Properties

Label 7728.2.a.bc.1.2
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +5.70156 q^{13} -3.70156 q^{15} +4.00000 q^{17} +5.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.70156 q^{25} -1.00000 q^{27} +0.298438 q^{29} +2.00000 q^{31} -4.00000 q^{33} -3.70156 q^{35} +4.29844 q^{37} -5.70156 q^{39} -0.298438 q^{41} +1.70156 q^{43} +3.70156 q^{45} +11.1047 q^{47} +1.00000 q^{49} -4.00000 q^{51} -9.40312 q^{53} +14.8062 q^{55} -5.40312 q^{57} +7.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +21.1047 q^{65} -14.8062 q^{67} +1.00000 q^{69} +7.40312 q^{71} +1.40312 q^{73} -8.70156 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -13.4031 q^{83} +14.8062 q^{85} -0.298438 q^{87} -11.4031 q^{89} -5.70156 q^{91} -2.00000 q^{93} +20.0000 q^{95} +6.29844 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 8 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 4 q^{31} - 8 q^{33} - q^{35} + 15 q^{37} - 5 q^{39} - 7 q^{41} - 3 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 2 q^{59} - 4 q^{61} - 2 q^{63} + 23 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} - 10 q^{73} - 11 q^{75} - 8 q^{77} - 16 q^{79} + 2 q^{81} - 14 q^{83} + 4 q^{85} - 7 q^{87} - 10 q^{89} - 5 q^{91} - 4 q^{93} + 40 q^{95} + 19 q^{97} + 8 q^{99} + O(q^{100}) \)

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
\(5\) 3.70156 1.65539 0.827694 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 5.70156 1.58133 0.790664 0.612250i \(-0.209735\pi\)
0.790664 + 0.612250i \(0.209735\pi\)
\(14\) 0 0
\(15\) −3.70156 −0.955739
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 5.40312 1.23956 0.619781 0.784775i \(-0.287221\pi\)
0.619781 + 0.784775i \(0.287221\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.70156 1.74031
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.298438 0.0554185 0.0277093 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −3.70156 −0.625678
\(36\) 0 0
\(37\) 4.29844 0.706659 0.353329 0.935499i \(-0.385049\pi\)
0.353329 + 0.935499i \(0.385049\pi\)
\(38\) 0 0
\(39\) −5.70156 −0.912981
\(40\) 0 0
\(41\) −0.298438 −0.0466082 −0.0233041 0.999728i \(-0.507419\pi\)
−0.0233041 + 0.999728i \(0.507419\pi\)
\(42\) 0 0
\(43\) 1.70156 0.259486 0.129743 0.991548i \(-0.458585\pi\)
0.129743 + 0.991548i \(0.458585\pi\)
\(44\) 0 0
\(45\) 3.70156 0.551796
\(46\) 0 0
\(47\) 11.1047 1.61978 0.809892 0.586578i \(-0.199525\pi\)
0.809892 + 0.586578i \(0.199525\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −9.40312 −1.29162 −0.645809 0.763499i \(-0.723480\pi\)
−0.645809 + 0.763499i \(0.723480\pi\)
\(54\) 0 0
\(55\) 14.8062 1.99647
\(56\) 0 0
\(57\) −5.40312 −0.715661
\(58\) 0 0
\(59\) 7.40312 0.963805 0.481902 0.876225i \(-0.339946\pi\)
0.481902 + 0.876225i \(0.339946\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 21.1047 2.61771
\(66\) 0 0
\(67\) −14.8062 −1.80887 −0.904436 0.426610i \(-0.859708\pi\)
−0.904436 + 0.426610i \(0.859708\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.40312 0.878589 0.439295 0.898343i \(-0.355228\pi\)
0.439295 + 0.898343i \(0.355228\pi\)
\(72\) 0 0
\(73\) 1.40312 0.164223 0.0821116 0.996623i \(-0.473834\pi\)
0.0821116 + 0.996623i \(0.473834\pi\)
\(74\) 0 0
\(75\) −8.70156 −1.00477
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.4031 −1.47118 −0.735592 0.677425i \(-0.763096\pi\)
−0.735592 + 0.677425i \(0.763096\pi\)
\(84\) 0 0
\(85\) 14.8062 1.60596
\(86\) 0 0
\(87\) −0.298438 −0.0319959
\(88\) 0 0
\(89\) −11.4031 −1.20873 −0.604364 0.796708i \(-0.706573\pi\)
−0.604364 + 0.796708i \(0.706573\pi\)
\(90\) 0 0
\(91\) −5.70156 −0.597686
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 6.29844 0.639509 0.319755 0.947500i \(-0.396399\pi\)
0.319755 + 0.947500i \(0.396399\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −3.40312 −0.338624 −0.169312 0.985563i \(-0.554154\pi\)
−0.169312 + 0.985563i \(0.554154\pi\)
\(102\) 0 0
\(103\) −2.29844 −0.226472 −0.113236 0.993568i \(-0.536122\pi\)
−0.113236 + 0.993568i \(0.536122\pi\)
\(104\) 0 0
\(105\) 3.70156 0.361235
\(106\) 0 0
\(107\) −18.8062 −1.81807 −0.909034 0.416721i \(-0.863179\pi\)
−0.909034 + 0.416721i \(0.863179\pi\)
\(108\) 0 0
\(109\) −7.70156 −0.737676 −0.368838 0.929494i \(-0.620244\pi\)
−0.368838 + 0.929494i \(0.620244\pi\)
\(110\) 0 0
\(111\) −4.29844 −0.407990
\(112\) 0 0
\(113\) 7.10469 0.668353 0.334176 0.942511i \(-0.391542\pi\)
0.334176 + 0.942511i \(0.391542\pi\)
\(114\) 0 0
\(115\) −3.70156 −0.345172
\(116\) 0 0
\(117\) 5.70156 0.527110
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0.298438 0.0269092
\(124\) 0 0
\(125\) 13.7016 1.22550
\(126\) 0 0
\(127\) 5.70156 0.505932 0.252966 0.967475i \(-0.418594\pi\)
0.252966 + 0.967475i \(0.418594\pi\)
\(128\) 0 0
\(129\) −1.70156 −0.149814
\(130\) 0 0
\(131\) 7.40312 0.646814 0.323407 0.946260i \(-0.395172\pi\)
0.323407 + 0.946260i \(0.395172\pi\)
\(132\) 0 0
\(133\) −5.40312 −0.468510
\(134\) 0 0
\(135\) −3.70156 −0.318580
\(136\) 0 0
\(137\) −19.7016 −1.68322 −0.841609 0.540087i \(-0.818391\pi\)
−0.841609 + 0.540087i \(0.818391\pi\)
\(138\) 0 0
\(139\) −17.7016 −1.50143 −0.750713 0.660628i \(-0.770290\pi\)
−0.750713 + 0.660628i \(0.770290\pi\)
\(140\) 0 0
\(141\) −11.1047 −0.935183
\(142\) 0 0
\(143\) 22.8062 1.90715
\(144\) 0 0
\(145\) 1.10469 0.0917392
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −20.2094 −1.65562 −0.827808 0.561011i \(-0.810412\pi\)
−0.827808 + 0.561011i \(0.810412\pi\)
\(150\) 0 0
\(151\) −23.9109 −1.94584 −0.972922 0.231133i \(-0.925757\pi\)
−0.972922 + 0.231133i \(0.925757\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 7.40312 0.594633
\(156\) 0 0
\(157\) −1.40312 −0.111982 −0.0559908 0.998431i \(-0.517832\pi\)
−0.0559908 + 0.998431i \(0.517832\pi\)
\(158\) 0 0
\(159\) 9.40312 0.745716
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 10.8062 0.846411 0.423205 0.906034i \(-0.360905\pi\)
0.423205 + 0.906034i \(0.360905\pi\)
\(164\) 0 0
\(165\) −14.8062 −1.15266
\(166\) 0 0
\(167\) 13.4031 1.03716 0.518582 0.855028i \(-0.326460\pi\)
0.518582 + 0.855028i \(0.326460\pi\)
\(168\) 0 0
\(169\) 19.5078 1.50060
\(170\) 0 0
\(171\) 5.40312 0.413187
\(172\) 0 0
\(173\) 19.4031 1.47519 0.737596 0.675242i \(-0.235961\pi\)
0.737596 + 0.675242i \(0.235961\pi\)
\(174\) 0 0
\(175\) −8.70156 −0.657776
\(176\) 0 0
\(177\) −7.40312 −0.556453
\(178\) 0 0
\(179\) 25.1047 1.87641 0.938206 0.346077i \(-0.112486\pi\)
0.938206 + 0.346077i \(0.112486\pi\)
\(180\) 0 0
\(181\) −16.8062 −1.24920 −0.624599 0.780945i \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 15.9109 1.16980
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −22.8062 −1.65020 −0.825101 0.564985i \(-0.808882\pi\)
−0.825101 + 0.564985i \(0.808882\pi\)
\(192\) 0 0
\(193\) −23.1047 −1.66311 −0.831556 0.555441i \(-0.812549\pi\)
−0.831556 + 0.555441i \(0.812549\pi\)
\(194\) 0 0
\(195\) −21.1047 −1.51134
\(196\) 0 0
\(197\) 22.5078 1.60362 0.801808 0.597582i \(-0.203872\pi\)
0.801808 + 0.597582i \(0.203872\pi\)
\(198\) 0 0
\(199\) 2.29844 0.162932 0.0814660 0.996676i \(-0.474040\pi\)
0.0814660 + 0.996676i \(0.474040\pi\)
\(200\) 0 0
\(201\) 14.8062 1.04435
\(202\) 0 0
\(203\) −0.298438 −0.0209462
\(204\) 0 0
\(205\) −1.10469 −0.0771546
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 21.6125 1.49497
\(210\) 0 0
\(211\) −10.8062 −0.743933 −0.371966 0.928246i \(-0.621316\pi\)
−0.371966 + 0.928246i \(0.621316\pi\)
\(212\) 0 0
\(213\) −7.40312 −0.507254
\(214\) 0 0
\(215\) 6.29844 0.429550
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) −1.40312 −0.0948143
\(220\) 0 0
\(221\) 22.8062 1.53411
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 8.70156 0.580104
\(226\) 0 0
\(227\) 23.1047 1.53351 0.766756 0.641939i \(-0.221870\pi\)
0.766756 + 0.641939i \(0.221870\pi\)
\(228\) 0 0
\(229\) −14.5969 −0.964589 −0.482294 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 41.1047 2.68137
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −10.8062 −0.698998 −0.349499 0.936937i \(-0.613648\pi\)
−0.349499 + 0.936937i \(0.613648\pi\)
\(240\) 0 0
\(241\) −15.9109 −1.02491 −0.512457 0.858713i \(-0.671264\pi\)
−0.512457 + 0.858713i \(0.671264\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.70156 0.236484
\(246\) 0 0
\(247\) 30.8062 1.96015
\(248\) 0 0
\(249\) 13.4031 0.849388
\(250\) 0 0
\(251\) −19.7016 −1.24355 −0.621776 0.783195i \(-0.713588\pi\)
−0.621776 + 0.783195i \(0.713588\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −14.8062 −0.927203
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −4.29844 −0.267092
\(260\) 0 0
\(261\) 0.298438 0.0184728
\(262\) 0 0
\(263\) −9.70156 −0.598224 −0.299112 0.954218i \(-0.596690\pi\)
−0.299112 + 0.954218i \(0.596690\pi\)
\(264\) 0 0
\(265\) −34.8062 −2.13813
\(266\) 0 0
\(267\) 11.4031 0.697860
\(268\) 0 0
\(269\) −10.2094 −0.622476 −0.311238 0.950332i \(-0.600744\pi\)
−0.311238 + 0.950332i \(0.600744\pi\)
\(270\) 0 0
\(271\) −1.40312 −0.0852337 −0.0426169 0.999091i \(-0.513569\pi\)
−0.0426169 + 0.999091i \(0.513569\pi\)
\(272\) 0 0
\(273\) 5.70156 0.345074
\(274\) 0 0
\(275\) 34.8062 2.09890
\(276\) 0 0
\(277\) −21.4031 −1.28599 −0.642995 0.765871i \(-0.722308\pi\)
−0.642995 + 0.765871i \(0.722308\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 14.5078 0.865463 0.432732 0.901523i \(-0.357550\pi\)
0.432732 + 0.901523i \(0.357550\pi\)
\(282\) 0 0
\(283\) −4.80625 −0.285702 −0.142851 0.989744i \(-0.545627\pi\)
−0.142851 + 0.989744i \(0.545627\pi\)
\(284\) 0 0
\(285\) −20.0000 −1.18470
\(286\) 0 0
\(287\) 0.298438 0.0176162
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.29844 −0.369221
\(292\) 0 0
\(293\) −20.8062 −1.21551 −0.607757 0.794123i \(-0.707931\pi\)
−0.607757 + 0.794123i \(0.707931\pi\)
\(294\) 0 0
\(295\) 27.4031 1.59547
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −5.70156 −0.329730
\(300\) 0 0
\(301\) −1.70156 −0.0980764
\(302\) 0 0
\(303\) 3.40312 0.195504
\(304\) 0 0
\(305\) −7.40312 −0.423902
\(306\) 0 0
\(307\) −11.9109 −0.679793 −0.339896 0.940463i \(-0.610392\pi\)
−0.339896 + 0.940463i \(0.610392\pi\)
\(308\) 0 0
\(309\) 2.29844 0.130754
\(310\) 0 0
\(311\) −13.4031 −0.760021 −0.380011 0.924982i \(-0.624080\pi\)
−0.380011 + 0.924982i \(0.624080\pi\)
\(312\) 0 0
\(313\) −6.20937 −0.350974 −0.175487 0.984482i \(-0.556150\pi\)
−0.175487 + 0.984482i \(0.556150\pi\)
\(314\) 0 0
\(315\) −3.70156 −0.208559
\(316\) 0 0
\(317\) −8.29844 −0.466087 −0.233043 0.972466i \(-0.574868\pi\)
−0.233043 + 0.972466i \(0.574868\pi\)
\(318\) 0 0
\(319\) 1.19375 0.0668373
\(320\) 0 0
\(321\) 18.8062 1.04966
\(322\) 0 0
\(323\) 21.6125 1.20255
\(324\) 0 0
\(325\) 49.6125 2.75201
\(326\) 0 0
\(327\) 7.70156 0.425897
\(328\) 0 0
\(329\) −11.1047 −0.612221
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 4.29844 0.235553
\(334\) 0 0
\(335\) −54.8062 −2.99439
\(336\) 0 0
\(337\) −4.20937 −0.229299 −0.114650 0.993406i \(-0.536575\pi\)
−0.114650 + 0.993406i \(0.536575\pi\)
\(338\) 0 0
\(339\) −7.10469 −0.385874
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.70156 0.199285
\(346\) 0 0
\(347\) 10.2984 0.552849 0.276425 0.961036i \(-0.410850\pi\)
0.276425 + 0.961036i \(0.410850\pi\)
\(348\) 0 0
\(349\) 12.5969 0.674295 0.337148 0.941452i \(-0.390538\pi\)
0.337148 + 0.941452i \(0.390538\pi\)
\(350\) 0 0
\(351\) −5.70156 −0.304327
\(352\) 0 0
\(353\) 4.29844 0.228783 0.114391 0.993436i \(-0.463508\pi\)
0.114391 + 0.993436i \(0.463508\pi\)
\(354\) 0 0
\(355\) 27.4031 1.45441
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 2.89531 0.152809 0.0764044 0.997077i \(-0.475656\pi\)
0.0764044 + 0.997077i \(0.475656\pi\)
\(360\) 0 0
\(361\) 10.1938 0.536513
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 5.19375 0.271853
\(366\) 0 0
\(367\) 17.1047 0.892857 0.446429 0.894819i \(-0.352696\pi\)
0.446429 + 0.894819i \(0.352696\pi\)
\(368\) 0 0
\(369\) −0.298438 −0.0155361
\(370\) 0 0
\(371\) 9.40312 0.488186
\(372\) 0 0
\(373\) −0.806248 −0.0417460 −0.0208730 0.999782i \(-0.506645\pi\)
−0.0208730 + 0.999782i \(0.506645\pi\)
\(374\) 0 0
\(375\) −13.7016 −0.707546
\(376\) 0 0
\(377\) 1.70156 0.0876349
\(378\) 0 0
\(379\) −13.7016 −0.703802 −0.351901 0.936037i \(-0.614465\pi\)
−0.351901 + 0.936037i \(0.614465\pi\)
\(380\) 0 0
\(381\) −5.70156 −0.292100
\(382\) 0 0
\(383\) −29.6125 −1.51313 −0.756564 0.653920i \(-0.773123\pi\)
−0.756564 + 0.653920i \(0.773123\pi\)
\(384\) 0 0
\(385\) −14.8062 −0.754596
\(386\) 0 0
\(387\) 1.70156 0.0864953
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −7.40312 −0.373438
\(394\) 0 0
\(395\) −29.6125 −1.48997
\(396\) 0 0
\(397\) 8.59688 0.431465 0.215732 0.976453i \(-0.430786\pi\)
0.215732 + 0.976453i \(0.430786\pi\)
\(398\) 0 0
\(399\) 5.40312 0.270495
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 11.4031 0.568030
\(404\) 0 0
\(405\) 3.70156 0.183932
\(406\) 0 0
\(407\) 17.1938 0.852263
\(408\) 0 0
\(409\) 5.40312 0.267167 0.133584 0.991038i \(-0.457352\pi\)
0.133584 + 0.991038i \(0.457352\pi\)
\(410\) 0 0
\(411\) 19.7016 0.971806
\(412\) 0 0
\(413\) −7.40312 −0.364284
\(414\) 0 0
\(415\) −49.6125 −2.43538
\(416\) 0 0
\(417\) 17.7016 0.866849
\(418\) 0 0
\(419\) 0.209373 0.0102285 0.00511426 0.999987i \(-0.498372\pi\)
0.00511426 + 0.999987i \(0.498372\pi\)
\(420\) 0 0
\(421\) −15.7016 −0.765247 −0.382624 0.923904i \(-0.624979\pi\)
−0.382624 + 0.923904i \(0.624979\pi\)
\(422\) 0 0
\(423\) 11.1047 0.539928
\(424\) 0 0
\(425\) 34.8062 1.68835
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) −22.8062 −1.10110
\(430\) 0 0
\(431\) 3.91093 0.188383 0.0941916 0.995554i \(-0.469973\pi\)
0.0941916 + 0.995554i \(0.469973\pi\)
\(432\) 0 0
\(433\) 31.3141 1.50486 0.752429 0.658674i \(-0.228882\pi\)
0.752429 + 0.658674i \(0.228882\pi\)
\(434\) 0 0
\(435\) −1.10469 −0.0529657
\(436\) 0 0
\(437\) −5.40312 −0.258466
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 13.7016 0.650981 0.325490 0.945545i \(-0.394471\pi\)
0.325490 + 0.945545i \(0.394471\pi\)
\(444\) 0 0
\(445\) −42.2094 −2.00092
\(446\) 0 0
\(447\) 20.2094 0.955871
\(448\) 0 0
\(449\) −38.4187 −1.81309 −0.906546 0.422106i \(-0.861291\pi\)
−0.906546 + 0.422106i \(0.861291\pi\)
\(450\) 0 0
\(451\) −1.19375 −0.0562116
\(452\) 0 0
\(453\) 23.9109 1.12343
\(454\) 0 0
\(455\) −21.1047 −0.989403
\(456\) 0 0
\(457\) 28.2094 1.31958 0.659789 0.751451i \(-0.270645\pi\)
0.659789 + 0.751451i \(0.270645\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −2.20937 −0.102901 −0.0514504 0.998676i \(-0.516384\pi\)
−0.0514504 + 0.998676i \(0.516384\pi\)
\(462\) 0 0
\(463\) 22.2984 1.03630 0.518148 0.855291i \(-0.326622\pi\)
0.518148 + 0.855291i \(0.326622\pi\)
\(464\) 0 0
\(465\) −7.40312 −0.343312
\(466\) 0 0
\(467\) 3.70156 0.171288 0.0856439 0.996326i \(-0.472705\pi\)
0.0856439 + 0.996326i \(0.472705\pi\)
\(468\) 0 0
\(469\) 14.8062 0.683689
\(470\) 0 0
\(471\) 1.40312 0.0646526
\(472\) 0 0
\(473\) 6.80625 0.312952
\(474\) 0 0
\(475\) 47.0156 2.15722
\(476\) 0 0
\(477\) −9.40312 −0.430539
\(478\) 0 0
\(479\) −16.5969 −0.758331 −0.379165 0.925329i \(-0.623789\pi\)
−0.379165 + 0.925329i \(0.623789\pi\)
\(480\) 0 0
\(481\) 24.5078 1.11746
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 23.3141 1.05864
\(486\) 0 0
\(487\) 6.29844 0.285409 0.142705 0.989765i \(-0.454420\pi\)
0.142705 + 0.989765i \(0.454420\pi\)
\(488\) 0 0
\(489\) −10.8062 −0.488675
\(490\) 0 0
\(491\) 9.61250 0.433806 0.216903 0.976193i \(-0.430404\pi\)
0.216903 + 0.976193i \(0.430404\pi\)
\(492\) 0 0
\(493\) 1.19375 0.0537639
\(494\) 0 0
\(495\) 14.8062 0.665491
\(496\) 0 0
\(497\) −7.40312 −0.332076
\(498\) 0 0
\(499\) 23.4031 1.04767 0.523834 0.851820i \(-0.324501\pi\)
0.523834 + 0.851820i \(0.324501\pi\)
\(500\) 0 0
\(501\) −13.4031 −0.598807
\(502\) 0 0
\(503\) −4.59688 −0.204965 −0.102482 0.994735i \(-0.532678\pi\)
−0.102482 + 0.994735i \(0.532678\pi\)
\(504\) 0 0
\(505\) −12.5969 −0.560554
\(506\) 0 0
\(507\) −19.5078 −0.866372
\(508\) 0 0
\(509\) 37.6125 1.66714 0.833572 0.552410i \(-0.186292\pi\)
0.833572 + 0.552410i \(0.186292\pi\)
\(510\) 0 0
\(511\) −1.40312 −0.0620706
\(512\) 0 0
\(513\) −5.40312 −0.238554
\(514\) 0 0
\(515\) −8.50781 −0.374899
\(516\) 0 0
\(517\) 44.4187 1.95353
\(518\) 0 0
\(519\) −19.4031 −0.851703
\(520\) 0 0
\(521\) −20.5969 −0.902366 −0.451183 0.892432i \(-0.648998\pi\)
−0.451183 + 0.892432i \(0.648998\pi\)
\(522\) 0 0
\(523\) −20.8062 −0.909794 −0.454897 0.890544i \(-0.650324\pi\)
−0.454897 + 0.890544i \(0.650324\pi\)
\(524\) 0 0
\(525\) 8.70156 0.379767
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.40312 0.321268
\(532\) 0 0
\(533\) −1.70156 −0.0737028
\(534\) 0 0
\(535\) −69.6125 −3.00961
\(536\) 0 0
\(537\) −25.1047 −1.08335
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 33.4031 1.43611 0.718056 0.695985i \(-0.245032\pi\)
0.718056 + 0.695985i \(0.245032\pi\)
\(542\) 0 0
\(543\) 16.8062 0.721225
\(544\) 0 0
\(545\) −28.5078 −1.22114
\(546\) 0 0
\(547\) −29.0156 −1.24062 −0.620309 0.784357i \(-0.712993\pi\)
−0.620309 + 0.784357i \(0.712993\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 1.61250 0.0686947
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −15.9109 −0.675382
\(556\) 0 0
\(557\) −37.4031 −1.58482 −0.792411 0.609988i \(-0.791174\pi\)
−0.792411 + 0.609988i \(0.791174\pi\)
\(558\) 0 0
\(559\) 9.70156 0.410332
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) 3.10469 0.130847 0.0654235 0.997858i \(-0.479160\pi\)
0.0654235 + 0.997858i \(0.479160\pi\)
\(564\) 0 0
\(565\) 26.2984 1.10638
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 31.7016 1.32900 0.664499 0.747289i \(-0.268645\pi\)
0.664499 + 0.747289i \(0.268645\pi\)
\(570\) 0 0
\(571\) 14.8062 0.619622 0.309811 0.950798i \(-0.399734\pi\)
0.309811 + 0.950798i \(0.399734\pi\)
\(572\) 0 0
\(573\) 22.8062 0.952745
\(574\) 0 0
\(575\) −8.70156 −0.362880
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 23.1047 0.960198
\(580\) 0 0
\(581\) 13.4031 0.556055
\(582\) 0 0
\(583\) −37.6125 −1.55775
\(584\) 0 0
\(585\) 21.1047 0.872571
\(586\) 0 0
\(587\) −14.8062 −0.611119 −0.305560 0.952173i \(-0.598844\pi\)
−0.305560 + 0.952173i \(0.598844\pi\)
\(588\) 0 0
\(589\) 10.8062 0.445264
\(590\) 0 0
\(591\) −22.5078 −0.925848
\(592\) 0 0
\(593\) 43.7016 1.79461 0.897304 0.441413i \(-0.145523\pi\)
0.897304 + 0.441413i \(0.145523\pi\)
\(594\) 0 0
\(595\) −14.8062 −0.606997
\(596\) 0 0
\(597\) −2.29844 −0.0940688
\(598\) 0 0
\(599\) −13.6125 −0.556192 −0.278096 0.960553i \(-0.589703\pi\)
−0.278096 + 0.960553i \(0.589703\pi\)
\(600\) 0 0
\(601\) −13.4031 −0.546725 −0.273362 0.961911i \(-0.588136\pi\)
−0.273362 + 0.961911i \(0.588136\pi\)
\(602\) 0 0
\(603\) −14.8062 −0.602957
\(604\) 0 0
\(605\) 18.5078 0.752450
\(606\) 0 0
\(607\) 39.6125 1.60782 0.803911 0.594750i \(-0.202749\pi\)
0.803911 + 0.594750i \(0.202749\pi\)
\(608\) 0 0
\(609\) 0.298438 0.0120933
\(610\) 0 0
\(611\) 63.3141 2.56141
\(612\) 0 0
\(613\) 3.70156 0.149505 0.0747523 0.997202i \(-0.476183\pi\)
0.0747523 + 0.997202i \(0.476183\pi\)
\(614\) 0 0
\(615\) 1.10469 0.0445453
\(616\) 0 0
\(617\) −19.1938 −0.772711 −0.386356 0.922350i \(-0.626266\pi\)
−0.386356 + 0.922350i \(0.626266\pi\)
\(618\) 0 0
\(619\) 7.19375 0.289141 0.144571 0.989494i \(-0.453820\pi\)
0.144571 + 0.989494i \(0.453820\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 11.4031 0.456857
\(624\) 0 0
\(625\) 7.20937 0.288375
\(626\) 0 0
\(627\) −21.6125 −0.863120
\(628\) 0 0
\(629\) 17.1938 0.685560
\(630\) 0 0
\(631\) 29.6125 1.17885 0.589427 0.807821i \(-0.299353\pi\)
0.589427 + 0.807821i \(0.299353\pi\)
\(632\) 0 0
\(633\) 10.8062 0.429510
\(634\) 0 0
\(635\) 21.1047 0.837514
\(636\) 0 0
\(637\) 5.70156 0.225904
\(638\) 0 0
\(639\) 7.40312 0.292863
\(640\) 0 0
\(641\) −20.7172 −0.818280 −0.409140 0.912472i \(-0.634171\pi\)
−0.409140 + 0.912472i \(0.634171\pi\)
\(642\) 0 0
\(643\) −0.806248 −0.0317953 −0.0158977 0.999874i \(-0.505061\pi\)
−0.0158977 + 0.999874i \(0.505061\pi\)
\(644\) 0 0
\(645\) −6.29844 −0.248001
\(646\) 0 0
\(647\) 6.59688 0.259350 0.129675 0.991557i \(-0.458607\pi\)
0.129675 + 0.991557i \(0.458607\pi\)
\(648\) 0 0
\(649\) 29.6125 1.16239
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 7.10469 0.278028 0.139014 0.990290i \(-0.455607\pi\)
0.139014 + 0.990290i \(0.455607\pi\)
\(654\) 0 0
\(655\) 27.4031 1.07073
\(656\) 0 0
\(657\) 1.40312 0.0547411
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 42.4187 1.64990 0.824949 0.565207i \(-0.191204\pi\)
0.824949 + 0.565207i \(0.191204\pi\)
\(662\) 0 0
\(663\) −22.8062 −0.885721
\(664\) 0 0
\(665\) −20.0000 −0.775567
\(666\) 0 0
\(667\) −0.298438 −0.0115556
\(668\) 0 0
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 7.70156 0.296873 0.148437 0.988922i \(-0.452576\pi\)
0.148437 + 0.988922i \(0.452576\pi\)
\(674\) 0 0
\(675\) −8.70156 −0.334923
\(676\) 0 0
\(677\) −50.4187 −1.93775 −0.968875 0.247551i \(-0.920374\pi\)
−0.968875 + 0.247551i \(0.920374\pi\)
\(678\) 0 0
\(679\) −6.29844 −0.241712
\(680\) 0 0
\(681\) −23.1047 −0.885374
\(682\) 0 0
\(683\) −33.6125 −1.28615 −0.643073 0.765805i \(-0.722341\pi\)
−0.643073 + 0.765805i \(0.722341\pi\)
\(684\) 0 0
\(685\) −72.9266 −2.78638
\(686\) 0 0
\(687\) 14.5969 0.556906
\(688\) 0 0
\(689\) −53.6125 −2.04247
\(690\) 0 0
\(691\) 0.507811 0.0193180 0.00965901 0.999953i \(-0.496925\pi\)
0.00965901 + 0.999953i \(0.496925\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −65.5234 −2.48545
\(696\) 0 0
\(697\) −1.19375 −0.0452166
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 47.0156 1.77576 0.887878 0.460079i \(-0.152179\pi\)
0.887878 + 0.460079i \(0.152179\pi\)
\(702\) 0 0
\(703\) 23.2250 0.875947
\(704\) 0 0
\(705\) −41.1047 −1.54809
\(706\) 0 0
\(707\) 3.40312 0.127988
\(708\) 0 0
\(709\) 23.1938 0.871060 0.435530 0.900174i \(-0.356561\pi\)
0.435530 + 0.900174i \(0.356561\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 84.4187 3.15708
\(716\) 0 0
\(717\) 10.8062 0.403567
\(718\) 0 0
\(719\) 5.91093 0.220441 0.110220 0.993907i \(-0.464844\pi\)
0.110220 + 0.993907i \(0.464844\pi\)
\(720\) 0 0
\(721\) 2.29844 0.0855983
\(722\) 0 0
\(723\) 15.9109 0.591734
\(724\) 0 0
\(725\) 2.59688 0.0964455
\(726\) 0 0
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.80625 0.251738
\(732\) 0 0
\(733\) 52.2094 1.92840 0.964199 0.265181i \(-0.0854318\pi\)
0.964199 + 0.265181i \(0.0854318\pi\)
\(734\) 0 0
\(735\) −3.70156 −0.136534
\(736\) 0 0
\(737\) −59.2250 −2.18158
\(738\) 0 0
\(739\) 29.0156 1.06736 0.533678 0.845687i \(-0.320809\pi\)
0.533678 + 0.845687i \(0.320809\pi\)
\(740\) 0 0
\(741\) −30.8062 −1.13170
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −74.8062 −2.74069
\(746\) 0 0
\(747\) −13.4031 −0.490395
\(748\) 0 0
\(749\) 18.8062 0.687165
\(750\) 0 0
\(751\) −29.6125 −1.08058 −0.540288 0.841480i \(-0.681685\pi\)
−0.540288 + 0.841480i \(0.681685\pi\)
\(752\) 0 0
\(753\) 19.7016 0.717965
\(754\) 0 0
\(755\) −88.5078 −3.22113
\(756\) 0 0
\(757\) −26.4187 −0.960206 −0.480103 0.877212i \(-0.659401\pi\)
−0.480103 + 0.877212i \(0.659401\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −35.6125 −1.29095 −0.645476 0.763781i \(-0.723341\pi\)
−0.645476 + 0.763781i \(0.723341\pi\)
\(762\) 0 0
\(763\) 7.70156 0.278815
\(764\) 0 0
\(765\) 14.8062 0.535321
\(766\) 0 0
\(767\) 42.2094 1.52409
\(768\) 0 0
\(769\) 15.9109 0.573763 0.286881 0.957966i \(-0.407381\pi\)
0.286881 + 0.957966i \(0.407381\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) −7.10469 −0.255538 −0.127769 0.991804i \(-0.540782\pi\)
−0.127769 + 0.991804i \(0.540782\pi\)
\(774\) 0 0
\(775\) 17.4031 0.625139
\(776\) 0 0
\(777\) 4.29844 0.154206
\(778\) 0 0
\(779\) −1.61250 −0.0577737
\(780\) 0 0
\(781\) 29.6125 1.05962
\(782\) 0 0
\(783\) −0.298438 −0.0106653
\(784\) 0 0
\(785\) −5.19375 −0.185373
\(786\) 0 0
\(787\) 18.5969 0.662907 0.331454 0.943472i \(-0.392461\pi\)
0.331454 + 0.943472i \(0.392461\pi\)
\(788\) 0 0
\(789\) 9.70156 0.345385
\(790\) 0 0
\(791\) −7.10469 −0.252614
\(792\) 0 0
\(793\) −11.4031 −0.404937
\(794\) 0 0
\(795\) 34.8062 1.23445
\(796\) 0 0
\(797\) 15.7016 0.556178 0.278089 0.960555i \(-0.410299\pi\)
0.278089 + 0.960555i \(0.410299\pi\)
\(798\) 0 0
\(799\) 44.4187 1.57142
\(800\) 0 0
\(801\) −11.4031 −0.402910
\(802\) 0 0
\(803\) 5.61250 0.198061
\(804\) 0 0
\(805\) 3.70156 0.130463
\(806\) 0 0
\(807\) 10.2094 0.359387
\(808\) 0 0
\(809\) 31.0156 1.09045 0.545226 0.838289i \(-0.316444\pi\)
0.545226 + 0.838289i \(0.316444\pi\)
\(810\) 0 0
\(811\) −52.5078 −1.84380 −0.921899 0.387430i \(-0.873363\pi\)
−0.921899 + 0.387430i \(0.873363\pi\)
\(812\) 0 0
\(813\) 1.40312 0.0492097
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 9.19375 0.321649
\(818\) 0 0
\(819\) −5.70156 −0.199229
\(820\) 0 0
\(821\) 27.6125 0.963683 0.481841 0.876258i \(-0.339968\pi\)
0.481841 + 0.876258i \(0.339968\pi\)
\(822\) 0 0
\(823\) 13.7016 0.477606 0.238803 0.971068i \(-0.423245\pi\)
0.238803 + 0.971068i \(0.423245\pi\)
\(824\) 0 0
\(825\) −34.8062 −1.21180
\(826\) 0 0
\(827\) 47.4031 1.64837 0.824184 0.566322i \(-0.191634\pi\)
0.824184 + 0.566322i \(0.191634\pi\)
\(828\) 0 0
\(829\) 55.4031 1.92423 0.962115 0.272644i \(-0.0878981\pi\)
0.962115 + 0.272644i \(0.0878981\pi\)
\(830\) 0 0
\(831\) 21.4031 0.742466
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 49.6125 1.71691
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −42.2094 −1.45723 −0.728615 0.684924i \(-0.759835\pi\)
−0.728615 + 0.684924i \(0.759835\pi\)
\(840\) 0 0
\(841\) −28.9109 −0.996929
\(842\) 0 0
\(843\) −14.5078 −0.499676
\(844\) 0 0
\(845\) 72.2094 2.48408
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) 4.80625 0.164950
\(850\) 0 0
\(851\) −4.29844 −0.147349
\(852\) 0 0
\(853\) 9.10469 0.311739 0.155869 0.987778i \(-0.450182\pi\)
0.155869 + 0.987778i \(0.450182\pi\)
\(854\) 0 0
\(855\) 20.0000 0.683986
\(856\) 0 0
\(857\) −0.298438 −0.0101944 −0.00509722 0.999987i \(-0.501623\pi\)
−0.00509722 + 0.999987i \(0.501623\pi\)
\(858\) 0 0
\(859\) 15.4922 0.528587 0.264293 0.964442i \(-0.414861\pi\)
0.264293 + 0.964442i \(0.414861\pi\)
\(860\) 0 0
\(861\) −0.298438 −0.0101707
\(862\) 0 0
\(863\) 6.38750 0.217433 0.108717 0.994073i \(-0.465326\pi\)
0.108717 + 0.994073i \(0.465326\pi\)
\(864\) 0 0
\(865\) 71.8219 2.44202
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −84.4187 −2.86042
\(872\) 0 0
\(873\) 6.29844 0.213170
\(874\) 0 0
\(875\) −13.7016 −0.463197
\(876\) 0 0
\(877\) −17.4031 −0.587662 −0.293831 0.955857i \(-0.594930\pi\)
−0.293831 + 0.955857i \(0.594930\pi\)
\(878\) 0 0
\(879\) 20.8062 0.701777
\(880\) 0 0
\(881\) 6.20937 0.209199 0.104600 0.994514i \(-0.466644\pi\)
0.104600 + 0.994514i \(0.466644\pi\)
\(882\) 0 0
\(883\) 24.5969 0.827751 0.413875 0.910334i \(-0.364175\pi\)
0.413875 + 0.910334i \(0.364175\pi\)
\(884\) 0 0
\(885\) −27.4031 −0.921146
\(886\) 0 0
\(887\) 28.2094 0.947178 0.473589 0.880746i \(-0.342958\pi\)
0.473589 + 0.880746i \(0.342958\pi\)
\(888\) 0 0
\(889\) −5.70156 −0.191224
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 60.0000 2.00782
\(894\) 0 0
\(895\) 92.9266 3.10619
\(896\) 0 0
\(897\) 5.70156 0.190370
\(898\) 0 0
\(899\) 0.596876 0.0199069
\(900\) 0 0
\(901\) −37.6125 −1.25305
\(902\) 0 0
\(903\) 1.70156 0.0566244
\(904\) 0 0
\(905\) −62.2094 −2.06791
\(906\) 0 0
\(907\) −20.5078 −0.680951 −0.340475 0.940253i \(-0.610588\pi\)
−0.340475 + 0.940253i \(0.610588\pi\)
\(908\) 0 0
\(909\) −3.40312 −0.112875
\(910\) 0 0
\(911\) −53.1047 −1.75944 −0.879718 0.475495i \(-0.842269\pi\)
−0.879718 + 0.475495i \(0.842269\pi\)
\(912\) 0 0
\(913\) −53.6125 −1.77431
\(914\) 0 0
\(915\) 7.40312 0.244740
\(916\) 0 0
\(917\) −7.40312 −0.244473
\(918\) 0 0
\(919\) 52.4187 1.72913 0.864567 0.502517i \(-0.167593\pi\)
0.864567 + 0.502517i \(0.167593\pi\)
\(920\) 0 0
\(921\) 11.9109 0.392479
\(922\) 0 0
\(923\) 42.2094 1.38934
\(924\) 0 0
\(925\) 37.4031 1.22981
\(926\) 0 0
\(927\) −2.29844 −0.0754906
\(928\) 0 0
\(929\) −45.9109 −1.50629 −0.753144 0.657855i \(-0.771464\pi\)
−0.753144 + 0.657855i \(0.771464\pi\)
\(930\) 0 0
\(931\) 5.40312 0.177080
\(932\) 0 0
\(933\) 13.4031 0.438799
\(934\) 0 0
\(935\) 59.2250 1.93686
\(936\) 0 0
\(937\) 25.1047 0.820134 0.410067 0.912055i \(-0.365505\pi\)
0.410067 + 0.912055i \(0.365505\pi\)
\(938\) 0 0
\(939\) 6.20937 0.202635
\(940\) 0 0
\(941\) 25.3141 0.825215 0.412607 0.910909i \(-0.364618\pi\)
0.412607 + 0.910909i \(0.364618\pi\)
\(942\) 0 0
\(943\) 0.298438 0.00971847
\(944\) 0 0
\(945\) 3.70156 0.120412
\(946\) 0 0
\(947\) −15.9109 −0.517036 −0.258518 0.966006i \(-0.583234\pi\)
−0.258518 + 0.966006i \(0.583234\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 8.29844 0.269095
\(952\) 0 0
\(953\) 61.2250 1.98327 0.991636 0.129066i \(-0.0411978\pi\)
0.991636 + 0.129066i \(0.0411978\pi\)
\(954\) 0 0
\(955\) −84.4187 −2.73173
\(956\) 0 0
\(957\) −1.19375 −0.0385885
\(958\) 0 0
\(959\) 19.7016 0.636197
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −18.8062 −0.606023
\(964\) 0 0
\(965\) −85.5234 −2.75310
\(966\) 0 0
\(967\) 14.8062 0.476137 0.238068 0.971248i \(-0.423486\pi\)
0.238068 + 0.971248i \(0.423486\pi\)
\(968\) 0 0
\(969\) −21.6125 −0.694293
\(970\) 0 0
\(971\) −49.8219 −1.59886 −0.799430 0.600759i \(-0.794865\pi\)
−0.799430 + 0.600759i \(0.794865\pi\)
\(972\) 0 0
\(973\) 17.7016 0.567486
\(974\) 0 0
\(975\) −49.6125 −1.58887
\(976\) 0 0
\(977\) 16.7172 0.534830 0.267415 0.963581i \(-0.413831\pi\)
0.267415 + 0.963581i \(0.413831\pi\)
\(978\) 0 0
\(979\) −45.6125 −1.45778
\(980\) 0 0
\(981\) −7.70156 −0.245892
\(982\) 0 0
\(983\) −42.2094 −1.34627 −0.673135 0.739520i \(-0.735053\pi\)
−0.673135 + 0.739520i \(0.735053\pi\)
\(984\) 0 0
\(985\) 83.3141 2.65461
\(986\) 0 0
\(987\) 11.1047 0.353466
\(988\) 0 0
\(989\) −1.70156 −0.0541065
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 8.50781 0.269716
\(996\) 0 0
\(997\) −23.4031 −0.741184 −0.370592 0.928796i \(-0.620845\pi\)
−0.370592 + 0.928796i \(0.620845\pi\)
\(998\) 0 0
\(999\) −4.29844 −0.135997
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.bc.1.2 2
4.3 odd 2 966.2.a.n.1.2 2
12.11 even 2 2898.2.a.bb.1.1 2
28.27 even 2 6762.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.2 2 4.3 odd 2
2898.2.a.bb.1.1 2 12.11 even 2
6762.2.a.bo.1.1 2 28.27 even 2
7728.2.a.bc.1.2 2 1.1 even 1 trivial