Properties

Label 7728.2.a.bj.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.37228 q^{11} -0.372281 q^{13} -2.37228 q^{15} +4.00000 q^{17} -7.37228 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.627719 q^{25} +1.00000 q^{27} -0.372281 q^{29} +6.00000 q^{31} -3.37228 q^{33} +2.37228 q^{35} -8.37228 q^{37} -0.372281 q^{39} +5.74456 q^{41} +2.37228 q^{43} -2.37228 q^{45} +7.74456 q^{47} +1.00000 q^{49} +4.00000 q^{51} -10.1168 q^{53} +8.00000 q^{55} -7.37228 q^{57} -9.37228 q^{59} -2.62772 q^{61} -1.00000 q^{63} +0.883156 q^{65} +4.00000 q^{67} -1.00000 q^{69} -12.7446 q^{71} +12.0000 q^{73} +0.627719 q^{75} +3.37228 q^{77} +2.74456 q^{79} +1.00000 q^{81} -2.74456 q^{83} -9.48913 q^{85} -0.372281 q^{87} +15.4891 q^{89} +0.372281 q^{91} +6.00000 q^{93} +17.4891 q^{95} +5.11684 q^{97} -3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 9 q^{19} - 2 q^{21} - 2 q^{23} + 7 q^{25} + 2 q^{27} + 5 q^{29} + 12 q^{31} - q^{33} - q^{35} - 11 q^{37} + 5 q^{39} - q^{43} + q^{45} + 4 q^{47} + 2 q^{49} + 8 q^{51} - 3 q^{53} + 16 q^{55} - 9 q^{57} - 13 q^{59} - 11 q^{61} - 2 q^{63} + 19 q^{65} + 8 q^{67} - 2 q^{69} - 14 q^{71} + 24 q^{73} + 7 q^{75} + q^{77} - 6 q^{79} + 2 q^{81} + 6 q^{83} + 4 q^{85} + 5 q^{87} + 8 q^{89} - 5 q^{91} + 12 q^{93} + 12 q^{95} - 7 q^{97} - q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.37228 −1.06092 −0.530458 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.37228 −1.01678 −0.508391 0.861127i \(-0.669759\pi\)
−0.508391 + 0.861127i \(0.669759\pi\)
\(12\) 0 0
\(13\) −0.372281 −0.103252 −0.0516261 0.998666i \(-0.516440\pi\)
−0.0516261 + 0.998666i \(0.516440\pi\)
\(14\) 0 0
\(15\) −2.37228 −0.612520
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −7.37228 −1.69132 −0.845659 0.533724i \(-0.820792\pi\)
−0.845659 + 0.533724i \(0.820792\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.372281 −0.0691309 −0.0345655 0.999402i \(-0.511005\pi\)
−0.0345655 + 0.999402i \(0.511005\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −3.37228 −0.587039
\(34\) 0 0
\(35\) 2.37228 0.400989
\(36\) 0 0
\(37\) −8.37228 −1.37639 −0.688197 0.725524i \(-0.741598\pi\)
−0.688197 + 0.725524i \(0.741598\pi\)
\(38\) 0 0
\(39\) −0.372281 −0.0596127
\(40\) 0 0
\(41\) 5.74456 0.897150 0.448575 0.893745i \(-0.351932\pi\)
0.448575 + 0.893745i \(0.351932\pi\)
\(42\) 0 0
\(43\) 2.37228 0.361770 0.180885 0.983504i \(-0.442104\pi\)
0.180885 + 0.983504i \(0.442104\pi\)
\(44\) 0 0
\(45\) −2.37228 −0.353639
\(46\) 0 0
\(47\) 7.74456 1.12966 0.564830 0.825207i \(-0.308942\pi\)
0.564830 + 0.825207i \(0.308942\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −10.1168 −1.38966 −0.694828 0.719176i \(-0.744519\pi\)
−0.694828 + 0.719176i \(0.744519\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −7.37228 −0.976483
\(58\) 0 0
\(59\) −9.37228 −1.22017 −0.610084 0.792337i \(-0.708864\pi\)
−0.610084 + 0.792337i \(0.708864\pi\)
\(60\) 0 0
\(61\) −2.62772 −0.336445 −0.168222 0.985749i \(-0.553803\pi\)
−0.168222 + 0.985749i \(0.553803\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0.883156 0.109542
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −12.7446 −1.51250 −0.756251 0.654282i \(-0.772971\pi\)
−0.756251 + 0.654282i \(0.772971\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) 0.627719 0.0724827
\(76\) 0 0
\(77\) 3.37228 0.384307
\(78\) 0 0
\(79\) 2.74456 0.308787 0.154394 0.988009i \(-0.450658\pi\)
0.154394 + 0.988009i \(0.450658\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.74456 −0.301255 −0.150627 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) 0 0
\(85\) −9.48913 −1.02924
\(86\) 0 0
\(87\) −0.372281 −0.0399127
\(88\) 0 0
\(89\) 15.4891 1.64184 0.820922 0.571040i \(-0.193460\pi\)
0.820922 + 0.571040i \(0.193460\pi\)
\(90\) 0 0
\(91\) 0.372281 0.0390257
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 17.4891 1.79435
\(96\) 0 0
\(97\) 5.11684 0.519537 0.259768 0.965671i \(-0.416354\pi\)
0.259768 + 0.965671i \(0.416354\pi\)
\(98\) 0 0
\(99\) −3.37228 −0.338927
\(100\) 0 0
\(101\) 12.1168 1.20567 0.602836 0.797865i \(-0.294038\pi\)
0.602836 + 0.797865i \(0.294038\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 2.37228 0.231511
\(106\) 0 0
\(107\) 5.48913 0.530654 0.265327 0.964159i \(-0.414520\pi\)
0.265327 + 0.964159i \(0.414520\pi\)
\(108\) 0 0
\(109\) 10.3723 0.993484 0.496742 0.867898i \(-0.334529\pi\)
0.496742 + 0.867898i \(0.334529\pi\)
\(110\) 0 0
\(111\) −8.37228 −0.794662
\(112\) 0 0
\(113\) −7.62772 −0.717555 −0.358778 0.933423i \(-0.616806\pi\)
−0.358778 + 0.933423i \(0.616806\pi\)
\(114\) 0 0
\(115\) 2.37228 0.221216
\(116\) 0 0
\(117\) −0.372281 −0.0344174
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 0 0
\(123\) 5.74456 0.517970
\(124\) 0 0
\(125\) 10.3723 0.927725
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) 2.37228 0.208868
\(130\) 0 0
\(131\) −4.11684 −0.359690 −0.179845 0.983695i \(-0.557560\pi\)
−0.179845 + 0.983695i \(0.557560\pi\)
\(132\) 0 0
\(133\) 7.37228 0.639258
\(134\) 0 0
\(135\) −2.37228 −0.204173
\(136\) 0 0
\(137\) 18.4891 1.57963 0.789816 0.613343i \(-0.210176\pi\)
0.789816 + 0.613343i \(0.210176\pi\)
\(138\) 0 0
\(139\) 5.62772 0.477337 0.238668 0.971101i \(-0.423289\pi\)
0.238668 + 0.971101i \(0.423289\pi\)
\(140\) 0 0
\(141\) 7.74456 0.652210
\(142\) 0 0
\(143\) 1.25544 0.104985
\(144\) 0 0
\(145\) 0.883156 0.0733421
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 22.1168 1.81188 0.905941 0.423403i \(-0.139165\pi\)
0.905941 + 0.423403i \(0.139165\pi\)
\(150\) 0 0
\(151\) −4.48913 −0.365320 −0.182660 0.983176i \(-0.558471\pi\)
−0.182660 + 0.983176i \(0.558471\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −14.2337 −1.14328
\(156\) 0 0
\(157\) −12.8614 −1.02645 −0.513226 0.858254i \(-0.671550\pi\)
−0.513226 + 0.858254i \(0.671550\pi\)
\(158\) 0 0
\(159\) −10.1168 −0.802318
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 3.37228 0.264137 0.132069 0.991241i \(-0.457838\pi\)
0.132069 + 0.991241i \(0.457838\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) −1.88316 −0.145723 −0.0728615 0.997342i \(-0.523213\pi\)
−0.0728615 + 0.997342i \(0.523213\pi\)
\(168\) 0 0
\(169\) −12.8614 −0.989339
\(170\) 0 0
\(171\) −7.37228 −0.563772
\(172\) 0 0
\(173\) −7.48913 −0.569388 −0.284694 0.958618i \(-0.591892\pi\)
−0.284694 + 0.958618i \(0.591892\pi\)
\(174\) 0 0
\(175\) −0.627719 −0.0474511
\(176\) 0 0
\(177\) −9.37228 −0.704464
\(178\) 0 0
\(179\) 15.1168 1.12989 0.564943 0.825130i \(-0.308898\pi\)
0.564943 + 0.825130i \(0.308898\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −2.62772 −0.194247
\(184\) 0 0
\(185\) 19.8614 1.46024
\(186\) 0 0
\(187\) −13.4891 −0.986423
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 12.8614 0.930619 0.465309 0.885148i \(-0.345943\pi\)
0.465309 + 0.885148i \(0.345943\pi\)
\(192\) 0 0
\(193\) 27.2337 1.96032 0.980162 0.198199i \(-0.0635091\pi\)
0.980162 + 0.198199i \(0.0635091\pi\)
\(194\) 0 0
\(195\) 0.883156 0.0632441
\(196\) 0 0
\(197\) 9.86141 0.702596 0.351298 0.936264i \(-0.385740\pi\)
0.351298 + 0.936264i \(0.385740\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 0.372281 0.0261290
\(204\) 0 0
\(205\) −13.6277 −0.951801
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 24.8614 1.71970
\(210\) 0 0
\(211\) 11.3723 0.782900 0.391450 0.920199i \(-0.371974\pi\)
0.391450 + 0.920199i \(0.371974\pi\)
\(212\) 0 0
\(213\) −12.7446 −0.873243
\(214\) 0 0
\(215\) −5.62772 −0.383807
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) −1.48913 −0.100169
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 0.627719 0.0418479
\(226\) 0 0
\(227\) 9.11684 0.605106 0.302553 0.953133i \(-0.402161\pi\)
0.302553 + 0.953133i \(0.402161\pi\)
\(228\) 0 0
\(229\) −26.3505 −1.74129 −0.870646 0.491910i \(-0.836299\pi\)
−0.870646 + 0.491910i \(0.836299\pi\)
\(230\) 0 0
\(231\) 3.37228 0.221880
\(232\) 0 0
\(233\) 3.25544 0.213271 0.106635 0.994298i \(-0.465992\pi\)
0.106635 + 0.994298i \(0.465992\pi\)
\(234\) 0 0
\(235\) −18.3723 −1.19848
\(236\) 0 0
\(237\) 2.74456 0.178279
\(238\) 0 0
\(239\) 9.48913 0.613800 0.306900 0.951742i \(-0.400708\pi\)
0.306900 + 0.951742i \(0.400708\pi\)
\(240\) 0 0
\(241\) 22.4891 1.44865 0.724326 0.689458i \(-0.242151\pi\)
0.724326 + 0.689458i \(0.242151\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.37228 −0.151559
\(246\) 0 0
\(247\) 2.74456 0.174632
\(248\) 0 0
\(249\) −2.74456 −0.173930
\(250\) 0 0
\(251\) −9.86141 −0.622446 −0.311223 0.950337i \(-0.600739\pi\)
−0.311223 + 0.950337i \(0.600739\pi\)
\(252\) 0 0
\(253\) 3.37228 0.212014
\(254\) 0 0
\(255\) −9.48913 −0.594232
\(256\) 0 0
\(257\) 28.1168 1.75388 0.876940 0.480599i \(-0.159581\pi\)
0.876940 + 0.480599i \(0.159581\pi\)
\(258\) 0 0
\(259\) 8.37228 0.520228
\(260\) 0 0
\(261\) −0.372281 −0.0230436
\(262\) 0 0
\(263\) −1.00000 −0.0616626 −0.0308313 0.999525i \(-0.509815\pi\)
−0.0308313 + 0.999525i \(0.509815\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 15.4891 0.947919
\(268\) 0 0
\(269\) 23.4891 1.43216 0.716079 0.698020i \(-0.245935\pi\)
0.716079 + 0.698020i \(0.245935\pi\)
\(270\) 0 0
\(271\) −21.4891 −1.30537 −0.652686 0.757629i \(-0.726358\pi\)
−0.652686 + 0.757629i \(0.726358\pi\)
\(272\) 0 0
\(273\) 0.372281 0.0225315
\(274\) 0 0
\(275\) −2.11684 −0.127650
\(276\) 0 0
\(277\) 18.1168 1.08854 0.544268 0.838912i \(-0.316808\pi\)
0.544268 + 0.838912i \(0.316808\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 7.62772 0.455032 0.227516 0.973774i \(-0.426940\pi\)
0.227516 + 0.973774i \(0.426940\pi\)
\(282\) 0 0
\(283\) −16.7446 −0.995361 −0.497680 0.867360i \(-0.665815\pi\)
−0.497680 + 0.867360i \(0.665815\pi\)
\(284\) 0 0
\(285\) 17.4891 1.03597
\(286\) 0 0
\(287\) −5.74456 −0.339091
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 5.11684 0.299955
\(292\) 0 0
\(293\) 22.2337 1.29891 0.649453 0.760402i \(-0.274998\pi\)
0.649453 + 0.760402i \(0.274998\pi\)
\(294\) 0 0
\(295\) 22.2337 1.29450
\(296\) 0 0
\(297\) −3.37228 −0.195680
\(298\) 0 0
\(299\) 0.372281 0.0215296
\(300\) 0 0
\(301\) −2.37228 −0.136736
\(302\) 0 0
\(303\) 12.1168 0.696094
\(304\) 0 0
\(305\) 6.23369 0.356940
\(306\) 0 0
\(307\) 26.3723 1.50515 0.752573 0.658509i \(-0.228813\pi\)
0.752573 + 0.658509i \(0.228813\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −16.6277 −0.942871 −0.471436 0.881900i \(-0.656264\pi\)
−0.471436 + 0.881900i \(0.656264\pi\)
\(312\) 0 0
\(313\) −31.0951 −1.75760 −0.878799 0.477192i \(-0.841655\pi\)
−0.878799 + 0.477192i \(0.841655\pi\)
\(314\) 0 0
\(315\) 2.37228 0.133663
\(316\) 0 0
\(317\) 10.8832 0.611259 0.305629 0.952151i \(-0.401133\pi\)
0.305629 + 0.952151i \(0.401133\pi\)
\(318\) 0 0
\(319\) 1.25544 0.0702910
\(320\) 0 0
\(321\) 5.48913 0.306373
\(322\) 0 0
\(323\) −29.4891 −1.64082
\(324\) 0 0
\(325\) −0.233688 −0.0129627
\(326\) 0 0
\(327\) 10.3723 0.573588
\(328\) 0 0
\(329\) −7.74456 −0.426972
\(330\) 0 0
\(331\) 3.37228 0.185357 0.0926787 0.995696i \(-0.470457\pi\)
0.0926787 + 0.995696i \(0.470457\pi\)
\(332\) 0 0
\(333\) −8.37228 −0.458798
\(334\) 0 0
\(335\) −9.48913 −0.518446
\(336\) 0 0
\(337\) −30.7446 −1.67476 −0.837382 0.546619i \(-0.815915\pi\)
−0.837382 + 0.546619i \(0.815915\pi\)
\(338\) 0 0
\(339\) −7.62772 −0.414281
\(340\) 0 0
\(341\) −20.2337 −1.09572
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.37228 0.127719
\(346\) 0 0
\(347\) 12.3723 0.664179 0.332089 0.943248i \(-0.392246\pi\)
0.332089 + 0.943248i \(0.392246\pi\)
\(348\) 0 0
\(349\) −8.51087 −0.455577 −0.227788 0.973711i \(-0.573149\pi\)
−0.227788 + 0.973711i \(0.573149\pi\)
\(350\) 0 0
\(351\) −0.372281 −0.0198709
\(352\) 0 0
\(353\) 10.8832 0.579252 0.289626 0.957140i \(-0.406469\pi\)
0.289626 + 0.957140i \(0.406469\pi\)
\(354\) 0 0
\(355\) 30.2337 1.60464
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −3.86141 −0.203797 −0.101899 0.994795i \(-0.532492\pi\)
−0.101899 + 0.994795i \(0.532492\pi\)
\(360\) 0 0
\(361\) 35.3505 1.86055
\(362\) 0 0
\(363\) 0.372281 0.0195397
\(364\) 0 0
\(365\) −28.4674 −1.49005
\(366\) 0 0
\(367\) −26.4891 −1.38272 −0.691361 0.722510i \(-0.742988\pi\)
−0.691361 + 0.722510i \(0.742988\pi\)
\(368\) 0 0
\(369\) 5.74456 0.299050
\(370\) 0 0
\(371\) 10.1168 0.525240
\(372\) 0 0
\(373\) 3.48913 0.180660 0.0903300 0.995912i \(-0.471208\pi\)
0.0903300 + 0.995912i \(0.471208\pi\)
\(374\) 0 0
\(375\) 10.3723 0.535622
\(376\) 0 0
\(377\) 0.138593 0.00713792
\(378\) 0 0
\(379\) 13.8614 0.712013 0.356006 0.934484i \(-0.384138\pi\)
0.356006 + 0.934484i \(0.384138\pi\)
\(380\) 0 0
\(381\) 1.00000 0.0512316
\(382\) 0 0
\(383\) 8.23369 0.420722 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 2.37228 0.120590
\(388\) 0 0
\(389\) −3.48913 −0.176906 −0.0884528 0.996080i \(-0.528192\pi\)
−0.0884528 + 0.996080i \(0.528192\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −4.11684 −0.207667
\(394\) 0 0
\(395\) −6.51087 −0.327598
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 7.37228 0.369076
\(400\) 0 0
\(401\) −4.11684 −0.205585 −0.102793 0.994703i \(-0.532778\pi\)
−0.102793 + 0.994703i \(0.532778\pi\)
\(402\) 0 0
\(403\) −2.23369 −0.111268
\(404\) 0 0
\(405\) −2.37228 −0.117880
\(406\) 0 0
\(407\) 28.2337 1.39949
\(408\) 0 0
\(409\) 16.2337 0.802704 0.401352 0.915924i \(-0.368540\pi\)
0.401352 + 0.915924i \(0.368540\pi\)
\(410\) 0 0
\(411\) 18.4891 0.912001
\(412\) 0 0
\(413\) 9.37228 0.461180
\(414\) 0 0
\(415\) 6.51087 0.319606
\(416\) 0 0
\(417\) 5.62772 0.275591
\(418\) 0 0
\(419\) 34.7446 1.69738 0.848691 0.528888i \(-0.177391\pi\)
0.848691 + 0.528888i \(0.177391\pi\)
\(420\) 0 0
\(421\) 26.3723 1.28531 0.642653 0.766157i \(-0.277834\pi\)
0.642653 + 0.766157i \(0.277834\pi\)
\(422\) 0 0
\(423\) 7.74456 0.376554
\(424\) 0 0
\(425\) 2.51087 0.121795
\(426\) 0 0
\(427\) 2.62772 0.127164
\(428\) 0 0
\(429\) 1.25544 0.0606131
\(430\) 0 0
\(431\) −38.4891 −1.85396 −0.926978 0.375116i \(-0.877603\pi\)
−0.926978 + 0.375116i \(0.877603\pi\)
\(432\) 0 0
\(433\) −5.51087 −0.264836 −0.132418 0.991194i \(-0.542274\pi\)
−0.132418 + 0.991194i \(0.542274\pi\)
\(434\) 0 0
\(435\) 0.883156 0.0423441
\(436\) 0 0
\(437\) 7.37228 0.352664
\(438\) 0 0
\(439\) −7.25544 −0.346283 −0.173142 0.984897i \(-0.555392\pi\)
−0.173142 + 0.984897i \(0.555392\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 29.3505 1.39449 0.697243 0.716835i \(-0.254410\pi\)
0.697243 + 0.716835i \(0.254410\pi\)
\(444\) 0 0
\(445\) −36.7446 −1.74186
\(446\) 0 0
\(447\) 22.1168 1.04609
\(448\) 0 0
\(449\) 30.4674 1.43784 0.718922 0.695091i \(-0.244636\pi\)
0.718922 + 0.695091i \(0.244636\pi\)
\(450\) 0 0
\(451\) −19.3723 −0.912205
\(452\) 0 0
\(453\) −4.48913 −0.210918
\(454\) 0 0
\(455\) −0.883156 −0.0414030
\(456\) 0 0
\(457\) 12.2337 0.572268 0.286134 0.958190i \(-0.407630\pi\)
0.286134 + 0.958190i \(0.407630\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −32.9783 −1.53595 −0.767975 0.640480i \(-0.778736\pi\)
−0.767975 + 0.640480i \(0.778736\pi\)
\(462\) 0 0
\(463\) −9.23369 −0.429126 −0.214563 0.976710i \(-0.568833\pi\)
−0.214563 + 0.976710i \(0.568833\pi\)
\(464\) 0 0
\(465\) −14.2337 −0.660071
\(466\) 0 0
\(467\) −41.1168 −1.90266 −0.951330 0.308173i \(-0.900282\pi\)
−0.951330 + 0.308173i \(0.900282\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −12.8614 −0.592622
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −4.62772 −0.212334
\(476\) 0 0
\(477\) −10.1168 −0.463218
\(478\) 0 0
\(479\) 31.7228 1.44945 0.724726 0.689037i \(-0.241966\pi\)
0.724726 + 0.689037i \(0.241966\pi\)
\(480\) 0 0
\(481\) 3.11684 0.142116
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −12.1386 −0.551185
\(486\) 0 0
\(487\) 6.37228 0.288756 0.144378 0.989523i \(-0.453882\pi\)
0.144378 + 0.989523i \(0.453882\pi\)
\(488\) 0 0
\(489\) 3.37228 0.152500
\(490\) 0 0
\(491\) 40.2337 1.81572 0.907860 0.419272i \(-0.137715\pi\)
0.907860 + 0.419272i \(0.137715\pi\)
\(492\) 0 0
\(493\) −1.48913 −0.0670668
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 12.7446 0.571672
\(498\) 0 0
\(499\) 7.25544 0.324798 0.162399 0.986725i \(-0.448077\pi\)
0.162399 + 0.986725i \(0.448077\pi\)
\(500\) 0 0
\(501\) −1.88316 −0.0841332
\(502\) 0 0
\(503\) 1.76631 0.0787560 0.0393780 0.999224i \(-0.487462\pi\)
0.0393780 + 0.999224i \(0.487462\pi\)
\(504\) 0 0
\(505\) −28.7446 −1.27912
\(506\) 0 0
\(507\) −12.8614 −0.571195
\(508\) 0 0
\(509\) 13.8832 0.615360 0.307680 0.951490i \(-0.400447\pi\)
0.307680 + 0.951490i \(0.400447\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) −7.37228 −0.325494
\(514\) 0 0
\(515\) 16.6060 0.731746
\(516\) 0 0
\(517\) −26.1168 −1.14862
\(518\) 0 0
\(519\) −7.48913 −0.328736
\(520\) 0 0
\(521\) 11.4891 0.503348 0.251674 0.967812i \(-0.419019\pi\)
0.251674 + 0.967812i \(0.419019\pi\)
\(522\) 0 0
\(523\) 2.39403 0.104684 0.0523418 0.998629i \(-0.483331\pi\)
0.0523418 + 0.998629i \(0.483331\pi\)
\(524\) 0 0
\(525\) −0.627719 −0.0273959
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.37228 −0.406722
\(532\) 0 0
\(533\) −2.13859 −0.0926328
\(534\) 0 0
\(535\) −13.0217 −0.562979
\(536\) 0 0
\(537\) 15.1168 0.652340
\(538\) 0 0
\(539\) −3.37228 −0.145254
\(540\) 0 0
\(541\) 33.0951 1.42287 0.711435 0.702752i \(-0.248046\pi\)
0.711435 + 0.702752i \(0.248046\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −24.6060 −1.05400
\(546\) 0 0
\(547\) −20.7446 −0.886973 −0.443487 0.896281i \(-0.646259\pi\)
−0.443487 + 0.896281i \(0.646259\pi\)
\(548\) 0 0
\(549\) −2.62772 −0.112148
\(550\) 0 0
\(551\) 2.74456 0.116922
\(552\) 0 0
\(553\) −2.74456 −0.116711
\(554\) 0 0
\(555\) 19.8614 0.843070
\(556\) 0 0
\(557\) −3.76631 −0.159584 −0.0797919 0.996812i \(-0.525426\pi\)
−0.0797919 + 0.996812i \(0.525426\pi\)
\(558\) 0 0
\(559\) −0.883156 −0.0373535
\(560\) 0 0
\(561\) −13.4891 −0.569511
\(562\) 0 0
\(563\) 22.3723 0.942879 0.471440 0.881898i \(-0.343735\pi\)
0.471440 + 0.881898i \(0.343735\pi\)
\(564\) 0 0
\(565\) 18.0951 0.761266
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 12.8614 0.537293
\(574\) 0 0
\(575\) −0.627719 −0.0261777
\(576\) 0 0
\(577\) −12.5109 −0.520835 −0.260417 0.965496i \(-0.583860\pi\)
−0.260417 + 0.965496i \(0.583860\pi\)
\(578\) 0 0
\(579\) 27.2337 1.13179
\(580\) 0 0
\(581\) 2.74456 0.113864
\(582\) 0 0
\(583\) 34.1168 1.41298
\(584\) 0 0
\(585\) 0.883156 0.0365140
\(586\) 0 0
\(587\) −14.3505 −0.592310 −0.296155 0.955140i \(-0.595704\pi\)
−0.296155 + 0.955140i \(0.595704\pi\)
\(588\) 0 0
\(589\) −44.2337 −1.82262
\(590\) 0 0
\(591\) 9.86141 0.405644
\(592\) 0 0
\(593\) 29.1168 1.19569 0.597843 0.801613i \(-0.296025\pi\)
0.597843 + 0.801613i \(0.296025\pi\)
\(594\) 0 0
\(595\) 9.48913 0.389016
\(596\) 0 0
\(597\) −1.00000 −0.0409273
\(598\) 0 0
\(599\) 10.7446 0.439011 0.219505 0.975611i \(-0.429556\pi\)
0.219505 + 0.975611i \(0.429556\pi\)
\(600\) 0 0
\(601\) −14.9783 −0.610976 −0.305488 0.952196i \(-0.598820\pi\)
−0.305488 + 0.952196i \(0.598820\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −0.883156 −0.0359054
\(606\) 0 0
\(607\) −16.7446 −0.679641 −0.339820 0.940490i \(-0.610366\pi\)
−0.339820 + 0.940490i \(0.610366\pi\)
\(608\) 0 0
\(609\) 0.372281 0.0150856
\(610\) 0 0
\(611\) −2.88316 −0.116640
\(612\) 0 0
\(613\) 19.3505 0.781561 0.390780 0.920484i \(-0.372205\pi\)
0.390780 + 0.920484i \(0.372205\pi\)
\(614\) 0 0
\(615\) −13.6277 −0.549523
\(616\) 0 0
\(617\) −16.9783 −0.683519 −0.341759 0.939788i \(-0.611023\pi\)
−0.341759 + 0.939788i \(0.611023\pi\)
\(618\) 0 0
\(619\) −35.7228 −1.43582 −0.717911 0.696135i \(-0.754901\pi\)
−0.717911 + 0.696135i \(0.754901\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −15.4891 −0.620559
\(624\) 0 0
\(625\) −27.7446 −1.10978
\(626\) 0 0
\(627\) 24.8614 0.992869
\(628\) 0 0
\(629\) −33.4891 −1.33530
\(630\) 0 0
\(631\) −29.7228 −1.18325 −0.591623 0.806215i \(-0.701513\pi\)
−0.591623 + 0.806215i \(0.701513\pi\)
\(632\) 0 0
\(633\) 11.3723 0.452008
\(634\) 0 0
\(635\) −2.37228 −0.0941411
\(636\) 0 0
\(637\) −0.372281 −0.0147503
\(638\) 0 0
\(639\) −12.7446 −0.504167
\(640\) 0 0
\(641\) 32.4891 1.28324 0.641622 0.767021i \(-0.278262\pi\)
0.641622 + 0.767021i \(0.278262\pi\)
\(642\) 0 0
\(643\) 0.116844 0.00460788 0.00230394 0.999997i \(-0.499267\pi\)
0.00230394 + 0.999997i \(0.499267\pi\)
\(644\) 0 0
\(645\) −5.62772 −0.221591
\(646\) 0 0
\(647\) −48.4674 −1.90545 −0.952725 0.303835i \(-0.901733\pi\)
−0.952725 + 0.303835i \(0.901733\pi\)
\(648\) 0 0
\(649\) 31.6060 1.24064
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −43.1168 −1.68729 −0.843646 0.536899i \(-0.819595\pi\)
−0.843646 + 0.536899i \(0.819595\pi\)
\(654\) 0 0
\(655\) 9.76631 0.381601
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −36.1168 −1.40478 −0.702391 0.711791i \(-0.747884\pi\)
−0.702391 + 0.711791i \(0.747884\pi\)
\(662\) 0 0
\(663\) −1.48913 −0.0578328
\(664\) 0 0
\(665\) −17.4891 −0.678199
\(666\) 0 0
\(667\) 0.372281 0.0144148
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 8.86141 0.342091
\(672\) 0 0
\(673\) 4.25544 0.164035 0.0820175 0.996631i \(-0.473864\pi\)
0.0820175 + 0.996631i \(0.473864\pi\)
\(674\) 0 0
\(675\) 0.627719 0.0241609
\(676\) 0 0
\(677\) 11.2554 0.432582 0.216291 0.976329i \(-0.430604\pi\)
0.216291 + 0.976329i \(0.430604\pi\)
\(678\) 0 0
\(679\) −5.11684 −0.196366
\(680\) 0 0
\(681\) 9.11684 0.349358
\(682\) 0 0
\(683\) −30.9783 −1.18535 −0.592675 0.805442i \(-0.701928\pi\)
−0.592675 + 0.805442i \(0.701928\pi\)
\(684\) 0 0
\(685\) −43.8614 −1.67586
\(686\) 0 0
\(687\) −26.3505 −1.00534
\(688\) 0 0
\(689\) 3.76631 0.143485
\(690\) 0 0
\(691\) 3.39403 0.129115 0.0645575 0.997914i \(-0.479436\pi\)
0.0645575 + 0.997914i \(0.479436\pi\)
\(692\) 0 0
\(693\) 3.37228 0.128102
\(694\) 0 0
\(695\) −13.3505 −0.506415
\(696\) 0 0
\(697\) 22.9783 0.870363
\(698\) 0 0
\(699\) 3.25544 0.123132
\(700\) 0 0
\(701\) 42.3505 1.59956 0.799779 0.600295i \(-0.204950\pi\)
0.799779 + 0.600295i \(0.204950\pi\)
\(702\) 0 0
\(703\) 61.7228 2.32792
\(704\) 0 0
\(705\) −18.3723 −0.691940
\(706\) 0 0
\(707\) −12.1168 −0.455701
\(708\) 0 0
\(709\) 3.48913 0.131037 0.0655184 0.997851i \(-0.479130\pi\)
0.0655184 + 0.997851i \(0.479130\pi\)
\(710\) 0 0
\(711\) 2.74456 0.102929
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −2.97825 −0.111380
\(716\) 0 0
\(717\) 9.48913 0.354378
\(718\) 0 0
\(719\) −2.37228 −0.0884712 −0.0442356 0.999021i \(-0.514085\pi\)
−0.0442356 + 0.999021i \(0.514085\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 0 0
\(723\) 22.4891 0.836380
\(724\) 0 0
\(725\) −0.233688 −0.00867895
\(726\) 0 0
\(727\) 18.3505 0.680584 0.340292 0.940320i \(-0.389474\pi\)
0.340292 + 0.940320i \(0.389474\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 9.48913 0.350968
\(732\) 0 0
\(733\) 35.2119 1.30058 0.650291 0.759685i \(-0.274647\pi\)
0.650291 + 0.759685i \(0.274647\pi\)
\(734\) 0 0
\(735\) −2.37228 −0.0875029
\(736\) 0 0
\(737\) −13.4891 −0.496878
\(738\) 0 0
\(739\) 24.7446 0.910243 0.455122 0.890429i \(-0.349596\pi\)
0.455122 + 0.890429i \(0.349596\pi\)
\(740\) 0 0
\(741\) 2.74456 0.100824
\(742\) 0 0
\(743\) −19.3723 −0.710700 −0.355350 0.934733i \(-0.615638\pi\)
−0.355350 + 0.934733i \(0.615638\pi\)
\(744\) 0 0
\(745\) −52.4674 −1.92226
\(746\) 0 0
\(747\) −2.74456 −0.100418
\(748\) 0 0
\(749\) −5.48913 −0.200568
\(750\) 0 0
\(751\) −22.9783 −0.838488 −0.419244 0.907874i \(-0.637705\pi\)
−0.419244 + 0.907874i \(0.637705\pi\)
\(752\) 0 0
\(753\) −9.86141 −0.359370
\(754\) 0 0
\(755\) 10.6495 0.387574
\(756\) 0 0
\(757\) 14.2337 0.517332 0.258666 0.965967i \(-0.416717\pi\)
0.258666 + 0.965967i \(0.416717\pi\)
\(758\) 0 0
\(759\) 3.37228 0.122406
\(760\) 0 0
\(761\) −35.0951 −1.27220 −0.636098 0.771608i \(-0.719453\pi\)
−0.636098 + 0.771608i \(0.719453\pi\)
\(762\) 0 0
\(763\) −10.3723 −0.375502
\(764\) 0 0
\(765\) −9.48913 −0.343080
\(766\) 0 0
\(767\) 3.48913 0.125985
\(768\) 0 0
\(769\) −31.3505 −1.13053 −0.565265 0.824910i \(-0.691226\pi\)
−0.565265 + 0.824910i \(0.691226\pi\)
\(770\) 0 0
\(771\) 28.1168 1.01260
\(772\) 0 0
\(773\) −34.3723 −1.23629 −0.618143 0.786066i \(-0.712115\pi\)
−0.618143 + 0.786066i \(0.712115\pi\)
\(774\) 0 0
\(775\) 3.76631 0.135290
\(776\) 0 0
\(777\) 8.37228 0.300354
\(778\) 0 0
\(779\) −42.3505 −1.51737
\(780\) 0 0
\(781\) 42.9783 1.53788
\(782\) 0 0
\(783\) −0.372281 −0.0133042
\(784\) 0 0
\(785\) 30.5109 1.08898
\(786\) 0 0
\(787\) 47.8397 1.70530 0.852650 0.522483i \(-0.174994\pi\)
0.852650 + 0.522483i \(0.174994\pi\)
\(788\) 0 0
\(789\) −1.00000 −0.0356009
\(790\) 0 0
\(791\) 7.62772 0.271210
\(792\) 0 0
\(793\) 0.978251 0.0347387
\(794\) 0 0
\(795\) 24.0000 0.851192
\(796\) 0 0
\(797\) −10.6060 −0.375683 −0.187841 0.982199i \(-0.560149\pi\)
−0.187841 + 0.982199i \(0.560149\pi\)
\(798\) 0 0
\(799\) 30.9783 1.09593
\(800\) 0 0
\(801\) 15.4891 0.547281
\(802\) 0 0
\(803\) −40.4674 −1.42806
\(804\) 0 0
\(805\) −2.37228 −0.0836119
\(806\) 0 0
\(807\) 23.4891 0.826856
\(808\) 0 0
\(809\) 34.7446 1.22155 0.610777 0.791803i \(-0.290857\pi\)
0.610777 + 0.791803i \(0.290857\pi\)
\(810\) 0 0
\(811\) 17.8614 0.627199 0.313599 0.949555i \(-0.398465\pi\)
0.313599 + 0.949555i \(0.398465\pi\)
\(812\) 0 0
\(813\) −21.4891 −0.753657
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −17.4891 −0.611867
\(818\) 0 0
\(819\) 0.372281 0.0130086
\(820\) 0 0
\(821\) −9.76631 −0.340847 −0.170423 0.985371i \(-0.554514\pi\)
−0.170423 + 0.985371i \(0.554514\pi\)
\(822\) 0 0
\(823\) 20.7663 0.723868 0.361934 0.932204i \(-0.382117\pi\)
0.361934 + 0.932204i \(0.382117\pi\)
\(824\) 0 0
\(825\) −2.11684 −0.0736990
\(826\) 0 0
\(827\) −9.60597 −0.334032 −0.167016 0.985954i \(-0.553413\pi\)
−0.167016 + 0.985954i \(0.553413\pi\)
\(828\) 0 0
\(829\) 42.4674 1.47495 0.737476 0.675373i \(-0.236017\pi\)
0.737476 + 0.675373i \(0.236017\pi\)
\(830\) 0 0
\(831\) 18.1168 0.628466
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 4.46738 0.154600
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −28.8614 −0.995221
\(842\) 0 0
\(843\) 7.62772 0.262713
\(844\) 0 0
\(845\) 30.5109 1.04961
\(846\) 0 0
\(847\) −0.372281 −0.0127917
\(848\) 0 0
\(849\) −16.7446 −0.574672
\(850\) 0 0
\(851\) 8.37228 0.286998
\(852\) 0 0
\(853\) −37.3505 −1.27886 −0.639429 0.768850i \(-0.720829\pi\)
−0.639429 + 0.768850i \(0.720829\pi\)
\(854\) 0 0
\(855\) 17.4891 0.598115
\(856\) 0 0
\(857\) −15.7446 −0.537824 −0.268912 0.963165i \(-0.586664\pi\)
−0.268912 + 0.963165i \(0.586664\pi\)
\(858\) 0 0
\(859\) −0.883156 −0.0301329 −0.0150664 0.999886i \(-0.504796\pi\)
−0.0150664 + 0.999886i \(0.504796\pi\)
\(860\) 0 0
\(861\) −5.74456 −0.195774
\(862\) 0 0
\(863\) 25.7228 0.875615 0.437807 0.899069i \(-0.355755\pi\)
0.437807 + 0.899069i \(0.355755\pi\)
\(864\) 0 0
\(865\) 17.7663 0.604073
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −9.25544 −0.313969
\(870\) 0 0
\(871\) −1.48913 −0.0504571
\(872\) 0 0
\(873\) 5.11684 0.173179
\(874\) 0 0
\(875\) −10.3723 −0.350647
\(876\) 0 0
\(877\) −50.3505 −1.70022 −0.850108 0.526608i \(-0.823464\pi\)
−0.850108 + 0.526608i \(0.823464\pi\)
\(878\) 0 0
\(879\) 22.2337 0.749924
\(880\) 0 0
\(881\) −0.510875 −0.0172118 −0.00860590 0.999963i \(-0.502739\pi\)
−0.00860590 + 0.999963i \(0.502739\pi\)
\(882\) 0 0
\(883\) 31.7228 1.06756 0.533779 0.845624i \(-0.320771\pi\)
0.533779 + 0.845624i \(0.320771\pi\)
\(884\) 0 0
\(885\) 22.2337 0.747377
\(886\) 0 0
\(887\) −45.9565 −1.54307 −0.771534 0.636188i \(-0.780510\pi\)
−0.771534 + 0.636188i \(0.780510\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) −3.37228 −0.112976
\(892\) 0 0
\(893\) −57.0951 −1.91061
\(894\) 0 0
\(895\) −35.8614 −1.19871
\(896\) 0 0
\(897\) 0.372281 0.0124301
\(898\) 0 0
\(899\) −2.23369 −0.0744977
\(900\) 0 0
\(901\) −40.4674 −1.34816
\(902\) 0 0
\(903\) −2.37228 −0.0789446
\(904\) 0 0
\(905\) 14.2337 0.473144
\(906\) 0 0
\(907\) −23.8614 −0.792305 −0.396153 0.918185i \(-0.629655\pi\)
−0.396153 + 0.918185i \(0.629655\pi\)
\(908\) 0 0
\(909\) 12.1168 0.401890
\(910\) 0 0
\(911\) −16.8832 −0.559364 −0.279682 0.960093i \(-0.590229\pi\)
−0.279682 + 0.960093i \(0.590229\pi\)
\(912\) 0 0
\(913\) 9.25544 0.306310
\(914\) 0 0
\(915\) 6.23369 0.206079
\(916\) 0 0
\(917\) 4.11684 0.135950
\(918\) 0 0
\(919\) 11.7663 0.388135 0.194067 0.980988i \(-0.437832\pi\)
0.194067 + 0.980988i \(0.437832\pi\)
\(920\) 0 0
\(921\) 26.3723 0.868996
\(922\) 0 0
\(923\) 4.74456 0.156169
\(924\) 0 0
\(925\) −5.25544 −0.172798
\(926\) 0 0
\(927\) −7.00000 −0.229910
\(928\) 0 0
\(929\) 9.11684 0.299114 0.149557 0.988753i \(-0.452215\pi\)
0.149557 + 0.988753i \(0.452215\pi\)
\(930\) 0 0
\(931\) −7.37228 −0.241617
\(932\) 0 0
\(933\) −16.6277 −0.544367
\(934\) 0 0
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −43.7446 −1.42907 −0.714536 0.699598i \(-0.753362\pi\)
−0.714536 + 0.699598i \(0.753362\pi\)
\(938\) 0 0
\(939\) −31.0951 −1.01475
\(940\) 0 0
\(941\) 54.0951 1.76345 0.881725 0.471764i \(-0.156383\pi\)
0.881725 + 0.471764i \(0.156383\pi\)
\(942\) 0 0
\(943\) −5.74456 −0.187069
\(944\) 0 0
\(945\) 2.37228 0.0771703
\(946\) 0 0
\(947\) −22.0951 −0.717994 −0.358997 0.933339i \(-0.616881\pi\)
−0.358997 + 0.933339i \(0.616881\pi\)
\(948\) 0 0
\(949\) −4.46738 −0.145017
\(950\) 0 0
\(951\) 10.8832 0.352911
\(952\) 0 0
\(953\) −9.37228 −0.303598 −0.151799 0.988411i \(-0.548507\pi\)
−0.151799 + 0.988411i \(0.548507\pi\)
\(954\) 0 0
\(955\) −30.5109 −0.987309
\(956\) 0 0
\(957\) 1.25544 0.0405825
\(958\) 0 0
\(959\) −18.4891 −0.597045
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 5.48913 0.176885
\(964\) 0 0
\(965\) −64.6060 −2.07974
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) −29.4891 −0.947327
\(970\) 0 0
\(971\) 27.7663 0.891063 0.445532 0.895266i \(-0.353015\pi\)
0.445532 + 0.895266i \(0.353015\pi\)
\(972\) 0 0
\(973\) −5.62772 −0.180416
\(974\) 0 0
\(975\) −0.233688 −0.00748400
\(976\) 0 0
\(977\) 38.4891 1.23138 0.615688 0.787990i \(-0.288878\pi\)
0.615688 + 0.787990i \(0.288878\pi\)
\(978\) 0 0
\(979\) −52.2337 −1.66940
\(980\) 0 0
\(981\) 10.3723 0.331161
\(982\) 0 0
\(983\) 3.48913 0.111286 0.0556429 0.998451i \(-0.482279\pi\)
0.0556429 + 0.998451i \(0.482279\pi\)
\(984\) 0 0
\(985\) −23.3940 −0.745396
\(986\) 0 0
\(987\) −7.74456 −0.246512
\(988\) 0 0
\(989\) −2.37228 −0.0754342
\(990\) 0 0
\(991\) −5.64947 −0.179461 −0.0897306 0.995966i \(-0.528601\pi\)
−0.0897306 + 0.995966i \(0.528601\pi\)
\(992\) 0 0
\(993\) 3.37228 0.107016
\(994\) 0 0
\(995\) 2.37228 0.0752064
\(996\) 0 0
\(997\) −30.2337 −0.957511 −0.478755 0.877948i \(-0.658912\pi\)
−0.478755 + 0.877948i \(0.658912\pi\)
\(998\) 0 0
\(999\) −8.37228 −0.264887
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 7728.2.a.bj.1.1 2
4.3 odd 2 3864.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.i.1.1 2 4.3 odd 2
7728.2.a.bj.1.1 2 1.1 even 1 trivial