Properties

Label 6762.2.a.bd.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -1.00000 q^{12} -5.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -8.00000 q^{19} +3.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -5.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -3.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.00000 q^{36} -7.00000 q^{37} -8.00000 q^{38} +5.00000 q^{39} +3.00000 q^{40} -9.00000 q^{41} -1.00000 q^{43} +3.00000 q^{45} -1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -5.00000 q^{52} -12.0000 q^{53} -1.00000 q^{54} +8.00000 q^{57} +3.00000 q^{58} +6.00000 q^{59} -3.00000 q^{60} -14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -15.0000 q^{65} -4.00000 q^{67} +1.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} -7.00000 q^{74} -4.00000 q^{75} -8.00000 q^{76} +5.00000 q^{78} -16.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} +12.0000 q^{83} -1.00000 q^{86} -3.00000 q^{87} -6.00000 q^{89} +3.00000 q^{90} -1.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} -24.0000 q^{95} -1.00000 q^{96} +1.00000 q^{97} +O(q^{100})\)

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 \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 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
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −5.00000 −0.980581
\(27\) −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
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −8.00000 −1.29777
\(39\) 5.00000 0.800641
\(40\) 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
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 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
\(52\) −5.00000 −0.693375
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −3.00000 −0.387298
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −15.0000 −1.86052
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −7.00000 −0.813733
\(75\) −4.00000 −0.461880
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 5.00000 0.566139
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 2.00000 0.207390
\(94\) 3.00000 0.309426
\(95\) −24.0000 −2.46235
\(96\) −1.00000 −0.102062
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 8.00000 0.749269
\(115\) −3.00000 −0.279751
\(116\) 3.00000 0.278543
\(117\) −5.00000 −0.462250
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) 9.00000 0.811503
\(124\) −2.00000 −0.179605
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00000 0.0880451
\(130\) −15.0000 −1.31559
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 1.00000 0.0851257
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −4.00000 −0.326599
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 5.00000 0.400320
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −16.0000 −1.27289
\(159\) 12.0000 0.951662
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −1.00000 −0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) −6.00000 −0.449719
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 3.00000 0.223607
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) −21.0000 −1.54395
\(186\) 2.00000 0.146647
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 1.00000 0.0717958
\(195\) 15.0000 1.07417
\(196\) 0 0
\(197\) 9.00000 0.641223 0.320612 0.947211i \(-0.396112\pi\)
0.320612 + 0.947211i \(0.396112\pi\)
\(198\) 0 0
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 4.00000 0.282843
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) −27.0000 −1.88576
\(206\) 1.00000 0.0696733
\(207\) −1.00000 −0.0695048
\(208\) −5.00000 −0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −12.0000 −0.824163
\(213\) −6.00000 −0.411113
\(214\) 12.0000 0.820303
\(215\) −3.00000 −0.204598
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −19.0000 −1.28684
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 15.0000 0.997785
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 8.00000 0.529813
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −5.00000 −0.326860
\(235\) 9.00000 0.587095
\(236\) 6.00000 0.390567
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −3.00000 −0.193649
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) 40.0000 2.54514
\(248\) −2.00000 −0.127000
\(249\) −12.0000 −0.760469
\(250\) −3.00000 −0.189737
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −13.0000 −0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 1.00000 0.0622573
\(259\) 0 0
\(260\) −15.0000 −0.930261
\(261\) 3.00000 0.185695
\(262\) 18.0000 1.11204
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −3.00000 −0.182574
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −5.00000 −0.299880
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) −3.00000 −0.178647
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 6.00000 0.356034
\(285\) 24.0000 1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 9.00000 0.528498
\(291\) −1.00000 −0.0586210
\(292\) 4.00000 0.234082
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 11.0000 0.632979
\(303\) −6.00000 −0.344691
\(304\) −8.00000 −0.458831
\(305\) −42.0000 −2.40491
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −6.00000 −0.340777
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 5.00000 0.283069
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) 3.00000 0.167705
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −20.0000 −1.10940
\(326\) 20.0000 1.10770
\(327\) 19.0000 1.05070
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 12.0000 0.658586
\(333\) −7.00000 −0.383598
\(334\) 12.0000 0.656611
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 12.0000 0.652714
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) 3.00000 0.161515
\(346\) −6.00000 −0.322562
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) −3.00000 −0.160817
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −27.0000 −1.43706 −0.718532 0.695493i \(-0.755186\pi\)
−0.718532 + 0.695493i \(0.755186\pi\)
\(354\) −6.00000 −0.318896
\(355\) 18.0000 0.955341
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) 21.0000 1.10834 0.554169 0.832404i \(-0.313036\pi\)
0.554169 + 0.832404i \(0.313036\pi\)
\(360\) 3.00000 0.158114
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 14.0000 0.731792
\(367\) −29.0000 −1.51379 −0.756894 0.653538i \(-0.773284\pi\)
−0.756894 + 0.653538i \(0.773284\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −9.00000 −0.468521
\(370\) −21.0000 −1.09174
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 3.00000 0.154713
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) −24.0000 −1.23117
\(381\) 13.0000 0.666010
\(382\) −12.0000 −0.613973
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −1.00000 −0.0508987
\(387\) −1.00000 −0.0508329
\(388\) 1.00000 0.0507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 15.0000 0.759555
\(391\) 0 0
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 9.00000 0.453413
\(395\) −48.0000 −2.41514
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −17.0000 −0.852133
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 10.0000 0.498135
\(404\) 6.00000 0.298511
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) −27.0000 −1.33343
\(411\) 9.00000 0.443937
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 36.0000 1.76717
\(416\) −5.00000 −0.245145
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 8.00000 0.389434
\(423\) 3.00000 0.145865
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −3.00000 −0.144673
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) −19.0000 −0.909935
\(437\) 8.00000 0.382692
\(438\) −4.00000 −0.191127
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 7.00000 0.332205
\(445\) −18.0000 −0.853282
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 4.00000 0.188562
\(451\) 0 0
\(452\) 15.0000 0.705541
\(453\) −11.0000 −0.516825
\(454\) 15.0000 0.703985
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 4.00000 0.186908
\(459\) 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
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 3.00000 0.139272
\(465\) 6.00000 0.278243
\(466\) −6.00000 −0.277945
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −5.00000 −0.231125
\(469\) 0 0
\(470\) 9.00000 0.415139
\(471\) −4.00000 −0.184310
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) −32.0000 −1.46826
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 12.0000 0.548867
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) −3.00000 −0.136931
\(481\) 35.0000 1.59586
\(482\) 19.0000 0.865426
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 3.00000 0.136223
\(486\) −1.00000 −0.0453609
\(487\) −31.0000 −1.40474 −0.702372 0.711810i \(-0.747876\pi\)
−0.702372 + 0.711810i \(0.747876\pi\)
\(488\) −14.0000 −0.633750
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 9.00000 0.405751
\(493\) 0 0
\(494\) 40.0000 1.79969
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −3.00000 −0.134164
\(501\) −12.0000 −0.536120
\(502\) 15.0000 0.669483
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −13.0000 −0.576782
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −6.00000 −0.264649
\(515\) 3.00000 0.132196
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) −15.0000 −0.657794
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 3.00000 0.131306
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −36.0000 −1.56374
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 45.0000 1.94917
\(534\) 6.00000 0.259645
\(535\) 36.0000 1.55642
\(536\) −4.00000 −0.172774
\(537\) −15.0000 −0.647298
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) −3.00000 −0.129099
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 28.0000 1.20270
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) −57.0000 −2.44161
\(546\) 0 0
\(547\) −10.0000 −0.427569 −0.213785 0.976881i \(-0.568579\pi\)
−0.213785 + 0.976881i \(0.568579\pi\)
\(548\) −9.00000 −0.384461
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) 21.0000 0.891400
\(556\) −5.00000 −0.212047
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) −3.00000 −0.126323
\(565\) 45.0000 1.89316
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 45.0000 1.88650 0.943249 0.332086i \(-0.107752\pi\)
0.943249 + 0.332086i \(0.107752\pi\)
\(570\) 24.0000 1.00525
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −17.0000 −0.707107
\(579\) 1.00000 0.0415586
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) −1.00000 −0.0414513
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) −15.0000 −0.620174
\(586\) 30.0000 1.23929
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 18.0000 0.741048
\(591\) −9.00000 −0.370211
\(592\) −7.00000 −0.287698
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.0000 0.695764
\(598\) 5.00000 0.204465
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −4.00000 −0.163299
\(601\) 16.0000 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 11.0000 0.447584
\(605\) −33.0000 −1.34164
\(606\) −6.00000 −0.243733
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −42.0000 −1.70053
\(611\) −15.0000 −0.606835
\(612\) 0 0
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) −11.0000 −0.443924
\(615\) 27.0000 1.08875
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) −6.00000 −0.240966
\(621\) 1.00000 0.0401286
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 5.00000 0.200160
\(625\) −29.0000 −1.16000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −16.0000 −0.636446
\(633\) −8.00000 −0.317971
\(634\) −27.0000 −1.07231
\(635\) −39.0000 −1.54767
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 3.00000 0.118585
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) −12.0000 −0.473602
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −9.00000 −0.352197 −0.176099 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 19.0000 0.742959
\(655\) 54.0000 2.10995
\(656\) −9.00000 −0.351391
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −7.00000 −0.271244
\(667\) −3.00000 −0.116160
\(668\) 12.0000 0.464294
\(669\) 14.0000 0.541271
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) −4.00000 −0.154074
\(675\) −4.00000 −0.153960
\(676\) 12.0000 0.461538
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −15.0000 −0.576072
\(679\) 0 0
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −8.00000 −0.305888
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) −1.00000 −0.0381246
\(689\) 60.0000 2.28582
\(690\) 3.00000 0.114208
\(691\) 49.0000 1.86405 0.932024 0.362397i \(-0.118041\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) −15.0000 −0.568982
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 5.00000 0.188713
\(703\) 56.0000 2.11208
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) −27.0000 −1.01616
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 18.0000 0.675528
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) 15.0000 0.560576
\(717\) −12.0000 −0.448148
\(718\) 21.0000 0.783713
\(719\) −45.0000 −1.67822 −0.839108 0.543964i \(-0.816923\pi\)
−0.839108 + 0.543964i \(0.816923\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) −19.0000 −0.706618
\(724\) −2.00000 −0.0743294
\(725\) 12.0000 0.445669
\(726\) 11.0000 0.408248
\(727\) 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
\(730\) 12.0000 0.444140
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) −29.0000 −1.07041
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −9.00000 −0.331295
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) −21.0000 −0.771975
\(741\) −40.0000 −1.46944
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 3.00000 0.109399
\(753\) −15.0000 −0.546630
\(754\) −15.0000 −0.546268
\(755\) 33.0000 1.20099
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 29.0000 1.05333
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 13.0000 0.470940
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −30.0000 −1.08324
\(768\) −1.00000 −0.0360844
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −1.00000 −0.0359908
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) −1.00000 −0.0359443
\(775\) −8.00000 −0.287368
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 72.0000 2.57967
\(780\) 15.0000 0.537086
\(781\) 0 0
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) −18.0000 −0.642039
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 9.00000 0.320612
\(789\) 9.00000 0.320408
\(790\) −48.0000 −1.70776
\(791\) 0 0
\(792\) 0 0
\(793\) 70.0000 2.48577
\(794\) 22.0000 0.780751
\(795\) 36.0000 1.27679
\(796\) −17.0000 −0.602549
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 3.00000 0.105409
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) −27.0000 −0.942881
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 9.00000 0.313911
\(823\) −13.0000 −0.453152 −0.226576 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 36.0000 1.24958
\(831\) 4.00000 0.138758
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 5.00000 0.173136
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) −36.0000 −1.24360
\(839\) −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
\(842\) −19.0000 −0.654783
\(843\) 15.0000 0.516627
\(844\) 8.00000 0.275371
\(845\) 36.0000 1.23844
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) −6.00000 −0.205557
\(853\) 49.0000 1.67773 0.838864 0.544341i \(-0.183220\pi\)
0.838864 + 0.544341i \(0.183220\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 12.0000 0.410152
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 0 0
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) 15.0000 0.510902
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −17.0000 −0.577684
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) −9.00000 −0.305129
\(871\) 20.0000 0.677674
\(872\) −19.0000 −0.643421
\(873\) 1.00000 0.0338449
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −40.0000 −1.35070 −0.675352 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) −26.0000 −0.877457
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 0 0
\(885\) −18.0000 −0.605063
\(886\) −27.0000 −0.907083
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 7.00000 0.234905
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 45.0000 1.50418
\(896\) 0 0
\(897\) −5.00000 −0.166945
\(898\) 18.0000 0.600668
\(899\) −6.00000 −0.200111
\(900\) 4.00000 0.133333
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 15.0000 0.498893
\(905\) −6.00000 −0.199447
\(906\) −11.0000 −0.365451
\(907\) 41.0000 1.36138 0.680691 0.732570i \(-0.261680\pi\)
0.680691 + 0.732570i \(0.261680\pi\)
\(908\) 15.0000 0.497792
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) 42.0000 1.38848
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 11.0000 0.362462
\(922\) 30.0000 0.987997
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −31.0000 −1.01872
\(927\) 1.00000 0.0328443
\(928\) 3.00000 0.0984798
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 6.00000 0.196748
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 12.0000 0.392862
\(934\) 33.0000 1.07979
\(935\) 0 0
\(936\) −5.00000 −0.163430
\(937\) 25.0000 0.816714 0.408357 0.912822i \(-0.366102\pi\)
0.408357 + 0.912822i \(0.366102\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 9.00000 0.293548
\(941\) 51.0000 1.66255 0.831276 0.555860i \(-0.187611\pi\)
0.831276 + 0.555860i \(0.187611\pi\)
\(942\) −4.00000 −0.130327
\(943\) 9.00000 0.293080
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −3.00000 −0.0974869 −0.0487435 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) 16.0000 0.519656
\(949\) −20.0000 −0.649227
\(950\) −32.0000 −1.03822
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) −12.0000 −0.388514
\(955\) −36.0000 −1.16493
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) 0 0
\(960\) −3.00000 −0.0968246
\(961\) −27.0000 −0.870968
\(962\) 35.0000 1.12845
\(963\) 12.0000 0.386695
\(964\) 19.0000 0.611949
\(965\) −3.00000 −0.0965734
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 3.00000 0.0963242
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −31.0000 −0.993304
\(975\) 20.0000 0.640513
\(976\) −14.0000 −0.448129
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) −19.0000 −0.606623
\(982\) 12.0000 0.382935
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 9.00000 0.286910
\(985\) 27.0000 0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) 40.0000 1.27257
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) −51.0000 −1.61681
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 14.0000 0.443162
\(999\) 7.00000 0.221470
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.bd.1.1 1
7.6 odd 2 966.2.a.i.1.1 1
21.20 even 2 2898.2.a.j.1.1 1
28.27 even 2 7728.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.i.1.1 1 7.6 odd 2
2898.2.a.j.1.1 1 21.20 even 2
6762.2.a.bd.1.1 1 1.1 even 1 trivial
7728.2.a.a.1.1 1 28.27 even 2