Properties

Label 4864.2.a.ba.1.3
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.24914 q^{3} -1.47068 q^{5} -1.47068 q^{7} +2.05863 q^{9} +O(q^{10})\) \(q+2.24914 q^{3} -1.47068 q^{5} -1.47068 q^{7} +2.05863 q^{9} +4.77846 q^{11} -4.49828 q^{13} -3.30777 q^{15} +1.71982 q^{17} +1.00000 q^{19} -3.30777 q^{21} -4.00000 q^{23} -2.83709 q^{25} -2.11727 q^{27} -7.80605 q^{29} +4.13187 q^{31} +10.7474 q^{33} +2.16291 q^{35} -3.30777 q^{37} -10.1173 q^{39} -9.80605 q^{41} -1.83709 q^{43} -3.02760 q^{45} +5.14486 q^{47} -4.83709 q^{49} +3.86813 q^{51} +12.8647 q^{53} -7.02760 q^{55} +2.24914 q^{57} -9.05863 q^{59} -8.91033 q^{61} -3.02760 q^{63} +6.61555 q^{65} -5.92332 q^{67} -8.99656 q^{69} +6.05520 q^{71} -0.335371 q^{73} -6.38101 q^{75} -7.02760 q^{77} +0.366407 q^{79} -10.9379 q^{81} +18.0552 q^{83} -2.52932 q^{85} -17.5569 q^{87} +2.36641 q^{89} +6.61555 q^{91} +9.29317 q^{93} -1.47068 q^{95} -7.05863 q^{97} +9.83709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} - 4q^{5} - 4q^{7} + 7q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - 4q^{5} - 4q^{7} + 7q^{9} + 6q^{11} + 4q^{13} - 2q^{15} - 4q^{17} + 3q^{19} - 2q^{21} - 12q^{23} - q^{25} - 8q^{27} + 2q^{29} + 2q^{31} + 6q^{33} + 14q^{35} - 2q^{37} - 32q^{39} - 4q^{41} + 2q^{43} + 8q^{45} - 7q^{49} + 22q^{51} + 14q^{53} - 4q^{55} - 2q^{57} - 28q^{59} - 8q^{61} + 8q^{63} + 4q^{65} + 6q^{67} + 8q^{69} - 16q^{71} + 24q^{73} - 4q^{77} - 6q^{79} + 3q^{81} + 20q^{83} - 8q^{85} - 36q^{87} + 4q^{91} + 32q^{93} - 4q^{95} - 22q^{97} + 22q^{99} + O(q^{100}) \)

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
\(3\) 2.24914 1.29854 0.649271 0.760557i \(-0.275074\pi\)
0.649271 + 0.760557i \(0.275074\pi\)
\(4\) 0 0
\(5\) −1.47068 −0.657710 −0.328855 0.944380i \(-0.606663\pi\)
−0.328855 + 0.944380i \(0.606663\pi\)
\(6\) 0 0
\(7\) −1.47068 −0.555866 −0.277933 0.960600i \(-0.589649\pi\)
−0.277933 + 0.960600i \(0.589649\pi\)
\(8\) 0 0
\(9\) 2.05863 0.686211
\(10\) 0 0
\(11\) 4.77846 1.44076 0.720380 0.693580i \(-0.243968\pi\)
0.720380 + 0.693580i \(0.243968\pi\)
\(12\) 0 0
\(13\) −4.49828 −1.24760 −0.623799 0.781585i \(-0.714412\pi\)
−0.623799 + 0.781585i \(0.714412\pi\)
\(14\) 0 0
\(15\) −3.30777 −0.854063
\(16\) 0 0
\(17\) 1.71982 0.417119 0.208559 0.978010i \(-0.433123\pi\)
0.208559 + 0.978010i \(0.433123\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.30777 −0.721815
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −2.83709 −0.567418
\(26\) 0 0
\(27\) −2.11727 −0.407468
\(28\) 0 0
\(29\) −7.80605 −1.44955 −0.724774 0.688987i \(-0.758056\pi\)
−0.724774 + 0.688987i \(0.758056\pi\)
\(30\) 0 0
\(31\) 4.13187 0.742107 0.371053 0.928612i \(-0.378997\pi\)
0.371053 + 0.928612i \(0.378997\pi\)
\(32\) 0 0
\(33\) 10.7474 1.87089
\(34\) 0 0
\(35\) 2.16291 0.365598
\(36\) 0 0
\(37\) −3.30777 −0.543795 −0.271897 0.962326i \(-0.587651\pi\)
−0.271897 + 0.962326i \(0.587651\pi\)
\(38\) 0 0
\(39\) −10.1173 −1.62006
\(40\) 0 0
\(41\) −9.80605 −1.53145 −0.765724 0.643169i \(-0.777619\pi\)
−0.765724 + 0.643169i \(0.777619\pi\)
\(42\) 0 0
\(43\) −1.83709 −0.280154 −0.140077 0.990141i \(-0.544735\pi\)
−0.140077 + 0.990141i \(0.544735\pi\)
\(44\) 0 0
\(45\) −3.02760 −0.451328
\(46\) 0 0
\(47\) 5.14486 0.750456 0.375228 0.926933i \(-0.377564\pi\)
0.375228 + 0.926933i \(0.377564\pi\)
\(48\) 0 0
\(49\) −4.83709 −0.691013
\(50\) 0 0
\(51\) 3.86813 0.541646
\(52\) 0 0
\(53\) 12.8647 1.76710 0.883550 0.468336i \(-0.155146\pi\)
0.883550 + 0.468336i \(0.155146\pi\)
\(54\) 0 0
\(55\) −7.02760 −0.947601
\(56\) 0 0
\(57\) 2.24914 0.297906
\(58\) 0 0
\(59\) −9.05863 −1.17933 −0.589667 0.807647i \(-0.700741\pi\)
−0.589667 + 0.807647i \(0.700741\pi\)
\(60\) 0 0
\(61\) −8.91033 −1.14085 −0.570426 0.821349i \(-0.693222\pi\)
−0.570426 + 0.821349i \(0.693222\pi\)
\(62\) 0 0
\(63\) −3.02760 −0.381441
\(64\) 0 0
\(65\) 6.61555 0.820558
\(66\) 0 0
\(67\) −5.92332 −0.723649 −0.361824 0.932246i \(-0.617846\pi\)
−0.361824 + 0.932246i \(0.617846\pi\)
\(68\) 0 0
\(69\) −8.99656 −1.08306
\(70\) 0 0
\(71\) 6.05520 0.718619 0.359310 0.933218i \(-0.383012\pi\)
0.359310 + 0.933218i \(0.383012\pi\)
\(72\) 0 0
\(73\) −0.335371 −0.0392522 −0.0196261 0.999807i \(-0.506248\pi\)
−0.0196261 + 0.999807i \(0.506248\pi\)
\(74\) 0 0
\(75\) −6.38101 −0.736816
\(76\) 0 0
\(77\) −7.02760 −0.800869
\(78\) 0 0
\(79\) 0.366407 0.0412240 0.0206120 0.999788i \(-0.493439\pi\)
0.0206120 + 0.999788i \(0.493439\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) 0 0
\(83\) 18.0552 1.98182 0.990908 0.134544i \(-0.0429571\pi\)
0.990908 + 0.134544i \(0.0429571\pi\)
\(84\) 0 0
\(85\) −2.52932 −0.274343
\(86\) 0 0
\(87\) −17.5569 −1.88230
\(88\) 0 0
\(89\) 2.36641 0.250839 0.125419 0.992104i \(-0.459972\pi\)
0.125419 + 0.992104i \(0.459972\pi\)
\(90\) 0 0
\(91\) 6.61555 0.693498
\(92\) 0 0
\(93\) 9.29317 0.963656
\(94\) 0 0
\(95\) −1.47068 −0.150889
\(96\) 0 0
\(97\) −7.05863 −0.716696 −0.358348 0.933588i \(-0.616660\pi\)
−0.358348 + 0.933588i \(0.616660\pi\)
\(98\) 0 0
\(99\) 9.83709 0.988665
\(100\) 0 0
\(101\) 6.94137 0.690692 0.345346 0.938475i \(-0.387762\pi\)
0.345346 + 0.938475i \(0.387762\pi\)
\(102\) 0 0
\(103\) −0.824101 −0.0812010 −0.0406005 0.999175i \(-0.512927\pi\)
−0.0406005 + 0.999175i \(0.512927\pi\)
\(104\) 0 0
\(105\) 4.86469 0.474745
\(106\) 0 0
\(107\) −11.4396 −1.10591 −0.552956 0.833210i \(-0.686500\pi\)
−0.552956 + 0.833210i \(0.686500\pi\)
\(108\) 0 0
\(109\) 4.13187 0.395762 0.197881 0.980226i \(-0.436594\pi\)
0.197881 + 0.980226i \(0.436594\pi\)
\(110\) 0 0
\(111\) −7.43965 −0.706140
\(112\) 0 0
\(113\) −7.92332 −0.745363 −0.372682 0.927959i \(-0.621562\pi\)
−0.372682 + 0.927959i \(0.621562\pi\)
\(114\) 0 0
\(115\) 5.88273 0.548568
\(116\) 0 0
\(117\) −9.26031 −0.856116
\(118\) 0 0
\(119\) −2.52932 −0.231862
\(120\) 0 0
\(121\) 11.8337 1.07579
\(122\) 0 0
\(123\) −22.0552 −1.98865
\(124\) 0 0
\(125\) 11.5259 1.03091
\(126\) 0 0
\(127\) −1.19051 −0.105640 −0.0528202 0.998604i \(-0.516821\pi\)
−0.0528202 + 0.998604i \(0.516821\pi\)
\(128\) 0 0
\(129\) −4.13187 −0.363791
\(130\) 0 0
\(131\) −2.39744 −0.209466 −0.104733 0.994500i \(-0.533399\pi\)
−0.104733 + 0.994500i \(0.533399\pi\)
\(132\) 0 0
\(133\) −1.47068 −0.127524
\(134\) 0 0
\(135\) 3.11383 0.267996
\(136\) 0 0
\(137\) −9.39400 −0.802584 −0.401292 0.915950i \(-0.631439\pi\)
−0.401292 + 0.915950i \(0.631439\pi\)
\(138\) 0 0
\(139\) 12.2181 1.03633 0.518163 0.855282i \(-0.326616\pi\)
0.518163 + 0.855282i \(0.326616\pi\)
\(140\) 0 0
\(141\) 11.5715 0.974498
\(142\) 0 0
\(143\) −21.4948 −1.79749
\(144\) 0 0
\(145\) 11.4802 0.953382
\(146\) 0 0
\(147\) −10.8793 −0.897309
\(148\) 0 0
\(149\) −16.8190 −1.37787 −0.688935 0.724823i \(-0.741921\pi\)
−0.688935 + 0.724823i \(0.741921\pi\)
\(150\) 0 0
\(151\) 7.17590 0.583966 0.291983 0.956423i \(-0.405685\pi\)
0.291983 + 0.956423i \(0.405685\pi\)
\(152\) 0 0
\(153\) 3.54049 0.286231
\(154\) 0 0
\(155\) −6.07668 −0.488091
\(156\) 0 0
\(157\) −15.6742 −1.25094 −0.625468 0.780250i \(-0.715092\pi\)
−0.625468 + 0.780250i \(0.715092\pi\)
\(158\) 0 0
\(159\) 28.9345 2.29465
\(160\) 0 0
\(161\) 5.88273 0.463624
\(162\) 0 0
\(163\) −9.32238 −0.730185 −0.365093 0.930971i \(-0.618963\pi\)
−0.365093 + 0.930971i \(0.618963\pi\)
\(164\) 0 0
\(165\) −15.8061 −1.23050
\(166\) 0 0
\(167\) −20.3043 −1.57120 −0.785598 0.618737i \(-0.787645\pi\)
−0.785598 + 0.618737i \(0.787645\pi\)
\(168\) 0 0
\(169\) 7.23453 0.556503
\(170\) 0 0
\(171\) 2.05863 0.157428
\(172\) 0 0
\(173\) 0.560352 0.0426028 0.0213014 0.999773i \(-0.493219\pi\)
0.0213014 + 0.999773i \(0.493219\pi\)
\(174\) 0 0
\(175\) 4.17246 0.315408
\(176\) 0 0
\(177\) −20.3741 −1.53141
\(178\) 0 0
\(179\) −4.36641 −0.326361 −0.163180 0.986596i \(-0.552175\pi\)
−0.163180 + 0.986596i \(0.552175\pi\)
\(180\) 0 0
\(181\) −5.05863 −0.376005 −0.188003 0.982169i \(-0.560201\pi\)
−0.188003 + 0.982169i \(0.560201\pi\)
\(182\) 0 0
\(183\) −20.0406 −1.48144
\(184\) 0 0
\(185\) 4.86469 0.357659
\(186\) 0 0
\(187\) 8.21811 0.600967
\(188\) 0 0
\(189\) 3.11383 0.226498
\(190\) 0 0
\(191\) 2.46725 0.178524 0.0892618 0.996008i \(-0.471549\pi\)
0.0892618 + 0.996008i \(0.471549\pi\)
\(192\) 0 0
\(193\) 12.2491 0.881712 0.440856 0.897578i \(-0.354675\pi\)
0.440856 + 0.897578i \(0.354675\pi\)
\(194\) 0 0
\(195\) 14.8793 1.06553
\(196\) 0 0
\(197\) 1.23109 0.0877119 0.0438559 0.999038i \(-0.486036\pi\)
0.0438559 + 0.999038i \(0.486036\pi\)
\(198\) 0 0
\(199\) −27.0276 −1.91594 −0.957968 0.286876i \(-0.907383\pi\)
−0.957968 + 0.286876i \(0.907383\pi\)
\(200\) 0 0
\(201\) −13.3224 −0.939688
\(202\) 0 0
\(203\) 11.4802 0.805755
\(204\) 0 0
\(205\) 14.4216 1.00725
\(206\) 0 0
\(207\) −8.23453 −0.572340
\(208\) 0 0
\(209\) 4.77846 0.330533
\(210\) 0 0
\(211\) −0.325819 −0.0224303 −0.0112152 0.999937i \(-0.503570\pi\)
−0.0112152 + 0.999937i \(0.503570\pi\)
\(212\) 0 0
\(213\) 13.6190 0.933157
\(214\) 0 0
\(215\) 2.70178 0.184260
\(216\) 0 0
\(217\) −6.07668 −0.412512
\(218\) 0 0
\(219\) −0.754297 −0.0509707
\(220\) 0 0
\(221\) −7.73625 −0.520397
\(222\) 0 0
\(223\) −23.4182 −1.56820 −0.784098 0.620637i \(-0.786874\pi\)
−0.784098 + 0.620637i \(0.786874\pi\)
\(224\) 0 0
\(225\) −5.84053 −0.389369
\(226\) 0 0
\(227\) 23.8466 1.58276 0.791379 0.611326i \(-0.209364\pi\)
0.791379 + 0.611326i \(0.209364\pi\)
\(228\) 0 0
\(229\) 13.9690 0.923095 0.461548 0.887115i \(-0.347294\pi\)
0.461548 + 0.887115i \(0.347294\pi\)
\(230\) 0 0
\(231\) −15.8061 −1.03996
\(232\) 0 0
\(233\) −12.8337 −0.840761 −0.420380 0.907348i \(-0.638103\pi\)
−0.420380 + 0.907348i \(0.638103\pi\)
\(234\) 0 0
\(235\) −7.56647 −0.493582
\(236\) 0 0
\(237\) 0.824101 0.0535311
\(238\) 0 0
\(239\) −24.2587 −1.56916 −0.784582 0.620025i \(-0.787123\pi\)
−0.784582 + 0.620025i \(0.787123\pi\)
\(240\) 0 0
\(241\) −26.0698 −1.67930 −0.839652 0.543125i \(-0.817241\pi\)
−0.839652 + 0.543125i \(0.817241\pi\)
\(242\) 0 0
\(243\) −18.2491 −1.17068
\(244\) 0 0
\(245\) 7.11383 0.454486
\(246\) 0 0
\(247\) −4.49828 −0.286219
\(248\) 0 0
\(249\) 40.6087 2.57347
\(250\) 0 0
\(251\) −23.3319 −1.47270 −0.736349 0.676602i \(-0.763452\pi\)
−0.736349 + 0.676602i \(0.763452\pi\)
\(252\) 0 0
\(253\) −19.1138 −1.20168
\(254\) 0 0
\(255\) −5.68879 −0.356246
\(256\) 0 0
\(257\) 2.82410 0.176163 0.0880813 0.996113i \(-0.471926\pi\)
0.0880813 + 0.996113i \(0.471926\pi\)
\(258\) 0 0
\(259\) 4.86469 0.302277
\(260\) 0 0
\(261\) −16.0698 −0.994696
\(262\) 0 0
\(263\) −7.08967 −0.437168 −0.218584 0.975818i \(-0.570144\pi\)
−0.218584 + 0.975818i \(0.570144\pi\)
\(264\) 0 0
\(265\) −18.9199 −1.16224
\(266\) 0 0
\(267\) 5.32238 0.325724
\(268\) 0 0
\(269\) 20.8647 1.27214 0.636071 0.771630i \(-0.280558\pi\)
0.636071 + 0.771630i \(0.280558\pi\)
\(270\) 0 0
\(271\) −0.0620710 −0.00377055 −0.00188527 0.999998i \(-0.500600\pi\)
−0.00188527 + 0.999998i \(0.500600\pi\)
\(272\) 0 0
\(273\) 14.8793 0.900536
\(274\) 0 0
\(275\) −13.5569 −0.817513
\(276\) 0 0
\(277\) −7.82248 −0.470007 −0.235004 0.971994i \(-0.575510\pi\)
−0.235004 + 0.971994i \(0.575510\pi\)
\(278\) 0 0
\(279\) 8.50601 0.509242
\(280\) 0 0
\(281\) 20.6448 1.23156 0.615782 0.787917i \(-0.288840\pi\)
0.615782 + 0.787917i \(0.288840\pi\)
\(282\) 0 0
\(283\) −21.3871 −1.27133 −0.635666 0.771964i \(-0.719275\pi\)
−0.635666 + 0.771964i \(0.719275\pi\)
\(284\) 0 0
\(285\) −3.30777 −0.195936
\(286\) 0 0
\(287\) 14.4216 0.851280
\(288\) 0 0
\(289\) −14.0422 −0.826012
\(290\) 0 0
\(291\) −15.8759 −0.930659
\(292\) 0 0
\(293\) 24.6087 1.43765 0.718827 0.695189i \(-0.244679\pi\)
0.718827 + 0.695189i \(0.244679\pi\)
\(294\) 0 0
\(295\) 13.3224 0.775659
\(296\) 0 0
\(297\) −10.1173 −0.587063
\(298\) 0 0
\(299\) 17.9931 1.04057
\(300\) 0 0
\(301\) 2.70178 0.155728
\(302\) 0 0
\(303\) 15.6121 0.896892
\(304\) 0 0
\(305\) 13.1043 0.750349
\(306\) 0 0
\(307\) 10.4216 0.594792 0.297396 0.954754i \(-0.403882\pi\)
0.297396 + 0.954754i \(0.403882\pi\)
\(308\) 0 0
\(309\) −1.85352 −0.105443
\(310\) 0 0
\(311\) −7.91377 −0.448749 −0.224374 0.974503i \(-0.572034\pi\)
−0.224374 + 0.974503i \(0.572034\pi\)
\(312\) 0 0
\(313\) −17.7846 −1.00524 −0.502622 0.864506i \(-0.667631\pi\)
−0.502622 + 0.864506i \(0.667631\pi\)
\(314\) 0 0
\(315\) 4.45264 0.250878
\(316\) 0 0
\(317\) 30.0552 1.68807 0.844034 0.536290i \(-0.180175\pi\)
0.844034 + 0.536290i \(0.180175\pi\)
\(318\) 0 0
\(319\) −37.3009 −2.08845
\(320\) 0 0
\(321\) −25.7294 −1.43607
\(322\) 0 0
\(323\) 1.71982 0.0956936
\(324\) 0 0
\(325\) 12.7620 0.707910
\(326\) 0 0
\(327\) 9.29317 0.513913
\(328\) 0 0
\(329\) −7.56647 −0.417153
\(330\) 0 0
\(331\) 18.8095 1.03386 0.516932 0.856027i \(-0.327074\pi\)
0.516932 + 0.856027i \(0.327074\pi\)
\(332\) 0 0
\(333\) −6.80949 −0.373158
\(334\) 0 0
\(335\) 8.71133 0.475951
\(336\) 0 0
\(337\) 30.2017 1.64519 0.822595 0.568628i \(-0.192525\pi\)
0.822595 + 0.568628i \(0.192525\pi\)
\(338\) 0 0
\(339\) −17.8207 −0.967886
\(340\) 0 0
\(341\) 19.7440 1.06920
\(342\) 0 0
\(343\) 17.4086 0.939977
\(344\) 0 0
\(345\) 13.2311 0.712338
\(346\) 0 0
\(347\) −9.36802 −0.502902 −0.251451 0.967870i \(-0.580908\pi\)
−0.251451 + 0.967870i \(0.580908\pi\)
\(348\) 0 0
\(349\) 21.3173 1.14109 0.570545 0.821266i \(-0.306732\pi\)
0.570545 + 0.821266i \(0.306732\pi\)
\(350\) 0 0
\(351\) 9.52406 0.508357
\(352\) 0 0
\(353\) 18.4362 0.981260 0.490630 0.871368i \(-0.336767\pi\)
0.490630 + 0.871368i \(0.336767\pi\)
\(354\) 0 0
\(355\) −8.90528 −0.472643
\(356\) 0 0
\(357\) −5.68879 −0.301083
\(358\) 0 0
\(359\) −37.1932 −1.96298 −0.981491 0.191510i \(-0.938661\pi\)
−0.981491 + 0.191510i \(0.938661\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 26.6155 1.39695
\(364\) 0 0
\(365\) 0.493225 0.0258166
\(366\) 0 0
\(367\) −11.6121 −0.606147 −0.303074 0.952967i \(-0.598013\pi\)
−0.303074 + 0.952967i \(0.598013\pi\)
\(368\) 0 0
\(369\) −20.1871 −1.05090
\(370\) 0 0
\(371\) −18.9199 −0.982271
\(372\) 0 0
\(373\) 24.6087 1.27419 0.637094 0.770786i \(-0.280136\pi\)
0.637094 + 0.770786i \(0.280136\pi\)
\(374\) 0 0
\(375\) 25.9233 1.33867
\(376\) 0 0
\(377\) 35.1138 1.80845
\(378\) 0 0
\(379\) 1.05863 0.0543783 0.0271892 0.999630i \(-0.491344\pi\)
0.0271892 + 0.999630i \(0.491344\pi\)
\(380\) 0 0
\(381\) −2.67762 −0.137179
\(382\) 0 0
\(383\) −18.4837 −0.944472 −0.472236 0.881472i \(-0.656553\pi\)
−0.472236 + 0.881472i \(0.656553\pi\)
\(384\) 0 0
\(385\) 10.3354 0.526739
\(386\) 0 0
\(387\) −3.78189 −0.192245
\(388\) 0 0
\(389\) 2.46725 0.125094 0.0625472 0.998042i \(-0.480078\pi\)
0.0625472 + 0.998042i \(0.480078\pi\)
\(390\) 0 0
\(391\) −6.87930 −0.347901
\(392\) 0 0
\(393\) −5.39218 −0.272000
\(394\) 0 0
\(395\) −0.538868 −0.0271134
\(396\) 0 0
\(397\) 6.91721 0.347165 0.173582 0.984819i \(-0.444466\pi\)
0.173582 + 0.984819i \(0.444466\pi\)
\(398\) 0 0
\(399\) −3.30777 −0.165596
\(400\) 0 0
\(401\) 20.8578 1.04159 0.520795 0.853682i \(-0.325636\pi\)
0.520795 + 0.853682i \(0.325636\pi\)
\(402\) 0 0
\(403\) −18.5863 −0.925851
\(404\) 0 0
\(405\) 16.0862 0.799331
\(406\) 0 0
\(407\) −15.8061 −0.783477
\(408\) 0 0
\(409\) −24.9820 −1.23528 −0.617639 0.786462i \(-0.711911\pi\)
−0.617639 + 0.786462i \(0.711911\pi\)
\(410\) 0 0
\(411\) −21.1284 −1.04219
\(412\) 0 0
\(413\) 13.3224 0.655552
\(414\) 0 0
\(415\) −26.5535 −1.30346
\(416\) 0 0
\(417\) 27.4802 1.34571
\(418\) 0 0
\(419\) 31.6742 1.54738 0.773692 0.633561i \(-0.218408\pi\)
0.773692 + 0.633561i \(0.218408\pi\)
\(420\) 0 0
\(421\) 38.5941 1.88096 0.940480 0.339850i \(-0.110376\pi\)
0.940480 + 0.339850i \(0.110376\pi\)
\(422\) 0 0
\(423\) 10.5914 0.514971
\(424\) 0 0
\(425\) −4.87930 −0.236681
\(426\) 0 0
\(427\) 13.1043 0.634160
\(428\) 0 0
\(429\) −48.3449 −2.33411
\(430\) 0 0
\(431\) −37.4948 −1.80606 −0.903032 0.429574i \(-0.858664\pi\)
−0.903032 + 0.429574i \(0.858664\pi\)
\(432\) 0 0
\(433\) 13.4182 0.644836 0.322418 0.946597i \(-0.395504\pi\)
0.322418 + 0.946597i \(0.395504\pi\)
\(434\) 0 0
\(435\) 25.8207 1.23801
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 36.3043 1.73271 0.866356 0.499427i \(-0.166456\pi\)
0.866356 + 0.499427i \(0.166456\pi\)
\(440\) 0 0
\(441\) −9.95779 −0.474181
\(442\) 0 0
\(443\) 27.9215 1.32659 0.663295 0.748358i \(-0.269157\pi\)
0.663295 + 0.748358i \(0.269157\pi\)
\(444\) 0 0
\(445\) −3.48024 −0.164979
\(446\) 0 0
\(447\) −37.8284 −1.78922
\(448\) 0 0
\(449\) −12.2086 −0.576157 −0.288079 0.957607i \(-0.593016\pi\)
−0.288079 + 0.957607i \(0.593016\pi\)
\(450\) 0 0
\(451\) −46.8578 −2.20645
\(452\) 0 0
\(453\) 16.1396 0.758305
\(454\) 0 0
\(455\) −9.72938 −0.456120
\(456\) 0 0
\(457\) −4.42666 −0.207070 −0.103535 0.994626i \(-0.533015\pi\)
−0.103535 + 0.994626i \(0.533015\pi\)
\(458\) 0 0
\(459\) −3.64133 −0.169963
\(460\) 0 0
\(461\) 6.96553 0.324417 0.162208 0.986757i \(-0.448138\pi\)
0.162208 + 0.986757i \(0.448138\pi\)
\(462\) 0 0
\(463\) 30.7501 1.42908 0.714539 0.699595i \(-0.246636\pi\)
0.714539 + 0.699595i \(0.246636\pi\)
\(464\) 0 0
\(465\) −13.6673 −0.633806
\(466\) 0 0
\(467\) −5.16635 −0.239070 −0.119535 0.992830i \(-0.538140\pi\)
−0.119535 + 0.992830i \(0.538140\pi\)
\(468\) 0 0
\(469\) 8.71133 0.402252
\(470\) 0 0
\(471\) −35.2534 −1.62439
\(472\) 0 0
\(473\) −8.77846 −0.403634
\(474\) 0 0
\(475\) −2.83709 −0.130175
\(476\) 0 0
\(477\) 26.4837 1.21260
\(478\) 0 0
\(479\) 23.3776 1.06815 0.534074 0.845437i \(-0.320660\pi\)
0.534074 + 0.845437i \(0.320660\pi\)
\(480\) 0 0
\(481\) 14.8793 0.678437
\(482\) 0 0
\(483\) 13.2311 0.602036
\(484\) 0 0
\(485\) 10.3810 0.471378
\(486\) 0 0
\(487\) 41.9018 1.89875 0.949377 0.314140i \(-0.101716\pi\)
0.949377 + 0.314140i \(0.101716\pi\)
\(488\) 0 0
\(489\) −20.9673 −0.948176
\(490\) 0 0
\(491\) 4.90528 0.221372 0.110686 0.993855i \(-0.464695\pi\)
0.110686 + 0.993855i \(0.464695\pi\)
\(492\) 0 0
\(493\) −13.4250 −0.604633
\(494\) 0 0
\(495\) −14.4672 −0.650254
\(496\) 0 0
\(497\) −8.90528 −0.399456
\(498\) 0 0
\(499\) 5.77502 0.258525 0.129263 0.991610i \(-0.458739\pi\)
0.129263 + 0.991610i \(0.458739\pi\)
\(500\) 0 0
\(501\) −45.6673 −2.04026
\(502\) 0 0
\(503\) 31.3776 1.39906 0.699529 0.714605i \(-0.253393\pi\)
0.699529 + 0.714605i \(0.253393\pi\)
\(504\) 0 0
\(505\) −10.2086 −0.454275
\(506\) 0 0
\(507\) 16.2715 0.722642
\(508\) 0 0
\(509\) −17.2603 −0.765050 −0.382525 0.923945i \(-0.624945\pi\)
−0.382525 + 0.923945i \(0.624945\pi\)
\(510\) 0 0
\(511\) 0.493225 0.0218190
\(512\) 0 0
\(513\) −2.11727 −0.0934796
\(514\) 0 0
\(515\) 1.21199 0.0534067
\(516\) 0 0
\(517\) 24.5845 1.08123
\(518\) 0 0
\(519\) 1.26031 0.0553215
\(520\) 0 0
\(521\) −18.8939 −0.827757 −0.413878 0.910332i \(-0.635826\pi\)
−0.413878 + 0.910332i \(0.635826\pi\)
\(522\) 0 0
\(523\) 40.2017 1.75790 0.878948 0.476917i \(-0.158246\pi\)
0.878948 + 0.476917i \(0.158246\pi\)
\(524\) 0 0
\(525\) 9.38445 0.409571
\(526\) 0 0
\(527\) 7.10610 0.309546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −18.6484 −0.809272
\(532\) 0 0
\(533\) 44.1104 1.91063
\(534\) 0 0
\(535\) 16.8241 0.727369
\(536\) 0 0
\(537\) −9.82066 −0.423793
\(538\) 0 0
\(539\) −23.1138 −0.995583
\(540\) 0 0
\(541\) 22.4052 0.963274 0.481637 0.876371i \(-0.340042\pi\)
0.481637 + 0.876371i \(0.340042\pi\)
\(542\) 0 0
\(543\) −11.3776 −0.488259
\(544\) 0 0
\(545\) −6.07668 −0.260296
\(546\) 0 0
\(547\) 37.9931 1.62447 0.812234 0.583331i \(-0.198251\pi\)
0.812234 + 0.583331i \(0.198251\pi\)
\(548\) 0 0
\(549\) −18.3431 −0.782865
\(550\) 0 0
\(551\) −7.80605 −0.332549
\(552\) 0 0
\(553\) −0.538868 −0.0229150
\(554\) 0 0
\(555\) 10.9414 0.464435
\(556\) 0 0
\(557\) −16.1775 −0.685463 −0.342732 0.939433i \(-0.611352\pi\)
−0.342732 + 0.939433i \(0.611352\pi\)
\(558\) 0 0
\(559\) 8.26375 0.349519
\(560\) 0 0
\(561\) 18.4837 0.780381
\(562\) 0 0
\(563\) 9.59750 0.404486 0.202243 0.979335i \(-0.435177\pi\)
0.202243 + 0.979335i \(0.435177\pi\)
\(564\) 0 0
\(565\) 11.6527 0.490233
\(566\) 0 0
\(567\) 16.0862 0.675558
\(568\) 0 0
\(569\) 23.9018 1.00202 0.501009 0.865442i \(-0.332963\pi\)
0.501009 + 0.865442i \(0.332963\pi\)
\(570\) 0 0
\(571\) 36.1725 1.51377 0.756885 0.653548i \(-0.226720\pi\)
0.756885 + 0.653548i \(0.226720\pi\)
\(572\) 0 0
\(573\) 5.54918 0.231820
\(574\) 0 0
\(575\) 11.3484 0.473259
\(576\) 0 0
\(577\) −12.5699 −0.523292 −0.261646 0.965164i \(-0.584265\pi\)
−0.261646 + 0.965164i \(0.584265\pi\)
\(578\) 0 0
\(579\) 27.5500 1.14494
\(580\) 0 0
\(581\) −26.5535 −1.10162
\(582\) 0 0
\(583\) 61.4734 2.54597
\(584\) 0 0
\(585\) 13.6190 0.563076
\(586\) 0 0
\(587\) 0.107714 0.00444585 0.00222292 0.999998i \(-0.499292\pi\)
0.00222292 + 0.999998i \(0.499292\pi\)
\(588\) 0 0
\(589\) 4.13187 0.170251
\(590\) 0 0
\(591\) 2.76891 0.113898
\(592\) 0 0
\(593\) 25.7846 1.05885 0.529423 0.848358i \(-0.322409\pi\)
0.529423 + 0.848358i \(0.322409\pi\)
\(594\) 0 0
\(595\) 3.71982 0.152498
\(596\) 0 0
\(597\) −60.7889 −2.48792
\(598\) 0 0
\(599\) −23.6336 −0.965642 −0.482821 0.875719i \(-0.660388\pi\)
−0.482821 + 0.875719i \(0.660388\pi\)
\(600\) 0 0
\(601\) 35.3415 1.44161 0.720805 0.693138i \(-0.243772\pi\)
0.720805 + 0.693138i \(0.243772\pi\)
\(602\) 0 0
\(603\) −12.1939 −0.496576
\(604\) 0 0
\(605\) −17.4036 −0.707555
\(606\) 0 0
\(607\) 28.8432 1.17071 0.585355 0.810777i \(-0.300955\pi\)
0.585355 + 0.810777i \(0.300955\pi\)
\(608\) 0 0
\(609\) 25.8207 1.04631
\(610\) 0 0
\(611\) −23.1430 −0.936267
\(612\) 0 0
\(613\) −27.7604 −1.12123 −0.560616 0.828076i \(-0.689436\pi\)
−0.560616 + 0.828076i \(0.689436\pi\)
\(614\) 0 0
\(615\) 32.4362 1.30795
\(616\) 0 0
\(617\) 32.5370 1.30989 0.654946 0.755676i \(-0.272691\pi\)
0.654946 + 0.755676i \(0.272691\pi\)
\(618\) 0 0
\(619\) 6.20855 0.249543 0.124771 0.992186i \(-0.460180\pi\)
0.124771 + 0.992186i \(0.460180\pi\)
\(620\) 0 0
\(621\) 8.46907 0.339852
\(622\) 0 0
\(623\) −3.48024 −0.139433
\(624\) 0 0
\(625\) −2.76547 −0.110619
\(626\) 0 0
\(627\) 10.7474 0.429211
\(628\) 0 0
\(629\) −5.68879 −0.226827
\(630\) 0 0
\(631\) −2.67256 −0.106393 −0.0531965 0.998584i \(-0.516941\pi\)
−0.0531965 + 0.998584i \(0.516941\pi\)
\(632\) 0 0
\(633\) −0.732814 −0.0291267
\(634\) 0 0
\(635\) 1.75086 0.0694807
\(636\) 0 0
\(637\) 21.7586 0.862107
\(638\) 0 0
\(639\) 12.4654 0.493125
\(640\) 0 0
\(641\) −24.6639 −0.974164 −0.487082 0.873356i \(-0.661939\pi\)
−0.487082 + 0.873356i \(0.661939\pi\)
\(642\) 0 0
\(643\) −25.5405 −1.00722 −0.503609 0.863932i \(-0.667995\pi\)
−0.503609 + 0.863932i \(0.667995\pi\)
\(644\) 0 0
\(645\) 6.07668 0.239269
\(646\) 0 0
\(647\) 21.7535 0.855220 0.427610 0.903963i \(-0.359356\pi\)
0.427610 + 0.903963i \(0.359356\pi\)
\(648\) 0 0
\(649\) −43.2863 −1.69914
\(650\) 0 0
\(651\) −13.6673 −0.535664
\(652\) 0 0
\(653\) −9.37940 −0.367044 −0.183522 0.983016i \(-0.558750\pi\)
−0.183522 + 0.983016i \(0.558750\pi\)
\(654\) 0 0
\(655\) 3.52588 0.137767
\(656\) 0 0
\(657\) −0.690407 −0.0269353
\(658\) 0 0
\(659\) −39.8138 −1.55092 −0.775462 0.631394i \(-0.782483\pi\)
−0.775462 + 0.631394i \(0.782483\pi\)
\(660\) 0 0
\(661\) −19.1138 −0.743442 −0.371721 0.928345i \(-0.621232\pi\)
−0.371721 + 0.928345i \(0.621232\pi\)
\(662\) 0 0
\(663\) −17.3999 −0.675757
\(664\) 0 0
\(665\) 2.16291 0.0838740
\(666\) 0 0
\(667\) 31.2242 1.20901
\(668\) 0 0
\(669\) −52.6707 −2.03637
\(670\) 0 0
\(671\) −42.5776 −1.64369
\(672\) 0 0
\(673\) −21.1138 −0.813878 −0.406939 0.913455i \(-0.633404\pi\)
−0.406939 + 0.913455i \(0.633404\pi\)
\(674\) 0 0
\(675\) 6.00688 0.231205
\(676\) 0 0
\(677\) −36.3855 −1.39841 −0.699204 0.714922i \(-0.746462\pi\)
−0.699204 + 0.714922i \(0.746462\pi\)
\(678\) 0 0
\(679\) 10.3810 0.398387
\(680\) 0 0
\(681\) 53.6344 2.05528
\(682\) 0 0
\(683\) −12.6087 −0.482457 −0.241229 0.970468i \(-0.577550\pi\)
−0.241229 + 0.970468i \(0.577550\pi\)
\(684\) 0 0
\(685\) 13.8156 0.527867
\(686\) 0 0
\(687\) 31.4182 1.19868
\(688\) 0 0
\(689\) −57.8690 −2.20463
\(690\) 0 0
\(691\) −16.8405 −0.640644 −0.320322 0.947309i \(-0.603791\pi\)
−0.320322 + 0.947309i \(0.603791\pi\)
\(692\) 0 0
\(693\) −14.4672 −0.549565
\(694\) 0 0
\(695\) −17.9690 −0.681602
\(696\) 0 0
\(697\) −16.8647 −0.638796
\(698\) 0 0
\(699\) −28.8647 −1.09176
\(700\) 0 0
\(701\) 33.9311 1.28156 0.640779 0.767725i \(-0.278611\pi\)
0.640779 + 0.767725i \(0.278611\pi\)
\(702\) 0 0
\(703\) −3.30777 −0.124755
\(704\) 0 0
\(705\) −17.0180 −0.640937
\(706\) 0 0
\(707\) −10.2086 −0.383932
\(708\) 0 0
\(709\) −30.8984 −1.16041 −0.580207 0.814469i \(-0.697028\pi\)
−0.580207 + 0.814469i \(0.697028\pi\)
\(710\) 0 0
\(711\) 0.754297 0.0282884
\(712\) 0 0
\(713\) −16.5275 −0.618960
\(714\) 0 0
\(715\) 31.6121 1.18223
\(716\) 0 0
\(717\) −54.5612 −2.03763
\(718\) 0 0
\(719\) −26.8122 −0.999925 −0.499963 0.866047i \(-0.666653\pi\)
−0.499963 + 0.866047i \(0.666653\pi\)
\(720\) 0 0
\(721\) 1.21199 0.0451369
\(722\) 0 0
\(723\) −58.6347 −2.18065
\(724\) 0 0
\(725\) 22.1465 0.822500
\(726\) 0 0
\(727\) −30.1897 −1.11968 −0.559838 0.828602i \(-0.689137\pi\)
−0.559838 + 0.828602i \(0.689137\pi\)
\(728\) 0 0
\(729\) −8.23109 −0.304855
\(730\) 0 0
\(731\) −3.15947 −0.116857
\(732\) 0 0
\(733\) −37.8207 −1.39694 −0.698469 0.715640i \(-0.746135\pi\)
−0.698469 + 0.715640i \(0.746135\pi\)
\(734\) 0 0
\(735\) 16.0000 0.590169
\(736\) 0 0
\(737\) −28.3043 −1.04260
\(738\) 0 0
\(739\) 39.3803 1.44863 0.724313 0.689471i \(-0.242157\pi\)
0.724313 + 0.689471i \(0.242157\pi\)
\(740\) 0 0
\(741\) −10.1173 −0.371667
\(742\) 0 0
\(743\) −13.9854 −0.513074 −0.256537 0.966534i \(-0.582582\pi\)
−0.256537 + 0.966534i \(0.582582\pi\)
\(744\) 0 0
\(745\) 24.7355 0.906238
\(746\) 0 0
\(747\) 37.1690 1.35994
\(748\) 0 0
\(749\) 16.8241 0.614739
\(750\) 0 0
\(751\) −25.0777 −0.915100 −0.457550 0.889184i \(-0.651273\pi\)
−0.457550 + 0.889184i \(0.651273\pi\)
\(752\) 0 0
\(753\) −52.4768 −1.91236
\(754\) 0 0
\(755\) −10.5535 −0.384080
\(756\) 0 0
\(757\) −24.6137 −0.894601 −0.447301 0.894384i \(-0.647615\pi\)
−0.447301 + 0.894384i \(0.647615\pi\)
\(758\) 0 0
\(759\) −42.9897 −1.56043
\(760\) 0 0
\(761\) −37.8010 −1.37029 −0.685143 0.728409i \(-0.740260\pi\)
−0.685143 + 0.728409i \(0.740260\pi\)
\(762\) 0 0
\(763\) −6.07668 −0.219991
\(764\) 0 0
\(765\) −5.20693 −0.188257
\(766\) 0 0
\(767\) 40.7483 1.47134
\(768\) 0 0
\(769\) 9.98357 0.360017 0.180008 0.983665i \(-0.442388\pi\)
0.180008 + 0.983665i \(0.442388\pi\)
\(770\) 0 0
\(771\) 6.35180 0.228754
\(772\) 0 0
\(773\) −12.5896 −0.452815 −0.226408 0.974033i \(-0.572698\pi\)
−0.226408 + 0.974033i \(0.572698\pi\)
\(774\) 0 0
\(775\) −11.7225 −0.421085
\(776\) 0 0
\(777\) 10.9414 0.392519
\(778\) 0 0
\(779\) −9.80605 −0.351338
\(780\) 0 0
\(781\) 28.9345 1.03536
\(782\) 0 0
\(783\) 16.5275 0.590645
\(784\) 0 0
\(785\) 23.0518 0.822753
\(786\) 0 0
\(787\) −29.2603 −1.04302 −0.521509 0.853246i \(-0.674631\pi\)
−0.521509 + 0.853246i \(0.674631\pi\)
\(788\) 0 0
\(789\) −15.9457 −0.567681
\(790\) 0 0
\(791\) 11.6527 0.414322
\(792\) 0 0
\(793\) 40.0812 1.42332
\(794\) 0 0
\(795\) −42.5535 −1.50922
\(796\) 0 0
\(797\) −46.8578 −1.65979 −0.829894 0.557920i \(-0.811599\pi\)
−0.829894 + 0.557920i \(0.811599\pi\)
\(798\) 0 0
\(799\) 8.84826 0.313029
\(800\) 0 0
\(801\) 4.87156 0.172128
\(802\) 0 0
\(803\) −1.60256 −0.0565530
\(804\) 0 0
\(805\) −8.65164 −0.304930
\(806\) 0 0
\(807\) 46.9276 1.65193
\(808\) 0 0
\(809\) −22.6681 −0.796967 −0.398483 0.917176i \(-0.630463\pi\)
−0.398483 + 0.917176i \(0.630463\pi\)
\(810\) 0 0
\(811\) 32.7620 1.15043 0.575215 0.818002i \(-0.304918\pi\)
0.575215 + 0.818002i \(0.304918\pi\)
\(812\) 0 0
\(813\) −0.139606 −0.00489621
\(814\) 0 0
\(815\) 13.7103 0.480250
\(816\) 0 0
\(817\) −1.83709 −0.0642717
\(818\) 0 0
\(819\) 13.6190 0.475886
\(820\) 0 0
\(821\) −3.20006 −0.111683 −0.0558414 0.998440i \(-0.517784\pi\)
−0.0558414 + 0.998440i \(0.517784\pi\)
\(822\) 0 0
\(823\) 24.4741 0.853114 0.426557 0.904461i \(-0.359726\pi\)
0.426557 + 0.904461i \(0.359726\pi\)
\(824\) 0 0
\(825\) −30.4914 −1.06157
\(826\) 0 0
\(827\) −24.5681 −0.854316 −0.427158 0.904177i \(-0.640485\pi\)
−0.427158 + 0.904177i \(0.640485\pi\)
\(828\) 0 0
\(829\) 11.2380 0.390311 0.195155 0.980772i \(-0.437479\pi\)
0.195155 + 0.980772i \(0.437479\pi\)
\(830\) 0 0
\(831\) −17.5939 −0.610324
\(832\) 0 0
\(833\) −8.31894 −0.288234
\(834\) 0 0
\(835\) 29.8613 1.03339
\(836\) 0 0
\(837\) −8.74828 −0.302385
\(838\) 0 0
\(839\) −1.94480 −0.0671421 −0.0335711 0.999436i \(-0.510688\pi\)
−0.0335711 + 0.999436i \(0.510688\pi\)
\(840\) 0 0
\(841\) 31.9345 1.10119
\(842\) 0 0
\(843\) 46.4330 1.59924
\(844\) 0 0
\(845\) −10.6397 −0.366017
\(846\) 0 0
\(847\) −17.4036 −0.597993
\(848\) 0 0
\(849\) −48.1027 −1.65088
\(850\) 0 0
\(851\) 13.2311 0.453556
\(852\) 0 0
\(853\) 29.8207 1.02104 0.510520 0.859866i \(-0.329453\pi\)
0.510520 + 0.859866i \(0.329453\pi\)
\(854\) 0 0
\(855\) −3.02760 −0.103542
\(856\) 0 0
\(857\) −6.49828 −0.221977 −0.110989 0.993822i \(-0.535402\pi\)
−0.110989 + 0.993822i \(0.535402\pi\)
\(858\) 0 0
\(859\) −21.0422 −0.717951 −0.358975 0.933347i \(-0.616874\pi\)
−0.358975 + 0.933347i \(0.616874\pi\)
\(860\) 0 0
\(861\) 32.4362 1.10542
\(862\) 0 0
\(863\) −7.80605 −0.265721 −0.132861 0.991135i \(-0.542416\pi\)
−0.132861 + 0.991135i \(0.542416\pi\)
\(864\) 0 0
\(865\) −0.824101 −0.0280203
\(866\) 0 0
\(867\) −31.5829 −1.07261
\(868\) 0 0
\(869\) 1.75086 0.0593938
\(870\) 0 0
\(871\) 26.6448 0.902823
\(872\) 0 0
\(873\) −14.5311 −0.491804
\(874\) 0 0
\(875\) −16.9509 −0.573046
\(876\) 0 0
\(877\) 4.32582 0.146073 0.0730363 0.997329i \(-0.476731\pi\)
0.0730363 + 0.997329i \(0.476731\pi\)
\(878\) 0 0
\(879\) 55.3484 1.86685
\(880\) 0 0
\(881\) −5.65775 −0.190615 −0.0953073 0.995448i \(-0.530383\pi\)
−0.0953073 + 0.995448i \(0.530383\pi\)
\(882\) 0 0
\(883\) −19.6578 −0.661536 −0.330768 0.943712i \(-0.607308\pi\)
−0.330768 + 0.943712i \(0.607308\pi\)
\(884\) 0 0
\(885\) 29.9639 1.00723
\(886\) 0 0
\(887\) 40.7811 1.36930 0.684648 0.728874i \(-0.259956\pi\)
0.684648 + 0.728874i \(0.259956\pi\)
\(888\) 0 0
\(889\) 1.75086 0.0587219
\(890\) 0 0
\(891\) −52.2664 −1.75099
\(892\) 0 0
\(893\) 5.14486 0.172166
\(894\) 0 0
\(895\) 6.42160 0.214650
\(896\) 0 0
\(897\) 40.4691 1.35122
\(898\) 0 0
\(899\) −32.2536 −1.07572
\(900\) 0 0
\(901\) 22.1250 0.737091
\(902\) 0 0
\(903\) 6.07668 0.202219
\(904\) 0 0
\(905\) 7.43965 0.247302
\(906\) 0 0
\(907\) 37.1475 1.23346 0.616732 0.787173i \(-0.288456\pi\)
0.616732 + 0.787173i \(0.288456\pi\)
\(908\) 0 0
\(909\) 14.2897 0.473960
\(910\) 0 0
\(911\) −20.4070 −0.676114 −0.338057 0.941126i \(-0.609770\pi\)
−0.338057 + 0.941126i \(0.609770\pi\)
\(912\) 0 0
\(913\) 86.2760 2.85532
\(914\) 0 0
\(915\) 29.4734 0.974359
\(916\) 0 0
\(917\) 3.52588 0.116435
\(918\) 0 0
\(919\) 52.9536 1.74678 0.873389 0.487023i \(-0.161917\pi\)
0.873389 + 0.487023i \(0.161917\pi\)
\(920\) 0 0
\(921\) 23.4396 0.772363
\(922\) 0 0
\(923\) −27.2380 −0.896549
\(924\) 0 0
\(925\) 9.38445 0.308559
\(926\) 0 0
\(927\) −1.69652 −0.0557210
\(928\) 0 0
\(929\) 56.8984 1.86678 0.933388 0.358869i \(-0.116837\pi\)
0.933388 + 0.358869i \(0.116837\pi\)
\(930\) 0 0
\(931\) −4.83709 −0.158529
\(932\) 0 0
\(933\) −17.7992 −0.582719
\(934\) 0 0
\(935\) −12.0862 −0.395262
\(936\) 0 0
\(937\) 40.9148 1.33663 0.668315 0.743879i \(-0.267016\pi\)
0.668315 + 0.743879i \(0.267016\pi\)
\(938\) 0 0
\(939\) −40.0000 −1.30535
\(940\) 0 0
\(941\) −2.59644 −0.0846416 −0.0423208 0.999104i \(-0.513475\pi\)
−0.0423208 + 0.999104i \(0.513475\pi\)
\(942\) 0 0
\(943\) 39.2242 1.27732
\(944\) 0 0
\(945\) −4.57946 −0.148970
\(946\) 0 0
\(947\) −25.7586 −0.837042 −0.418521 0.908207i \(-0.637451\pi\)
−0.418521 + 0.908207i \(0.637451\pi\)
\(948\) 0 0
\(949\) 1.50859 0.0489711
\(950\) 0 0
\(951\) 67.5984 2.19203
\(952\) 0 0
\(953\) 2.25602 0.0730795 0.0365398 0.999332i \(-0.488366\pi\)
0.0365398 + 0.999332i \(0.488366\pi\)
\(954\) 0 0
\(955\) −3.62854 −0.117417
\(956\) 0 0
\(957\) −83.8950 −2.71194
\(958\) 0 0
\(959\) 13.8156 0.446129
\(960\) 0 0
\(961\) −13.9276 −0.449278
\(962\) 0 0
\(963\) −23.5500 −0.758889
\(964\) 0 0
\(965\) −18.0146 −0.579911
\(966\) 0 0
\(967\) −45.0449 −1.44855 −0.724273 0.689513i \(-0.757824\pi\)
−0.724273 + 0.689513i \(0.757824\pi\)
\(968\) 0 0
\(969\) 3.86813 0.124262
\(970\) 0 0
\(971\) −17.5760 −0.564041 −0.282021 0.959408i \(-0.591005\pi\)
−0.282021 + 0.959408i \(0.591005\pi\)
\(972\) 0 0
\(973\) −17.9690 −0.576059
\(974\) 0 0
\(975\) 28.7036 0.919251
\(976\) 0 0
\(977\) −15.9042 −0.508821 −0.254410 0.967096i \(-0.581881\pi\)
−0.254410 + 0.967096i \(0.581881\pi\)
\(978\) 0 0
\(979\) 11.3078 0.361398
\(980\) 0 0
\(981\) 8.50601 0.271576
\(982\) 0 0
\(983\) −41.7294 −1.33096 −0.665480 0.746416i \(-0.731773\pi\)
−0.665480 + 0.746416i \(0.731773\pi\)
\(984\) 0 0
\(985\) −1.81055 −0.0576889
\(986\) 0 0
\(987\) −17.0180 −0.541690
\(988\) 0 0
\(989\) 7.34836 0.233664
\(990\) 0 0
\(991\) 0.193945 0.00616087 0.00308044 0.999995i \(-0.499019\pi\)
0.00308044 + 0.999995i \(0.499019\pi\)
\(992\) 0 0
\(993\) 42.3052 1.34251
\(994\) 0 0
\(995\) 39.7490 1.26013
\(996\) 0 0
\(997\) −40.6949 −1.28882 −0.644410 0.764680i \(-0.722897\pi\)
−0.644410 + 0.764680i \(0.722897\pi\)
\(998\) 0 0
\(999\) 7.00344 0.221579
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.ba.1.3 3
4.3 odd 2 4864.2.a.bg.1.1 3
8.3 odd 2 4864.2.a.bb.1.3 3
8.5 even 2 4864.2.a.bh.1.1 3
16.3 odd 4 2432.2.c.e.1217.2 6
16.5 even 4 2432.2.c.h.1217.2 yes 6
16.11 odd 4 2432.2.c.e.1217.5 yes 6
16.13 even 4 2432.2.c.h.1217.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.e.1217.2 6 16.3 odd 4
2432.2.c.e.1217.5 yes 6 16.11 odd 4
2432.2.c.h.1217.2 yes 6 16.5 even 4
2432.2.c.h.1217.5 yes 6 16.13 even 4
4864.2.a.ba.1.3 3 1.1 even 1 trivial
4864.2.a.bb.1.3 3 8.3 odd 2
4864.2.a.bg.1.1 3 4.3 odd 2
4864.2.a.bh.1.1 3 8.5 even 2