Properties

Label 7728.2.a.cg.1.4
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 13 x^{4} + 7 x^{3} + 31 x^{2} - 17 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.52361\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.16465 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.16465 q^{5} +1.00000 q^{7} +1.00000 q^{9} -6.56356 q^{11} -1.77879 q^{13} -1.16465 q^{15} +4.94344 q^{17} -1.64398 q^{19} -1.00000 q^{21} -1.00000 q^{23} -3.64359 q^{25} -1.00000 q^{27} -0.671372 q^{29} -2.94344 q^{31} +6.56356 q^{33} +1.16465 q^{35} -0.671372 q^{37} +1.77879 q^{39} +3.91605 q^{41} +10.3156 q^{43} +1.16465 q^{45} -8.81177 q^{47} +1.00000 q^{49} -4.94344 q^{51} +5.05684 q^{53} -7.64426 q^{55} +1.64398 q^{57} +3.05684 q^{59} -12.2078 q^{61} +1.00000 q^{63} -2.07167 q^{65} -6.98566 q^{67} +1.00000 q^{69} +9.62041 q^{71} +9.47894 q^{73} +3.64359 q^{75} -6.56356 q^{77} +9.48029 q^{79} +1.00000 q^{81} +0.614137 q^{83} +5.75738 q^{85} +0.671372 q^{87} -1.08356 q^{89} -1.77879 q^{91} +2.94344 q^{93} -1.91466 q^{95} +0.463829 q^{97} -6.56356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99} + O(q^{100}) \)

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
\(5\) 1.16465 0.520848 0.260424 0.965494i \(-0.416138\pi\)
0.260424 + 0.965494i \(0.416138\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.56356 −1.97899 −0.989494 0.144571i \(-0.953820\pi\)
−0.989494 + 0.144571i \(0.953820\pi\)
\(12\) 0 0
\(13\) −1.77879 −0.493347 −0.246674 0.969099i \(-0.579338\pi\)
−0.246674 + 0.969099i \(0.579338\pi\)
\(14\) 0 0
\(15\) −1.16465 −0.300712
\(16\) 0 0
\(17\) 4.94344 1.19896 0.599480 0.800390i \(-0.295374\pi\)
0.599480 + 0.800390i \(0.295374\pi\)
\(18\) 0 0
\(19\) −1.64398 −0.377155 −0.188577 0.982058i \(-0.560388\pi\)
−0.188577 + 0.982058i \(0.560388\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.64359 −0.728717
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.671372 −0.124671 −0.0623353 0.998055i \(-0.519855\pi\)
−0.0623353 + 0.998055i \(0.519855\pi\)
\(30\) 0 0
\(31\) −2.94344 −0.528657 −0.264329 0.964433i \(-0.585150\pi\)
−0.264329 + 0.964433i \(0.585150\pi\)
\(32\) 0 0
\(33\) 6.56356 1.14257
\(34\) 0 0
\(35\) 1.16465 0.196862
\(36\) 0 0
\(37\) −0.671372 −0.110373 −0.0551865 0.998476i \(-0.517575\pi\)
−0.0551865 + 0.998476i \(0.517575\pi\)
\(38\) 0 0
\(39\) 1.77879 0.284834
\(40\) 0 0
\(41\) 3.91605 0.611584 0.305792 0.952098i \(-0.401079\pi\)
0.305792 + 0.952098i \(0.401079\pi\)
\(42\) 0 0
\(43\) 10.3156 1.57312 0.786560 0.617514i \(-0.211860\pi\)
0.786560 + 0.617514i \(0.211860\pi\)
\(44\) 0 0
\(45\) 1.16465 0.173616
\(46\) 0 0
\(47\) −8.81177 −1.28533 −0.642665 0.766148i \(-0.722171\pi\)
−0.642665 + 0.766148i \(0.722171\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.94344 −0.692220
\(52\) 0 0
\(53\) 5.05684 0.694611 0.347305 0.937752i \(-0.387097\pi\)
0.347305 + 0.937752i \(0.387097\pi\)
\(54\) 0 0
\(55\) −7.64426 −1.03075
\(56\) 0 0
\(57\) 1.64398 0.217751
\(58\) 0 0
\(59\) 3.05684 0.397967 0.198983 0.980003i \(-0.436236\pi\)
0.198983 + 0.980003i \(0.436236\pi\)
\(60\) 0 0
\(61\) −12.2078 −1.56305 −0.781526 0.623873i \(-0.785558\pi\)
−0.781526 + 0.623873i \(0.785558\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −2.07167 −0.256959
\(66\) 0 0
\(67\) −6.98566 −0.853434 −0.426717 0.904385i \(-0.640330\pi\)
−0.426717 + 0.904385i \(0.640330\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 9.62041 1.14173 0.570866 0.821043i \(-0.306608\pi\)
0.570866 + 0.821043i \(0.306608\pi\)
\(72\) 0 0
\(73\) 9.47894 1.10943 0.554713 0.832042i \(-0.312828\pi\)
0.554713 + 0.832042i \(0.312828\pi\)
\(74\) 0 0
\(75\) 3.64359 0.420725
\(76\) 0 0
\(77\) −6.56356 −0.747987
\(78\) 0 0
\(79\) 9.48029 1.06662 0.533308 0.845921i \(-0.320949\pi\)
0.533308 + 0.845921i \(0.320949\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.614137 0.0674103 0.0337051 0.999432i \(-0.489269\pi\)
0.0337051 + 0.999432i \(0.489269\pi\)
\(84\) 0 0
\(85\) 5.75738 0.624476
\(86\) 0 0
\(87\) 0.671372 0.0719786
\(88\) 0 0
\(89\) −1.08356 −0.114857 −0.0574285 0.998350i \(-0.518290\pi\)
−0.0574285 + 0.998350i \(0.518290\pi\)
\(90\) 0 0
\(91\) −1.77879 −0.186468
\(92\) 0 0
\(93\) 2.94344 0.305220
\(94\) 0 0
\(95\) −1.91466 −0.196440
\(96\) 0 0
\(97\) 0.463829 0.0470947 0.0235473 0.999723i \(-0.492504\pi\)
0.0235473 + 0.999723i \(0.492504\pi\)
\(98\) 0 0
\(99\) −6.56356 −0.659663
\(100\) 0 0
\(101\) 11.3669 1.13105 0.565524 0.824732i \(-0.308674\pi\)
0.565524 + 0.824732i \(0.308674\pi\)
\(102\) 0 0
\(103\) 15.3486 1.51234 0.756172 0.654373i \(-0.227067\pi\)
0.756172 + 0.654373i \(0.227067\pi\)
\(104\) 0 0
\(105\) −1.16465 −0.113658
\(106\) 0 0
\(107\) −15.2911 −1.47825 −0.739123 0.673570i \(-0.764760\pi\)
−0.739123 + 0.673570i \(0.764760\pi\)
\(108\) 0 0
\(109\) 17.5797 1.68383 0.841917 0.539607i \(-0.181427\pi\)
0.841917 + 0.539607i \(0.181427\pi\)
\(110\) 0 0
\(111\) 0.671372 0.0637238
\(112\) 0 0
\(113\) 8.29043 0.779898 0.389949 0.920836i \(-0.372493\pi\)
0.389949 + 0.920836i \(0.372493\pi\)
\(114\) 0 0
\(115\) −1.16465 −0.108604
\(116\) 0 0
\(117\) −1.77879 −0.164449
\(118\) 0 0
\(119\) 4.94344 0.453164
\(120\) 0 0
\(121\) 32.0804 2.91640
\(122\) 0 0
\(123\) −3.91605 −0.353098
\(124\) 0 0
\(125\) −10.0668 −0.900399
\(126\) 0 0
\(127\) 5.34413 0.474215 0.237107 0.971483i \(-0.423801\pi\)
0.237107 + 0.971483i \(0.423801\pi\)
\(128\) 0 0
\(129\) −10.3156 −0.908241
\(130\) 0 0
\(131\) −8.46107 −0.739247 −0.369623 0.929182i \(-0.620513\pi\)
−0.369623 + 0.929182i \(0.620513\pi\)
\(132\) 0 0
\(133\) −1.64398 −0.142551
\(134\) 0 0
\(135\) −1.16465 −0.100237
\(136\) 0 0
\(137\) −5.25879 −0.449289 −0.224644 0.974441i \(-0.572122\pi\)
−0.224644 + 0.974441i \(0.572122\pi\)
\(138\) 0 0
\(139\) 5.96248 0.505731 0.252865 0.967502i \(-0.418627\pi\)
0.252865 + 0.967502i \(0.418627\pi\)
\(140\) 0 0
\(141\) 8.81177 0.742085
\(142\) 0 0
\(143\) 11.6752 0.976328
\(144\) 0 0
\(145\) −0.781914 −0.0649345
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 10.5397 0.863446 0.431723 0.902006i \(-0.357906\pi\)
0.431723 + 0.902006i \(0.357906\pi\)
\(150\) 0 0
\(151\) 8.55080 0.695854 0.347927 0.937522i \(-0.386886\pi\)
0.347927 + 0.937522i \(0.386886\pi\)
\(152\) 0 0
\(153\) 4.94344 0.399653
\(154\) 0 0
\(155\) −3.42808 −0.275350
\(156\) 0 0
\(157\) 5.41258 0.431971 0.215985 0.976397i \(-0.430704\pi\)
0.215985 + 0.976397i \(0.430704\pi\)
\(158\) 0 0
\(159\) −5.05684 −0.401034
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −21.1426 −1.65602 −0.828009 0.560715i \(-0.810527\pi\)
−0.828009 + 0.560715i \(0.810527\pi\)
\(164\) 0 0
\(165\) 7.64426 0.595105
\(166\) 0 0
\(167\) 2.35602 0.182314 0.0911571 0.995837i \(-0.470943\pi\)
0.0911571 + 0.995837i \(0.470943\pi\)
\(168\) 0 0
\(169\) −9.83591 −0.756609
\(170\) 0 0
\(171\) −1.64398 −0.125718
\(172\) 0 0
\(173\) 1.85438 0.140986 0.0704931 0.997512i \(-0.477543\pi\)
0.0704931 + 0.997512i \(0.477543\pi\)
\(174\) 0 0
\(175\) −3.64359 −0.275429
\(176\) 0 0
\(177\) −3.05684 −0.229766
\(178\) 0 0
\(179\) 6.40490 0.478725 0.239362 0.970930i \(-0.423062\pi\)
0.239362 + 0.970930i \(0.423062\pi\)
\(180\) 0 0
\(181\) −12.9354 −0.961478 −0.480739 0.876864i \(-0.659632\pi\)
−0.480739 + 0.876864i \(0.659632\pi\)
\(182\) 0 0
\(183\) 12.2078 0.902428
\(184\) 0 0
\(185\) −0.781914 −0.0574875
\(186\) 0 0
\(187\) −32.4466 −2.37273
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −1.87403 −0.135600 −0.0678001 0.997699i \(-0.521598\pi\)
−0.0678001 + 0.997699i \(0.521598\pi\)
\(192\) 0 0
\(193\) 24.9062 1.79279 0.896394 0.443259i \(-0.146178\pi\)
0.896394 + 0.443259i \(0.146178\pi\)
\(194\) 0 0
\(195\) 2.07167 0.148355
\(196\) 0 0
\(197\) −1.41396 −0.100741 −0.0503704 0.998731i \(-0.516040\pi\)
−0.0503704 + 0.998731i \(0.516040\pi\)
\(198\) 0 0
\(199\) 2.92556 0.207388 0.103694 0.994609i \(-0.466934\pi\)
0.103694 + 0.994609i \(0.466934\pi\)
\(200\) 0 0
\(201\) 6.98566 0.492730
\(202\) 0 0
\(203\) −0.671372 −0.0471211
\(204\) 0 0
\(205\) 4.56083 0.318542
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 10.7904 0.746385
\(210\) 0 0
\(211\) 6.56356 0.451854 0.225927 0.974144i \(-0.427459\pi\)
0.225927 + 0.974144i \(0.427459\pi\)
\(212\) 0 0
\(213\) −9.62041 −0.659179
\(214\) 0 0
\(215\) 12.0141 0.819356
\(216\) 0 0
\(217\) −2.94344 −0.199814
\(218\) 0 0
\(219\) −9.47894 −0.640527
\(220\) 0 0
\(221\) −8.79333 −0.591504
\(222\) 0 0
\(223\) −14.5652 −0.975358 −0.487679 0.873023i \(-0.662156\pi\)
−0.487679 + 0.873023i \(0.662156\pi\)
\(224\) 0 0
\(225\) −3.64359 −0.242906
\(226\) 0 0
\(227\) 18.8153 1.24882 0.624408 0.781099i \(-0.285340\pi\)
0.624408 + 0.781099i \(0.285340\pi\)
\(228\) 0 0
\(229\) 9.18899 0.607226 0.303613 0.952796i \(-0.401807\pi\)
0.303613 + 0.952796i \(0.401807\pi\)
\(230\) 0 0
\(231\) 6.56356 0.431851
\(232\) 0 0
\(233\) −29.8388 −1.95481 −0.977403 0.211383i \(-0.932203\pi\)
−0.977403 + 0.211383i \(0.932203\pi\)
\(234\) 0 0
\(235\) −10.2626 −0.669461
\(236\) 0 0
\(237\) −9.48029 −0.615811
\(238\) 0 0
\(239\) −8.87254 −0.573917 −0.286958 0.957943i \(-0.592644\pi\)
−0.286958 + 0.957943i \(0.592644\pi\)
\(240\) 0 0
\(241\) 29.5203 1.90157 0.950786 0.309849i \(-0.100278\pi\)
0.950786 + 0.309849i \(0.100278\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.16465 0.0744068
\(246\) 0 0
\(247\) 2.92429 0.186068
\(248\) 0 0
\(249\) −0.614137 −0.0389194
\(250\) 0 0
\(251\) −13.7833 −0.869993 −0.434996 0.900432i \(-0.643250\pi\)
−0.434996 + 0.900432i \(0.643250\pi\)
\(252\) 0 0
\(253\) 6.56356 0.412648
\(254\) 0 0
\(255\) −5.75738 −0.360541
\(256\) 0 0
\(257\) −17.2531 −1.07622 −0.538109 0.842875i \(-0.680861\pi\)
−0.538109 + 0.842875i \(0.680861\pi\)
\(258\) 0 0
\(259\) −0.671372 −0.0417170
\(260\) 0 0
\(261\) −0.671372 −0.0415569
\(262\) 0 0
\(263\) 22.7629 1.40362 0.701811 0.712363i \(-0.252375\pi\)
0.701811 + 0.712363i \(0.252375\pi\)
\(264\) 0 0
\(265\) 5.88946 0.361787
\(266\) 0 0
\(267\) 1.08356 0.0663128
\(268\) 0 0
\(269\) −4.80332 −0.292864 −0.146432 0.989221i \(-0.546779\pi\)
−0.146432 + 0.989221i \(0.546779\pi\)
\(270\) 0 0
\(271\) 11.8544 0.720103 0.360051 0.932933i \(-0.382759\pi\)
0.360051 + 0.932933i \(0.382759\pi\)
\(272\) 0 0
\(273\) 1.77879 0.107657
\(274\) 0 0
\(275\) 23.9149 1.44212
\(276\) 0 0
\(277\) 23.7654 1.42792 0.713962 0.700184i \(-0.246899\pi\)
0.713962 + 0.700184i \(0.246899\pi\)
\(278\) 0 0
\(279\) −2.94344 −0.176219
\(280\) 0 0
\(281\) 18.5534 1.10680 0.553402 0.832914i \(-0.313329\pi\)
0.553402 + 0.832914i \(0.313329\pi\)
\(282\) 0 0
\(283\) 29.0913 1.72930 0.864648 0.502378i \(-0.167541\pi\)
0.864648 + 0.502378i \(0.167541\pi\)
\(284\) 0 0
\(285\) 1.91466 0.113415
\(286\) 0 0
\(287\) 3.91605 0.231157
\(288\) 0 0
\(289\) 7.43759 0.437506
\(290\) 0 0
\(291\) −0.463829 −0.0271901
\(292\) 0 0
\(293\) −23.5422 −1.37535 −0.687675 0.726018i \(-0.741369\pi\)
−0.687675 + 0.726018i \(0.741369\pi\)
\(294\) 0 0
\(295\) 3.56016 0.207280
\(296\) 0 0
\(297\) 6.56356 0.380857
\(298\) 0 0
\(299\) 1.77879 0.102870
\(300\) 0 0
\(301\) 10.3156 0.594583
\(302\) 0 0
\(303\) −11.3669 −0.653010
\(304\) 0 0
\(305\) −14.2179 −0.814112
\(306\) 0 0
\(307\) 9.08511 0.518515 0.259257 0.965808i \(-0.416522\pi\)
0.259257 + 0.965808i \(0.416522\pi\)
\(308\) 0 0
\(309\) −15.3486 −0.873152
\(310\) 0 0
\(311\) 6.98628 0.396155 0.198078 0.980186i \(-0.436530\pi\)
0.198078 + 0.980186i \(0.436530\pi\)
\(312\) 0 0
\(313\) −34.9157 −1.97355 −0.986777 0.162087i \(-0.948178\pi\)
−0.986777 + 0.162087i \(0.948178\pi\)
\(314\) 0 0
\(315\) 1.16465 0.0656207
\(316\) 0 0
\(317\) 4.29043 0.240974 0.120487 0.992715i \(-0.461554\pi\)
0.120487 + 0.992715i \(0.461554\pi\)
\(318\) 0 0
\(319\) 4.40659 0.246722
\(320\) 0 0
\(321\) 15.2911 0.853466
\(322\) 0 0
\(323\) −8.12692 −0.452194
\(324\) 0 0
\(325\) 6.48117 0.359511
\(326\) 0 0
\(327\) −17.5797 −0.972162
\(328\) 0 0
\(329\) −8.81177 −0.485809
\(330\) 0 0
\(331\) 18.0319 0.991122 0.495561 0.868573i \(-0.334963\pi\)
0.495561 + 0.868573i \(0.334963\pi\)
\(332\) 0 0
\(333\) −0.671372 −0.0367910
\(334\) 0 0
\(335\) −8.13585 −0.444509
\(336\) 0 0
\(337\) 22.9618 1.25081 0.625404 0.780301i \(-0.284934\pi\)
0.625404 + 0.780301i \(0.284934\pi\)
\(338\) 0 0
\(339\) −8.29043 −0.450274
\(340\) 0 0
\(341\) 19.3195 1.04621
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.16465 0.0627027
\(346\) 0 0
\(347\) −10.7196 −0.575461 −0.287730 0.957711i \(-0.592901\pi\)
−0.287730 + 0.957711i \(0.592901\pi\)
\(348\) 0 0
\(349\) 20.6235 1.10395 0.551976 0.833860i \(-0.313874\pi\)
0.551976 + 0.833860i \(0.313874\pi\)
\(350\) 0 0
\(351\) 1.77879 0.0949447
\(352\) 0 0
\(353\) −0.999325 −0.0531887 −0.0265944 0.999646i \(-0.508466\pi\)
−0.0265944 + 0.999646i \(0.508466\pi\)
\(354\) 0 0
\(355\) 11.2044 0.594669
\(356\) 0 0
\(357\) −4.94344 −0.261635
\(358\) 0 0
\(359\) −19.7172 −1.04063 −0.520317 0.853973i \(-0.674186\pi\)
−0.520317 + 0.853973i \(0.674186\pi\)
\(360\) 0 0
\(361\) −16.2973 −0.857754
\(362\) 0 0
\(363\) −32.0804 −1.68378
\(364\) 0 0
\(365\) 11.0397 0.577842
\(366\) 0 0
\(367\) −16.9023 −0.882291 −0.441146 0.897436i \(-0.645428\pi\)
−0.441146 + 0.897436i \(0.645428\pi\)
\(368\) 0 0
\(369\) 3.91605 0.203861
\(370\) 0 0
\(371\) 5.05684 0.262538
\(372\) 0 0
\(373\) −21.3505 −1.10548 −0.552742 0.833352i \(-0.686419\pi\)
−0.552742 + 0.833352i \(0.686419\pi\)
\(374\) 0 0
\(375\) 10.0668 0.519846
\(376\) 0 0
\(377\) 1.19423 0.0615059
\(378\) 0 0
\(379\) −1.13552 −0.0583276 −0.0291638 0.999575i \(-0.509284\pi\)
−0.0291638 + 0.999575i \(0.509284\pi\)
\(380\) 0 0
\(381\) −5.34413 −0.273788
\(382\) 0 0
\(383\) 0.201599 0.0103013 0.00515063 0.999987i \(-0.498360\pi\)
0.00515063 + 0.999987i \(0.498360\pi\)
\(384\) 0 0
\(385\) −7.64426 −0.389588
\(386\) 0 0
\(387\) 10.3156 0.524373
\(388\) 0 0
\(389\) −7.86167 −0.398603 −0.199301 0.979938i \(-0.563867\pi\)
−0.199301 + 0.979938i \(0.563867\pi\)
\(390\) 0 0
\(391\) −4.94344 −0.250000
\(392\) 0 0
\(393\) 8.46107 0.426804
\(394\) 0 0
\(395\) 11.0412 0.555544
\(396\) 0 0
\(397\) −14.9506 −0.750348 −0.375174 0.926954i \(-0.622417\pi\)
−0.375174 + 0.926954i \(0.622417\pi\)
\(398\) 0 0
\(399\) 1.64398 0.0823020
\(400\) 0 0
\(401\) 21.9652 1.09689 0.548445 0.836187i \(-0.315220\pi\)
0.548445 + 0.836187i \(0.315220\pi\)
\(402\) 0 0
\(403\) 5.23576 0.260812
\(404\) 0 0
\(405\) 1.16465 0.0578720
\(406\) 0 0
\(407\) 4.40659 0.218427
\(408\) 0 0
\(409\) 13.4334 0.664237 0.332118 0.943238i \(-0.392237\pi\)
0.332118 + 0.943238i \(0.392237\pi\)
\(410\) 0 0
\(411\) 5.25879 0.259397
\(412\) 0 0
\(413\) 3.05684 0.150417
\(414\) 0 0
\(415\) 0.715255 0.0351105
\(416\) 0 0
\(417\) −5.96248 −0.291984
\(418\) 0 0
\(419\) 17.6348 0.861518 0.430759 0.902467i \(-0.358246\pi\)
0.430759 + 0.902467i \(0.358246\pi\)
\(420\) 0 0
\(421\) −21.4428 −1.04506 −0.522528 0.852622i \(-0.675011\pi\)
−0.522528 + 0.852622i \(0.675011\pi\)
\(422\) 0 0
\(423\) −8.81177 −0.428443
\(424\) 0 0
\(425\) −18.0119 −0.873703
\(426\) 0 0
\(427\) −12.2078 −0.590778
\(428\) 0 0
\(429\) −11.6752 −0.563683
\(430\) 0 0
\(431\) −14.5274 −0.699760 −0.349880 0.936795i \(-0.613778\pi\)
−0.349880 + 0.936795i \(0.613778\pi\)
\(432\) 0 0
\(433\) 9.11498 0.438038 0.219019 0.975721i \(-0.429714\pi\)
0.219019 + 0.975721i \(0.429714\pi\)
\(434\) 0 0
\(435\) 0.781914 0.0374899
\(436\) 0 0
\(437\) 1.64398 0.0786422
\(438\) 0 0
\(439\) −21.9574 −1.04797 −0.523986 0.851727i \(-0.675556\pi\)
−0.523986 + 0.851727i \(0.675556\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −8.69523 −0.413123 −0.206561 0.978434i \(-0.566227\pi\)
−0.206561 + 0.978434i \(0.566227\pi\)
\(444\) 0 0
\(445\) −1.26197 −0.0598231
\(446\) 0 0
\(447\) −10.5397 −0.498511
\(448\) 0 0
\(449\) −22.3569 −1.05509 −0.527543 0.849528i \(-0.676887\pi\)
−0.527543 + 0.849528i \(0.676887\pi\)
\(450\) 0 0
\(451\) −25.7032 −1.21032
\(452\) 0 0
\(453\) −8.55080 −0.401751
\(454\) 0 0
\(455\) −2.07167 −0.0971213
\(456\) 0 0
\(457\) −3.14429 −0.147084 −0.0735418 0.997292i \(-0.523430\pi\)
−0.0735418 + 0.997292i \(0.523430\pi\)
\(458\) 0 0
\(459\) −4.94344 −0.230740
\(460\) 0 0
\(461\) 22.0935 1.02900 0.514499 0.857491i \(-0.327978\pi\)
0.514499 + 0.857491i \(0.327978\pi\)
\(462\) 0 0
\(463\) 20.2259 0.939976 0.469988 0.882673i \(-0.344258\pi\)
0.469988 + 0.882673i \(0.344258\pi\)
\(464\) 0 0
\(465\) 3.42808 0.158973
\(466\) 0 0
\(467\) 40.3656 1.86790 0.933948 0.357410i \(-0.116340\pi\)
0.933948 + 0.357410i \(0.116340\pi\)
\(468\) 0 0
\(469\) −6.98566 −0.322568
\(470\) 0 0
\(471\) −5.41258 −0.249399
\(472\) 0 0
\(473\) −67.7073 −3.11319
\(474\) 0 0
\(475\) 5.98999 0.274839
\(476\) 0 0
\(477\) 5.05684 0.231537
\(478\) 0 0
\(479\) 36.9367 1.68768 0.843841 0.536594i \(-0.180289\pi\)
0.843841 + 0.536594i \(0.180289\pi\)
\(480\) 0 0
\(481\) 1.19423 0.0544522
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 0.540199 0.0245292
\(486\) 0 0
\(487\) 30.0616 1.36222 0.681111 0.732180i \(-0.261497\pi\)
0.681111 + 0.732180i \(0.261497\pi\)
\(488\) 0 0
\(489\) 21.1426 0.956103
\(490\) 0 0
\(491\) 1.88374 0.0850119 0.0425059 0.999096i \(-0.486466\pi\)
0.0425059 + 0.999096i \(0.486466\pi\)
\(492\) 0 0
\(493\) −3.31889 −0.149475
\(494\) 0 0
\(495\) −7.64426 −0.343584
\(496\) 0 0
\(497\) 9.62041 0.431534
\(498\) 0 0
\(499\) −22.0103 −0.985316 −0.492658 0.870223i \(-0.663975\pi\)
−0.492658 + 0.870223i \(0.663975\pi\)
\(500\) 0 0
\(501\) −2.35602 −0.105259
\(502\) 0 0
\(503\) 41.7473 1.86142 0.930711 0.365757i \(-0.119190\pi\)
0.930711 + 0.365757i \(0.119190\pi\)
\(504\) 0 0
\(505\) 13.2385 0.589103
\(506\) 0 0
\(507\) 9.83591 0.436828
\(508\) 0 0
\(509\) 32.2029 1.42737 0.713684 0.700468i \(-0.247025\pi\)
0.713684 + 0.700468i \(0.247025\pi\)
\(510\) 0 0
\(511\) 9.47894 0.419323
\(512\) 0 0
\(513\) 1.64398 0.0725835
\(514\) 0 0
\(515\) 17.8758 0.787701
\(516\) 0 0
\(517\) 57.8366 2.54365
\(518\) 0 0
\(519\) −1.85438 −0.0813984
\(520\) 0 0
\(521\) 42.3487 1.85533 0.927665 0.373415i \(-0.121813\pi\)
0.927665 + 0.373415i \(0.121813\pi\)
\(522\) 0 0
\(523\) 30.1185 1.31699 0.658494 0.752586i \(-0.271194\pi\)
0.658494 + 0.752586i \(0.271194\pi\)
\(524\) 0 0
\(525\) 3.64359 0.159019
\(526\) 0 0
\(527\) −14.5507 −0.633839
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.05684 0.132656
\(532\) 0 0
\(533\) −6.96582 −0.301723
\(534\) 0 0
\(535\) −17.8088 −0.769942
\(536\) 0 0
\(537\) −6.40490 −0.276392
\(538\) 0 0
\(539\) −6.56356 −0.282713
\(540\) 0 0
\(541\) 17.6549 0.759042 0.379521 0.925183i \(-0.376089\pi\)
0.379521 + 0.925183i \(0.376089\pi\)
\(542\) 0 0
\(543\) 12.9354 0.555110
\(544\) 0 0
\(545\) 20.4743 0.877021
\(546\) 0 0
\(547\) 9.31226 0.398164 0.199082 0.979983i \(-0.436204\pi\)
0.199082 + 0.979983i \(0.436204\pi\)
\(548\) 0 0
\(549\) −12.2078 −0.521017
\(550\) 0 0
\(551\) 1.10372 0.0470202
\(552\) 0 0
\(553\) 9.48029 0.403143
\(554\) 0 0
\(555\) 0.781914 0.0331904
\(556\) 0 0
\(557\) −18.8575 −0.799018 −0.399509 0.916729i \(-0.630819\pi\)
−0.399509 + 0.916729i \(0.630819\pi\)
\(558\) 0 0
\(559\) −18.3493 −0.776094
\(560\) 0 0
\(561\) 32.4466 1.36990
\(562\) 0 0
\(563\) −11.5942 −0.488636 −0.244318 0.969695i \(-0.578564\pi\)
−0.244318 + 0.969695i \(0.578564\pi\)
\(564\) 0 0
\(565\) 9.65546 0.406208
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 27.8808 1.16882 0.584412 0.811457i \(-0.301325\pi\)
0.584412 + 0.811457i \(0.301325\pi\)
\(570\) 0 0
\(571\) 18.1923 0.761325 0.380663 0.924714i \(-0.375696\pi\)
0.380663 + 0.924714i \(0.375696\pi\)
\(572\) 0 0
\(573\) 1.87403 0.0782888
\(574\) 0 0
\(575\) 3.64359 0.151948
\(576\) 0 0
\(577\) −29.0990 −1.21141 −0.605704 0.795690i \(-0.707108\pi\)
−0.605704 + 0.795690i \(0.707108\pi\)
\(578\) 0 0
\(579\) −24.9062 −1.03507
\(580\) 0 0
\(581\) 0.614137 0.0254787
\(582\) 0 0
\(583\) −33.1909 −1.37463
\(584\) 0 0
\(585\) −2.07167 −0.0856529
\(586\) 0 0
\(587\) −10.1911 −0.420632 −0.210316 0.977633i \(-0.567449\pi\)
−0.210316 + 0.977633i \(0.567449\pi\)
\(588\) 0 0
\(589\) 4.83896 0.199386
\(590\) 0 0
\(591\) 1.41396 0.0581627
\(592\) 0 0
\(593\) 10.4629 0.429660 0.214830 0.976651i \(-0.431080\pi\)
0.214830 + 0.976651i \(0.431080\pi\)
\(594\) 0 0
\(595\) 5.75738 0.236030
\(596\) 0 0
\(597\) −2.92556 −0.119735
\(598\) 0 0
\(599\) 17.0401 0.696240 0.348120 0.937450i \(-0.386820\pi\)
0.348120 + 0.937450i \(0.386820\pi\)
\(600\) 0 0
\(601\) 24.0270 0.980082 0.490041 0.871699i \(-0.336982\pi\)
0.490041 + 0.871699i \(0.336982\pi\)
\(602\) 0 0
\(603\) −6.98566 −0.284478
\(604\) 0 0
\(605\) 37.3624 1.51900
\(606\) 0 0
\(607\) 17.2171 0.698819 0.349409 0.936970i \(-0.386382\pi\)
0.349409 + 0.936970i \(0.386382\pi\)
\(608\) 0 0
\(609\) 0.671372 0.0272054
\(610\) 0 0
\(611\) 15.6743 0.634113
\(612\) 0 0
\(613\) −3.65928 −0.147797 −0.0738985 0.997266i \(-0.523544\pi\)
−0.0738985 + 0.997266i \(0.523544\pi\)
\(614\) 0 0
\(615\) −4.56083 −0.183910
\(616\) 0 0
\(617\) 15.5059 0.624245 0.312123 0.950042i \(-0.398960\pi\)
0.312123 + 0.950042i \(0.398960\pi\)
\(618\) 0 0
\(619\) 27.4731 1.10424 0.552119 0.833765i \(-0.313819\pi\)
0.552119 + 0.833765i \(0.313819\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −1.08356 −0.0434119
\(624\) 0 0
\(625\) 6.49367 0.259747
\(626\) 0 0
\(627\) −10.7904 −0.430926
\(628\) 0 0
\(629\) −3.31889 −0.132333
\(630\) 0 0
\(631\) −16.1579 −0.643237 −0.321618 0.946869i \(-0.604227\pi\)
−0.321618 + 0.946869i \(0.604227\pi\)
\(632\) 0 0
\(633\) −6.56356 −0.260878
\(634\) 0 0
\(635\) 6.22405 0.246994
\(636\) 0 0
\(637\) −1.77879 −0.0704782
\(638\) 0 0
\(639\) 9.62041 0.380577
\(640\) 0 0
\(641\) −15.1536 −0.598533 −0.299266 0.954170i \(-0.596742\pi\)
−0.299266 + 0.954170i \(0.596742\pi\)
\(642\) 0 0
\(643\) 16.0551 0.633150 0.316575 0.948567i \(-0.397467\pi\)
0.316575 + 0.948567i \(0.397467\pi\)
\(644\) 0 0
\(645\) −12.0141 −0.473055
\(646\) 0 0
\(647\) 35.5239 1.39659 0.698293 0.715812i \(-0.253943\pi\)
0.698293 + 0.715812i \(0.253943\pi\)
\(648\) 0 0
\(649\) −20.0638 −0.787572
\(650\) 0 0
\(651\) 2.94344 0.115362
\(652\) 0 0
\(653\) 15.5511 0.608560 0.304280 0.952583i \(-0.401584\pi\)
0.304280 + 0.952583i \(0.401584\pi\)
\(654\) 0 0
\(655\) −9.85419 −0.385035
\(656\) 0 0
\(657\) 9.47894 0.369809
\(658\) 0 0
\(659\) −33.5364 −1.30639 −0.653197 0.757188i \(-0.726573\pi\)
−0.653197 + 0.757188i \(0.726573\pi\)
\(660\) 0 0
\(661\) 30.6772 1.19321 0.596603 0.802536i \(-0.296517\pi\)
0.596603 + 0.802536i \(0.296517\pi\)
\(662\) 0 0
\(663\) 8.79333 0.341505
\(664\) 0 0
\(665\) −1.91466 −0.0742475
\(666\) 0 0
\(667\) 0.671372 0.0259956
\(668\) 0 0
\(669\) 14.5652 0.563123
\(670\) 0 0
\(671\) 80.1268 3.09326
\(672\) 0 0
\(673\) −2.11360 −0.0814734 −0.0407367 0.999170i \(-0.512970\pi\)
−0.0407367 + 0.999170i \(0.512970\pi\)
\(674\) 0 0
\(675\) 3.64359 0.140242
\(676\) 0 0
\(677\) −37.0481 −1.42388 −0.711938 0.702242i \(-0.752182\pi\)
−0.711938 + 0.702242i \(0.752182\pi\)
\(678\) 0 0
\(679\) 0.463829 0.0178001
\(680\) 0 0
\(681\) −18.8153 −0.721004
\(682\) 0 0
\(683\) −4.12692 −0.157912 −0.0789561 0.996878i \(-0.525159\pi\)
−0.0789561 + 0.996878i \(0.525159\pi\)
\(684\) 0 0
\(685\) −6.12466 −0.234011
\(686\) 0 0
\(687\) −9.18899 −0.350582
\(688\) 0 0
\(689\) −8.99505 −0.342684
\(690\) 0 0
\(691\) 22.4364 0.853520 0.426760 0.904365i \(-0.359655\pi\)
0.426760 + 0.904365i \(0.359655\pi\)
\(692\) 0 0
\(693\) −6.56356 −0.249329
\(694\) 0 0
\(695\) 6.94420 0.263409
\(696\) 0 0
\(697\) 19.3587 0.733265
\(698\) 0 0
\(699\) 29.8388 1.12861
\(700\) 0 0
\(701\) −47.2272 −1.78375 −0.891874 0.452284i \(-0.850609\pi\)
−0.891874 + 0.452284i \(0.850609\pi\)
\(702\) 0 0
\(703\) 1.10372 0.0416277
\(704\) 0 0
\(705\) 10.2626 0.386514
\(706\) 0 0
\(707\) 11.3669 0.427496
\(708\) 0 0
\(709\) 0.595776 0.0223748 0.0111874 0.999937i \(-0.496439\pi\)
0.0111874 + 0.999937i \(0.496439\pi\)
\(710\) 0 0
\(711\) 9.48029 0.355538
\(712\) 0 0
\(713\) 2.94344 0.110233
\(714\) 0 0
\(715\) 13.5975 0.508519
\(716\) 0 0
\(717\) 8.87254 0.331351
\(718\) 0 0
\(719\) −22.2121 −0.828370 −0.414185 0.910193i \(-0.635933\pi\)
−0.414185 + 0.910193i \(0.635933\pi\)
\(720\) 0 0
\(721\) 15.3486 0.571612
\(722\) 0 0
\(723\) −29.5203 −1.09787
\(724\) 0 0
\(725\) 2.44620 0.0908497
\(726\) 0 0
\(727\) 21.6161 0.801696 0.400848 0.916144i \(-0.368715\pi\)
0.400848 + 0.916144i \(0.368715\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.9947 1.88611
\(732\) 0 0
\(733\) −28.8306 −1.06488 −0.532442 0.846467i \(-0.678725\pi\)
−0.532442 + 0.846467i \(0.678725\pi\)
\(734\) 0 0
\(735\) −1.16465 −0.0429588
\(736\) 0 0
\(737\) 45.8508 1.68894
\(738\) 0 0
\(739\) 52.5595 1.93343 0.966716 0.255853i \(-0.0823562\pi\)
0.966716 + 0.255853i \(0.0823562\pi\)
\(740\) 0 0
\(741\) −2.92429 −0.107427
\(742\) 0 0
\(743\) −5.00834 −0.183738 −0.0918691 0.995771i \(-0.529284\pi\)
−0.0918691 + 0.995771i \(0.529284\pi\)
\(744\) 0 0
\(745\) 12.2751 0.449724
\(746\) 0 0
\(747\) 0.614137 0.0224701
\(748\) 0 0
\(749\) −15.2911 −0.558725
\(750\) 0 0
\(751\) 1.62343 0.0592399 0.0296199 0.999561i \(-0.490570\pi\)
0.0296199 + 0.999561i \(0.490570\pi\)
\(752\) 0 0
\(753\) 13.7833 0.502291
\(754\) 0 0
\(755\) 9.95870 0.362434
\(756\) 0 0
\(757\) 19.8414 0.721147 0.360574 0.932731i \(-0.382581\pi\)
0.360574 + 0.932731i \(0.382581\pi\)
\(758\) 0 0
\(759\) −6.56356 −0.238242
\(760\) 0 0
\(761\) −25.9904 −0.942152 −0.471076 0.882093i \(-0.656134\pi\)
−0.471076 + 0.882093i \(0.656134\pi\)
\(762\) 0 0
\(763\) 17.5797 0.636429
\(764\) 0 0
\(765\) 5.75738 0.208159
\(766\) 0 0
\(767\) −5.43748 −0.196336
\(768\) 0 0
\(769\) 50.8662 1.83428 0.917142 0.398561i \(-0.130490\pi\)
0.917142 + 0.398561i \(0.130490\pi\)
\(770\) 0 0
\(771\) 17.2531 0.621355
\(772\) 0 0
\(773\) −41.0843 −1.47770 −0.738850 0.673870i \(-0.764631\pi\)
−0.738850 + 0.673870i \(0.764631\pi\)
\(774\) 0 0
\(775\) 10.7247 0.385242
\(776\) 0 0
\(777\) 0.671372 0.0240853
\(778\) 0 0
\(779\) −6.43791 −0.230662
\(780\) 0 0
\(781\) −63.1441 −2.25947
\(782\) 0 0
\(783\) 0.671372 0.0239929
\(784\) 0 0
\(785\) 6.30377 0.224991
\(786\) 0 0
\(787\) −7.53400 −0.268558 −0.134279 0.990944i \(-0.542872\pi\)
−0.134279 + 0.990944i \(0.542872\pi\)
\(788\) 0 0
\(789\) −22.7629 −0.810382
\(790\) 0 0
\(791\) 8.29043 0.294774
\(792\) 0 0
\(793\) 21.7151 0.771127
\(794\) 0 0
\(795\) −5.88946 −0.208878
\(796\) 0 0
\(797\) 15.8202 0.560381 0.280190 0.959944i \(-0.409602\pi\)
0.280190 + 0.959944i \(0.409602\pi\)
\(798\) 0 0
\(799\) −43.5605 −1.54106
\(800\) 0 0
\(801\) −1.08356 −0.0382857
\(802\) 0 0
\(803\) −62.2156 −2.19554
\(804\) 0 0
\(805\) −1.16465 −0.0410486
\(806\) 0 0
\(807\) 4.80332 0.169085
\(808\) 0 0
\(809\) −17.7335 −0.623478 −0.311739 0.950168i \(-0.600911\pi\)
−0.311739 + 0.950168i \(0.600911\pi\)
\(810\) 0 0
\(811\) 8.01688 0.281511 0.140755 0.990044i \(-0.455047\pi\)
0.140755 + 0.990044i \(0.455047\pi\)
\(812\) 0 0
\(813\) −11.8544 −0.415751
\(814\) 0 0
\(815\) −24.6238 −0.862534
\(816\) 0 0
\(817\) −16.9587 −0.593310
\(818\) 0 0
\(819\) −1.77879 −0.0621559
\(820\) 0 0
\(821\) −20.9584 −0.731454 −0.365727 0.930722i \(-0.619180\pi\)
−0.365727 + 0.930722i \(0.619180\pi\)
\(822\) 0 0
\(823\) 45.0419 1.57006 0.785032 0.619456i \(-0.212647\pi\)
0.785032 + 0.619456i \(0.212647\pi\)
\(824\) 0 0
\(825\) −23.9149 −0.832611
\(826\) 0 0
\(827\) 12.1066 0.420987 0.210494 0.977595i \(-0.432493\pi\)
0.210494 + 0.977595i \(0.432493\pi\)
\(828\) 0 0
\(829\) −27.8045 −0.965688 −0.482844 0.875706i \(-0.660396\pi\)
−0.482844 + 0.875706i \(0.660396\pi\)
\(830\) 0 0
\(831\) −23.7654 −0.824413
\(832\) 0 0
\(833\) 4.94344 0.171280
\(834\) 0 0
\(835\) 2.74394 0.0949580
\(836\) 0 0
\(837\) 2.94344 0.101740
\(838\) 0 0
\(839\) 23.3694 0.806802 0.403401 0.915023i \(-0.367828\pi\)
0.403401 + 0.915023i \(0.367828\pi\)
\(840\) 0 0
\(841\) −28.5493 −0.984457
\(842\) 0 0
\(843\) −18.5534 −0.639014
\(844\) 0 0
\(845\) −11.4554 −0.394078
\(846\) 0 0
\(847\) 32.0804 1.10229
\(848\) 0 0
\(849\) −29.0913 −0.998410
\(850\) 0 0
\(851\) 0.671372 0.0230143
\(852\) 0 0
\(853\) −16.7158 −0.572339 −0.286170 0.958179i \(-0.592382\pi\)
−0.286170 + 0.958179i \(0.592382\pi\)
\(854\) 0 0
\(855\) −1.91466 −0.0654801
\(856\) 0 0
\(857\) −18.2207 −0.622408 −0.311204 0.950343i \(-0.600732\pi\)
−0.311204 + 0.950343i \(0.600732\pi\)
\(858\) 0 0
\(859\) 9.83298 0.335497 0.167748 0.985830i \(-0.446350\pi\)
0.167748 + 0.985830i \(0.446350\pi\)
\(860\) 0 0
\(861\) −3.91605 −0.133459
\(862\) 0 0
\(863\) 0.648383 0.0220712 0.0110356 0.999939i \(-0.496487\pi\)
0.0110356 + 0.999939i \(0.496487\pi\)
\(864\) 0 0
\(865\) 2.15971 0.0734324
\(866\) 0 0
\(867\) −7.43759 −0.252594
\(868\) 0 0
\(869\) −62.2245 −2.11082
\(870\) 0 0
\(871\) 12.4260 0.421039
\(872\) 0 0
\(873\) 0.463829 0.0156982
\(874\) 0 0
\(875\) −10.0668 −0.340319
\(876\) 0 0
\(877\) 51.2854 1.73179 0.865893 0.500229i \(-0.166751\pi\)
0.865893 + 0.500229i \(0.166751\pi\)
\(878\) 0 0
\(879\) 23.5422 0.794059
\(880\) 0 0
\(881\) −15.9911 −0.538755 −0.269378 0.963035i \(-0.586818\pi\)
−0.269378 + 0.963035i \(0.586818\pi\)
\(882\) 0 0
\(883\) 26.8120 0.902294 0.451147 0.892450i \(-0.351015\pi\)
0.451147 + 0.892450i \(0.351015\pi\)
\(884\) 0 0
\(885\) −3.56016 −0.119673
\(886\) 0 0
\(887\) 52.6201 1.76681 0.883404 0.468612i \(-0.155246\pi\)
0.883404 + 0.468612i \(0.155246\pi\)
\(888\) 0 0
\(889\) 5.34413 0.179236
\(890\) 0 0
\(891\) −6.56356 −0.219888
\(892\) 0 0
\(893\) 14.4864 0.484768
\(894\) 0 0
\(895\) 7.45947 0.249343
\(896\) 0 0
\(897\) −1.77879 −0.0593920
\(898\) 0 0
\(899\) 1.97614 0.0659081
\(900\) 0 0
\(901\) 24.9982 0.832811
\(902\) 0 0
\(903\) −10.3156 −0.343283
\(904\) 0 0
\(905\) −15.0652 −0.500784
\(906\) 0 0
\(907\) 11.3547 0.377028 0.188514 0.982071i \(-0.439633\pi\)
0.188514 + 0.982071i \(0.439633\pi\)
\(908\) 0 0
\(909\) 11.3669 0.377016
\(910\) 0 0
\(911\) −56.2394 −1.86329 −0.931646 0.363366i \(-0.881627\pi\)
−0.931646 + 0.363366i \(0.881627\pi\)
\(912\) 0 0
\(913\) −4.03093 −0.133404
\(914\) 0 0
\(915\) 14.2179 0.470028
\(916\) 0 0
\(917\) −8.46107 −0.279409
\(918\) 0 0
\(919\) −45.2603 −1.49300 −0.746500 0.665385i \(-0.768267\pi\)
−0.746500 + 0.665385i \(0.768267\pi\)
\(920\) 0 0
\(921\) −9.08511 −0.299365
\(922\) 0 0
\(923\) −17.1127 −0.563270
\(924\) 0 0
\(925\) 2.44620 0.0804307
\(926\) 0 0
\(927\) 15.3486 0.504115
\(928\) 0 0
\(929\) −50.0431 −1.64186 −0.820930 0.571028i \(-0.806545\pi\)
−0.820930 + 0.571028i \(0.806545\pi\)
\(930\) 0 0
\(931\) −1.64398 −0.0538793
\(932\) 0 0
\(933\) −6.98628 −0.228720
\(934\) 0 0
\(935\) −37.7889 −1.23583
\(936\) 0 0
\(937\) 30.4113 0.993493 0.496746 0.867896i \(-0.334528\pi\)
0.496746 + 0.867896i \(0.334528\pi\)
\(938\) 0 0
\(939\) 34.9157 1.13943
\(940\) 0 0
\(941\) −27.8445 −0.907706 −0.453853 0.891077i \(-0.649951\pi\)
−0.453853 + 0.891077i \(0.649951\pi\)
\(942\) 0 0
\(943\) −3.91605 −0.127524
\(944\) 0 0
\(945\) −1.16465 −0.0378861
\(946\) 0 0
\(947\) −4.96761 −0.161426 −0.0807129 0.996737i \(-0.525720\pi\)
−0.0807129 + 0.996737i \(0.525720\pi\)
\(948\) 0 0
\(949\) −16.8610 −0.547332
\(950\) 0 0
\(951\) −4.29043 −0.139127
\(952\) 0 0
\(953\) −37.1476 −1.20333 −0.601664 0.798749i \(-0.705495\pi\)
−0.601664 + 0.798749i \(0.705495\pi\)
\(954\) 0 0
\(955\) −2.18259 −0.0706270
\(956\) 0 0
\(957\) −4.40659 −0.142445
\(958\) 0 0
\(959\) −5.25879 −0.169815
\(960\) 0 0
\(961\) −22.3362 −0.720521
\(962\) 0 0
\(963\) −15.2911 −0.492749
\(964\) 0 0
\(965\) 29.0070 0.933769
\(966\) 0 0
\(967\) −30.9579 −0.995538 −0.497769 0.867310i \(-0.665847\pi\)
−0.497769 + 0.867310i \(0.665847\pi\)
\(968\) 0 0
\(969\) 8.12692 0.261074
\(970\) 0 0
\(971\) −29.6060 −0.950103 −0.475052 0.879958i \(-0.657571\pi\)
−0.475052 + 0.879958i \(0.657571\pi\)
\(972\) 0 0
\(973\) 5.96248 0.191148
\(974\) 0 0
\(975\) −6.48117 −0.207564
\(976\) 0 0
\(977\) −19.2597 −0.616172 −0.308086 0.951358i \(-0.599688\pi\)
−0.308086 + 0.951358i \(0.599688\pi\)
\(978\) 0 0
\(979\) 7.11201 0.227301
\(980\) 0 0
\(981\) 17.5797 0.561278
\(982\) 0 0
\(983\) 47.8490 1.52615 0.763073 0.646312i \(-0.223690\pi\)
0.763073 + 0.646312i \(0.223690\pi\)
\(984\) 0 0
\(985\) −1.64678 −0.0524706
\(986\) 0 0
\(987\) 8.81177 0.280482
\(988\) 0 0
\(989\) −10.3156 −0.328018
\(990\) 0 0
\(991\) −27.6629 −0.878741 −0.439371 0.898306i \(-0.644798\pi\)
−0.439371 + 0.898306i \(0.644798\pi\)
\(992\) 0 0
\(993\) −18.0319 −0.572224
\(994\) 0 0
\(995\) 3.40726 0.108017
\(996\) 0 0
\(997\) −11.7434 −0.371917 −0.185959 0.982558i \(-0.559539\pi\)
−0.185959 + 0.982558i \(0.559539\pi\)
\(998\) 0 0
\(999\) 0.671372 0.0212413
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.cg.1.4 6
4.3 odd 2 3864.2.a.x.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.x.1.4 6 4.3 odd 2
7728.2.a.cg.1.4 6 1.1 even 1 trivial