.964579
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) −9.00000 −0.455733
\(391\) 0 0
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 21.0000 1.05796
\(395\) −12.0000 −0.603786
\(396\) 4.00000 0.201008
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −23.0000 −1.15289
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 12.0000 0.598506
\(403\) −6.00000 −0.298881
\(404\) −14.0000 −0.696526
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) −15.0000 −0.740797
\(411\) −17.0000 −0.838548
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 12.0000 0.589057
\(416\) 3.00000 0.147087
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) −12.0000 −0.579365
\(430\) −21.0000 −1.01271
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 1.00000 0.0481125
\(433\) 27.0000 1.29754 0.648769 0.760986i \(-0.275284\pi\)
0.648769 + 0.760986i \(0.275284\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 12.0000 0.572078
\(441\) 0 0
\(442\) 0 0
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) −5.00000 −0.237289
\(445\) 30.0000 1.42214
\(446\) 2.00000 0.0947027
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −4.00000 −0.188562
\(451\) −20.0000 −0.941763
\(452\) 7.00000 0.329252
\(453\) −17.0000 −0.798730
\(454\) −5.00000 −0.234662
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 1.00000 0.0464238
\(465\) −6.00000 −0.278243
\(466\) 10.0000 0.463241
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) −3.00000 −0.138675
\(469\) 0 0
\(470\) 9.00000 0.415139
\(471\) 24.0000 1.10586
\(472\) −2.00000 −0.0920575
\(473\) −28.0000 −1.28744
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) −24.0000 −1.09773
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 3.00000 0.136931
\(481\) 15.0000 0.683941
\(482\) 9.00000 0.409939
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 57.0000 2.58824
\(486\) −1.00000 −0.0453609
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) −6.00000 −0.271607
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −5.00000 −0.225417
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 3.00000 0.134164
\(501\) 12.0000 0.536120
\(502\) 19.0000 0.848012
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 42.0000 1.86898
\(506\) 4.00000 0.177822
\(507\) −4.00000 −0.177646
\(508\) 7.00000 0.310575
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −3.00000 −0.132196
\(516\) −7.00000 −0.308158
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) −9.00000 −0.394676
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 36.0000 1.56374
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 10.0000 0.432742
\(535\) −12.0000 −0.518805
\(536\) 12.0000 0.518321
\(537\) 9.00000 0.388379
\(538\) −2.00000 −0.0862261
\(539\) 0 0
\(540\) −3.00000 −0.129099
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) −12.0000 −0.515444
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −17.0000 −0.726204
\(549\) 6.00000 0.256074
\(550\) −16.0000 −0.682242
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 24.0000 1.01966
\(555\) 15.0000 0.636715
\(556\) −7.00000 −0.296866
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) 0 0
\(562\) −1.00000 −0.0421825
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 3.00000 0.126323
\(565\) −21.0000 −0.883477
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −12.0000 −0.501745
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 17.0000 0.707107
\(579\) −1.00000 −0.0415586
\(580\) −3.00000 −0.124568
\(581\) 0 0
\(582\) 19.0000 0.787575
\(583\) 48.0000 1.98796
\(584\) 0 0
\(585\) 9.00000 0.372104
\(586\) 14.0000 0.578335
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) −21.0000 −0.863825
\(592\) −5.00000 −0.205499
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 23.0000 0.941327
\(598\) −3.00000 −0.122679
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −4.00000 −0.163299
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −17.0000 −0.691720
\(605\) −15.0000 −0.609837
\(606\) 14.0000 0.568711
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 18.0000 0.728799
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 21.0000 0.848182 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(614\) 25.0000 1.00892
\(615\) 15.0000 0.604858
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −6.00000 −0.240966
\(621\) −1.00000 −0.0401286
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) −3.00000 −0.120096
\(625\) −29.0000 −1.16000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 24.0000 0.957704
\(629\) 0 0
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 33.0000 1.31060
\(635\) −21.0000 −0.833360
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 10.0000 0.395594
\(640\) 3.00000 0.118585
\(641\) 1.00000 0.0394976 0.0197488 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) −4.00000 −0.157867
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 21.0000 0.826874
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.00000 0.314027
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 13.0000 0.508729 0.254365 0.967108i \(-0.418134\pi\)
0.254365 + 0.967108i \(0.418134\pi\)
\(654\) 1.00000 0.0391031
\(655\) −54.0000 −2.10995
\(656\) −5.00000 −0.195217
\(657\) 0 0
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) −12.0000 −0.467099
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) −1.00000 −0.0387202
\(668\) 12.0000 0.464294
\(669\) −2.00000 −0.0773245
\(670\) −36.0000 −1.39080
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 20.0000 0.770371
\(675\) 4.00000 0.153960
\(676\) −4.00000 −0.153846
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −7.00000 −0.268833
\(679\) 0 0
\(680\) 0 0
\(681\) 5.00000 0.191600
\(682\) −8.00000 −0.306336
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 51.0000 1.94861
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) −7.00000 −0.266872
\(689\) −36.0000 −1.37149
\(690\) −3.00000 −0.114208
\(691\) −21.0000 −0.798878 −0.399439 0.916760i \(-0.630795\pi\)
−0.399439 + 0.916760i \(0.630795\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) 21.0000 0.796575
\(696\) −1.00000 −0.0379049
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 3.00000 0.113228
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) −9.00000 −0.338960
\(706\) −17.0000 −0.639803
\(707\) 0 0
\(708\) 2.00000 0.0751646
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 30.0000 1.12588
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 9.00000 0.336346
\(717\) 24.0000 0.896296
\(718\) −9.00000 −0.335877
\(719\) 43.0000 1.60363 0.801815 0.597573i \(-0.203868\pi\)
0.801815 + 0.597573i \(0.203868\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) −9.00000 −0.334714
\(724\) −18.0000 −0.668965
\(725\) 4.00000 0.148556
\(726\) −5.00000 −0.185567
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 6.00000 0.221766
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 21.0000 0.775124
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −48.0000 −1.76810
\(738\) 5.00000 0.184053
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 15.0000 0.551411
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −2.00000 −0.0733236
\(745\) −36.0000 −1.31894
\(746\) 2.00000 0.0732252
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 3.00000 0.109399
\(753\) −19.0000 −0.692398
\(754\) 3.00000 0.109254
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 5.00000 0.181608
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −7.00000 −0.253583
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −6.00000 −0.216647
\(768\) 1.00000 0.0360844
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −1.00000 −0.0359908
\(773\) −35.0000 −1.25886 −0.629431 0.777056i \(-0.716712\pi\)
−0.629431 + 0.777056i \(0.716712\pi\)
\(774\) 7.00000 0.251610
\(775\) 8.00000 0.287368
\(776\) 19.0000 0.682060
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 0 0
\(780\) 9.00000 0.322252
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −72.0000 −2.56979
\(786\) −18.0000 −0.642039
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −21.0000 −0.748094
\(789\) −21.0000 −0.747620
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −18.0000 −0.639199
\(794\) 6.00000 0.212932
\(795\) −36.0000 −1.27679
\(796\) 23.0000 0.815213
\(797\) −47.0000 −1.66483 −0.832413 0.554156i \(-0.813041\pi\)
−0.832413 + 0.554156i \(0.813041\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) −10.0000 −0.353333
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 2.00000 0.0704033
\(808\) 14.0000 0.492518
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 3.00000 0.105409
\(811\) −23.0000 −0.807639 −0.403820 0.914839i \(-0.632318\pi\)
−0.403820 + 0.914839i \(0.632318\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 20.0000 0.701000
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 15.0000 0.523823
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 17.0000 0.592943
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 16.0000 0.557048
\(826\) 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
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −12.0000 −0.416526
\(831\) −24.0000 −0.832551
\(832\) −3.00000 −0.104006
\(833\) 0 0
\(834\) 7.00000 0.242390
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 4.00000 0.138178
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 1.00000 0.0344623
\(843\) 1.00000 0.0344418
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 5.00000 0.171398
\(852\) 10.0000 0.342594
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 12.0000 0.409673
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 21.0000 0.716094
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −27.0000 −0.917497
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 3.00000 0.101710
\(871\) 36.0000 1.21981
\(872\) 1.00000 0.0338643
\(873\) −19.0000 −0.643053
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 14.0000 0.472477
\(879\) −14.0000 −0.472208
\(880\) −12.0000 −0.404520
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) 6.00000 0.201916 0.100958 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(884\) 0 0
\(885\) −6.00000 −0.201688
\(886\) 29.0000 0.974274
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 5.00000 0.167789
\(889\) 0 0
\(890\) −30.0000 −1.00560
\(891\) 4.00000 0.134005
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) −27.0000 −0.902510
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 18.0000 0.600668
\(899\) 2.00000 0.0667037
\(900\) 4.00000 0.133333
\(901\) 0 0
\(902\) 20.0000 0.665927
\(903\) 0 0
\(904\) −7.00000 −0.232817
\(905\) 54.0000 1.79502
\(906\) 17.0000 0.564787
\(907\) −49.0000 −1.62702 −0.813509 0.581552i \(-0.802446\pi\)
−0.813509 + 0.581552i \(0.802446\pi\)
\(908\) 5.00000 0.165931
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −55.0000 −1.82223 −0.911116 0.412151i \(-0.864778\pi\)
−0.911116 + 0.412151i \(0.864778\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 4.00000 0.132308
\(915\) −18.0000 −0.595062
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) 60.0000 1.97922 0.989609 0.143787i \(-0.0459280\pi\)
0.989609 + 0.143787i \(0.0459280\pi\)
\(920\) −3.00000 −0.0989071
\(921\) −25.0000 −0.823778
\(922\) −30.0000 −0.987997
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) −5.00000 −0.164310
\(927\) 1.00000 0.0328443
\(928\) −1.00000 −0.0328266
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 6.00000 0.196748
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 4.00000 0.130954
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) 53.0000 1.73143 0.865717 0.500533i \(-0.166863\pi\)
0.865717 + 0.500533i \(0.166863\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) −9.00000 −0.293548
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) −24.0000 −0.781962
\(943\) 5.00000 0.162822
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 28.0000 0.910359
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) 0 0
\(951\) −33.0000 −1.07010
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −12.0000 −0.388514
\(955\) 60.0000 1.94155
\(956\) 24.0000 0.776215
\(957\) 4.00000 0.129302
\(958\) −6.00000 −0.193851
\(959\) 0 0
\(960\) −3.00000 −0.0968246
\(961\) −27.0000 −0.870968
\(962\) −15.0000 −0.483619
\(963\) 4.00000 0.128898
\(964\) −9.00000 −0.289870
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −57.0000 −1.83016
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −13.0000 −0.416547
\(975\) −12.0000 −0.384308
\(976\) 6.00000 0.192055
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) 12.0000 0.383718
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) −28.0000 −0.893516
\(983\) −2.00000 −0.0637901 −0.0318950 0.999491i \(-0.510154\pi\)
−0.0318950 + 0.999491i \(0.510154\pi\)
\(984\) 5.00000 0.159394
\(985\) 63.0000 2.00735
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.00000 0.222587
\(990\) 12.0000 0.381385
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −69.0000 −2.18745
\(996\) −4.00000 −0.126745
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) 10.0000 0.316544
\(999\) −5.00000 −0.158193
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.l.1.1 1
7.6 odd 2 966.2.a.d.1.1 1
21.20 even 2 2898.2.a.k.1.1 1
28.27 even 2 7728.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.d.1.1 1 7.6 odd 2
2898.2.a.k.1.1 1 21.20 even 2
6762.2.a.l.1.1 1 1.1 even 1 trivial
7728.2.a.u.1.1 1 28.27 even 2