Properties

Label 77.2.a.a.1.1
Level $77$
Weight $2$
Character 77.1
Self dual yes
Analytic conductor $0.615$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.614848095564\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 77.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +6.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -5.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} +2.00000 q^{28} +10.0000 q^{29} +1.00000 q^{31} +3.00000 q^{33} +1.00000 q^{35} -12.0000 q^{36} -5.00000 q^{37} +12.0000 q^{39} -2.00000 q^{41} -8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} +8.00000 q^{47} -12.0000 q^{48} +1.00000 q^{49} -6.00000 q^{51} +8.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} +18.0000 q^{57} +3.00000 q^{59} -6.00000 q^{60} -2.00000 q^{61} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -3.00000 q^{67} -4.00000 q^{68} +15.0000 q^{69} +1.00000 q^{71} +10.0000 q^{73} +12.0000 q^{75} +12.0000 q^{76} +1.00000 q^{77} +6.00000 q^{79} -4.00000 q^{80} +9.00000 q^{81} +12.0000 q^{83} -6.00000 q^{84} -2.00000 q^{85} -30.0000 q^{87} -15.0000 q^{89} +4.00000 q^{91} +10.0000 q^{92} -3.00000 q^{93} +6.00000 q^{95} -5.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 6.00000 1.73205
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 4.00000 1.00000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 2.00000 0.447214
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 2.00000 0.377964
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −12.0000 −2.00000
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 2.00000 0.301511
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −12.0000 −1.73205
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 8.00000 1.10940
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −6.00000 −0.774597
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −4.00000 −0.485071
\(69\) 15.0000 1.80579
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 12.0000 1.37649
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −4.00000 −0.447214
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −6.00000 −0.654654
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −30.0000 −3.21634
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 10.0000 1.04257
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 8.00000 0.800000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 18.0000 1.73205
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 15.0000 1.42374
\(112\) −4.00000 −0.377964
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) 5.00000 0.466252
\(116\) −20.0000 −1.85695
\(117\) −24.0000 −2.21880
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) −2.00000 −0.179605
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) −6.00000 −0.522233
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) −2.00000 −0.169031
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 24.0000 2.00000
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 10.0000 0.821995
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) −24.0000 −1.92154
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 4.00000 0.312348
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −36.0000 −2.75299
\(172\) 16.0000 1.21999
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 1.00000 0.0747435 0.0373718 0.999301i \(-0.488101\pi\)
0.0373718 + 0.999301i \(0.488101\pi\)
\(180\) 12.0000 0.894427
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −16.0000 −1.16692
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 24.0000 1.73205
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) −2.00000 −0.142857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) −10.0000 −0.701862
\(204\) 12.0000 0.840168
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −30.0000 −2.08514
\(208\) −16.0000 −1.10940
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 12.0000 0.824163
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) −30.0000 −2.02721
\(220\) −2.00000 −0.134840
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −36.0000 −2.38416
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −6.00000 −0.390567
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 12.0000 0.774597
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 12.0000 0.755929
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) −8.00000 −0.496139
\(261\) 60.0000 3.71391
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 45.0000 2.75396
\(268\) 6.00000 0.366508
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 8.00000 0.485071
\(273\) −12.0000 −0.726273
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) −30.0000 −1.80579
\(277\) 24.0000 1.44202 0.721010 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.00000 −0.118678
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 15.0000 0.879316
\(292\) −20.0000 −1.17041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 20.0000 1.15663
\(300\) −24.0000 −1.38564
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 36.0000 2.06815
\(304\) −24.0000 −1.37649
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −2.00000 −0.113961
\(309\) 36.0000 2.04797
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) 0 0
\(315\) 6.00000 0.338062
\(316\) −12.0000 −0.675053
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 8.00000 0.447214
\(321\) 30.0000 1.67444
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −18.0000 −1.00000
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) −12.0000 −0.663602
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −24.0000 −1.31717
\(333\) −30.0000 −1.64399
\(334\) 0 0
\(335\) 3.00000 0.163908
\(336\) 12.0000 0.654654
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 57.0000 3.09582
\(340\) 4.00000 0.216930
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −15.0000 −0.807573
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 60.0000 3.21634
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 36.0000 1.92154
\(352\) 0 0
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) −1.00000 −0.0530745
\(356\) 30.0000 1.59000
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −3.00000 −0.157459
\(364\) −8.00000 −0.419314
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −20.0000 −1.04257
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 6.00000 0.311086
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −27.0000 −1.39427
\(376\) 0 0
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) −12.0000 −0.615587
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 17.0000 0.868659 0.434330 0.900754i \(-0.356985\pi\)
0.434330 + 0.900754i \(0.356985\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −48.0000 −2.43998
\(388\) 10.0000 0.507673
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) −54.0000 −2.72394
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 12.0000 0.603023
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −18.0000 −0.901127
\(400\) −16.0000 −0.800000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 24.0000 1.19404
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 24.0000 1.18240
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 30.0000 1.46911
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 6.00000 0.292770
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 48.0000 2.33384
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 20.0000 0.966736
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) −36.0000 −1.73205
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 30.0000 1.43839
\(436\) −8.00000 −0.383131
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) −30.0000 −1.42374
\(445\) 15.0000 0.711068
\(446\) 0 0
\(447\) 66.0000 3.12169
\(448\) 8.00000 0.377964
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 38.0000 1.78737
\(453\) −18.0000 −0.845714
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) −18.0000 −0.840168
\(460\) −10.0000 −0.466252
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 40.0000 1.85695
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 48.0000 2.21880
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) −21.0000 −0.967629
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 4.00000 0.183340
\(477\) −36.0000 −1.64833
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) −15.0000 −0.682524
\(484\) −2.00000 −0.0909091
\(485\) 5.00000 0.227038
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −12.0000 −0.541002
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 4.00000 0.179605
\(497\) −1.00000 −0.0448561
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) −18.0000 −0.804984
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −4.00000 −0.177471
\(509\) −31.0000 −1.37405 −0.687025 0.726633i \(-0.741084\pi\)
−0.687025 + 0.726633i \(0.741084\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) −48.0000 −2.11308
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) −48.0000 −2.10697
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) −36.0000 −1.57267
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 12.0000 0.522233
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) −12.0000 −0.520266
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 10.0000 0.432338
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) −18.0000 −0.774597
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 0 0
\(543\) −15.0000 −0.643712
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) 6.00000 0.256307
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) −15.0000 −0.636715
\(556\) 20.0000 0.848189
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 4.00000 0.169031
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 48.0000 2.02116
\(565\) 19.0000 0.799336
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −8.00000 −0.334497
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) −48.0000 −2.00000
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) −42.0000 −1.74546
\(580\) 20.0000 0.830455
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 6.00000 0.247436
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) −20.0000 −0.821995
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 44.0000 1.80231
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) −18.0000 −0.733017
\(604\) −12.0000 −0.488273
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0 0
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) −24.0000 −0.970143
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 2.00000 0.0803219
\(621\) 45.0000 1.80579
\(622\) 0 0
\(623\) 15.0000 0.600962
\(624\) 48.0000 1.92154
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −18.0000 −0.718851
\(628\) −14.0000 −0.558661
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 6.00000 0.238479
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) −36.0000 −1.42749
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) −10.0000 −0.394055
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) −8.00000 −0.313304
\(653\) −17.0000 −0.665261 −0.332631 0.943057i \(-0.607936\pi\)
−0.332631 + 0.943057i \(0.607936\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) −8.00000 −0.312348
\(657\) 60.0000 2.34082
\(658\) 0 0
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 6.00000 0.233550
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 0 0
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) −50.0000 −1.93601
\(668\) 4.00000 0.154765
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) −6.00000 −0.230769
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 72.0000 2.75299
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) 21.0000 0.801200
\(688\) −32.0000 −1.21999
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) −32.0000 −1.21646
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) −8.00000 −0.302372
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 8.00000 0.301511
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 18.0000 0.676481
\(709\) 39.0000 1.46468 0.732338 0.680941i \(-0.238429\pi\)
0.732338 + 0.680941i \(0.238429\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −2.00000 −0.0747435
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −11.0000 −0.410231 −0.205115 0.978738i \(-0.565757\pi\)
−0.205115 + 0.978738i \(0.565757\pi\)
\(720\) −24.0000 −0.894427
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 36.0000 1.33885
\(724\) −10.0000 −0.371647
\(725\) −40.0000 −1.48556
\(726\) 0 0
\(727\) −19.0000 −0.704671 −0.352335 0.935874i \(-0.614612\pi\)
−0.352335 + 0.935874i \(0.614612\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) −12.0000 −0.443533
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 3.00000 0.110506
\(738\) 0 0
\(739\) −18.0000 −0.662141 −0.331070 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) −10.0000 −0.367607
\(741\) −72.0000 −2.64499
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) 72.0000 2.63434
\(748\) 4.00000 0.146254
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 32.0000 1.16692
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) −18.0000 −0.654654
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −10.0000 −0.361787
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) −48.0000 −1.73205
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −28.0000 −1.00774
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) −15.0000 −0.538122
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 24.0000 0.859338
\(781\) −1.00000 −0.0357828
\(782\) 0 0
\(783\) −90.0000 −3.21634
\(784\) 4.00000 0.142857
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −36.0000 −1.28245
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) 19.0000 0.675562
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 16.0000 0.567105
\(797\) 23.0000 0.814702 0.407351 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −90.0000 −3.17999
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) −18.0000 −0.634811
\(805\) −5.00000 −0.176227
\(806\) 0 0
\(807\) 54.0000 1.90089
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 20.0000 0.701862
\(813\) −48.0000 −1.68343
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) −24.0000 −0.840168
\(817\) 48.0000 1.67931
\(818\) 0 0
\(819\) 24.0000 0.838628
\(820\) −4.00000 −0.139686
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 60.0000 2.08514
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) −72.0000 −2.49765
\(832\) 32.0000 1.10940
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) −12.0000 −0.415029
\(837\) −9.00000 −0.311086
\(838\) 0 0
\(839\) 45.0000 1.55357 0.776786 0.629764i \(-0.216849\pi\)
0.776786 + 0.629764i \(0.216849\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 12.0000 0.413302
\(844\) 4.00000 0.137686
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 0 0
\(851\) 25.0000 0.856989
\(852\) 6.00000 0.205557
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) 36.0000 1.23117
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) 55.0000 1.87658 0.938288 0.345855i \(-0.112411\pi\)
0.938288 + 0.345855i \(0.112411\pi\)
\(860\) −16.0000 −0.545595
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 52.0000 1.77010 0.885050 0.465495i \(-0.154124\pi\)
0.885050 + 0.465495i \(0.154124\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) 39.0000 1.32451
\(868\) 2.00000 0.0678844
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) −30.0000 −1.01535
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 60.0000 2.02721
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 4.00000 0.134840
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 16.0000 0.538138
\(885\) 9.00000 0.302532
\(886\) 0 0
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) −2.00000 −0.0669650
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −1.00000 −0.0334263
\(896\) 0 0
\(897\) −60.0000 −2.00334
\(898\) 0 0
\(899\) 10.0000 0.333519
\(900\) 48.0000 1.60000
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 0 0
\(905\) −5.00000 −0.166206
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −8.00000 −0.265489
\(909\) −72.0000 −2.38809
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 72.0000 2.38416
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 14.0000 0.462573
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 84.0000 2.76789
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 6.00000 0.197386
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) −72.0000 −2.36479
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −12.0000 −0.393073
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 0 0
\(939\) 69.0000 2.25173
\(940\) 16.0000 0.521862
\(941\) 58.0000 1.89075 0.945373 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) 10.0000 0.325645
\(944\) 12.0000 0.390567
\(945\) −9.00000 −0.292770
\(946\) 0 0
\(947\) 5.00000 0.162478 0.0812391 0.996695i \(-0.474112\pi\)
0.0812391 + 0.996695i \(0.474112\pi\)
\(948\) 36.0000 1.16923
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 0 0
\(955\) −5.00000 −0.161796
\(956\) −8.00000 −0.258738
\(957\) 30.0000 0.969762
\(958\) 0 0
\(959\) 3.00000 0.0968751
\(960\) −24.0000 −0.774597
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −60.0000 −1.93347
\(964\) 24.0000 0.772988
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) 29.0000 0.930654 0.465327 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) 0 0
\(973\) 10.0000 0.320585
\(974\) 0 0
\(975\) −48.0000 −1.53723
\(976\) −8.00000 −0.256074
\(977\) −31.0000 −0.991778 −0.495889 0.868386i \(-0.665158\pi\)
−0.495889 + 0.868386i \(0.665158\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) 2.00000 0.0638877
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) −27.0000 −0.861166 −0.430583 0.902551i \(-0.641692\pi\)
−0.430583 + 0.902551i \(0.641692\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) −48.0000 −1.52708
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 51.0000 1.61844
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 72.0000 2.28141
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) 45.0000 1.42374
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 77.2.a.a.1.1 1
3.2 odd 2 693.2.a.c.1.1 1
4.3 odd 2 1232.2.a.l.1.1 1
5.2 odd 4 1925.2.b.e.1849.2 2
5.3 odd 4 1925.2.b.e.1849.1 2
5.4 even 2 1925.2.a.h.1.1 1
7.2 even 3 539.2.e.f.67.1 2
7.3 odd 6 539.2.e.c.177.1 2
7.4 even 3 539.2.e.f.177.1 2
7.5 odd 6 539.2.e.c.67.1 2
7.6 odd 2 539.2.a.c.1.1 1
8.3 odd 2 4928.2.a.a.1.1 1
8.5 even 2 4928.2.a.bj.1.1 1
11.2 odd 10 847.2.f.h.323.1 4
11.3 even 5 847.2.f.i.372.1 4
11.4 even 5 847.2.f.i.148.1 4
11.5 even 5 847.2.f.i.729.1 4
11.6 odd 10 847.2.f.h.729.1 4
11.7 odd 10 847.2.f.h.148.1 4
11.8 odd 10 847.2.f.h.372.1 4
11.9 even 5 847.2.f.i.323.1 4
11.10 odd 2 847.2.a.b.1.1 1
21.20 even 2 4851.2.a.j.1.1 1
28.27 even 2 8624.2.a.a.1.1 1
33.32 even 2 7623.2.a.j.1.1 1
77.76 even 2 5929.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.a.1.1 1 1.1 even 1 trivial
539.2.a.c.1.1 1 7.6 odd 2
539.2.e.c.67.1 2 7.5 odd 6
539.2.e.c.177.1 2 7.3 odd 6
539.2.e.f.67.1 2 7.2 even 3
539.2.e.f.177.1 2 7.4 even 3
693.2.a.c.1.1 1 3.2 odd 2
847.2.a.b.1.1 1 11.10 odd 2
847.2.f.h.148.1 4 11.7 odd 10
847.2.f.h.323.1 4 11.2 odd 10
847.2.f.h.372.1 4 11.8 odd 10
847.2.f.h.729.1 4 11.6 odd 10
847.2.f.i.148.1 4 11.4 even 5
847.2.f.i.323.1 4 11.9 even 5
847.2.f.i.372.1 4 11.3 even 5
847.2.f.i.729.1 4 11.5 even 5
1232.2.a.l.1.1 1 4.3 odd 2
1925.2.a.h.1.1 1 5.4 even 2
1925.2.b.e.1849.1 2 5.3 odd 4
1925.2.b.e.1849.2 2 5.2 odd 4
4851.2.a.j.1.1 1 21.20 even 2
4928.2.a.a.1.1 1 8.3 odd 2
4928.2.a.bj.1.1 1 8.5 even 2
5929.2.a.f.1.1 1 77.76 even 2
7623.2.a.j.1.1 1 33.32 even 2
8624.2.a.a.1.1 1 28.27 even 2