Properties

Label 4864.2.a.z.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} -1.73205 q^{5} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -1.73205 q^{5} +1.73205 q^{7} +1.00000 q^{9} -3.00000 q^{11} -3.46410 q^{15} -3.00000 q^{17} +1.00000 q^{19} +3.46410 q^{21} +3.46410 q^{23} -2.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -6.92820 q^{31} -6.00000 q^{33} -3.00000 q^{35} +10.3923 q^{37} -6.00000 q^{41} -1.00000 q^{43} -1.73205 q^{45} -5.19615 q^{47} -4.00000 q^{49} -6.00000 q^{51} +5.19615 q^{55} +2.00000 q^{57} -6.00000 q^{59} -5.19615 q^{61} +1.73205 q^{63} +4.00000 q^{67} +6.92820 q^{69} -3.46410 q^{71} +1.00000 q^{73} -4.00000 q^{75} -5.19615 q^{77} -11.0000 q^{81} -12.0000 q^{83} +5.19615 q^{85} +6.92820 q^{87} -13.8564 q^{93} -1.73205 q^{95} +4.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 4q^{25} - 8q^{27} - 12q^{33} - 6q^{35} - 12q^{41} - 2q^{43} - 8q^{49} - 12q^{51} + 4q^{57} - 12q^{59} + 8q^{67} + 2q^{73} - 8q^{75} - 22q^{81} - 24q^{83} + 8q^{97} - 6q^{99} + O(q^{100}) \)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 10.3923 1.70848 0.854242 0.519875i \(-0.174022\pi\)
0.854242 + 0.519875i \(0.174022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −1.73205 −0.258199
\(46\) 0 0
\(47\) −5.19615 −0.757937 −0.378968 0.925410i \(-0.623721\pi\)
−0.378968 + 0.925410i \(0.623721\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 5.19615 0.700649
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −5.19615 −0.665299 −0.332650 0.943051i \(-0.607943\pi\)
−0.332650 + 0.943051i \(0.607943\pi\)
\(62\) 0 0
\(63\) 1.73205 0.218218
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −5.19615 −0.592157
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 5.19615 0.563602
\(86\) 0 0
\(87\) 6.92820 0.742781
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −13.8564 −1.43684
\(94\) 0 0
\(95\) −1.73205 −0.177705
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −17.3205 −1.70664 −0.853320 0.521387i \(-0.825415\pi\)
−0.853320 + 0.521387i \(0.825415\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 20.7846 1.97279
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.19615 −0.476331
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 1.73205 0.150188
\(134\) 0 0
\(135\) 6.92820 0.596285
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −10.3923 −0.875190
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) −8.00000 −0.659829
\(148\) 0 0
\(149\) −15.5885 −1.27706 −0.638528 0.769599i \(-0.720456\pi\)
−0.638528 + 0.769599i \(0.720456\pi\)
\(150\) 0 0
\(151\) 17.3205 1.40952 0.704761 0.709444i \(-0.251054\pi\)
0.704761 + 0.709444i \(0.251054\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 13.8564 1.10586 0.552931 0.833227i \(-0.313509\pi\)
0.552931 + 0.833227i \(0.313509\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 10.3923 0.809040
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 10.3923 0.790112 0.395056 0.918657i \(-0.370725\pi\)
0.395056 + 0.918657i \(0.370725\pi\)
\(174\) 0 0
\(175\) −3.46410 −0.261861
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −24.2487 −1.80239 −0.901196 0.433411i \(-0.857310\pi\)
−0.901196 + 0.433411i \(0.857310\pi\)
\(182\) 0 0
\(183\) −10.3923 −0.768221
\(184\) 0 0
\(185\) −18.0000 −1.32339
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) −6.92820 −0.503953
\(190\) 0 0
\(191\) 22.5167 1.62925 0.814624 0.579989i \(-0.196943\pi\)
0.814624 + 0.579989i \(0.196943\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −27.7128 −1.97446 −0.987228 0.159313i \(-0.949072\pi\)
−0.987228 + 0.159313i \(0.949072\pi\)
\(198\) 0 0
\(199\) 19.0526 1.35060 0.675300 0.737543i \(-0.264014\pi\)
0.675300 + 0.737543i \(0.264014\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 10.3923 0.725830
\(206\) 0 0
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 0 0
\(213\) −6.92820 −0.474713
\(214\) 0 0
\(215\) 1.73205 0.118125
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.73205 −0.114457 −0.0572286 0.998361i \(-0.518226\pi\)
−0.0572286 + 0.998361i \(0.518226\pi\)
\(230\) 0 0
\(231\) −10.3923 −0.683763
\(232\) 0 0
\(233\) −27.0000 −1.76883 −0.884414 0.466702i \(-0.845442\pi\)
−0.884414 + 0.466702i \(0.845442\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.73205 −0.112037 −0.0560185 0.998430i \(-0.517841\pi\)
−0.0560185 + 0.998430i \(0.517841\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 6.92820 0.442627
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −10.3923 −0.653359
\(254\) 0 0
\(255\) 10.3923 0.650791
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 18.0000 1.11847
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) 0 0
\(263\) −29.4449 −1.81565 −0.907824 0.419351i \(-0.862258\pi\)
−0.907824 + 0.419351i \(0.862258\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.7128 1.68968 0.844840 0.535019i \(-0.179696\pi\)
0.844840 + 0.535019i \(0.179696\pi\)
\(270\) 0 0
\(271\) 3.46410 0.210429 0.105215 0.994450i \(-0.466447\pi\)
0.105215 + 0.994450i \(0.466447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) 15.5885 0.936620 0.468310 0.883564i \(-0.344863\pi\)
0.468310 + 0.883564i \(0.344863\pi\)
\(278\) 0 0
\(279\) −6.92820 −0.414781
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) 0 0
\(285\) −3.46410 −0.205196
\(286\) 0 0
\(287\) −10.3923 −0.613438
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 10.3923 0.605063
\(296\) 0 0
\(297\) 12.0000 0.696311
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.73205 −0.0998337
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.00000 0.515339
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −34.6410 −1.97066
\(310\) 0 0
\(311\) −22.5167 −1.27680 −0.638401 0.769704i \(-0.720404\pi\)
−0.638401 + 0.769704i \(0.720404\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −17.3205 −0.972817 −0.486408 0.873732i \(-0.661693\pi\)
−0.486408 + 0.873732i \(0.661693\pi\)
\(318\) 0 0
\(319\) −10.3923 −0.581857
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 0 0
\(333\) 10.3923 0.569495
\(334\) 0 0
\(335\) −6.92820 −0.378528
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −36.0000 −1.95525
\(340\) 0 0
\(341\) 20.7846 1.12555
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) 36.3731 1.94701 0.973503 0.228675i \(-0.0734393\pi\)
0.973503 + 0.228675i \(0.0734393\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) −10.3923 −0.550019
\(358\) 0 0
\(359\) −29.4449 −1.55404 −0.777020 0.629476i \(-0.783270\pi\)
−0.777020 + 0.629476i \(0.783270\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −4.00000 −0.209946
\(364\) 0 0
\(365\) −1.73205 −0.0906597
\(366\) 0 0
\(367\) 10.3923 0.542474 0.271237 0.962513i \(-0.412567\pi\)
0.271237 + 0.962513i \(0.412567\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.8564 0.717458 0.358729 0.933442i \(-0.383210\pi\)
0.358729 + 0.933442i \(0.383210\pi\)
\(374\) 0 0
\(375\) 24.2487 1.25220
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −20.7846 −1.06483
\(382\) 0 0
\(383\) 27.7128 1.41606 0.708029 0.706183i \(-0.249584\pi\)
0.708029 + 0.706183i \(0.249584\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) −22.5167 −1.14164 −0.570820 0.821075i \(-0.693375\pi\)
−0.570820 + 0.821075i \(0.693375\pi\)
\(390\) 0 0
\(391\) −10.3923 −0.525561
\(392\) 0 0
\(393\) 30.0000 1.51330
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.5167 −1.13008 −0.565039 0.825064i \(-0.691139\pi\)
−0.565039 + 0.825064i \(0.691139\pi\)
\(398\) 0 0
\(399\) 3.46410 0.173422
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 19.0526 0.946729
\(406\) 0 0
\(407\) −31.1769 −1.54538
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) −10.3923 −0.511372
\(414\) 0 0
\(415\) 20.7846 1.02028
\(416\) 0 0
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 13.8564 0.675320 0.337660 0.941268i \(-0.390365\pi\)
0.337660 + 0.941268i \(0.390365\pi\)
\(422\) 0 0
\(423\) −5.19615 −0.252646
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −9.00000 −0.435541
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.7128 −1.33488 −0.667440 0.744664i \(-0.732610\pi\)
−0.667440 + 0.744664i \(0.732610\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 3.46410 0.165710
\(438\) 0 0
\(439\) 24.2487 1.15733 0.578664 0.815566i \(-0.303574\pi\)
0.578664 + 0.815566i \(0.303574\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 27.0000 1.28281 0.641404 0.767203i \(-0.278352\pi\)
0.641404 + 0.767203i \(0.278352\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −31.1769 −1.47462
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 34.6410 1.62758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 0 0
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) 22.5167 1.04871 0.524353 0.851501i \(-0.324307\pi\)
0.524353 + 0.851501i \(0.324307\pi\)
\(462\) 0 0
\(463\) 25.9808 1.20743 0.603714 0.797201i \(-0.293687\pi\)
0.603714 + 0.797201i \(0.293687\pi\)
\(464\) 0 0
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 6.92820 0.319915
\(470\) 0 0
\(471\) 27.7128 1.27694
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) −6.92820 −0.314594
\(486\) 0 0
\(487\) 24.2487 1.09881 0.549407 0.835555i \(-0.314854\pi\)
0.549407 + 0.835555i \(0.314854\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −10.3923 −0.468046
\(494\) 0 0
\(495\) 5.19615 0.233550
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −29.0000 −1.29822 −0.649109 0.760695i \(-0.724858\pi\)
−0.649109 + 0.760695i \(0.724858\pi\)
\(500\) 0 0
\(501\) 6.92820 0.309529
\(502\) 0 0
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −26.0000 −1.15470
\(508\) 0 0
\(509\) 27.7128 1.22835 0.614174 0.789170i \(-0.289489\pi\)
0.614174 + 0.789170i \(0.289489\pi\)
\(510\) 0 0
\(511\) 1.73205 0.0766214
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 30.0000 1.32196
\(516\) 0 0
\(517\) 15.5885 0.685580
\(518\) 0 0
\(519\) 20.7846 0.912343
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) −6.92820 −0.302372
\(526\) 0 0
\(527\) 20.7846 0.905392
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.3923 0.449299
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −5.19615 −0.223400 −0.111700 0.993742i \(-0.535630\pi\)
−0.111700 + 0.993742i \(0.535630\pi\)
\(542\) 0 0
\(543\) −48.4974 −2.08122
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) −5.19615 −0.221766
\(550\) 0 0
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −36.0000 −1.52811
\(556\) 0 0
\(557\) −29.4449 −1.24762 −0.623809 0.781576i \(-0.714416\pi\)
−0.623809 + 0.781576i \(0.714416\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 31.1769 1.31162
\(566\) 0 0
\(567\) −19.0526 −0.800132
\(568\) 0 0
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 45.0333 1.88129
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) 37.0000 1.54033 0.770165 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 0 0
\(579\) 40.0000 1.66234
\(580\) 0 0
\(581\) −20.7846 −0.862291
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) −6.92820 −0.285472
\(590\) 0 0
\(591\) −55.4256 −2.27991
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) 0 0
\(597\) 38.1051 1.55954
\(598\) 0 0
\(599\) 27.7128 1.13231 0.566157 0.824297i \(-0.308429\pi\)
0.566157 + 0.824297i \(0.308429\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) −27.7128 −1.12483 −0.562414 0.826856i \(-0.690127\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.73205 0.0699569 0.0349784 0.999388i \(-0.488864\pi\)
0.0349784 + 0.999388i \(0.488864\pi\)
\(614\) 0 0
\(615\) 20.7846 0.838116
\(616\) 0 0
\(617\) −15.0000 −0.603877 −0.301939 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −13.8564 −0.556038
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 0 0
\(629\) −31.1769 −1.24310
\(630\) 0 0
\(631\) −36.3731 −1.44799 −0.723994 0.689806i \(-0.757696\pi\)
−0.723994 + 0.689806i \(0.757696\pi\)
\(632\) 0 0
\(633\) 44.0000 1.74884
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.46410 −0.137038
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 0 0
\(645\) 3.46410 0.136399
\(646\) 0 0
\(647\) 19.0526 0.749033 0.374517 0.927220i \(-0.377809\pi\)
0.374517 + 0.927220i \(0.377809\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) −22.5167 −0.881145 −0.440573 0.897717i \(-0.645225\pi\)
−0.440573 + 0.897717i \(0.645225\pi\)
\(654\) 0 0
\(655\) −25.9808 −1.01515
\(656\) 0 0
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −34.6410 −1.34738 −0.673690 0.739014i \(-0.735292\pi\)
−0.673690 + 0.739014i \(0.735292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) −6.92820 −0.267860
\(670\) 0 0
\(671\) 15.5885 0.601786
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 8.00000 0.307920
\(676\) 0 0
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 0 0
\(679\) 6.92820 0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −5.19615 −0.198535
\(686\) 0 0
\(687\) −3.46410 −0.132164
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −7.00000 −0.266293 −0.133146 0.991096i \(-0.542508\pi\)
−0.133146 + 0.991096i \(0.542508\pi\)
\(692\) 0 0
\(693\) −5.19615 −0.197386
\(694\) 0 0
\(695\) 8.66025 0.328502
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 10.3923 0.391953
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.92820 0.260194 0.130097 0.991501i \(-0.458471\pi\)
0.130097 + 0.991501i \(0.458471\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.46410 −0.129369
\(718\) 0 0
\(719\) 29.4449 1.09811 0.549054 0.835787i \(-0.314988\pi\)
0.549054 + 0.835787i \(0.314988\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) 0 0
\(725\) −6.92820 −0.257307
\(726\) 0 0
\(727\) 19.0526 0.706620 0.353310 0.935506i \(-0.385056\pi\)
0.353310 + 0.935506i \(0.385056\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 13.8564 0.511101
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.92820 −0.254171 −0.127086 0.991892i \(-0.540562\pi\)
−0.127086 + 0.991892i \(0.540562\pi\)
\(744\) 0 0
\(745\) 27.0000 0.989203
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −10.3923 −0.379727
\(750\) 0 0
\(751\) −20.7846 −0.758441 −0.379221 0.925306i \(-0.623808\pi\)
−0.379221 + 0.925306i \(0.623808\pi\)
\(752\) 0 0
\(753\) 42.0000 1.53057
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) 36.3731 1.32200 0.661001 0.750385i \(-0.270132\pi\)
0.661001 + 0.750385i \(0.270132\pi\)
\(758\) 0 0
\(759\) −20.7846 −0.754434
\(760\) 0 0
\(761\) 39.0000 1.41375 0.706874 0.707339i \(-0.250105\pi\)
0.706874 + 0.707339i \(0.250105\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5.19615 0.187867
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.2487 0.872166 0.436083 0.899907i \(-0.356365\pi\)
0.436083 + 0.899907i \(0.356365\pi\)
\(774\) 0 0
\(775\) 13.8564 0.497737
\(776\) 0 0
\(777\) 36.0000 1.29149
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 10.3923 0.371866
\(782\) 0 0
\(783\) −13.8564 −0.495188
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 0 0
\(789\) −58.8897 −2.09653
\(790\) 0 0
\(791\) −31.1769 −1.10852
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.6410 −1.22705 −0.613524 0.789676i \(-0.710249\pi\)
−0.613524 + 0.789676i \(0.710249\pi\)
\(798\) 0 0
\(799\) 15.5885 0.551480
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) 55.4256 1.95107
\(808\) 0 0
\(809\) −33.0000 −1.16022 −0.580109 0.814539i \(-0.696990\pi\)
−0.580109 + 0.814539i \(0.696990\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) 6.92820 0.242983
\(814\) 0 0
\(815\) 6.92820 0.242684
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.19615 0.181347 0.0906735 0.995881i \(-0.471098\pi\)
0.0906735 + 0.995881i \(0.471098\pi\)
\(822\) 0 0
\(823\) −29.4449 −1.02638 −0.513192 0.858274i \(-0.671537\pi\)
−0.513192 + 0.858274i \(0.671537\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −13.8564 −0.481253 −0.240626 0.970618i \(-0.577353\pi\)
−0.240626 + 0.970618i \(0.577353\pi\)
\(830\) 0 0
\(831\) 31.1769 1.08152
\(832\) 0 0
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) 27.7128 0.957895
\(838\) 0 0
\(839\) 24.2487 0.837158 0.418579 0.908180i \(-0.362528\pi\)
0.418579 + 0.908180i \(0.362528\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) −60.0000 −2.06651
\(844\) 0 0
\(845\) 22.5167 0.774597
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) 38.0000 1.30416
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) 27.7128 0.948869 0.474434 0.880291i \(-0.342653\pi\)
0.474434 + 0.880291i \(0.342653\pi\)
\(854\) 0 0
\(855\) −1.73205 −0.0592349
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −1.00000 −0.0341196 −0.0170598 0.999854i \(-0.505431\pi\)
−0.0170598 + 0.999854i \(0.505431\pi\)
\(860\) 0 0
\(861\) −20.7846 −0.708338
\(862\) 0 0
\(863\) 24.2487 0.825436 0.412718 0.910859i \(-0.364580\pi\)
0.412718 + 0.910859i \(0.364580\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 21.0000 0.709930
\(876\) 0 0
\(877\) −31.1769 −1.05277 −0.526385 0.850246i \(-0.676453\pi\)
−0.526385 + 0.850246i \(0.676453\pi\)
\(878\) 0 0
\(879\) −27.7128 −0.934730
\(880\) 0 0
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) 0 0
\(883\) −49.0000 −1.64898 −0.824491 0.565876i \(-0.808538\pi\)
−0.824491 + 0.565876i \(0.808538\pi\)
\(884\) 0 0
\(885\) 20.7846 0.698667
\(886\) 0 0
\(887\) −31.1769 −1.04682 −0.523409 0.852081i \(-0.675340\pi\)
−0.523409 + 0.852081i \(0.675340\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 33.0000 1.10554
\(892\) 0 0
\(893\) −5.19615 −0.173883
\(894\) 0 0
\(895\) 10.3923 0.347376
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −3.46410 −0.115278
\(904\) 0 0
\(905\) 42.0000 1.39613
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.9615 1.72156 0.860781 0.508975i \(-0.169976\pi\)
0.860781 + 0.508975i \(0.169976\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 18.0000 0.595062
\(916\) 0 0
\(917\) 25.9808 0.857960
\(918\) 0 0
\(919\) −3.46410 −0.114270 −0.0571351 0.998366i \(-0.518197\pi\)
−0.0571351 + 0.998366i \(0.518197\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.7846 −0.683394
\(926\) 0 0
\(927\) −17.3205 −0.568880
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) −45.0333 −1.47432
\(934\) 0 0
\(935\) −15.5885 −0.509797
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) 0 0
\(939\) 52.0000 1.69696
\(940\) 0 0
\(941\) 38.1051 1.24219 0.621096 0.783735i \(-0.286688\pi\)
0.621096 + 0.783735i \(0.286688\pi\)
\(942\) 0 0
\(943\) −20.7846 −0.676840
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −34.6410 −1.12331
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) −39.0000 −1.26201
\(956\) 0 0
\(957\) −20.7846 −0.671871
\(958\) 0 0
\(959\) 5.19615 0.167793
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) −34.6410 −1.11513
\(966\) 0 0
\(967\) −24.2487 −0.779786 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) −8.66025 −0.277635
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −34.6410 −1.10488 −0.552438 0.833554i \(-0.686303\pi\)
−0.552438 + 0.833554i \(0.686303\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) −18.0000 −0.572946
\(988\) 0 0
\(989\) −3.46410 −0.110152
\(990\) 0 0
\(991\) −31.1769 −0.990367 −0.495184 0.868788i \(-0.664899\pi\)
−0.495184 + 0.868788i \(0.664899\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) −33.0000 −1.04617
\(996\) 0 0
\(997\) −19.0526 −0.603401 −0.301700 0.953403i \(-0.597554\pi\)
−0.301700 + 0.953403i \(0.597554\pi\)
\(998\) 0 0
\(999\) −41.5692 −1.31519
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 4864.2.a.z.1.1 2
4.3 odd 2 4864.2.a.q.1.1 2
8.3 odd 2 inner 4864.2.a.z.1.2 2
8.5 even 2 4864.2.a.q.1.2 2
16.3 odd 4 1216.2.c.e.609.2 yes 4
16.5 even 4 1216.2.c.e.609.1 4
16.11 odd 4 1216.2.c.e.609.3 yes 4
16.13 even 4 1216.2.c.e.609.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.e.609.1 4 16.5 even 4
1216.2.c.e.609.2 yes 4 16.3 odd 4
1216.2.c.e.609.3 yes 4 16.11 odd 4
1216.2.c.e.609.4 yes 4 16.13 even 4
4864.2.a.q.1.1 2 4.3 odd 2
4864.2.a.q.1.2 2 8.5 even 2
4864.2.a.z.1.1 2 1.1 even 1 trivial
4864.2.a.z.1.2 2 8.3 odd 2 inner