Properties

Label 7728.2.a.br.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.43163\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.43163 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.43163 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.95044 q^{13} -1.43163 q^{15} +4.51882 q^{17} -3.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -2.95044 q^{25} -1.00000 q^{27} -10.4197 q^{29} -0.344438 q^{31} +1.00000 q^{33} +1.43163 q^{35} -7.55645 q^{37} -3.95044 q^{39} +5.17438 q^{41} -7.08719 q^{43} +1.43163 q^{45} -7.51882 q^{47} +1.00000 q^{49} -4.51882 q^{51} -4.95044 q^{53} -1.43163 q^{55} +3.00000 q^{57} -4.77606 q^{59} +4.60601 q^{61} +1.00000 q^{63} +5.65556 q^{65} -10.4693 q^{67} +1.00000 q^{69} -12.1248 q^{71} +0.518817 q^{73} +2.95044 q^{75} -1.00000 q^{77} -1.38207 q^{79} +1.00000 q^{81} +6.41970 q^{83} +6.46926 q^{85} +10.4197 q^{87} -2.81370 q^{89} +3.95044 q^{91} +0.344438 q^{93} -4.29488 q^{95} -6.51882 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - q^{15} + 4 q^{17} - 9 q^{19} - 3 q^{21} - 3 q^{23} + 4 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 3 q^{33} + q^{35} + 6 q^{37} + q^{39} + 3 q^{41} - 15 q^{43} + q^{45} - 13 q^{47} + 3 q^{49} - 4 q^{51} - 2 q^{53} - q^{55} + 9 q^{57} - 14 q^{59} - 2 q^{61} + 3 q^{63} + 14 q^{65} - 9 q^{67} + 3 q^{69} - 11 q^{71} - 8 q^{73} - 4 q^{75} - 3 q^{77} + 12 q^{79} + 3 q^{81} - 16 q^{83} - 3 q^{85} - 4 q^{87} + 11 q^{89} - q^{91} + 4 q^{93} - 3 q^{95} - 10 q^{97} - 3 q^{99} + O(q^{100}) \)

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
\(5\) 1.43163 0.640243 0.320122 0.947376i \(-0.396276\pi\)
0.320122 + 0.947376i \(0.396276\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 3.95044 1.09566 0.547828 0.836591i \(-0.315455\pi\)
0.547828 + 0.836591i \(0.315455\pi\)
\(14\) 0 0
\(15\) −1.43163 −0.369645
\(16\) 0 0
\(17\) 4.51882 1.09597 0.547987 0.836487i \(-0.315394\pi\)
0.547987 + 0.836487i \(0.315394\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.95044 −0.590089
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.4197 −1.93489 −0.967445 0.253080i \(-0.918556\pi\)
−0.967445 + 0.253080i \(0.918556\pi\)
\(30\) 0 0
\(31\) −0.344438 −0.0618628 −0.0309314 0.999522i \(-0.509847\pi\)
−0.0309314 + 0.999522i \(0.509847\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 1.43163 0.241989
\(36\) 0 0
\(37\) −7.55645 −1.24227 −0.621136 0.783703i \(-0.713329\pi\)
−0.621136 + 0.783703i \(0.713329\pi\)
\(38\) 0 0
\(39\) −3.95044 −0.632577
\(40\) 0 0
\(41\) 5.17438 0.808102 0.404051 0.914736i \(-0.367602\pi\)
0.404051 + 0.914736i \(0.367602\pi\)
\(42\) 0 0
\(43\) −7.08719 −1.08079 −0.540393 0.841413i \(-0.681724\pi\)
−0.540393 + 0.841413i \(0.681724\pi\)
\(44\) 0 0
\(45\) 1.43163 0.213414
\(46\) 0 0
\(47\) −7.51882 −1.09673 −0.548366 0.836238i \(-0.684750\pi\)
−0.548366 + 0.836238i \(0.684750\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.51882 −0.632761
\(52\) 0 0
\(53\) −4.95044 −0.679996 −0.339998 0.940426i \(-0.610426\pi\)
−0.339998 + 0.940426i \(0.610426\pi\)
\(54\) 0 0
\(55\) −1.43163 −0.193041
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −4.77606 −0.621791 −0.310895 0.950444i \(-0.600629\pi\)
−0.310895 + 0.950444i \(0.600629\pi\)
\(60\) 0 0
\(61\) 4.60601 0.589739 0.294869 0.955538i \(-0.404724\pi\)
0.294869 + 0.955538i \(0.404724\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 5.65556 0.701486
\(66\) 0 0
\(67\) −10.4693 −1.27902 −0.639512 0.768781i \(-0.720864\pi\)
−0.639512 + 0.768781i \(0.720864\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −12.1248 −1.43895 −0.719476 0.694517i \(-0.755618\pi\)
−0.719476 + 0.694517i \(0.755618\pi\)
\(72\) 0 0
\(73\) 0.518817 0.0607229 0.0303615 0.999539i \(-0.490334\pi\)
0.0303615 + 0.999539i \(0.490334\pi\)
\(74\) 0 0
\(75\) 2.95044 0.340688
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −1.38207 −0.155495 −0.0777476 0.996973i \(-0.524773\pi\)
−0.0777476 + 0.996973i \(0.524773\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.41970 0.704654 0.352327 0.935877i \(-0.385390\pi\)
0.352327 + 0.935877i \(0.385390\pi\)
\(84\) 0 0
\(85\) 6.46926 0.701690
\(86\) 0 0
\(87\) 10.4197 1.11711
\(88\) 0 0
\(89\) −2.81370 −0.298251 −0.149126 0.988818i \(-0.547646\pi\)
−0.149126 + 0.988818i \(0.547646\pi\)
\(90\) 0 0
\(91\) 3.95044 0.414119
\(92\) 0 0
\(93\) 0.344438 0.0357165
\(94\) 0 0
\(95\) −4.29488 −0.440646
\(96\) 0 0
\(97\) −6.51882 −0.661886 −0.330943 0.943651i \(-0.607367\pi\)
−0.330943 + 0.943651i \(0.607367\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 12.8513 1.27876 0.639378 0.768893i \(-0.279192\pi\)
0.639378 + 0.768893i \(0.279192\pi\)
\(102\) 0 0
\(103\) 11.4197 1.12522 0.562608 0.826723i \(-0.309798\pi\)
0.562608 + 0.826723i \(0.309798\pi\)
\(104\) 0 0
\(105\) −1.43163 −0.139713
\(106\) 0 0
\(107\) 1.95044 0.188557 0.0942783 0.995546i \(-0.469946\pi\)
0.0942783 + 0.995546i \(0.469946\pi\)
\(108\) 0 0
\(109\) −7.15814 −0.685625 −0.342813 0.939404i \(-0.611380\pi\)
−0.342813 + 0.939404i \(0.611380\pi\)
\(110\) 0 0
\(111\) 7.55645 0.717227
\(112\) 0 0
\(113\) −0.469261 −0.0441443 −0.0220722 0.999756i \(-0.507026\pi\)
−0.0220722 + 0.999756i \(0.507026\pi\)
\(114\) 0 0
\(115\) −1.43163 −0.133500
\(116\) 0 0
\(117\) 3.95044 0.365219
\(118\) 0 0
\(119\) 4.51882 0.414239
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −5.17438 −0.466558
\(124\) 0 0
\(125\) −11.3821 −1.01804
\(126\) 0 0
\(127\) −2.70512 −0.240040 −0.120020 0.992771i \(-0.538296\pi\)
−0.120020 + 0.992771i \(0.538296\pi\)
\(128\) 0 0
\(129\) 7.08719 0.623992
\(130\) 0 0
\(131\) −10.9009 −0.952415 −0.476207 0.879333i \(-0.657989\pi\)
−0.476207 + 0.879333i \(0.657989\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) −1.43163 −0.123215
\(136\) 0 0
\(137\) 8.31112 0.710067 0.355034 0.934854i \(-0.384469\pi\)
0.355034 + 0.934854i \(0.384469\pi\)
\(138\) 0 0
\(139\) −16.5445 −1.40329 −0.701644 0.712527i \(-0.747550\pi\)
−0.701644 + 0.712527i \(0.747550\pi\)
\(140\) 0 0
\(141\) 7.51882 0.633199
\(142\) 0 0
\(143\) −3.95044 −0.330353
\(144\) 0 0
\(145\) −14.9171 −1.23880
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 13.5941 1.11367 0.556835 0.830623i \(-0.312015\pi\)
0.556835 + 0.830623i \(0.312015\pi\)
\(150\) 0 0
\(151\) 10.5565 0.859072 0.429536 0.903050i \(-0.358677\pi\)
0.429536 + 0.903050i \(0.358677\pi\)
\(152\) 0 0
\(153\) 4.51882 0.365325
\(154\) 0 0
\(155\) −0.493106 −0.0396072
\(156\) 0 0
\(157\) 3.24533 0.259005 0.129503 0.991579i \(-0.458662\pi\)
0.129503 + 0.991579i \(0.458662\pi\)
\(158\) 0 0
\(159\) 4.95044 0.392596
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 17.1248 1.34132 0.670660 0.741765i \(-0.266011\pi\)
0.670660 + 0.741765i \(0.266011\pi\)
\(164\) 0 0
\(165\) 1.43163 0.111452
\(166\) 0 0
\(167\) 7.86325 0.608477 0.304238 0.952596i \(-0.401598\pi\)
0.304238 + 0.952596i \(0.401598\pi\)
\(168\) 0 0
\(169\) 2.60601 0.200462
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 0 0
\(173\) 3.20769 0.243876 0.121938 0.992538i \(-0.461089\pi\)
0.121938 + 0.992538i \(0.461089\pi\)
\(174\) 0 0
\(175\) −2.95044 −0.223033
\(176\) 0 0
\(177\) 4.77606 0.358991
\(178\) 0 0
\(179\) 16.6436 1.24400 0.622002 0.783016i \(-0.286320\pi\)
0.622002 + 0.783016i \(0.286320\pi\)
\(180\) 0 0
\(181\) −4.79231 −0.356209 −0.178105 0.984012i \(-0.556997\pi\)
−0.178105 + 0.984012i \(0.556997\pi\)
\(182\) 0 0
\(183\) −4.60601 −0.340486
\(184\) 0 0
\(185\) −10.8180 −0.795357
\(186\) 0 0
\(187\) −4.51882 −0.330449
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −5.24533 −0.379538 −0.189769 0.981829i \(-0.560774\pi\)
−0.189769 + 0.981829i \(0.560774\pi\)
\(192\) 0 0
\(193\) −18.2830 −1.31604 −0.658018 0.753002i \(-0.728605\pi\)
−0.658018 + 0.753002i \(0.728605\pi\)
\(194\) 0 0
\(195\) −5.65556 −0.405003
\(196\) 0 0
\(197\) 5.77606 0.411528 0.205764 0.978602i \(-0.434032\pi\)
0.205764 + 0.978602i \(0.434032\pi\)
\(198\) 0 0
\(199\) −9.29920 −0.659203 −0.329601 0.944120i \(-0.606914\pi\)
−0.329601 + 0.944120i \(0.606914\pi\)
\(200\) 0 0
\(201\) 10.4693 0.738445
\(202\) 0 0
\(203\) −10.4197 −0.731320
\(204\) 0 0
\(205\) 7.40778 0.517382
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 4.21201 0.289967 0.144983 0.989434i \(-0.453687\pi\)
0.144983 + 0.989434i \(0.453687\pi\)
\(212\) 0 0
\(213\) 12.1248 0.830779
\(214\) 0 0
\(215\) −10.1462 −0.691966
\(216\) 0 0
\(217\) −0.344438 −0.0233819
\(218\) 0 0
\(219\) −0.518817 −0.0350584
\(220\) 0 0
\(221\) 17.8513 1.20081
\(222\) 0 0
\(223\) 22.0967 1.47970 0.739851 0.672771i \(-0.234896\pi\)
0.739851 + 0.672771i \(0.234896\pi\)
\(224\) 0 0
\(225\) −2.95044 −0.196696
\(226\) 0 0
\(227\) −2.49311 −0.165473 −0.0827366 0.996571i \(-0.526366\pi\)
−0.0827366 + 0.996571i \(0.526366\pi\)
\(228\) 0 0
\(229\) −23.0257 −1.52158 −0.760791 0.648997i \(-0.775189\pi\)
−0.760791 + 0.648997i \(0.775189\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) −20.5445 −1.34592 −0.672958 0.739680i \(-0.734977\pi\)
−0.672958 + 0.739680i \(0.734977\pi\)
\(234\) 0 0
\(235\) −10.7641 −0.702175
\(236\) 0 0
\(237\) 1.38207 0.0897752
\(238\) 0 0
\(239\) −13.6813 −0.884968 −0.442484 0.896776i \(-0.645903\pi\)
−0.442484 + 0.896776i \(0.645903\pi\)
\(240\) 0 0
\(241\) −18.6274 −1.19990 −0.599948 0.800039i \(-0.704812\pi\)
−0.599948 + 0.800039i \(0.704812\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.43163 0.0914633
\(246\) 0 0
\(247\) −11.8513 −0.754082
\(248\) 0 0
\(249\) −6.41970 −0.406832
\(250\) 0 0
\(251\) −11.6274 −0.733915 −0.366957 0.930238i \(-0.619601\pi\)
−0.366957 + 0.930238i \(0.619601\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) 0 0
\(255\) −6.46926 −0.405121
\(256\) 0 0
\(257\) 9.24533 0.576708 0.288354 0.957524i \(-0.406892\pi\)
0.288354 + 0.957524i \(0.406892\pi\)
\(258\) 0 0
\(259\) −7.55645 −0.469535
\(260\) 0 0
\(261\) −10.4197 −0.644964
\(262\) 0 0
\(263\) 0.825621 0.0509100 0.0254550 0.999676i \(-0.491897\pi\)
0.0254550 + 0.999676i \(0.491897\pi\)
\(264\) 0 0
\(265\) −7.08719 −0.435363
\(266\) 0 0
\(267\) 2.81370 0.172196
\(268\) 0 0
\(269\) −15.2616 −0.930514 −0.465257 0.885176i \(-0.654038\pi\)
−0.465257 + 0.885176i \(0.654038\pi\)
\(270\) 0 0
\(271\) 23.9718 1.45619 0.728093 0.685479i \(-0.240407\pi\)
0.728093 + 0.685479i \(0.240407\pi\)
\(272\) 0 0
\(273\) −3.95044 −0.239092
\(274\) 0 0
\(275\) 2.95044 0.177918
\(276\) 0 0
\(277\) −5.29488 −0.318139 −0.159069 0.987267i \(-0.550849\pi\)
−0.159069 + 0.987267i \(0.550849\pi\)
\(278\) 0 0
\(279\) −0.344438 −0.0206209
\(280\) 0 0
\(281\) 16.1744 0.964883 0.482441 0.875928i \(-0.339750\pi\)
0.482441 + 0.875928i \(0.339750\pi\)
\(282\) 0 0
\(283\) −4.29488 −0.255304 −0.127652 0.991819i \(-0.540744\pi\)
−0.127652 + 0.991819i \(0.540744\pi\)
\(284\) 0 0
\(285\) 4.29488 0.254407
\(286\) 0 0
\(287\) 5.17438 0.305434
\(288\) 0 0
\(289\) 3.41970 0.201159
\(290\) 0 0
\(291\) 6.51882 0.382140
\(292\) 0 0
\(293\) −5.72651 −0.334546 −0.167273 0.985911i \(-0.553496\pi\)
−0.167273 + 0.985911i \(0.553496\pi\)
\(294\) 0 0
\(295\) −6.83754 −0.398097
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −3.95044 −0.228460
\(300\) 0 0
\(301\) −7.08719 −0.408499
\(302\) 0 0
\(303\) −12.8513 −0.738290
\(304\) 0 0
\(305\) 6.59408 0.377576
\(306\) 0 0
\(307\) 15.1838 0.866588 0.433294 0.901253i \(-0.357351\pi\)
0.433294 + 0.901253i \(0.357351\pi\)
\(308\) 0 0
\(309\) −11.4197 −0.649644
\(310\) 0 0
\(311\) −7.66749 −0.434783 −0.217392 0.976084i \(-0.569755\pi\)
−0.217392 + 0.976084i \(0.569755\pi\)
\(312\) 0 0
\(313\) 26.8061 1.51517 0.757585 0.652736i \(-0.226379\pi\)
0.757585 + 0.652736i \(0.226379\pi\)
\(314\) 0 0
\(315\) 1.43163 0.0806630
\(316\) 0 0
\(317\) 2.05388 0.115357 0.0576786 0.998335i \(-0.481630\pi\)
0.0576786 + 0.998335i \(0.481630\pi\)
\(318\) 0 0
\(319\) 10.4197 0.583391
\(320\) 0 0
\(321\) −1.95044 −0.108863
\(322\) 0 0
\(323\) −13.5565 −0.754301
\(324\) 0 0
\(325\) −11.6556 −0.646534
\(326\) 0 0
\(327\) 7.15814 0.395846
\(328\) 0 0
\(329\) −7.51882 −0.414526
\(330\) 0 0
\(331\) 28.8727 1.58699 0.793494 0.608578i \(-0.208260\pi\)
0.793494 + 0.608578i \(0.208260\pi\)
\(332\) 0 0
\(333\) −7.55645 −0.414091
\(334\) 0 0
\(335\) −14.9881 −0.818886
\(336\) 0 0
\(337\) −13.3368 −0.726504 −0.363252 0.931691i \(-0.618334\pi\)
−0.363252 + 0.931691i \(0.618334\pi\)
\(338\) 0 0
\(339\) 0.469261 0.0254867
\(340\) 0 0
\(341\) 0.344438 0.0186523
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.43163 0.0770762
\(346\) 0 0
\(347\) −21.2830 −1.14253 −0.571265 0.820766i \(-0.693547\pi\)
−0.571265 + 0.820766i \(0.693547\pi\)
\(348\) 0 0
\(349\) −31.7804 −1.70117 −0.850583 0.525842i \(-0.823750\pi\)
−0.850583 + 0.525842i \(0.823750\pi\)
\(350\) 0 0
\(351\) −3.95044 −0.210859
\(352\) 0 0
\(353\) 22.3206 1.18801 0.594003 0.804463i \(-0.297547\pi\)
0.594003 + 0.804463i \(0.297547\pi\)
\(354\) 0 0
\(355\) −17.3582 −0.921279
\(356\) 0 0
\(357\) −4.51882 −0.239161
\(358\) 0 0
\(359\) 31.0590 1.63923 0.819616 0.572913i \(-0.194187\pi\)
0.819616 + 0.572913i \(0.194187\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 0.742752 0.0388774
\(366\) 0 0
\(367\) −7.74275 −0.404168 −0.202084 0.979368i \(-0.564771\pi\)
−0.202084 + 0.979368i \(0.564771\pi\)
\(368\) 0 0
\(369\) 5.17438 0.269367
\(370\) 0 0
\(371\) −4.95044 −0.257014
\(372\) 0 0
\(373\) −15.5565 −0.805482 −0.402741 0.915314i \(-0.631943\pi\)
−0.402741 + 0.915314i \(0.631943\pi\)
\(374\) 0 0
\(375\) 11.3821 0.587768
\(376\) 0 0
\(377\) −41.1625 −2.11997
\(378\) 0 0
\(379\) −29.1129 −1.49543 −0.747715 0.664020i \(-0.768849\pi\)
−0.747715 + 0.664020i \(0.768849\pi\)
\(380\) 0 0
\(381\) 2.70512 0.138587
\(382\) 0 0
\(383\) 31.2591 1.59727 0.798633 0.601818i \(-0.205557\pi\)
0.798633 + 0.601818i \(0.205557\pi\)
\(384\) 0 0
\(385\) −1.43163 −0.0729625
\(386\) 0 0
\(387\) −7.08719 −0.360262
\(388\) 0 0
\(389\) −8.10343 −0.410860 −0.205430 0.978672i \(-0.565859\pi\)
−0.205430 + 0.978672i \(0.565859\pi\)
\(390\) 0 0
\(391\) −4.51882 −0.228526
\(392\) 0 0
\(393\) 10.9009 0.549877
\(394\) 0 0
\(395\) −1.97861 −0.0995547
\(396\) 0 0
\(397\) −1.06148 −0.0532741 −0.0266371 0.999645i \(-0.508480\pi\)
−0.0266371 + 0.999645i \(0.508480\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −25.3915 −1.26799 −0.633996 0.773336i \(-0.718587\pi\)
−0.633996 + 0.773336i \(0.718587\pi\)
\(402\) 0 0
\(403\) −1.36068 −0.0677804
\(404\) 0 0
\(405\) 1.43163 0.0711381
\(406\) 0 0
\(407\) 7.55645 0.374559
\(408\) 0 0
\(409\) −31.6556 −1.56527 −0.782633 0.622483i \(-0.786124\pi\)
−0.782633 + 0.622483i \(0.786124\pi\)
\(410\) 0 0
\(411\) −8.31112 −0.409958
\(412\) 0 0
\(413\) −4.77606 −0.235015
\(414\) 0 0
\(415\) 9.19062 0.451150
\(416\) 0 0
\(417\) 16.5445 0.810189
\(418\) 0 0
\(419\) 2.84186 0.138834 0.0694171 0.997588i \(-0.477886\pi\)
0.0694171 + 0.997588i \(0.477886\pi\)
\(420\) 0 0
\(421\) 24.5164 1.19485 0.597427 0.801923i \(-0.296190\pi\)
0.597427 + 0.801923i \(0.296190\pi\)
\(422\) 0 0
\(423\) −7.51882 −0.365577
\(424\) 0 0
\(425\) −13.3325 −0.646722
\(426\) 0 0
\(427\) 4.60601 0.222900
\(428\) 0 0
\(429\) 3.95044 0.190729
\(430\) 0 0
\(431\) 3.74275 0.180282 0.0901410 0.995929i \(-0.471268\pi\)
0.0901410 + 0.995929i \(0.471268\pi\)
\(432\) 0 0
\(433\) 32.7070 1.57180 0.785899 0.618355i \(-0.212201\pi\)
0.785899 + 0.618355i \(0.212201\pi\)
\(434\) 0 0
\(435\) 14.9171 0.715222
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 38.1462 1.82062 0.910310 0.413928i \(-0.135843\pi\)
0.910310 + 0.413928i \(0.135843\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 40.1505 1.90761 0.953805 0.300427i \(-0.0971293\pi\)
0.953805 + 0.300427i \(0.0971293\pi\)
\(444\) 0 0
\(445\) −4.02817 −0.190953
\(446\) 0 0
\(447\) −13.5941 −0.642978
\(448\) 0 0
\(449\) 23.2334 1.09645 0.548226 0.836330i \(-0.315303\pi\)
0.548226 + 0.836330i \(0.315303\pi\)
\(450\) 0 0
\(451\) −5.17438 −0.243652
\(452\) 0 0
\(453\) −10.5565 −0.495985
\(454\) 0 0
\(455\) 5.65556 0.265137
\(456\) 0 0
\(457\) 21.9975 1.02900 0.514501 0.857490i \(-0.327977\pi\)
0.514501 + 0.857490i \(0.327977\pi\)
\(458\) 0 0
\(459\) −4.51882 −0.210920
\(460\) 0 0
\(461\) −25.0590 −1.16712 −0.583558 0.812072i \(-0.698340\pi\)
−0.583558 + 0.812072i \(0.698340\pi\)
\(462\) 0 0
\(463\) −29.7641 −1.38326 −0.691628 0.722253i \(-0.743106\pi\)
−0.691628 + 0.722253i \(0.743106\pi\)
\(464\) 0 0
\(465\) 0.493106 0.0228672
\(466\) 0 0
\(467\) 3.30680 0.153021 0.0765103 0.997069i \(-0.475622\pi\)
0.0765103 + 0.997069i \(0.475622\pi\)
\(468\) 0 0
\(469\) −10.4693 −0.483426
\(470\) 0 0
\(471\) −3.24533 −0.149537
\(472\) 0 0
\(473\) 7.08719 0.325869
\(474\) 0 0
\(475\) 8.85133 0.406127
\(476\) 0 0
\(477\) −4.95044 −0.226665
\(478\) 0 0
\(479\) −31.2830 −1.42935 −0.714677 0.699454i \(-0.753426\pi\)
−0.714677 + 0.699454i \(0.753426\pi\)
\(480\) 0 0
\(481\) −29.8513 −1.36110
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −9.33251 −0.423768
\(486\) 0 0
\(487\) −21.1086 −0.956521 −0.478261 0.878218i \(-0.658733\pi\)
−0.478261 + 0.878218i \(0.658733\pi\)
\(488\) 0 0
\(489\) −17.1248 −0.774411
\(490\) 0 0
\(491\) −2.32305 −0.104838 −0.0524188 0.998625i \(-0.516693\pi\)
−0.0524188 + 0.998625i \(0.516693\pi\)
\(492\) 0 0
\(493\) −47.0847 −2.12059
\(494\) 0 0
\(495\) −1.43163 −0.0643469
\(496\) 0 0
\(497\) −12.1248 −0.543873
\(498\) 0 0
\(499\) 9.43163 0.422218 0.211109 0.977463i \(-0.432293\pi\)
0.211109 + 0.977463i \(0.432293\pi\)
\(500\) 0 0
\(501\) −7.86325 −0.351304
\(502\) 0 0
\(503\) −7.95476 −0.354685 −0.177343 0.984149i \(-0.556750\pi\)
−0.177343 + 0.984149i \(0.556750\pi\)
\(504\) 0 0
\(505\) 18.3983 0.818714
\(506\) 0 0
\(507\) −2.60601 −0.115737
\(508\) 0 0
\(509\) −4.48118 −0.198625 −0.0993125 0.995056i \(-0.531664\pi\)
−0.0993125 + 0.995056i \(0.531664\pi\)
\(510\) 0 0
\(511\) 0.518817 0.0229511
\(512\) 0 0
\(513\) 3.00000 0.132453
\(514\) 0 0
\(515\) 16.3488 0.720412
\(516\) 0 0
\(517\) 7.51882 0.330677
\(518\) 0 0
\(519\) −3.20769 −0.140802
\(520\) 0 0
\(521\) 32.4240 1.42052 0.710261 0.703938i \(-0.248577\pi\)
0.710261 + 0.703938i \(0.248577\pi\)
\(522\) 0 0
\(523\) 25.4197 1.11153 0.555763 0.831341i \(-0.312426\pi\)
0.555763 + 0.831341i \(0.312426\pi\)
\(524\) 0 0
\(525\) 2.95044 0.128768
\(526\) 0 0
\(527\) −1.55645 −0.0678000
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.77606 −0.207264
\(532\) 0 0
\(533\) 20.4411 0.885402
\(534\) 0 0
\(535\) 2.79231 0.120722
\(536\) 0 0
\(537\) −16.6436 −0.718226
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −14.0095 −0.602314 −0.301157 0.953575i \(-0.597373\pi\)
−0.301157 + 0.953575i \(0.597373\pi\)
\(542\) 0 0
\(543\) 4.79231 0.205658
\(544\) 0 0
\(545\) −10.2478 −0.438967
\(546\) 0 0
\(547\) 3.15814 0.135032 0.0675161 0.997718i \(-0.478493\pi\)
0.0675161 + 0.997718i \(0.478493\pi\)
\(548\) 0 0
\(549\) 4.60601 0.196580
\(550\) 0 0
\(551\) 31.2591 1.33168
\(552\) 0 0
\(553\) −1.38207 −0.0587716
\(554\) 0 0
\(555\) 10.8180 0.459199
\(556\) 0 0
\(557\) −18.2496 −0.773262 −0.386631 0.922234i \(-0.626361\pi\)
−0.386631 + 0.922234i \(0.626361\pi\)
\(558\) 0 0
\(559\) −27.9975 −1.18417
\(560\) 0 0
\(561\) 4.51882 0.190785
\(562\) 0 0
\(563\) −17.8513 −0.752344 −0.376172 0.926550i \(-0.622760\pi\)
−0.376172 + 0.926550i \(0.622760\pi\)
\(564\) 0 0
\(565\) −0.671806 −0.0282631
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 9.69320 0.406360 0.203180 0.979141i \(-0.434872\pi\)
0.203180 + 0.979141i \(0.434872\pi\)
\(570\) 0 0
\(571\) −39.3821 −1.64809 −0.824044 0.566526i \(-0.808287\pi\)
−0.824044 + 0.566526i \(0.808287\pi\)
\(572\) 0 0
\(573\) 5.24533 0.219127
\(574\) 0 0
\(575\) 2.95044 0.123042
\(576\) 0 0
\(577\) −17.4530 −0.726579 −0.363289 0.931676i \(-0.618346\pi\)
−0.363289 + 0.931676i \(0.618346\pi\)
\(578\) 0 0
\(579\) 18.2830 0.759814
\(580\) 0 0
\(581\) 6.41970 0.266334
\(582\) 0 0
\(583\) 4.95044 0.205026
\(584\) 0 0
\(585\) 5.65556 0.233829
\(586\) 0 0
\(587\) −33.4693 −1.38142 −0.690712 0.723130i \(-0.742703\pi\)
−0.690712 + 0.723130i \(0.742703\pi\)
\(588\) 0 0
\(589\) 1.03331 0.0425769
\(590\) 0 0
\(591\) −5.77606 −0.237596
\(592\) 0 0
\(593\) −2.09911 −0.0862002 −0.0431001 0.999071i \(-0.513723\pi\)
−0.0431001 + 0.999071i \(0.513723\pi\)
\(594\) 0 0
\(595\) 6.46926 0.265214
\(596\) 0 0
\(597\) 9.29920 0.380591
\(598\) 0 0
\(599\) 45.2805 1.85011 0.925056 0.379832i \(-0.124018\pi\)
0.925056 + 0.379832i \(0.124018\pi\)
\(600\) 0 0
\(601\) −28.9924 −1.18262 −0.591312 0.806443i \(-0.701390\pi\)
−0.591312 + 0.806443i \(0.701390\pi\)
\(602\) 0 0
\(603\) −10.4693 −0.426341
\(604\) 0 0
\(605\) −14.3163 −0.582039
\(606\) 0 0
\(607\) −2.29920 −0.0933217 −0.0466609 0.998911i \(-0.514858\pi\)
−0.0466609 + 0.998911i \(0.514858\pi\)
\(608\) 0 0
\(609\) 10.4197 0.422228
\(610\) 0 0
\(611\) −29.7027 −1.20164
\(612\) 0 0
\(613\) 19.4573 0.785874 0.392937 0.919565i \(-0.371459\pi\)
0.392937 + 0.919565i \(0.371459\pi\)
\(614\) 0 0
\(615\) −7.40778 −0.298711
\(616\) 0 0
\(617\) 8.73843 0.351796 0.175898 0.984408i \(-0.443717\pi\)
0.175898 + 0.984408i \(0.443717\pi\)
\(618\) 0 0
\(619\) 36.0257 1.44800 0.723998 0.689802i \(-0.242303\pi\)
0.723998 + 0.689802i \(0.242303\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −2.81370 −0.112728
\(624\) 0 0
\(625\) −1.54266 −0.0617065
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) −34.1462 −1.36150
\(630\) 0 0
\(631\) −20.4435 −0.813845 −0.406922 0.913463i \(-0.633398\pi\)
−0.406922 + 0.913463i \(0.633398\pi\)
\(632\) 0 0
\(633\) −4.21201 −0.167412
\(634\) 0 0
\(635\) −3.87272 −0.153684
\(636\) 0 0
\(637\) 3.95044 0.156522
\(638\) 0 0
\(639\) −12.1248 −0.479651
\(640\) 0 0
\(641\) 27.8846 1.10138 0.550689 0.834711i \(-0.314365\pi\)
0.550689 + 0.834711i \(0.314365\pi\)
\(642\) 0 0
\(643\) −41.6864 −1.64395 −0.821976 0.569522i \(-0.807128\pi\)
−0.821976 + 0.569522i \(0.807128\pi\)
\(644\) 0 0
\(645\) 10.1462 0.399507
\(646\) 0 0
\(647\) −17.2334 −0.677515 −0.338757 0.940874i \(-0.610007\pi\)
−0.338757 + 0.940874i \(0.610007\pi\)
\(648\) 0 0
\(649\) 4.77606 0.187477
\(650\) 0 0
\(651\) 0.344438 0.0134996
\(652\) 0 0
\(653\) −42.2710 −1.65419 −0.827097 0.562060i \(-0.810009\pi\)
−0.827097 + 0.562060i \(0.810009\pi\)
\(654\) 0 0
\(655\) −15.6060 −0.609777
\(656\) 0 0
\(657\) 0.518817 0.0202410
\(658\) 0 0
\(659\) −12.6179 −0.491525 −0.245762 0.969330i \(-0.579038\pi\)
−0.245762 + 0.969330i \(0.579038\pi\)
\(660\) 0 0
\(661\) −40.4292 −1.57251 −0.786256 0.617901i \(-0.787983\pi\)
−0.786256 + 0.617901i \(0.787983\pi\)
\(662\) 0 0
\(663\) −17.8513 −0.693288
\(664\) 0 0
\(665\) −4.29488 −0.166548
\(666\) 0 0
\(667\) 10.4197 0.403453
\(668\) 0 0
\(669\) −22.0967 −0.854306
\(670\) 0 0
\(671\) −4.60601 −0.177813
\(672\) 0 0
\(673\) −29.2258 −1.12657 −0.563286 0.826262i \(-0.690463\pi\)
−0.563286 + 0.826262i \(0.690463\pi\)
\(674\) 0 0
\(675\) 2.95044 0.113563
\(676\) 0 0
\(677\) −2.83754 −0.109056 −0.0545278 0.998512i \(-0.517365\pi\)
−0.0545278 + 0.998512i \(0.517365\pi\)
\(678\) 0 0
\(679\) −6.51882 −0.250169
\(680\) 0 0
\(681\) 2.49311 0.0955360
\(682\) 0 0
\(683\) −7.63172 −0.292020 −0.146010 0.989283i \(-0.546643\pi\)
−0.146010 + 0.989283i \(0.546643\pi\)
\(684\) 0 0
\(685\) 11.8984 0.454616
\(686\) 0 0
\(687\) 23.0257 0.878486
\(688\) 0 0
\(689\) −19.5565 −0.745041
\(690\) 0 0
\(691\) 2.14867 0.0817392 0.0408696 0.999164i \(-0.486987\pi\)
0.0408696 + 0.999164i \(0.486987\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −23.6856 −0.898446
\(696\) 0 0
\(697\) 23.3821 0.885659
\(698\) 0 0
\(699\) 20.5445 0.777065
\(700\) 0 0
\(701\) −24.5112 −0.925776 −0.462888 0.886417i \(-0.653187\pi\)
−0.462888 + 0.886417i \(0.653187\pi\)
\(702\) 0 0
\(703\) 22.6694 0.854991
\(704\) 0 0
\(705\) 10.7641 0.405401
\(706\) 0 0
\(707\) 12.8513 0.483324
\(708\) 0 0
\(709\) −44.5916 −1.67467 −0.837337 0.546687i \(-0.815889\pi\)
−0.837337 + 0.546687i \(0.815889\pi\)
\(710\) 0 0
\(711\) −1.38207 −0.0518317
\(712\) 0 0
\(713\) 0.344438 0.0128993
\(714\) 0 0
\(715\) −5.65556 −0.211506
\(716\) 0 0
\(717\) 13.6813 0.510937
\(718\) 0 0
\(719\) 33.1086 1.23474 0.617371 0.786672i \(-0.288198\pi\)
0.617371 + 0.786672i \(0.288198\pi\)
\(720\) 0 0
\(721\) 11.4197 0.425292
\(722\) 0 0
\(723\) 18.6274 0.692760
\(724\) 0 0
\(725\) 30.7428 1.14176
\(726\) 0 0
\(727\) 12.8770 0.477583 0.238792 0.971071i \(-0.423249\pi\)
0.238792 + 0.971071i \(0.423249\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0257 −1.18451
\(732\) 0 0
\(733\) 4.17438 0.154184 0.0770921 0.997024i \(-0.475436\pi\)
0.0770921 + 0.997024i \(0.475436\pi\)
\(734\) 0 0
\(735\) −1.43163 −0.0528064
\(736\) 0 0
\(737\) 10.4693 0.385640
\(738\) 0 0
\(739\) −37.5001 −1.37946 −0.689732 0.724065i \(-0.742272\pi\)
−0.689732 + 0.724065i \(0.742272\pi\)
\(740\) 0 0
\(741\) 11.8513 0.435370
\(742\) 0 0
\(743\) 41.1010 1.50785 0.753924 0.656961i \(-0.228159\pi\)
0.753924 + 0.656961i \(0.228159\pi\)
\(744\) 0 0
\(745\) 19.4617 0.713020
\(746\) 0 0
\(747\) 6.41970 0.234885
\(748\) 0 0
\(749\) 1.95044 0.0712677
\(750\) 0 0
\(751\) −46.3701 −1.69207 −0.846035 0.533127i \(-0.821017\pi\)
−0.846035 + 0.533127i \(0.821017\pi\)
\(752\) 0 0
\(753\) 11.6274 0.423726
\(754\) 0 0
\(755\) 15.1129 0.550015
\(756\) 0 0
\(757\) −18.3872 −0.668295 −0.334147 0.942521i \(-0.608448\pi\)
−0.334147 + 0.942521i \(0.608448\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) −5.49497 −0.199193 −0.0995963 0.995028i \(-0.531755\pi\)
−0.0995963 + 0.995028i \(0.531755\pi\)
\(762\) 0 0
\(763\) −7.15814 −0.259142
\(764\) 0 0
\(765\) 6.46926 0.233897
\(766\) 0 0
\(767\) −18.8676 −0.681269
\(768\) 0 0
\(769\) −33.7027 −1.21535 −0.607675 0.794186i \(-0.707897\pi\)
−0.607675 + 0.794186i \(0.707897\pi\)
\(770\) 0 0
\(771\) −9.24533 −0.332962
\(772\) 0 0
\(773\) 30.4950 1.09683 0.548414 0.836207i \(-0.315232\pi\)
0.548414 + 0.836207i \(0.315232\pi\)
\(774\) 0 0
\(775\) 1.01624 0.0365045
\(776\) 0 0
\(777\) 7.55645 0.271086
\(778\) 0 0
\(779\) −15.5231 −0.556174
\(780\) 0 0
\(781\) 12.1248 0.433860
\(782\) 0 0
\(783\) 10.4197 0.372370
\(784\) 0 0
\(785\) 4.64610 0.165826
\(786\) 0 0
\(787\) −21.9128 −0.781107 −0.390554 0.920580i \(-0.627716\pi\)
−0.390554 + 0.920580i \(0.627716\pi\)
\(788\) 0 0
\(789\) −0.825621 −0.0293929
\(790\) 0 0
\(791\) −0.469261 −0.0166850
\(792\) 0 0
\(793\) 18.1958 0.646151
\(794\) 0 0
\(795\) 7.08719 0.251357
\(796\) 0 0
\(797\) −27.2873 −0.966565 −0.483283 0.875464i \(-0.660556\pi\)
−0.483283 + 0.875464i \(0.660556\pi\)
\(798\) 0 0
\(799\) −33.9762 −1.20199
\(800\) 0 0
\(801\) −2.81370 −0.0994171
\(802\) 0 0
\(803\) −0.518817 −0.0183086
\(804\) 0 0
\(805\) −1.43163 −0.0504582
\(806\) 0 0
\(807\) 15.2616 0.537233
\(808\) 0 0
\(809\) −16.5402 −0.581523 −0.290761 0.956796i \(-0.593909\pi\)
−0.290761 + 0.956796i \(0.593909\pi\)
\(810\) 0 0
\(811\) 3.89657 0.136827 0.0684135 0.997657i \(-0.478206\pi\)
0.0684135 + 0.997657i \(0.478206\pi\)
\(812\) 0 0
\(813\) −23.9718 −0.840729
\(814\) 0 0
\(815\) 24.5164 0.858771
\(816\) 0 0
\(817\) 21.2616 0.743848
\(818\) 0 0
\(819\) 3.95044 0.138040
\(820\) 0 0
\(821\) 19.3539 0.675456 0.337728 0.941244i \(-0.390342\pi\)
0.337728 + 0.941244i \(0.390342\pi\)
\(822\) 0 0
\(823\) −6.16246 −0.214810 −0.107405 0.994215i \(-0.534254\pi\)
−0.107405 + 0.994215i \(0.534254\pi\)
\(824\) 0 0
\(825\) −2.95044 −0.102721
\(826\) 0 0
\(827\) −24.2377 −0.842828 −0.421414 0.906868i \(-0.638466\pi\)
−0.421414 + 0.906868i \(0.638466\pi\)
\(828\) 0 0
\(829\) −5.72219 −0.198740 −0.0993699 0.995051i \(-0.531683\pi\)
−0.0993699 + 0.995051i \(0.531683\pi\)
\(830\) 0 0
\(831\) 5.29488 0.183677
\(832\) 0 0
\(833\) 4.51882 0.156568
\(834\) 0 0
\(835\) 11.2572 0.389573
\(836\) 0 0
\(837\) 0.344438 0.0119055
\(838\) 0 0
\(839\) 31.2052 1.07732 0.538662 0.842522i \(-0.318930\pi\)
0.538662 + 0.842522i \(0.318930\pi\)
\(840\) 0 0
\(841\) 79.5702 2.74380
\(842\) 0 0
\(843\) −16.1744 −0.557075
\(844\) 0 0
\(845\) 3.73083 0.128344
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 4.29488 0.147400
\(850\) 0 0
\(851\) 7.55645 0.259032
\(852\) 0 0
\(853\) 11.7590 0.402620 0.201310 0.979528i \(-0.435480\pi\)
0.201310 + 0.979528i \(0.435480\pi\)
\(854\) 0 0
\(855\) −4.29488 −0.146882
\(856\) 0 0
\(857\) 14.1324 0.482754 0.241377 0.970431i \(-0.422401\pi\)
0.241377 + 0.970431i \(0.422401\pi\)
\(858\) 0 0
\(859\) 31.3111 1.06832 0.534161 0.845383i \(-0.320628\pi\)
0.534161 + 0.845383i \(0.320628\pi\)
\(860\) 0 0
\(861\) −5.17438 −0.176342
\(862\) 0 0
\(863\) −19.8256 −0.674872 −0.337436 0.941348i \(-0.609560\pi\)
−0.337436 + 0.941348i \(0.609560\pi\)
\(864\) 0 0
\(865\) 4.59222 0.156140
\(866\) 0 0
\(867\) −3.41970 −0.116139
\(868\) 0 0
\(869\) 1.38207 0.0468835
\(870\) 0 0
\(871\) −41.3582 −1.40137
\(872\) 0 0
\(873\) −6.51882 −0.220629
\(874\) 0 0
\(875\) −11.3821 −0.384784
\(876\) 0 0
\(877\) 6.04195 0.204022 0.102011 0.994783i \(-0.467472\pi\)
0.102011 + 0.994783i \(0.467472\pi\)
\(878\) 0 0
\(879\) 5.72651 0.193150
\(880\) 0 0
\(881\) −12.5231 −0.421915 −0.210958 0.977495i \(-0.567658\pi\)
−0.210958 + 0.977495i \(0.567658\pi\)
\(882\) 0 0
\(883\) 48.0771 1.61792 0.808962 0.587861i \(-0.200030\pi\)
0.808962 + 0.587861i \(0.200030\pi\)
\(884\) 0 0
\(885\) 6.83754 0.229842
\(886\) 0 0
\(887\) −2.69074 −0.0903462 −0.0451731 0.998979i \(-0.514384\pi\)
−0.0451731 + 0.998979i \(0.514384\pi\)
\(888\) 0 0
\(889\) −2.70512 −0.0907268
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 22.5565 0.754823
\(894\) 0 0
\(895\) 23.8275 0.796465
\(896\) 0 0
\(897\) 3.95044 0.131901
\(898\) 0 0
\(899\) 3.58894 0.119698
\(900\) 0 0
\(901\) −22.3701 −0.745258
\(902\) 0 0
\(903\) 7.08719 0.235847
\(904\) 0 0
\(905\) −6.86080 −0.228061
\(906\) 0 0
\(907\) −28.4931 −0.946098 −0.473049 0.881036i \(-0.656847\pi\)
−0.473049 + 0.881036i \(0.656847\pi\)
\(908\) 0 0
\(909\) 12.8513 0.426252
\(910\) 0 0
\(911\) 17.6274 0.584022 0.292011 0.956415i \(-0.405676\pi\)
0.292011 + 0.956415i \(0.405676\pi\)
\(912\) 0 0
\(913\) −6.41970 −0.212461
\(914\) 0 0
\(915\) −6.59408 −0.217994
\(916\) 0 0
\(917\) −10.9009 −0.359979
\(918\) 0 0
\(919\) 8.10343 0.267308 0.133654 0.991028i \(-0.457329\pi\)
0.133654 + 0.991028i \(0.457329\pi\)
\(920\) 0 0
\(921\) −15.1838 −0.500325
\(922\) 0 0
\(923\) −47.8984 −1.57660
\(924\) 0 0
\(925\) 22.2949 0.733051
\(926\) 0 0
\(927\) 11.4197 0.375072
\(928\) 0 0
\(929\) 14.0967 0.462496 0.231248 0.972895i \(-0.425719\pi\)
0.231248 + 0.972895i \(0.425719\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) 7.66749 0.251022
\(934\) 0 0
\(935\) −6.46926 −0.211567
\(936\) 0 0
\(937\) −36.6036 −1.19579 −0.597893 0.801576i \(-0.703995\pi\)
−0.597893 + 0.801576i \(0.703995\pi\)
\(938\) 0 0
\(939\) −26.8061 −0.874784
\(940\) 0 0
\(941\) −14.4240 −0.470210 −0.235105 0.971970i \(-0.575543\pi\)
−0.235105 + 0.971970i \(0.575543\pi\)
\(942\) 0 0
\(943\) −5.17438 −0.168501
\(944\) 0 0
\(945\) −1.43163 −0.0465708
\(946\) 0 0
\(947\) 17.7352 0.576315 0.288157 0.957583i \(-0.406957\pi\)
0.288157 + 0.957583i \(0.406957\pi\)
\(948\) 0 0
\(949\) 2.04956 0.0665314
\(950\) 0 0
\(951\) −2.05388 −0.0666015
\(952\) 0 0
\(953\) 44.1104 1.42888 0.714439 0.699698i \(-0.246682\pi\)
0.714439 + 0.699698i \(0.246682\pi\)
\(954\) 0 0
\(955\) −7.50935 −0.242997
\(956\) 0 0
\(957\) −10.4197 −0.336821
\(958\) 0 0
\(959\) 8.31112 0.268380
\(960\) 0 0
\(961\) −30.8814 −0.996173
\(962\) 0 0
\(963\) 1.95044 0.0628522
\(964\) 0 0
\(965\) −26.1744 −0.842583
\(966\) 0 0
\(967\) −16.6889 −0.536678 −0.268339 0.963325i \(-0.586475\pi\)
−0.268339 + 0.963325i \(0.586475\pi\)
\(968\) 0 0
\(969\) 13.5565 0.435496
\(970\) 0 0
\(971\) 44.1438 1.41664 0.708320 0.705891i \(-0.249453\pi\)
0.708320 + 0.705891i \(0.249453\pi\)
\(972\) 0 0
\(973\) −16.5445 −0.530393
\(974\) 0 0
\(975\) 11.6556 0.373277
\(976\) 0 0
\(977\) 25.7428 0.823584 0.411792 0.911278i \(-0.364903\pi\)
0.411792 + 0.911278i \(0.364903\pi\)
\(978\) 0 0
\(979\) 2.81370 0.0899262
\(980\) 0 0
\(981\) −7.15814 −0.228542
\(982\) 0 0
\(983\) −48.8437 −1.55787 −0.778937 0.627103i \(-0.784241\pi\)
−0.778937 + 0.627103i \(0.784241\pi\)
\(984\) 0 0
\(985\) 8.26917 0.263478
\(986\) 0 0
\(987\) 7.51882 0.239327
\(988\) 0 0
\(989\) 7.08719 0.225360
\(990\) 0 0
\(991\) −40.1343 −1.27491 −0.637454 0.770489i \(-0.720012\pi\)
−0.637454 + 0.770489i \(0.720012\pi\)
\(992\) 0 0
\(993\) −28.8727 −0.916248
\(994\) 0 0
\(995\) −13.3130 −0.422050
\(996\) 0 0
\(997\) 11.8727 0.376013 0.188006 0.982168i \(-0.439797\pi\)
0.188006 + 0.982168i \(0.439797\pi\)
\(998\) 0 0
\(999\) 7.55645 0.239076
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.br.1.2 3
4.3 odd 2 1932.2.a.j.1.2 3
12.11 even 2 5796.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.j.1.2 3 4.3 odd 2
5796.2.a.o.1.2 3 12.11 even 2
7728.2.a.br.1.2 3 1.1 even 1 trivial