Properties

Label 3192.2.a.ba.1.1
Level $3192$
Weight $2$
Character 3192.1
Self dual yes
Analytic conductor $25.488$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3192.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.4882483252\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.135076.1
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.94486\) of defining polynomial
Character \(\chi\) \(=\) 3192.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.88971 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.88971 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.21056 q^{11} +5.39794 q^{13} +3.88971 q^{15} -4.41108 q^{17} +1.00000 q^{19} -1.00000 q^{21} -3.39794 q^{23} +10.1299 q^{25} -1.00000 q^{27} -7.28765 q^{29} +0.187375 q^{31} -3.21056 q^{33} -3.88971 q^{35} -4.60850 q^{37} -5.39794 q^{39} -9.77943 q^{41} -3.88971 q^{45} +4.45071 q^{47} +1.00000 q^{49} +4.41108 q^{51} +11.2876 q^{53} -12.4882 q^{55} -1.00000 q^{57} -5.77156 q^{59} +6.00000 q^{61} +1.00000 q^{63} -20.9964 q^{65} +2.48173 q^{67} +3.39794 q^{69} +12.4175 q^{71} -4.08714 q^{73} -10.1299 q^{75} +3.21056 q^{77} +2.68920 q^{79} +1.00000 q^{81} +3.92935 q^{83} +17.1578 q^{85} +7.28765 q^{87} +2.70874 q^{89} +5.39794 q^{91} -0.187375 q^{93} -3.88971 q^{95} -0.227011 q^{97} +3.21056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 2 q^{5} + 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} - 2 q^{15} - 2 q^{17} + 5 q^{19} - 5 q^{21} + 2 q^{23} + 19 q^{25} - 5 q^{27} + 4 q^{29} - 4 q^{31} - 2 q^{33} + 2 q^{35} + 10 q^{37} - 8 q^{39} - 6 q^{41} + 2 q^{45} - 2 q^{47} + 5 q^{49} + 2 q^{51} + 16 q^{53} - 16 q^{55} - 5 q^{57} - 12 q^{59} + 30 q^{61} + 5 q^{63} + 4 q^{65} + 18 q^{67} - 2 q^{69} - 10 q^{71} + 14 q^{73} - 19 q^{75} + 2 q^{77} - 2 q^{79} + 5 q^{81} - 6 q^{83} + 12 q^{85} - 4 q^{87} + 10 q^{89} + 8 q^{91} + 4 q^{93} + 2 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −3.88971 −1.73953 −0.869766 0.493464i \(-0.835730\pi\)
−0.869766 + 0.493464i \(0.835730\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.21056 0.968021 0.484010 0.875062i \(-0.339180\pi\)
0.484010 + 0.875062i \(0.339180\pi\)
\(12\) 0 0
\(13\) 5.39794 1.49712 0.748559 0.663068i \(-0.230746\pi\)
0.748559 + 0.663068i \(0.230746\pi\)
\(14\) 0 0
\(15\) 3.88971 1.00432
\(16\) 0 0
\(17\) −4.41108 −1.06984 −0.534922 0.844902i \(-0.679659\pi\)
−0.534922 + 0.844902i \(0.679659\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −3.39794 −0.708519 −0.354259 0.935147i \(-0.615267\pi\)
−0.354259 + 0.935147i \(0.615267\pi\)
\(24\) 0 0
\(25\) 10.1299 2.02597
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.28765 −1.35328 −0.676641 0.736313i \(-0.736565\pi\)
−0.676641 + 0.736313i \(0.736565\pi\)
\(30\) 0 0
\(31\) 0.187375 0.0336535 0.0168268 0.999858i \(-0.494644\pi\)
0.0168268 + 0.999858i \(0.494644\pi\)
\(32\) 0 0
\(33\) −3.21056 −0.558887
\(34\) 0 0
\(35\) −3.88971 −0.657481
\(36\) 0 0
\(37\) −4.60850 −0.757633 −0.378816 0.925472i \(-0.623669\pi\)
−0.378816 + 0.925472i \(0.623669\pi\)
\(38\) 0 0
\(39\) −5.39794 −0.864362
\(40\) 0 0
\(41\) −9.77943 −1.52729 −0.763645 0.645637i \(-0.776592\pi\)
−0.763645 + 0.645637i \(0.776592\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −3.88971 −0.579844
\(46\) 0 0
\(47\) 4.45071 0.649203 0.324602 0.945851i \(-0.394770\pi\)
0.324602 + 0.945851i \(0.394770\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.41108 0.617674
\(52\) 0 0
\(53\) 11.2876 1.55048 0.775239 0.631668i \(-0.217629\pi\)
0.775239 + 0.631668i \(0.217629\pi\)
\(54\) 0 0
\(55\) −12.4882 −1.68390
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −5.77156 −0.751394 −0.375697 0.926743i \(-0.622597\pi\)
−0.375697 + 0.926743i \(0.622597\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −20.9964 −2.60429
\(66\) 0 0
\(67\) 2.48173 0.303191 0.151596 0.988443i \(-0.451559\pi\)
0.151596 + 0.988443i \(0.451559\pi\)
\(68\) 0 0
\(69\) 3.39794 0.409064
\(70\) 0 0
\(71\) 12.4175 1.47369 0.736844 0.676063i \(-0.236315\pi\)
0.736844 + 0.676063i \(0.236315\pi\)
\(72\) 0 0
\(73\) −4.08714 −0.478363 −0.239181 0.970975i \(-0.576879\pi\)
−0.239181 + 0.970975i \(0.576879\pi\)
\(74\) 0 0
\(75\) −10.1299 −1.16970
\(76\) 0 0
\(77\) 3.21056 0.365878
\(78\) 0 0
\(79\) 2.68920 0.302558 0.151279 0.988491i \(-0.451661\pi\)
0.151279 + 0.988491i \(0.451661\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.92935 0.431302 0.215651 0.976470i \(-0.430813\pi\)
0.215651 + 0.976470i \(0.430813\pi\)
\(84\) 0 0
\(85\) 17.1578 1.86103
\(86\) 0 0
\(87\) 7.28765 0.781318
\(88\) 0 0
\(89\) 2.70874 0.287126 0.143563 0.989641i \(-0.454144\pi\)
0.143563 + 0.989641i \(0.454144\pi\)
\(90\) 0 0
\(91\) 5.39794 0.565858
\(92\) 0 0
\(93\) −0.187375 −0.0194299
\(94\) 0 0
\(95\) −3.88971 −0.399076
\(96\) 0 0
\(97\) −0.227011 −0.0230495 −0.0115248 0.999934i \(-0.503669\pi\)
−0.0115248 + 0.999934i \(0.503669\pi\)
\(98\) 0 0
\(99\) 3.21056 0.322674
\(100\) 0 0
\(101\) 19.6691 1.95715 0.978576 0.205885i \(-0.0660073\pi\)
0.978576 + 0.205885i \(0.0660073\pi\)
\(102\) 0 0
\(103\) −14.7219 −1.45059 −0.725297 0.688436i \(-0.758297\pi\)
−0.725297 + 0.688436i \(0.758297\pi\)
\(104\) 0 0
\(105\) 3.88971 0.379597
\(106\) 0 0
\(107\) 8.95366 0.865583 0.432792 0.901494i \(-0.357529\pi\)
0.432792 + 0.901494i \(0.357529\pi\)
\(108\) 0 0
\(109\) 13.5921 1.30188 0.650941 0.759128i \(-0.274374\pi\)
0.650941 + 0.759128i \(0.274374\pi\)
\(110\) 0 0
\(111\) 4.60850 0.437419
\(112\) 0 0
\(113\) 8.35047 0.785546 0.392773 0.919635i \(-0.371516\pi\)
0.392773 + 0.919635i \(0.371516\pi\)
\(114\) 0 0
\(115\) 13.2170 1.23249
\(116\) 0 0
\(117\) 5.39794 0.499039
\(118\) 0 0
\(119\) −4.41108 −0.404363
\(120\) 0 0
\(121\) −0.692290 −0.0629355
\(122\) 0 0
\(123\) 9.77943 0.881781
\(124\) 0 0
\(125\) −19.9537 −1.78471
\(126\) 0 0
\(127\) −5.70565 −0.506294 −0.253147 0.967428i \(-0.581466\pi\)
−0.253147 + 0.967428i \(0.581466\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.3505 −0.904325 −0.452163 0.891936i \(-0.649347\pi\)
−0.452163 + 0.891936i \(0.649347\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 3.88971 0.334773
\(136\) 0 0
\(137\) 16.7959 1.43497 0.717484 0.696575i \(-0.245294\pi\)
0.717484 + 0.696575i \(0.245294\pi\)
\(138\) 0 0
\(139\) −16.3469 −1.38652 −0.693261 0.720686i \(-0.743827\pi\)
−0.693261 + 0.720686i \(0.743827\pi\)
\(140\) 0 0
\(141\) −4.45071 −0.374818
\(142\) 0 0
\(143\) 17.3304 1.44924
\(144\) 0 0
\(145\) 28.3469 2.35408
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 4.82907 0.395613 0.197807 0.980241i \(-0.436618\pi\)
0.197807 + 0.980241i \(0.436618\pi\)
\(150\) 0 0
\(151\) 21.2645 1.73048 0.865240 0.501358i \(-0.167166\pi\)
0.865240 + 0.501358i \(0.167166\pi\)
\(152\) 0 0
\(153\) −4.41108 −0.356614
\(154\) 0 0
\(155\) −0.728835 −0.0585414
\(156\) 0 0
\(157\) 23.1377 1.84659 0.923296 0.384090i \(-0.125485\pi\)
0.923296 + 0.384090i \(0.125485\pi\)
\(158\) 0 0
\(159\) −11.2876 −0.895169
\(160\) 0 0
\(161\) −3.39794 −0.267795
\(162\) 0 0
\(163\) 7.27117 0.569522 0.284761 0.958599i \(-0.408086\pi\)
0.284761 + 0.958599i \(0.408086\pi\)
\(164\) 0 0
\(165\) 12.4882 0.972202
\(166\) 0 0
\(167\) −7.12986 −0.551725 −0.275863 0.961197i \(-0.588964\pi\)
−0.275863 + 0.961197i \(0.588964\pi\)
\(168\) 0 0
\(169\) 16.1377 1.24136
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −3.08349 −0.234433 −0.117217 0.993106i \(-0.537397\pi\)
−0.117217 + 0.993106i \(0.537397\pi\)
\(174\) 0 0
\(175\) 10.1299 0.765746
\(176\) 0 0
\(177\) 5.77156 0.433817
\(178\) 0 0
\(179\) 13.3669 0.999091 0.499545 0.866288i \(-0.333500\pi\)
0.499545 + 0.866288i \(0.333500\pi\)
\(180\) 0 0
\(181\) 8.15305 0.606012 0.303006 0.952989i \(-0.402010\pi\)
0.303006 + 0.952989i \(0.402010\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 17.9257 1.31793
\(186\) 0 0
\(187\) −14.1620 −1.03563
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −15.5722 −1.12677 −0.563383 0.826196i \(-0.690500\pi\)
−0.563383 + 0.826196i \(0.690500\pi\)
\(192\) 0 0
\(193\) 8.95727 0.644759 0.322379 0.946611i \(-0.395517\pi\)
0.322379 + 0.946611i \(0.395517\pi\)
\(194\) 0 0
\(195\) 20.9964 1.50359
\(196\) 0 0
\(197\) 4.52923 0.322694 0.161347 0.986898i \(-0.448416\pi\)
0.161347 + 0.986898i \(0.448416\pi\)
\(198\) 0 0
\(199\) −6.08714 −0.431506 −0.215753 0.976448i \(-0.569221\pi\)
−0.215753 + 0.976448i \(0.569221\pi\)
\(200\) 0 0
\(201\) −2.48173 −0.175048
\(202\) 0 0
\(203\) −7.28765 −0.511493
\(204\) 0 0
\(205\) 38.0392 2.65677
\(206\) 0 0
\(207\) −3.39794 −0.236173
\(208\) 0 0
\(209\) 3.21056 0.222079
\(210\) 0 0
\(211\) 23.5574 1.62176 0.810880 0.585212i \(-0.198989\pi\)
0.810880 + 0.585212i \(0.198989\pi\)
\(212\) 0 0
\(213\) −12.4175 −0.850834
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.187375 0.0127198
\(218\) 0 0
\(219\) 4.08714 0.276183
\(220\) 0 0
\(221\) −23.8107 −1.60168
\(222\) 0 0
\(223\) 7.27648 0.487269 0.243635 0.969867i \(-0.421660\pi\)
0.243635 + 0.969867i \(0.421660\pi\)
\(224\) 0 0
\(225\) 10.1299 0.675324
\(226\) 0 0
\(227\) −10.4619 −0.694380 −0.347190 0.937795i \(-0.612864\pi\)
−0.347190 + 0.937795i \(0.612864\pi\)
\(228\) 0 0
\(229\) −4.20055 −0.277580 −0.138790 0.990322i \(-0.544321\pi\)
−0.138790 + 0.990322i \(0.544321\pi\)
\(230\) 0 0
\(231\) −3.21056 −0.211239
\(232\) 0 0
\(233\) −1.40468 −0.0920233 −0.0460117 0.998941i \(-0.514651\pi\)
−0.0460117 + 0.998941i \(0.514651\pi\)
\(234\) 0 0
\(235\) −17.3120 −1.12931
\(236\) 0 0
\(237\) −2.68920 −0.174682
\(238\) 0 0
\(239\) −20.2994 −1.31306 −0.656528 0.754301i \(-0.727976\pi\)
−0.656528 + 0.754301i \(0.727976\pi\)
\(240\) 0 0
\(241\) −23.6245 −1.52179 −0.760893 0.648878i \(-0.775239\pi\)
−0.760893 + 0.648878i \(0.775239\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.88971 −0.248505
\(246\) 0 0
\(247\) 5.39794 0.343463
\(248\) 0 0
\(249\) −3.92935 −0.249012
\(250\) 0 0
\(251\) 1.82380 0.115117 0.0575585 0.998342i \(-0.481668\pi\)
0.0575585 + 0.998342i \(0.481668\pi\)
\(252\) 0 0
\(253\) −10.9093 −0.685861
\(254\) 0 0
\(255\) −17.1578 −1.07446
\(256\) 0 0
\(257\) −6.72883 −0.419733 −0.209867 0.977730i \(-0.567303\pi\)
−0.209867 + 0.977730i \(0.567303\pi\)
\(258\) 0 0
\(259\) −4.60850 −0.286358
\(260\) 0 0
\(261\) −7.28765 −0.451094
\(262\) 0 0
\(263\) −13.4371 −0.828566 −0.414283 0.910148i \(-0.635968\pi\)
−0.414283 + 0.910148i \(0.635968\pi\)
\(264\) 0 0
\(265\) −43.9057 −2.69711
\(266\) 0 0
\(267\) −2.70874 −0.165772
\(268\) 0 0
\(269\) 15.8666 0.967401 0.483701 0.875234i \(-0.339292\pi\)
0.483701 + 0.875234i \(0.339292\pi\)
\(270\) 0 0
\(271\) −0.667979 −0.0405768 −0.0202884 0.999794i \(-0.506458\pi\)
−0.0202884 + 0.999794i \(0.506458\pi\)
\(272\) 0 0
\(273\) −5.39794 −0.326698
\(274\) 0 0
\(275\) 32.5226 1.96118
\(276\) 0 0
\(277\) 24.8501 1.49310 0.746549 0.665330i \(-0.231709\pi\)
0.746549 + 0.665330i \(0.231709\pi\)
\(278\) 0 0
\(279\) 0.187375 0.0112178
\(280\) 0 0
\(281\) 11.0671 0.660206 0.330103 0.943945i \(-0.392916\pi\)
0.330103 + 0.943945i \(0.392916\pi\)
\(282\) 0 0
\(283\) −29.0306 −1.72569 −0.862844 0.505470i \(-0.831319\pi\)
−0.862844 + 0.505470i \(0.831319\pi\)
\(284\) 0 0
\(285\) 3.88971 0.230407
\(286\) 0 0
\(287\) −9.77943 −0.577261
\(288\) 0 0
\(289\) 2.45760 0.144565
\(290\) 0 0
\(291\) 0.227011 0.0133076
\(292\) 0 0
\(293\) 10.8830 0.635792 0.317896 0.948126i \(-0.397024\pi\)
0.317896 + 0.948126i \(0.397024\pi\)
\(294\) 0 0
\(295\) 22.4497 1.30707
\(296\) 0 0
\(297\) −3.21056 −0.186296
\(298\) 0 0
\(299\) −18.3419 −1.06074
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −19.6691 −1.12996
\(304\) 0 0
\(305\) −23.3383 −1.33635
\(306\) 0 0
\(307\) 9.77156 0.557693 0.278846 0.960336i \(-0.410048\pi\)
0.278846 + 0.960336i \(0.410048\pi\)
\(308\) 0 0
\(309\) 14.7219 0.837500
\(310\) 0 0
\(311\) 26.4307 1.49875 0.749373 0.662148i \(-0.230355\pi\)
0.749373 + 0.662148i \(0.230355\pi\)
\(312\) 0 0
\(313\) 15.7873 0.892350 0.446175 0.894946i \(-0.352786\pi\)
0.446175 + 0.894946i \(0.352786\pi\)
\(314\) 0 0
\(315\) −3.88971 −0.219160
\(316\) 0 0
\(317\) 24.1707 1.35756 0.678780 0.734342i \(-0.262509\pi\)
0.678780 + 0.734342i \(0.262509\pi\)
\(318\) 0 0
\(319\) −23.3975 −1.31001
\(320\) 0 0
\(321\) −8.95366 −0.499745
\(322\) 0 0
\(323\) −4.41108 −0.245439
\(324\) 0 0
\(325\) 54.6804 3.03312
\(326\) 0 0
\(327\) −13.5921 −0.751642
\(328\) 0 0
\(329\) 4.45071 0.245376
\(330\) 0 0
\(331\) −7.32398 −0.402562 −0.201281 0.979534i \(-0.564510\pi\)
−0.201281 + 0.979534i \(0.564510\pi\)
\(332\) 0 0
\(333\) −4.60850 −0.252544
\(334\) 0 0
\(335\) −9.65321 −0.527411
\(336\) 0 0
\(337\) −8.73670 −0.475918 −0.237959 0.971275i \(-0.576478\pi\)
−0.237959 + 0.971275i \(0.576478\pi\)
\(338\) 0 0
\(339\) −8.35047 −0.453535
\(340\) 0 0
\(341\) 0.601579 0.0325773
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −13.2170 −0.711579
\(346\) 0 0
\(347\) −13.1179 −0.704205 −0.352102 0.935961i \(-0.614533\pi\)
−0.352102 + 0.935961i \(0.614533\pi\)
\(348\) 0 0
\(349\) −9.08910 −0.486529 −0.243264 0.969960i \(-0.578218\pi\)
−0.243264 + 0.969960i \(0.578218\pi\)
\(350\) 0 0
\(351\) −5.39794 −0.288121
\(352\) 0 0
\(353\) 35.7494 1.90275 0.951373 0.308041i \(-0.0996735\pi\)
0.951373 + 0.308041i \(0.0996735\pi\)
\(354\) 0 0
\(355\) −48.3006 −2.56353
\(356\) 0 0
\(357\) 4.41108 0.233459
\(358\) 0 0
\(359\) −20.5357 −1.08383 −0.541915 0.840433i \(-0.682301\pi\)
−0.541915 + 0.840433i \(0.682301\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.692290 0.0363358
\(364\) 0 0
\(365\) 15.8978 0.832128
\(366\) 0 0
\(367\) 7.85870 0.410221 0.205110 0.978739i \(-0.434245\pi\)
0.205110 + 0.978739i \(0.434245\pi\)
\(368\) 0 0
\(369\) −9.77943 −0.509097
\(370\) 0 0
\(371\) 11.2876 0.586026
\(372\) 0 0
\(373\) 17.8126 0.922303 0.461151 0.887321i \(-0.347437\pi\)
0.461151 + 0.887321i \(0.347437\pi\)
\(374\) 0 0
\(375\) 19.9537 1.03040
\(376\) 0 0
\(377\) −39.3383 −2.02602
\(378\) 0 0
\(379\) −1.06425 −0.0546668 −0.0273334 0.999626i \(-0.508702\pi\)
−0.0273334 + 0.999626i \(0.508702\pi\)
\(380\) 0 0
\(381\) 5.70565 0.292309
\(382\) 0 0
\(383\) 20.7009 1.05777 0.528884 0.848694i \(-0.322610\pi\)
0.528884 + 0.848694i \(0.322610\pi\)
\(384\) 0 0
\(385\) −12.4882 −0.636456
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.0260 1.62379 0.811893 0.583807i \(-0.198437\pi\)
0.811893 + 0.583807i \(0.198437\pi\)
\(390\) 0 0
\(391\) 14.9886 0.758004
\(392\) 0 0
\(393\) 10.3505 0.522112
\(394\) 0 0
\(395\) −10.4602 −0.526310
\(396\) 0 0
\(397\) 17.6381 0.885232 0.442616 0.896711i \(-0.354050\pi\)
0.442616 + 0.896711i \(0.354050\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −3.23341 −0.161469 −0.0807345 0.996736i \(-0.525727\pi\)
−0.0807345 + 0.996736i \(0.525727\pi\)
\(402\) 0 0
\(403\) 1.01144 0.0503833
\(404\) 0 0
\(405\) −3.88971 −0.193281
\(406\) 0 0
\(407\) −14.7959 −0.733404
\(408\) 0 0
\(409\) −37.9523 −1.87662 −0.938309 0.345797i \(-0.887609\pi\)
−0.938309 + 0.345797i \(0.887609\pi\)
\(410\) 0 0
\(411\) −16.7959 −0.828479
\(412\) 0 0
\(413\) −5.77156 −0.284000
\(414\) 0 0
\(415\) −15.2840 −0.750264
\(416\) 0 0
\(417\) 16.3469 0.800509
\(418\) 0 0
\(419\) 27.7016 1.35331 0.676655 0.736301i \(-0.263429\pi\)
0.676655 + 0.736301i \(0.263429\pi\)
\(420\) 0 0
\(421\) 13.8054 0.672834 0.336417 0.941713i \(-0.390785\pi\)
0.336417 + 0.941713i \(0.390785\pi\)
\(422\) 0 0
\(423\) 4.45071 0.216401
\(424\) 0 0
\(425\) −44.6836 −2.16747
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −17.3304 −0.836720
\(430\) 0 0
\(431\) −9.40106 −0.452833 −0.226417 0.974031i \(-0.572701\pi\)
−0.226417 + 0.974031i \(0.572701\pi\)
\(432\) 0 0
\(433\) 36.9236 1.77443 0.887217 0.461352i \(-0.152635\pi\)
0.887217 + 0.461352i \(0.152635\pi\)
\(434\) 0 0
\(435\) −28.3469 −1.35913
\(436\) 0 0
\(437\) −3.39794 −0.162545
\(438\) 0 0
\(439\) 10.9040 0.520418 0.260209 0.965552i \(-0.416208\pi\)
0.260209 + 0.965552i \(0.416208\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 33.1704 1.57598 0.787988 0.615691i \(-0.211123\pi\)
0.787988 + 0.615691i \(0.211123\pi\)
\(444\) 0 0
\(445\) −10.5362 −0.499465
\(446\) 0 0
\(447\) −4.82907 −0.228407
\(448\) 0 0
\(449\) −28.3711 −1.33892 −0.669458 0.742850i \(-0.733474\pi\)
−0.669458 + 0.742850i \(0.733474\pi\)
\(450\) 0 0
\(451\) −31.3975 −1.47845
\(452\) 0 0
\(453\) −21.2645 −0.999093
\(454\) 0 0
\(455\) −20.9964 −0.984328
\(456\) 0 0
\(457\) −1.70015 −0.0795298 −0.0397649 0.999209i \(-0.512661\pi\)
−0.0397649 + 0.999209i \(0.512661\pi\)
\(458\) 0 0
\(459\) 4.41108 0.205891
\(460\) 0 0
\(461\) −21.7221 −1.01170 −0.505850 0.862621i \(-0.668821\pi\)
−0.505850 + 0.862621i \(0.668821\pi\)
\(462\) 0 0
\(463\) 13.8186 0.642204 0.321102 0.947045i \(-0.395947\pi\)
0.321102 + 0.947045i \(0.395947\pi\)
\(464\) 0 0
\(465\) 0.728835 0.0337989
\(466\) 0 0
\(467\) 33.4481 1.54779 0.773896 0.633312i \(-0.218305\pi\)
0.773896 + 0.633312i \(0.218305\pi\)
\(468\) 0 0
\(469\) 2.48173 0.114596
\(470\) 0 0
\(471\) −23.1377 −1.06613
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.1299 0.464790
\(476\) 0 0
\(477\) 11.2876 0.516826
\(478\) 0 0
\(479\) −18.2039 −0.831756 −0.415878 0.909421i \(-0.636526\pi\)
−0.415878 + 0.909421i \(0.636526\pi\)
\(480\) 0 0
\(481\) −24.8764 −1.13427
\(482\) 0 0
\(483\) 3.39794 0.154611
\(484\) 0 0
\(485\) 0.883009 0.0400954
\(486\) 0 0
\(487\) −5.03105 −0.227979 −0.113989 0.993482i \(-0.536363\pi\)
−0.113989 + 0.993482i \(0.536363\pi\)
\(488\) 0 0
\(489\) −7.27117 −0.328813
\(490\) 0 0
\(491\) 10.3063 0.465116 0.232558 0.972583i \(-0.425290\pi\)
0.232558 + 0.972583i \(0.425290\pi\)
\(492\) 0 0
\(493\) 32.1464 1.44780
\(494\) 0 0
\(495\) −12.4882 −0.561301
\(496\) 0 0
\(497\) 12.4175 0.557002
\(498\) 0 0
\(499\) 40.3006 1.80410 0.902050 0.431631i \(-0.142062\pi\)
0.902050 + 0.431631i \(0.142062\pi\)
\(500\) 0 0
\(501\) 7.12986 0.318539
\(502\) 0 0
\(503\) 10.7133 0.477682 0.238841 0.971059i \(-0.423233\pi\)
0.238841 + 0.971059i \(0.423233\pi\)
\(504\) 0 0
\(505\) −76.5073 −3.40453
\(506\) 0 0
\(507\) −16.1377 −0.716702
\(508\) 0 0
\(509\) 24.8429 1.10114 0.550571 0.834788i \(-0.314410\pi\)
0.550571 + 0.834788i \(0.314410\pi\)
\(510\) 0 0
\(511\) −4.08714 −0.180804
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 57.2640 2.52335
\(516\) 0 0
\(517\) 14.2893 0.628442
\(518\) 0 0
\(519\) 3.08349 0.135350
\(520\) 0 0
\(521\) 20.0934 0.880307 0.440154 0.897922i \(-0.354924\pi\)
0.440154 + 0.897922i \(0.354924\pi\)
\(522\) 0 0
\(523\) −0.313894 −0.0137256 −0.00686281 0.999976i \(-0.502185\pi\)
−0.00686281 + 0.999976i \(0.502185\pi\)
\(524\) 0 0
\(525\) −10.1299 −0.442103
\(526\) 0 0
\(527\) −0.826525 −0.0360040
\(528\) 0 0
\(529\) −11.4540 −0.498001
\(530\) 0 0
\(531\) −5.77156 −0.250465
\(532\) 0 0
\(533\) −52.7887 −2.28653
\(534\) 0 0
\(535\) −34.8272 −1.50571
\(536\) 0 0
\(537\) −13.3669 −0.576825
\(538\) 0 0
\(539\) 3.21056 0.138289
\(540\) 0 0
\(541\) −28.1749 −1.21133 −0.605667 0.795718i \(-0.707094\pi\)
−0.605667 + 0.795718i \(0.707094\pi\)
\(542\) 0 0
\(543\) −8.15305 −0.349881
\(544\) 0 0
\(545\) −52.8692 −2.26467
\(546\) 0 0
\(547\) 9.89928 0.423262 0.211631 0.977350i \(-0.432122\pi\)
0.211631 + 0.977350i \(0.432122\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −7.28765 −0.310464
\(552\) 0 0
\(553\) 2.68920 0.114356
\(554\) 0 0
\(555\) −17.9257 −0.760905
\(556\) 0 0
\(557\) 26.6477 1.12910 0.564549 0.825400i \(-0.309050\pi\)
0.564549 + 0.825400i \(0.309050\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 14.1620 0.597922
\(562\) 0 0
\(563\) −26.0871 −1.09944 −0.549721 0.835348i \(-0.685266\pi\)
−0.549721 + 0.835348i \(0.685266\pi\)
\(564\) 0 0
\(565\) −32.4809 −1.36648
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −41.1726 −1.72605 −0.863023 0.505164i \(-0.831432\pi\)
−0.863023 + 0.505164i \(0.831432\pi\)
\(570\) 0 0
\(571\) 29.7052 1.24312 0.621561 0.783366i \(-0.286499\pi\)
0.621561 + 0.783366i \(0.286499\pi\)
\(572\) 0 0
\(573\) 15.5722 0.650538
\(574\) 0 0
\(575\) −34.4206 −1.43544
\(576\) 0 0
\(577\) −20.8094 −0.866305 −0.433152 0.901321i \(-0.642599\pi\)
−0.433152 + 0.901321i \(0.642599\pi\)
\(578\) 0 0
\(579\) −8.95727 −0.372251
\(580\) 0 0
\(581\) 3.92935 0.163017
\(582\) 0 0
\(583\) 36.2397 1.50090
\(584\) 0 0
\(585\) −20.9964 −0.868095
\(586\) 0 0
\(587\) −15.6887 −0.647541 −0.323771 0.946136i \(-0.604951\pi\)
−0.323771 + 0.946136i \(0.604951\pi\)
\(588\) 0 0
\(589\) 0.187375 0.00772065
\(590\) 0 0
\(591\) −4.52923 −0.186307
\(592\) 0 0
\(593\) 12.8786 0.528860 0.264430 0.964405i \(-0.414816\pi\)
0.264430 + 0.964405i \(0.414816\pi\)
\(594\) 0 0
\(595\) 17.1578 0.703402
\(596\) 0 0
\(597\) 6.08714 0.249130
\(598\) 0 0
\(599\) 39.2196 1.60247 0.801236 0.598349i \(-0.204176\pi\)
0.801236 + 0.598349i \(0.204176\pi\)
\(600\) 0 0
\(601\) 44.7757 1.82644 0.913219 0.407470i \(-0.133589\pi\)
0.913219 + 0.407470i \(0.133589\pi\)
\(602\) 0 0
\(603\) 2.48173 0.101064
\(604\) 0 0
\(605\) 2.69281 0.109478
\(606\) 0 0
\(607\) −21.5842 −0.876075 −0.438038 0.898957i \(-0.644326\pi\)
−0.438038 + 0.898957i \(0.644326\pi\)
\(608\) 0 0
\(609\) 7.28765 0.295310
\(610\) 0 0
\(611\) 24.0247 0.971934
\(612\) 0 0
\(613\) 8.91651 0.360134 0.180067 0.983654i \(-0.442368\pi\)
0.180067 + 0.983654i \(0.442368\pi\)
\(614\) 0 0
\(615\) −38.0392 −1.53389
\(616\) 0 0
\(617\) 7.97997 0.321262 0.160631 0.987015i \(-0.448647\pi\)
0.160631 + 0.987015i \(0.448647\pi\)
\(618\) 0 0
\(619\) −48.9484 −1.96740 −0.983702 0.179807i \(-0.942453\pi\)
−0.983702 + 0.179807i \(0.942453\pi\)
\(620\) 0 0
\(621\) 3.39794 0.136355
\(622\) 0 0
\(623\) 2.70874 0.108523
\(624\) 0 0
\(625\) 26.9648 1.07859
\(626\) 0 0
\(627\) −3.21056 −0.128218
\(628\) 0 0
\(629\) 20.3284 0.810548
\(630\) 0 0
\(631\) 17.7873 0.708101 0.354050 0.935226i \(-0.384804\pi\)
0.354050 + 0.935226i \(0.384804\pi\)
\(632\) 0 0
\(633\) −23.5574 −0.936324
\(634\) 0 0
\(635\) 22.1933 0.880715
\(636\) 0 0
\(637\) 5.39794 0.213874
\(638\) 0 0
\(639\) 12.4175 0.491229
\(640\) 0 0
\(641\) −47.3548 −1.87040 −0.935200 0.354119i \(-0.884781\pi\)
−0.935200 + 0.354119i \(0.884781\pi\)
\(642\) 0 0
\(643\) 31.6180 1.24689 0.623447 0.781866i \(-0.285732\pi\)
0.623447 + 0.781866i \(0.285732\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.8541 1.44888 0.724441 0.689336i \(-0.242098\pi\)
0.724441 + 0.689336i \(0.242098\pi\)
\(648\) 0 0
\(649\) −18.5300 −0.727365
\(650\) 0 0
\(651\) −0.187375 −0.00734380
\(652\) 0 0
\(653\) −17.2239 −0.674024 −0.337012 0.941500i \(-0.609416\pi\)
−0.337012 + 0.941500i \(0.609416\pi\)
\(654\) 0 0
\(655\) 40.2604 1.57310
\(656\) 0 0
\(657\) −4.08714 −0.159454
\(658\) 0 0
\(659\) −45.2939 −1.76440 −0.882200 0.470875i \(-0.843938\pi\)
−0.882200 + 0.470875i \(0.843938\pi\)
\(660\) 0 0
\(661\) 29.7862 1.15855 0.579274 0.815133i \(-0.303336\pi\)
0.579274 + 0.815133i \(0.303336\pi\)
\(662\) 0 0
\(663\) 23.8107 0.924732
\(664\) 0 0
\(665\) −3.88971 −0.150837
\(666\) 0 0
\(667\) 24.7630 0.958826
\(668\) 0 0
\(669\) −7.27648 −0.281325
\(670\) 0 0
\(671\) 19.2634 0.743654
\(672\) 0 0
\(673\) 12.3390 0.475634 0.237817 0.971310i \(-0.423568\pi\)
0.237817 + 0.971310i \(0.423568\pi\)
\(674\) 0 0
\(675\) −10.1299 −0.389899
\(676\) 0 0
\(677\) −22.1134 −0.849888 −0.424944 0.905220i \(-0.639706\pi\)
−0.424944 + 0.905220i \(0.639706\pi\)
\(678\) 0 0
\(679\) −0.227011 −0.00871190
\(680\) 0 0
\(681\) 10.4619 0.400900
\(682\) 0 0
\(683\) −8.10362 −0.310076 −0.155038 0.987908i \(-0.549550\pi\)
−0.155038 + 0.987908i \(0.549550\pi\)
\(684\) 0 0
\(685\) −65.3311 −2.49617
\(686\) 0 0
\(687\) 4.20055 0.160261
\(688\) 0 0
\(689\) 60.9300 2.32125
\(690\) 0 0
\(691\) 6.94343 0.264141 0.132070 0.991240i \(-0.457838\pi\)
0.132070 + 0.991240i \(0.457838\pi\)
\(692\) 0 0
\(693\) 3.21056 0.121959
\(694\) 0 0
\(695\) 63.5846 2.41190
\(696\) 0 0
\(697\) 43.1378 1.63396
\(698\) 0 0
\(699\) 1.40468 0.0531297
\(700\) 0 0
\(701\) −21.5329 −0.813285 −0.406643 0.913587i \(-0.633301\pi\)
−0.406643 + 0.913587i \(0.633301\pi\)
\(702\) 0 0
\(703\) −4.60850 −0.173813
\(704\) 0 0
\(705\) 17.3120 0.652007
\(706\) 0 0
\(707\) 19.6691 0.739734
\(708\) 0 0
\(709\) −38.7686 −1.45599 −0.727993 0.685584i \(-0.759547\pi\)
−0.727993 + 0.685584i \(0.759547\pi\)
\(710\) 0 0
\(711\) 2.68920 0.100853
\(712\) 0 0
\(713\) −0.636688 −0.0238442
\(714\) 0 0
\(715\) −67.4103 −2.52100
\(716\) 0 0
\(717\) 20.2994 0.758094
\(718\) 0 0
\(719\) −36.7725 −1.37138 −0.685692 0.727892i \(-0.740500\pi\)
−0.685692 + 0.727892i \(0.740500\pi\)
\(720\) 0 0
\(721\) −14.7219 −0.548273
\(722\) 0 0
\(723\) 23.6245 0.878603
\(724\) 0 0
\(725\) −73.8229 −2.74171
\(726\) 0 0
\(727\) 21.2853 0.789427 0.394714 0.918804i \(-0.370844\pi\)
0.394714 + 0.918804i \(0.370844\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.13845 0.226729 0.113364 0.993553i \(-0.463837\pi\)
0.113364 + 0.993553i \(0.463837\pi\)
\(734\) 0 0
\(735\) 3.88971 0.143474
\(736\) 0 0
\(737\) 7.96774 0.293496
\(738\) 0 0
\(739\) 7.89946 0.290586 0.145293 0.989389i \(-0.453587\pi\)
0.145293 + 0.989389i \(0.453587\pi\)
\(740\) 0 0
\(741\) −5.39794 −0.198298
\(742\) 0 0
\(743\) 7.80306 0.286267 0.143133 0.989703i \(-0.454282\pi\)
0.143133 + 0.989703i \(0.454282\pi\)
\(744\) 0 0
\(745\) −18.7837 −0.688182
\(746\) 0 0
\(747\) 3.92935 0.143767
\(748\) 0 0
\(749\) 8.95366 0.327160
\(750\) 0 0
\(751\) 4.19505 0.153080 0.0765399 0.997067i \(-0.475613\pi\)
0.0765399 + 0.997067i \(0.475613\pi\)
\(752\) 0 0
\(753\) −1.82380 −0.0664628
\(754\) 0 0
\(755\) −82.7128 −3.01023
\(756\) 0 0
\(757\) −53.7036 −1.95189 −0.975944 0.218019i \(-0.930040\pi\)
−0.975944 + 0.218019i \(0.930040\pi\)
\(758\) 0 0
\(759\) 10.9093 0.395982
\(760\) 0 0
\(761\) −9.55602 −0.346406 −0.173203 0.984886i \(-0.555412\pi\)
−0.173203 + 0.984886i \(0.555412\pi\)
\(762\) 0 0
\(763\) 13.5921 0.492065
\(764\) 0 0
\(765\) 17.1578 0.620342
\(766\) 0 0
\(767\) −31.1545 −1.12493
\(768\) 0 0
\(769\) 39.1819 1.41294 0.706468 0.707745i \(-0.250288\pi\)
0.706468 + 0.707745i \(0.250288\pi\)
\(770\) 0 0
\(771\) 6.72883 0.242333
\(772\) 0 0
\(773\) 24.0079 0.863503 0.431751 0.901993i \(-0.357896\pi\)
0.431751 + 0.901993i \(0.357896\pi\)
\(774\) 0 0
\(775\) 1.89808 0.0681811
\(776\) 0 0
\(777\) 4.60850 0.165329
\(778\) 0 0
\(779\) −9.77943 −0.350384
\(780\) 0 0
\(781\) 39.8672 1.42656
\(782\) 0 0
\(783\) 7.28765 0.260439
\(784\) 0 0
\(785\) −89.9991 −3.21221
\(786\) 0 0
\(787\) −31.5795 −1.12569 −0.562844 0.826563i \(-0.690293\pi\)
−0.562844 + 0.826563i \(0.690293\pi\)
\(788\) 0 0
\(789\) 13.4371 0.478373
\(790\) 0 0
\(791\) 8.35047 0.296909
\(792\) 0 0
\(793\) 32.3876 1.15012
\(794\) 0 0
\(795\) 43.9057 1.55718
\(796\) 0 0
\(797\) −43.9415 −1.55649 −0.778243 0.627963i \(-0.783889\pi\)
−0.778243 + 0.627963i \(0.783889\pi\)
\(798\) 0 0
\(799\) −19.6324 −0.694546
\(800\) 0 0
\(801\) 2.70874 0.0957086
\(802\) 0 0
\(803\) −13.1220 −0.463065
\(804\) 0 0
\(805\) 13.2170 0.465838
\(806\) 0 0
\(807\) −15.8666 −0.558529
\(808\) 0 0
\(809\) 29.9862 1.05426 0.527129 0.849786i \(-0.323269\pi\)
0.527129 + 0.849786i \(0.323269\pi\)
\(810\) 0 0
\(811\) −43.2495 −1.51870 −0.759348 0.650684i \(-0.774482\pi\)
−0.759348 + 0.650684i \(0.774482\pi\)
\(812\) 0 0
\(813\) 0.667979 0.0234270
\(814\) 0 0
\(815\) −28.2827 −0.990701
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 5.39794 0.188619
\(820\) 0 0
\(821\) 26.7141 0.932327 0.466163 0.884699i \(-0.345636\pi\)
0.466163 + 0.884699i \(0.345636\pi\)
\(822\) 0 0
\(823\) −34.3860 −1.19862 −0.599311 0.800517i \(-0.704559\pi\)
−0.599311 + 0.800517i \(0.704559\pi\)
\(824\) 0 0
\(825\) −32.5226 −1.13229
\(826\) 0 0
\(827\) −44.9941 −1.56460 −0.782298 0.622904i \(-0.785953\pi\)
−0.782298 + 0.622904i \(0.785953\pi\)
\(828\) 0 0
\(829\) 18.4814 0.641886 0.320943 0.947098i \(-0.396000\pi\)
0.320943 + 0.947098i \(0.396000\pi\)
\(830\) 0 0
\(831\) −24.8501 −0.862041
\(832\) 0 0
\(833\) −4.41108 −0.152835
\(834\) 0 0
\(835\) 27.7331 0.959744
\(836\) 0 0
\(837\) −0.187375 −0.00647662
\(838\) 0 0
\(839\) −3.79062 −0.130867 −0.0654334 0.997857i \(-0.520843\pi\)
−0.0654334 + 0.997857i \(0.520843\pi\)
\(840\) 0 0
\(841\) 24.1098 0.831374
\(842\) 0 0
\(843\) −11.0671 −0.381170
\(844\) 0 0
\(845\) −62.7711 −2.15939
\(846\) 0 0
\(847\) −0.692290 −0.0237874
\(848\) 0 0
\(849\) 29.0306 0.996326
\(850\) 0 0
\(851\) 15.6594 0.536797
\(852\) 0 0
\(853\) 22.2268 0.761032 0.380516 0.924774i \(-0.375746\pi\)
0.380516 + 0.924774i \(0.375746\pi\)
\(854\) 0 0
\(855\) −3.88971 −0.133025
\(856\) 0 0
\(857\) 47.8593 1.63484 0.817422 0.576039i \(-0.195403\pi\)
0.817422 + 0.576039i \(0.195403\pi\)
\(858\) 0 0
\(859\) 45.1825 1.54161 0.770803 0.637073i \(-0.219855\pi\)
0.770803 + 0.637073i \(0.219855\pi\)
\(860\) 0 0
\(861\) 9.77943 0.333282
\(862\) 0 0
\(863\) −57.7815 −1.96690 −0.983452 0.181169i \(-0.942012\pi\)
−0.983452 + 0.181169i \(0.942012\pi\)
\(864\) 0 0
\(865\) 11.9939 0.407804
\(866\) 0 0
\(867\) −2.45760 −0.0834644
\(868\) 0 0
\(869\) 8.63384 0.292883
\(870\) 0 0
\(871\) 13.3962 0.453913
\(872\) 0 0
\(873\) −0.227011 −0.00768318
\(874\) 0 0
\(875\) −19.9537 −0.674558
\(876\) 0 0
\(877\) 4.69062 0.158391 0.0791955 0.996859i \(-0.474765\pi\)
0.0791955 + 0.996859i \(0.474765\pi\)
\(878\) 0 0
\(879\) −10.8830 −0.367075
\(880\) 0 0
\(881\) 31.6008 1.06466 0.532329 0.846537i \(-0.321317\pi\)
0.532329 + 0.846537i \(0.321317\pi\)
\(882\) 0 0
\(883\) −53.3294 −1.79468 −0.897339 0.441341i \(-0.854503\pi\)
−0.897339 + 0.441341i \(0.854503\pi\)
\(884\) 0 0
\(885\) −22.4497 −0.754639
\(886\) 0 0
\(887\) −54.7883 −1.83961 −0.919805 0.392375i \(-0.871654\pi\)
−0.919805 + 0.392375i \(0.871654\pi\)
\(888\) 0 0
\(889\) −5.70565 −0.191361
\(890\) 0 0
\(891\) 3.21056 0.107558
\(892\) 0 0
\(893\) 4.45071 0.148937
\(894\) 0 0
\(895\) −51.9935 −1.73795
\(896\) 0 0
\(897\) 18.3419 0.612417
\(898\) 0 0
\(899\) −1.36552 −0.0455427
\(900\) 0 0
\(901\) −49.7907 −1.65877
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −31.7130 −1.05418
\(906\) 0 0
\(907\) −33.2313 −1.10343 −0.551714 0.834034i \(-0.686026\pi\)
−0.551714 + 0.834034i \(0.686026\pi\)
\(908\) 0 0
\(909\) 19.6691 0.652384
\(910\) 0 0
\(911\) 55.8193 1.84938 0.924689 0.380724i \(-0.124325\pi\)
0.924689 + 0.380724i \(0.124325\pi\)
\(912\) 0 0
\(913\) 12.6154 0.417509
\(914\) 0 0
\(915\) 23.3383 0.771540
\(916\) 0 0
\(917\) −10.3505 −0.341803
\(918\) 0 0
\(919\) −24.7495 −0.816411 −0.408205 0.912890i \(-0.633845\pi\)
−0.408205 + 0.912890i \(0.633845\pi\)
\(920\) 0 0
\(921\) −9.77156 −0.321984
\(922\) 0 0
\(923\) 67.0290 2.20629
\(924\) 0 0
\(925\) −46.6835 −1.53494
\(926\) 0 0
\(927\) −14.7219 −0.483531
\(928\) 0 0
\(929\) 21.4232 0.702873 0.351437 0.936212i \(-0.385693\pi\)
0.351437 + 0.936212i \(0.385693\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −26.4307 −0.865302
\(934\) 0 0
\(935\) 55.0863 1.80151
\(936\) 0 0
\(937\) −8.68246 −0.283644 −0.141822 0.989892i \(-0.545296\pi\)
−0.141822 + 0.989892i \(0.545296\pi\)
\(938\) 0 0
\(939\) −15.7873 −0.515199
\(940\) 0 0
\(941\) −1.10977 −0.0361774 −0.0180887 0.999836i \(-0.505758\pi\)
−0.0180887 + 0.999836i \(0.505758\pi\)
\(942\) 0 0
\(943\) 33.2299 1.08211
\(944\) 0 0
\(945\) 3.88971 0.126532
\(946\) 0 0
\(947\) 24.5890 0.799034 0.399517 0.916726i \(-0.369178\pi\)
0.399517 + 0.916726i \(0.369178\pi\)
\(948\) 0 0
\(949\) −22.0621 −0.716166
\(950\) 0 0
\(951\) −24.1707 −0.783787
\(952\) 0 0
\(953\) −52.5773 −1.70315 −0.851573 0.524236i \(-0.824351\pi\)
−0.851573 + 0.524236i \(0.824351\pi\)
\(954\) 0 0
\(955\) 60.5714 1.96004
\(956\) 0 0
\(957\) 23.3975 0.756332
\(958\) 0 0
\(959\) 16.7959 0.542367
\(960\) 0 0
\(961\) −30.9649 −0.998867
\(962\) 0 0
\(963\) 8.95366 0.288528
\(964\) 0 0
\(965\) −34.8412 −1.12158
\(966\) 0 0
\(967\) −17.7190 −0.569805 −0.284902 0.958557i \(-0.591961\pi\)
−0.284902 + 0.958557i \(0.591961\pi\)
\(968\) 0 0
\(969\) 4.41108 0.141704
\(970\) 0 0
\(971\) −18.8300 −0.604284 −0.302142 0.953263i \(-0.597702\pi\)
−0.302142 + 0.953263i \(0.597702\pi\)
\(972\) 0 0
\(973\) −16.3469 −0.524056
\(974\) 0 0
\(975\) −54.6804 −1.75117
\(976\) 0 0
\(977\) −24.3840 −0.780114 −0.390057 0.920791i \(-0.627545\pi\)
−0.390057 + 0.920791i \(0.627545\pi\)
\(978\) 0 0
\(979\) 8.69658 0.277944
\(980\) 0 0
\(981\) 13.5921 0.433961
\(982\) 0 0
\(983\) 57.6388 1.83839 0.919196 0.393801i \(-0.128840\pi\)
0.919196 + 0.393801i \(0.128840\pi\)
\(984\) 0 0
\(985\) −17.6174 −0.561337
\(986\) 0 0
\(987\) −4.45071 −0.141668
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 16.6557 0.529086 0.264543 0.964374i \(-0.414779\pi\)
0.264543 + 0.964374i \(0.414779\pi\)
\(992\) 0 0
\(993\) 7.32398 0.232419
\(994\) 0 0
\(995\) 23.6772 0.750618
\(996\) 0 0
\(997\) 16.8780 0.534532 0.267266 0.963623i \(-0.413880\pi\)
0.267266 + 0.963623i \(0.413880\pi\)
\(998\) 0 0
\(999\) 4.60850 0.145806
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 3192.2.a.ba.1.1 5
3.2 odd 2 9576.2.a.co.1.5 5
4.3 odd 2 6384.2.a.ce.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.ba.1.1 5 1.1 even 1 trivial
6384.2.a.ce.1.1 5 4.3 odd 2
9576.2.a.co.1.5 5 3.2 odd 2