Properties

Label 5520.2.a.bi.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.44949 q^{11} -0.449490 q^{13} +1.00000 q^{15} -0.550510 q^{17} +0.449490 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.34847 q^{29} -9.89898 q^{31} -2.44949 q^{33} -1.00000 q^{35} -5.89898 q^{37} +0.449490 q^{39} +0.550510 q^{41} -2.00000 q^{43} -1.00000 q^{45} -3.55051 q^{47} -6.00000 q^{49} +0.550510 q^{51} +5.44949 q^{53} -2.44949 q^{55} -0.449490 q^{57} +4.34847 q^{59} +15.3485 q^{61} +1.00000 q^{63} +0.449490 q^{65} +7.00000 q^{67} -1.00000 q^{69} -10.3485 q^{71} +9.34847 q^{73} -1.00000 q^{75} +2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} +9.24745 q^{83} +0.550510 q^{85} +4.34847 q^{87} -7.10102 q^{89} -0.449490 q^{91} +9.89898 q^{93} -0.449490 q^{95} +12.8990 q^{97} +2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} - 6q^{17} - 4q^{19} - 2q^{21} + 2q^{23} + 2q^{25} - 2q^{27} + 6q^{29} - 10q^{31} - 2q^{35} - 2q^{37} - 4q^{39} + 6q^{41} - 4q^{43} - 2q^{45} - 12q^{47} - 12q^{49} + 6q^{51} + 6q^{53} + 4q^{57} - 6q^{59} + 16q^{61} + 2q^{63} - 4q^{65} + 14q^{67} - 2q^{69} - 6q^{71} + 4q^{73} - 2q^{75} + 8q^{79} + 2q^{81} - 6q^{83} + 6q^{85} - 6q^{87} - 24q^{89} + 4q^{91} + 10q^{93} + 4q^{95} + 16q^{97} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) −0.449490 −0.124666 −0.0623330 0.998055i \(-0.519854\pi\)
−0.0623330 + 0.998055i \(0.519854\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −0.550510 −0.133518 −0.0667592 0.997769i \(-0.521266\pi\)
−0.0667592 + 0.997769i \(0.521266\pi\)
\(18\) 0 0
\(19\) 0.449490 0.103120 0.0515600 0.998670i \(-0.483581\pi\)
0.0515600 + 0.998670i \(0.483581\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.34847 −0.807490 −0.403745 0.914871i \(-0.632292\pi\)
−0.403745 + 0.914871i \(0.632292\pi\)
\(30\) 0 0
\(31\) −9.89898 −1.77791 −0.888955 0.457995i \(-0.848568\pi\)
−0.888955 + 0.457995i \(0.848568\pi\)
\(32\) 0 0
\(33\) −2.44949 −0.426401
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −5.89898 −0.969786 −0.484893 0.874573i \(-0.661142\pi\)
−0.484893 + 0.874573i \(0.661142\pi\)
\(38\) 0 0
\(39\) 0.449490 0.0719760
\(40\) 0 0
\(41\) 0.550510 0.0859753 0.0429876 0.999076i \(-0.486312\pi\)
0.0429876 + 0.999076i \(0.486312\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −3.55051 −0.517895 −0.258948 0.965891i \(-0.583376\pi\)
−0.258948 + 0.965891i \(0.583376\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0.550510 0.0770869
\(52\) 0 0
\(53\) 5.44949 0.748545 0.374272 0.927319i \(-0.377892\pi\)
0.374272 + 0.927319i \(0.377892\pi\)
\(54\) 0 0
\(55\) −2.44949 −0.330289
\(56\) 0 0
\(57\) −0.449490 −0.0595364
\(58\) 0 0
\(59\) 4.34847 0.566122 0.283061 0.959102i \(-0.408650\pi\)
0.283061 + 0.959102i \(0.408650\pi\)
\(60\) 0 0
\(61\) 15.3485 1.96517 0.982585 0.185813i \(-0.0594920\pi\)
0.982585 + 0.185813i \(0.0594920\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0.449490 0.0557523
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.3485 −1.22814 −0.614069 0.789253i \(-0.710468\pi\)
−0.614069 + 0.789253i \(0.710468\pi\)
\(72\) 0 0
\(73\) 9.34847 1.09416 0.547078 0.837082i \(-0.315740\pi\)
0.547078 + 0.837082i \(0.315740\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 2.44949 0.279145
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.24745 1.01504 0.507520 0.861640i \(-0.330562\pi\)
0.507520 + 0.861640i \(0.330562\pi\)
\(84\) 0 0
\(85\) 0.550510 0.0597112
\(86\) 0 0
\(87\) 4.34847 0.466205
\(88\) 0 0
\(89\) −7.10102 −0.752707 −0.376353 0.926476i \(-0.622822\pi\)
−0.376353 + 0.926476i \(0.622822\pi\)
\(90\) 0 0
\(91\) −0.449490 −0.0471193
\(92\) 0 0
\(93\) 9.89898 1.02648
\(94\) 0 0
\(95\) −0.449490 −0.0461167
\(96\) 0 0
\(97\) 12.8990 1.30969 0.654846 0.755762i \(-0.272733\pi\)
0.654846 + 0.755762i \(0.272733\pi\)
\(98\) 0 0
\(99\) 2.44949 0.246183
\(100\) 0 0
\(101\) −17.4495 −1.73629 −0.868145 0.496311i \(-0.834687\pi\)
−0.868145 + 0.496311i \(0.834687\pi\)
\(102\) 0 0
\(103\) −16.6969 −1.64520 −0.822599 0.568622i \(-0.807477\pi\)
−0.822599 + 0.568622i \(0.807477\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 0.550510 0.0532198 0.0266099 0.999646i \(-0.491529\pi\)
0.0266099 + 0.999646i \(0.491529\pi\)
\(108\) 0 0
\(109\) 16.4495 1.57558 0.787788 0.615947i \(-0.211226\pi\)
0.787788 + 0.615947i \(0.211226\pi\)
\(110\) 0 0
\(111\) 5.89898 0.559906
\(112\) 0 0
\(113\) −16.3485 −1.53793 −0.768967 0.639288i \(-0.779229\pi\)
−0.768967 + 0.639288i \(0.779229\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −0.449490 −0.0415553
\(118\) 0 0
\(119\) −0.550510 −0.0504652
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) −0.550510 −0.0496378
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.44949 0.572300 0.286150 0.958185i \(-0.407624\pi\)
0.286150 + 0.958185i \(0.407624\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −19.5959 −1.71210 −0.856052 0.516890i \(-0.827090\pi\)
−0.856052 + 0.516890i \(0.827090\pi\)
\(132\) 0 0
\(133\) 0.449490 0.0389757
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −7.10102 −0.606681 −0.303341 0.952882i \(-0.598102\pi\)
−0.303341 + 0.952882i \(0.598102\pi\)
\(138\) 0 0
\(139\) −14.7980 −1.25515 −0.627573 0.778558i \(-0.715952\pi\)
−0.627573 + 0.778558i \(0.715952\pi\)
\(140\) 0 0
\(141\) 3.55051 0.299007
\(142\) 0 0
\(143\) −1.10102 −0.0920720
\(144\) 0 0
\(145\) 4.34847 0.361121
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) 13.3485 1.09355 0.546775 0.837280i \(-0.315855\pi\)
0.546775 + 0.837280i \(0.315855\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) −0.550510 −0.0445061
\(154\) 0 0
\(155\) 9.89898 0.795105
\(156\) 0 0
\(157\) −20.5959 −1.64373 −0.821867 0.569680i \(-0.807067\pi\)
−0.821867 + 0.569680i \(0.807067\pi\)
\(158\) 0 0
\(159\) −5.44949 −0.432173
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 2.44949 0.190693
\(166\) 0 0
\(167\) 1.34847 0.104348 0.0521738 0.998638i \(-0.483385\pi\)
0.0521738 + 0.998638i \(0.483385\pi\)
\(168\) 0 0
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) 0.449490 0.0343733
\(172\) 0 0
\(173\) −19.5959 −1.48985 −0.744925 0.667148i \(-0.767515\pi\)
−0.744925 + 0.667148i \(0.767515\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.34847 −0.326851
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 0.898979 0.0668206 0.0334103 0.999442i \(-0.489363\pi\)
0.0334103 + 0.999442i \(0.489363\pi\)
\(182\) 0 0
\(183\) −15.3485 −1.13459
\(184\) 0 0
\(185\) 5.89898 0.433702
\(186\) 0 0
\(187\) −1.34847 −0.0986098
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.24745 −0.452050 −0.226025 0.974122i \(-0.572573\pi\)
−0.226025 + 0.974122i \(0.572573\pi\)
\(192\) 0 0
\(193\) 22.6969 1.63376 0.816881 0.576807i \(-0.195701\pi\)
0.816881 + 0.576807i \(0.195701\pi\)
\(194\) 0 0
\(195\) −0.449490 −0.0321886
\(196\) 0 0
\(197\) −13.1010 −0.933409 −0.466705 0.884413i \(-0.654559\pi\)
−0.466705 + 0.884413i \(0.654559\pi\)
\(198\) 0 0
\(199\) −6.89898 −0.489056 −0.244528 0.969642i \(-0.578633\pi\)
−0.244528 + 0.969642i \(0.578633\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) −4.34847 −0.305203
\(204\) 0 0
\(205\) −0.550510 −0.0384493
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 1.10102 0.0761592
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) 0 0
\(213\) 10.3485 0.709065
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −9.89898 −0.671987
\(218\) 0 0
\(219\) −9.34847 −0.631711
\(220\) 0 0
\(221\) 0.247449 0.0166452
\(222\) 0 0
\(223\) 17.5959 1.17831 0.589155 0.808020i \(-0.299461\pi\)
0.589155 + 0.808020i \(0.299461\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −2.20204 −0.146155 −0.0730773 0.997326i \(-0.523282\pi\)
−0.0730773 + 0.997326i \(0.523282\pi\)
\(228\) 0 0
\(229\) 20.4949 1.35434 0.677170 0.735826i \(-0.263206\pi\)
0.677170 + 0.735826i \(0.263206\pi\)
\(230\) 0 0
\(231\) −2.44949 −0.161165
\(232\) 0 0
\(233\) 2.20204 0.144261 0.0721303 0.997395i \(-0.477020\pi\)
0.0721303 + 0.997395i \(0.477020\pi\)
\(234\) 0 0
\(235\) 3.55051 0.231610
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −11.4495 −0.740606 −0.370303 0.928911i \(-0.620746\pi\)
−0.370303 + 0.928911i \(0.620746\pi\)
\(240\) 0 0
\(241\) −14.6515 −0.943788 −0.471894 0.881655i \(-0.656430\pi\)
−0.471894 + 0.881655i \(0.656430\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −0.202041 −0.0128556
\(248\) 0 0
\(249\) −9.24745 −0.586033
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 2.44949 0.153998
\(254\) 0 0
\(255\) −0.550510 −0.0344743
\(256\) 0 0
\(257\) −10.6515 −0.664424 −0.332212 0.943205i \(-0.607795\pi\)
−0.332212 + 0.943205i \(0.607795\pi\)
\(258\) 0 0
\(259\) −5.89898 −0.366545
\(260\) 0 0
\(261\) −4.34847 −0.269163
\(262\) 0 0
\(263\) −2.75255 −0.169730 −0.0848648 0.996392i \(-0.527046\pi\)
−0.0848648 + 0.996392i \(0.527046\pi\)
\(264\) 0 0
\(265\) −5.44949 −0.334759
\(266\) 0 0
\(267\) 7.10102 0.434575
\(268\) 0 0
\(269\) −0.550510 −0.0335652 −0.0167826 0.999859i \(-0.505342\pi\)
−0.0167826 + 0.999859i \(0.505342\pi\)
\(270\) 0 0
\(271\) −2.30306 −0.139901 −0.0699505 0.997550i \(-0.522284\pi\)
−0.0699505 + 0.997550i \(0.522284\pi\)
\(272\) 0 0
\(273\) 0.449490 0.0272044
\(274\) 0 0
\(275\) 2.44949 0.147710
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −9.89898 −0.592636
\(280\) 0 0
\(281\) −4.65153 −0.277487 −0.138744 0.990328i \(-0.544306\pi\)
−0.138744 + 0.990328i \(0.544306\pi\)
\(282\) 0 0
\(283\) 11.8990 0.707321 0.353660 0.935374i \(-0.384937\pi\)
0.353660 + 0.935374i \(0.384937\pi\)
\(284\) 0 0
\(285\) 0.449490 0.0266255
\(286\) 0 0
\(287\) 0.550510 0.0324956
\(288\) 0 0
\(289\) −16.6969 −0.982173
\(290\) 0 0
\(291\) −12.8990 −0.756152
\(292\) 0 0
\(293\) 3.24745 0.189718 0.0948590 0.995491i \(-0.469760\pi\)
0.0948590 + 0.995491i \(0.469760\pi\)
\(294\) 0 0
\(295\) −4.34847 −0.253178
\(296\) 0 0
\(297\) −2.44949 −0.142134
\(298\) 0 0
\(299\) −0.449490 −0.0259947
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 17.4495 1.00245
\(304\) 0 0
\(305\) −15.3485 −0.878851
\(306\) 0 0
\(307\) 17.3485 0.990129 0.495065 0.868856i \(-0.335144\pi\)
0.495065 + 0.868856i \(0.335144\pi\)
\(308\) 0 0
\(309\) 16.6969 0.949856
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −9.69694 −0.548103 −0.274052 0.961715i \(-0.588364\pi\)
−0.274052 + 0.961715i \(0.588364\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −18.2474 −1.02488 −0.512439 0.858723i \(-0.671258\pi\)
−0.512439 + 0.858723i \(0.671258\pi\)
\(318\) 0 0
\(319\) −10.6515 −0.596371
\(320\) 0 0
\(321\) −0.550510 −0.0307265
\(322\) 0 0
\(323\) −0.247449 −0.0137684
\(324\) 0 0
\(325\) −0.449490 −0.0249332
\(326\) 0 0
\(327\) −16.4495 −0.909659
\(328\) 0 0
\(329\) −3.55051 −0.195746
\(330\) 0 0
\(331\) 14.5959 0.802264 0.401132 0.916020i \(-0.368617\pi\)
0.401132 + 0.916020i \(0.368617\pi\)
\(332\) 0 0
\(333\) −5.89898 −0.323262
\(334\) 0 0
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) −0.202041 −0.0110059 −0.00550294 0.999985i \(-0.501752\pi\)
−0.00550294 + 0.999985i \(0.501752\pi\)
\(338\) 0 0
\(339\) 16.3485 0.887927
\(340\) 0 0
\(341\) −24.2474 −1.31307
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 19.1010 1.02540 0.512698 0.858569i \(-0.328646\pi\)
0.512698 + 0.858569i \(0.328646\pi\)
\(348\) 0 0
\(349\) −25.4949 −1.36471 −0.682355 0.731021i \(-0.739044\pi\)
−0.682355 + 0.731021i \(0.739044\pi\)
\(350\) 0 0
\(351\) 0.449490 0.0239920
\(352\) 0 0
\(353\) −20.4495 −1.08842 −0.544208 0.838950i \(-0.683170\pi\)
−0.544208 + 0.838950i \(0.683170\pi\)
\(354\) 0 0
\(355\) 10.3485 0.549240
\(356\) 0 0
\(357\) 0.550510 0.0291361
\(358\) 0 0
\(359\) 8.44949 0.445947 0.222974 0.974825i \(-0.428424\pi\)
0.222974 + 0.974825i \(0.428424\pi\)
\(360\) 0 0
\(361\) −18.7980 −0.989366
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −9.34847 −0.489321
\(366\) 0 0
\(367\) 15.6969 0.819374 0.409687 0.912226i \(-0.365638\pi\)
0.409687 + 0.912226i \(0.365638\pi\)
\(368\) 0 0
\(369\) 0.550510 0.0286584
\(370\) 0 0
\(371\) 5.44949 0.282923
\(372\) 0 0
\(373\) −11.5959 −0.600414 −0.300207 0.953874i \(-0.597056\pi\)
−0.300207 + 0.953874i \(0.597056\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.95459 0.100667
\(378\) 0 0
\(379\) 33.3939 1.71533 0.857664 0.514210i \(-0.171915\pi\)
0.857664 + 0.514210i \(0.171915\pi\)
\(380\) 0 0
\(381\) −6.44949 −0.330417
\(382\) 0 0
\(383\) −16.3485 −0.835368 −0.417684 0.908592i \(-0.637158\pi\)
−0.417684 + 0.908592i \(0.637158\pi\)
\(384\) 0 0
\(385\) −2.44949 −0.124838
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −32.6969 −1.65780 −0.828900 0.559396i \(-0.811033\pi\)
−0.828900 + 0.559396i \(0.811033\pi\)
\(390\) 0 0
\(391\) −0.550510 −0.0278405
\(392\) 0 0
\(393\) 19.5959 0.988483
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 5.79796 0.290991 0.145496 0.989359i \(-0.453522\pi\)
0.145496 + 0.989359i \(0.453522\pi\)
\(398\) 0 0
\(399\) −0.449490 −0.0225026
\(400\) 0 0
\(401\) −22.8990 −1.14352 −0.571760 0.820421i \(-0.693739\pi\)
−0.571760 + 0.820421i \(0.693739\pi\)
\(402\) 0 0
\(403\) 4.44949 0.221645
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −14.4495 −0.716235
\(408\) 0 0
\(409\) 12.1010 0.598357 0.299178 0.954197i \(-0.403287\pi\)
0.299178 + 0.954197i \(0.403287\pi\)
\(410\) 0 0
\(411\) 7.10102 0.350268
\(412\) 0 0
\(413\) 4.34847 0.213974
\(414\) 0 0
\(415\) −9.24745 −0.453939
\(416\) 0 0
\(417\) 14.7980 0.724659
\(418\) 0 0
\(419\) −38.4495 −1.87838 −0.939190 0.343397i \(-0.888422\pi\)
−0.939190 + 0.343397i \(0.888422\pi\)
\(420\) 0 0
\(421\) 2.24745 0.109534 0.0547670 0.998499i \(-0.482558\pi\)
0.0547670 + 0.998499i \(0.482558\pi\)
\(422\) 0 0
\(423\) −3.55051 −0.172632
\(424\) 0 0
\(425\) −0.550510 −0.0267037
\(426\) 0 0
\(427\) 15.3485 0.742764
\(428\) 0 0
\(429\) 1.10102 0.0531578
\(430\) 0 0
\(431\) −24.4949 −1.17988 −0.589939 0.807448i \(-0.700848\pi\)
−0.589939 + 0.807448i \(0.700848\pi\)
\(432\) 0 0
\(433\) −16.7980 −0.807258 −0.403629 0.914923i \(-0.632251\pi\)
−0.403629 + 0.914923i \(0.632251\pi\)
\(434\) 0 0
\(435\) −4.34847 −0.208493
\(436\) 0 0
\(437\) 0.449490 0.0215020
\(438\) 0 0
\(439\) 7.30306 0.348556 0.174278 0.984696i \(-0.444241\pi\)
0.174278 + 0.984696i \(0.444241\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 15.5505 0.738827 0.369414 0.929265i \(-0.379559\pi\)
0.369414 + 0.929265i \(0.379559\pi\)
\(444\) 0 0
\(445\) 7.10102 0.336621
\(446\) 0 0
\(447\) −13.3485 −0.631361
\(448\) 0 0
\(449\) 3.24745 0.153257 0.0766283 0.997060i \(-0.475585\pi\)
0.0766283 + 0.997060i \(0.475585\pi\)
\(450\) 0 0
\(451\) 1.34847 0.0634969
\(452\) 0 0
\(453\) 14.0000 0.657777
\(454\) 0 0
\(455\) 0.449490 0.0210724
\(456\) 0 0
\(457\) 9.89898 0.463055 0.231527 0.972828i \(-0.425628\pi\)
0.231527 + 0.972828i \(0.425628\pi\)
\(458\) 0 0
\(459\) 0.550510 0.0256956
\(460\) 0 0
\(461\) 30.4949 1.42029 0.710144 0.704056i \(-0.248630\pi\)
0.710144 + 0.704056i \(0.248630\pi\)
\(462\) 0 0
\(463\) −28.9444 −1.34516 −0.672580 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(464\) 0 0
\(465\) −9.89898 −0.459054
\(466\) 0 0
\(467\) −28.8434 −1.33471 −0.667356 0.744739i \(-0.732574\pi\)
−0.667356 + 0.744739i \(0.732574\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 20.5959 0.949010
\(472\) 0 0
\(473\) −4.89898 −0.225255
\(474\) 0 0
\(475\) 0.449490 0.0206240
\(476\) 0 0
\(477\) 5.44949 0.249515
\(478\) 0 0
\(479\) 20.9444 0.956973 0.478487 0.878095i \(-0.341185\pi\)
0.478487 + 0.878095i \(0.341185\pi\)
\(480\) 0 0
\(481\) 2.65153 0.120899
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −12.8990 −0.585712
\(486\) 0 0
\(487\) −34.9444 −1.58348 −0.791741 0.610857i \(-0.790825\pi\)
−0.791741 + 0.610857i \(0.790825\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −17.4495 −0.787484 −0.393742 0.919221i \(-0.628820\pi\)
−0.393742 + 0.919221i \(0.628820\pi\)
\(492\) 0 0
\(493\) 2.39388 0.107815
\(494\) 0 0
\(495\) −2.44949 −0.110096
\(496\) 0 0
\(497\) −10.3485 −0.464192
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) −1.34847 −0.0602452
\(502\) 0 0
\(503\) 37.0454 1.65177 0.825887 0.563836i \(-0.190675\pi\)
0.825887 + 0.563836i \(0.190675\pi\)
\(504\) 0 0
\(505\) 17.4495 0.776492
\(506\) 0 0
\(507\) 12.7980 0.568377
\(508\) 0 0
\(509\) 19.1010 0.846638 0.423319 0.905981i \(-0.360865\pi\)
0.423319 + 0.905981i \(0.360865\pi\)
\(510\) 0 0
\(511\) 9.34847 0.413552
\(512\) 0 0
\(513\) −0.449490 −0.0198455
\(514\) 0 0
\(515\) 16.6969 0.735755
\(516\) 0 0
\(517\) −8.69694 −0.382491
\(518\) 0 0
\(519\) 19.5959 0.860165
\(520\) 0 0
\(521\) 19.8434 0.869354 0.434677 0.900587i \(-0.356863\pi\)
0.434677 + 0.900587i \(0.356863\pi\)
\(522\) 0 0
\(523\) −25.3939 −1.11040 −0.555198 0.831718i \(-0.687358\pi\)
−0.555198 + 0.831718i \(0.687358\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 5.44949 0.237384
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.34847 0.188707
\(532\) 0 0
\(533\) −0.247449 −0.0107182
\(534\) 0 0
\(535\) −0.550510 −0.0238006
\(536\) 0 0
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) −14.6969 −0.633042
\(540\) 0 0
\(541\) 3.10102 0.133323 0.0666616 0.997776i \(-0.478765\pi\)
0.0666616 + 0.997776i \(0.478765\pi\)
\(542\) 0 0
\(543\) −0.898979 −0.0385789
\(544\) 0 0
\(545\) −16.4495 −0.704619
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 15.3485 0.655057
\(550\) 0 0
\(551\) −1.95459 −0.0832684
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −5.89898 −0.250398
\(556\) 0 0
\(557\) 25.0454 1.06121 0.530604 0.847620i \(-0.321965\pi\)
0.530604 + 0.847620i \(0.321965\pi\)
\(558\) 0 0
\(559\) 0.898979 0.0380228
\(560\) 0 0
\(561\) 1.34847 0.0569324
\(562\) 0 0
\(563\) −0.550510 −0.0232012 −0.0116006 0.999933i \(-0.503693\pi\)
−0.0116006 + 0.999933i \(0.503693\pi\)
\(564\) 0 0
\(565\) 16.3485 0.687785
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 40.2929 1.68916 0.844582 0.535426i \(-0.179849\pi\)
0.844582 + 0.535426i \(0.179849\pi\)
\(570\) 0 0
\(571\) 21.1464 0.884950 0.442475 0.896781i \(-0.354100\pi\)
0.442475 + 0.896781i \(0.354100\pi\)
\(572\) 0 0
\(573\) 6.24745 0.260991
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −28.0000 −1.16566 −0.582828 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 0 0
\(579\) −22.6969 −0.943253
\(580\) 0 0
\(581\) 9.24745 0.383649
\(582\) 0 0
\(583\) 13.3485 0.552837
\(584\) 0 0
\(585\) 0.449490 0.0185841
\(586\) 0 0
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) −4.44949 −0.183338
\(590\) 0 0
\(591\) 13.1010 0.538904
\(592\) 0 0
\(593\) 32.4495 1.33254 0.666270 0.745710i \(-0.267890\pi\)
0.666270 + 0.745710i \(0.267890\pi\)
\(594\) 0 0
\(595\) 0.550510 0.0225687
\(596\) 0 0
\(597\) 6.89898 0.282356
\(598\) 0 0
\(599\) −26.6969 −1.09081 −0.545404 0.838174i \(-0.683624\pi\)
−0.545404 + 0.838174i \(0.683624\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 7.00000 0.285062
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −8.24745 −0.334754 −0.167377 0.985893i \(-0.553530\pi\)
−0.167377 + 0.985893i \(0.553530\pi\)
\(608\) 0 0
\(609\) 4.34847 0.176209
\(610\) 0 0
\(611\) 1.59592 0.0645639
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 0.550510 0.0221987
\(616\) 0 0
\(617\) 17.4495 0.702490 0.351245 0.936284i \(-0.385758\pi\)
0.351245 + 0.936284i \(0.385758\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −7.10102 −0.284496
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.10102 −0.0439705
\(628\) 0 0
\(629\) 3.24745 0.129484
\(630\) 0 0
\(631\) −34.4495 −1.37141 −0.685706 0.727878i \(-0.740507\pi\)
−0.685706 + 0.727878i \(0.740507\pi\)
\(632\) 0 0
\(633\) 11.0000 0.437211
\(634\) 0 0
\(635\) −6.44949 −0.255940
\(636\) 0 0
\(637\) 2.69694 0.106857
\(638\) 0 0
\(639\) −10.3485 −0.409379
\(640\) 0 0
\(641\) −26.9444 −1.06424 −0.532120 0.846669i \(-0.678604\pi\)
−0.532120 + 0.846669i \(0.678604\pi\)
\(642\) 0 0
\(643\) −19.6969 −0.776771 −0.388386 0.921497i \(-0.626967\pi\)
−0.388386 + 0.921497i \(0.626967\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) −46.0454 −1.81023 −0.905116 0.425165i \(-0.860216\pi\)
−0.905116 + 0.425165i \(0.860216\pi\)
\(648\) 0 0
\(649\) 10.6515 0.418109
\(650\) 0 0
\(651\) 9.89898 0.387972
\(652\) 0 0
\(653\) −19.8434 −0.776531 −0.388265 0.921548i \(-0.626926\pi\)
−0.388265 + 0.921548i \(0.626926\pi\)
\(654\) 0 0
\(655\) 19.5959 0.765676
\(656\) 0 0
\(657\) 9.34847 0.364719
\(658\) 0 0
\(659\) −29.1464 −1.13538 −0.567692 0.823241i \(-0.692163\pi\)
−0.567692 + 0.823241i \(0.692163\pi\)
\(660\) 0 0
\(661\) −42.0908 −1.63714 −0.818571 0.574405i \(-0.805234\pi\)
−0.818571 + 0.574405i \(0.805234\pi\)
\(662\) 0 0
\(663\) −0.247449 −0.00961011
\(664\) 0 0
\(665\) −0.449490 −0.0174305
\(666\) 0 0
\(667\) −4.34847 −0.168373
\(668\) 0 0
\(669\) −17.5959 −0.680297
\(670\) 0 0
\(671\) 37.5959 1.45137
\(672\) 0 0
\(673\) 17.5505 0.676522 0.338261 0.941052i \(-0.390161\pi\)
0.338261 + 0.941052i \(0.390161\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −17.4495 −0.670638 −0.335319 0.942105i \(-0.608844\pi\)
−0.335319 + 0.942105i \(0.608844\pi\)
\(678\) 0 0
\(679\) 12.8990 0.495017
\(680\) 0 0
\(681\) 2.20204 0.0843824
\(682\) 0 0
\(683\) 6.24745 0.239052 0.119526 0.992831i \(-0.461863\pi\)
0.119526 + 0.992831i \(0.461863\pi\)
\(684\) 0 0
\(685\) 7.10102 0.271316
\(686\) 0 0
\(687\) −20.4949 −0.781929
\(688\) 0 0
\(689\) −2.44949 −0.0933181
\(690\) 0 0
\(691\) −35.7980 −1.36182 −0.680909 0.732368i \(-0.738415\pi\)
−0.680909 + 0.732368i \(0.738415\pi\)
\(692\) 0 0
\(693\) 2.44949 0.0930484
\(694\) 0 0
\(695\) 14.7980 0.561319
\(696\) 0 0
\(697\) −0.303062 −0.0114793
\(698\) 0 0
\(699\) −2.20204 −0.0832888
\(700\) 0 0
\(701\) −7.95459 −0.300441 −0.150220 0.988653i \(-0.547998\pi\)
−0.150220 + 0.988653i \(0.547998\pi\)
\(702\) 0 0
\(703\) −2.65153 −0.100004
\(704\) 0 0
\(705\) −3.55051 −0.133720
\(706\) 0 0
\(707\) −17.4495 −0.656256
\(708\) 0 0
\(709\) 44.7423 1.68033 0.840167 0.542328i \(-0.182457\pi\)
0.840167 + 0.542328i \(0.182457\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −9.89898 −0.370720
\(714\) 0 0
\(715\) 1.10102 0.0411758
\(716\) 0 0
\(717\) 11.4495 0.427589
\(718\) 0 0
\(719\) −41.9444 −1.56426 −0.782131 0.623114i \(-0.785867\pi\)
−0.782131 + 0.623114i \(0.785867\pi\)
\(720\) 0 0
\(721\) −16.6969 −0.621826
\(722\) 0 0
\(723\) 14.6515 0.544896
\(724\) 0 0
\(725\) −4.34847 −0.161498
\(726\) 0 0
\(727\) 45.6969 1.69481 0.847403 0.530951i \(-0.178165\pi\)
0.847403 + 0.530951i \(0.178165\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.10102 0.0407227
\(732\) 0 0
\(733\) 30.5959 1.13009 0.565043 0.825061i \(-0.308860\pi\)
0.565043 + 0.825061i \(0.308860\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 17.1464 0.631597
\(738\) 0 0
\(739\) 28.7980 1.05935 0.529675 0.848201i \(-0.322314\pi\)
0.529675 + 0.848201i \(0.322314\pi\)
\(740\) 0 0
\(741\) 0.202041 0.00742216
\(742\) 0 0
\(743\) 2.20204 0.0807851 0.0403925 0.999184i \(-0.487139\pi\)
0.0403925 + 0.999184i \(0.487139\pi\)
\(744\) 0 0
\(745\) −13.3485 −0.489050
\(746\) 0 0
\(747\) 9.24745 0.338346
\(748\) 0 0
\(749\) 0.550510 0.0201152
\(750\) 0 0
\(751\) −45.3485 −1.65479 −0.827395 0.561621i \(-0.810178\pi\)
−0.827395 + 0.561621i \(0.810178\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) 25.6969 0.933971 0.466986 0.884265i \(-0.345340\pi\)
0.466986 + 0.884265i \(0.345340\pi\)
\(758\) 0 0
\(759\) −2.44949 −0.0889108
\(760\) 0 0
\(761\) −20.1464 −0.730307 −0.365154 0.930947i \(-0.618984\pi\)
−0.365154 + 0.930947i \(0.618984\pi\)
\(762\) 0 0
\(763\) 16.4495 0.595512
\(764\) 0 0
\(765\) 0.550510 0.0199037
\(766\) 0 0
\(767\) −1.95459 −0.0705762
\(768\) 0 0
\(769\) 5.55051 0.200157 0.100078 0.994980i \(-0.468091\pi\)
0.100078 + 0.994980i \(0.468091\pi\)
\(770\) 0 0
\(771\) 10.6515 0.383606
\(772\) 0 0
\(773\) 52.2929 1.88084 0.940422 0.340010i \(-0.110431\pi\)
0.940422 + 0.340010i \(0.110431\pi\)
\(774\) 0 0
\(775\) −9.89898 −0.355582
\(776\) 0 0
\(777\) 5.89898 0.211625
\(778\) 0 0
\(779\) 0.247449 0.00886577
\(780\) 0 0
\(781\) −25.3485 −0.907040
\(782\) 0 0
\(783\) 4.34847 0.155402
\(784\) 0 0
\(785\) 20.5959 0.735100
\(786\) 0 0
\(787\) −7.69694 −0.274366 −0.137183 0.990546i \(-0.543805\pi\)
−0.137183 + 0.990546i \(0.543805\pi\)
\(788\) 0 0
\(789\) 2.75255 0.0979934
\(790\) 0 0
\(791\) −16.3485 −0.581285
\(792\) 0 0
\(793\) −6.89898 −0.244990
\(794\) 0 0
\(795\) 5.44949 0.193273
\(796\) 0 0
\(797\) −15.2474 −0.540092 −0.270046 0.962847i \(-0.587039\pi\)
−0.270046 + 0.962847i \(0.587039\pi\)
\(798\) 0 0
\(799\) 1.95459 0.0691485
\(800\) 0 0
\(801\) −7.10102 −0.250902
\(802\) 0 0
\(803\) 22.8990 0.808087
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) 0.550510 0.0193789
\(808\) 0 0
\(809\) 11.4495 0.402543 0.201271 0.979536i \(-0.435493\pi\)
0.201271 + 0.979536i \(0.435493\pi\)
\(810\) 0 0
\(811\) −16.3939 −0.575667 −0.287833 0.957680i \(-0.592935\pi\)
−0.287833 + 0.957680i \(0.592935\pi\)
\(812\) 0 0
\(813\) 2.30306 0.0807719
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) −0.898979 −0.0314513
\(818\) 0 0
\(819\) −0.449490 −0.0157064
\(820\) 0 0
\(821\) 50.6969 1.76934 0.884668 0.466222i \(-0.154385\pi\)
0.884668 + 0.466222i \(0.154385\pi\)
\(822\) 0 0
\(823\) 35.5959 1.24080 0.620398 0.784287i \(-0.286971\pi\)
0.620398 + 0.784287i \(0.286971\pi\)
\(824\) 0 0
\(825\) −2.44949 −0.0852803
\(826\) 0 0
\(827\) 2.14643 0.0746386 0.0373193 0.999303i \(-0.488118\pi\)
0.0373193 + 0.999303i \(0.488118\pi\)
\(828\) 0 0
\(829\) 1.69694 0.0589371 0.0294686 0.999566i \(-0.490619\pi\)
0.0294686 + 0.999566i \(0.490619\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 3.30306 0.114444
\(834\) 0 0
\(835\) −1.34847 −0.0466657
\(836\) 0 0
\(837\) 9.89898 0.342159
\(838\) 0 0
\(839\) −33.1918 −1.14591 −0.572955 0.819587i \(-0.694203\pi\)
−0.572955 + 0.819587i \(0.694203\pi\)
\(840\) 0 0
\(841\) −10.0908 −0.347959
\(842\) 0 0
\(843\) 4.65153 0.160207
\(844\) 0 0
\(845\) 12.7980 0.440263
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −11.8990 −0.408372
\(850\) 0 0
\(851\) −5.89898 −0.202214
\(852\) 0 0
\(853\) −42.0908 −1.44116 −0.720581 0.693371i \(-0.756125\pi\)
−0.720581 + 0.693371i \(0.756125\pi\)
\(854\) 0 0
\(855\) −0.449490 −0.0153722
\(856\) 0 0
\(857\) 25.5959 0.874340 0.437170 0.899379i \(-0.355981\pi\)
0.437170 + 0.899379i \(0.355981\pi\)
\(858\) 0 0
\(859\) 40.1918 1.37133 0.685664 0.727918i \(-0.259512\pi\)
0.685664 + 0.727918i \(0.259512\pi\)
\(860\) 0 0
\(861\) −0.550510 −0.0187613
\(862\) 0 0
\(863\) 42.4949 1.44654 0.723272 0.690564i \(-0.242637\pi\)
0.723272 + 0.690564i \(0.242637\pi\)
\(864\) 0 0
\(865\) 19.5959 0.666281
\(866\) 0 0
\(867\) 16.6969 0.567058
\(868\) 0 0
\(869\) 9.79796 0.332373
\(870\) 0 0
\(871\) −3.14643 −0.106613
\(872\) 0 0
\(873\) 12.8990 0.436564
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 38.4949 1.29988 0.649940 0.759985i \(-0.274794\pi\)
0.649940 + 0.759985i \(0.274794\pi\)
\(878\) 0 0
\(879\) −3.24745 −0.109534
\(880\) 0 0
\(881\) 3.55051 0.119620 0.0598099 0.998210i \(-0.480951\pi\)
0.0598099 + 0.998210i \(0.480951\pi\)
\(882\) 0 0
\(883\) 42.4495 1.42854 0.714270 0.699871i \(-0.246759\pi\)
0.714270 + 0.699871i \(0.246759\pi\)
\(884\) 0 0
\(885\) 4.34847 0.146172
\(886\) 0 0
\(887\) −42.4949 −1.42684 −0.713420 0.700737i \(-0.752855\pi\)
−0.713420 + 0.700737i \(0.752855\pi\)
\(888\) 0 0
\(889\) 6.44949 0.216309
\(890\) 0 0
\(891\) 2.44949 0.0820610
\(892\) 0 0
\(893\) −1.59592 −0.0534054
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 0.449490 0.0150080
\(898\) 0 0
\(899\) 43.0454 1.43564
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) −0.898979 −0.0298831
\(906\) 0 0
\(907\) −19.2020 −0.637593 −0.318797 0.947823i \(-0.603279\pi\)
−0.318797 + 0.947823i \(0.603279\pi\)
\(908\) 0 0
\(909\) −17.4495 −0.578763
\(910\) 0 0
\(911\) 33.7980 1.11978 0.559888 0.828568i \(-0.310844\pi\)
0.559888 + 0.828568i \(0.310844\pi\)
\(912\) 0 0
\(913\) 22.6515 0.749656
\(914\) 0 0
\(915\) 15.3485 0.507405
\(916\) 0 0
\(917\) −19.5959 −0.647114
\(918\) 0 0
\(919\) −13.3939 −0.441823 −0.220912 0.975294i \(-0.570903\pi\)
−0.220912 + 0.975294i \(0.570903\pi\)
\(920\) 0 0
\(921\) −17.3485 −0.571651
\(922\) 0 0
\(923\) 4.65153 0.153107
\(924\) 0 0
\(925\) −5.89898 −0.193957
\(926\) 0 0
\(927\) −16.6969 −0.548399
\(928\) 0 0
\(929\) −40.8434 −1.34003 −0.670014 0.742349i \(-0.733712\pi\)
−0.670014 + 0.742349i \(0.733712\pi\)
\(930\) 0 0
\(931\) −2.69694 −0.0883886
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 1.34847 0.0440997
\(936\) 0 0
\(937\) 0.898979 0.0293684 0.0146842 0.999892i \(-0.495326\pi\)
0.0146842 + 0.999892i \(0.495326\pi\)
\(938\) 0 0
\(939\) 9.69694 0.316448
\(940\) 0 0
\(941\) −59.1464 −1.92812 −0.964059 0.265687i \(-0.914401\pi\)
−0.964059 + 0.265687i \(0.914401\pi\)
\(942\) 0 0
\(943\) 0.550510 0.0179271
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −48.4949 −1.57587 −0.787936 0.615757i \(-0.788850\pi\)
−0.787936 + 0.615757i \(0.788850\pi\)
\(948\) 0 0
\(949\) −4.20204 −0.136404
\(950\) 0 0
\(951\) 18.2474 0.591714
\(952\) 0 0
\(953\) 49.5959 1.60657 0.803285 0.595595i \(-0.203084\pi\)
0.803285 + 0.595595i \(0.203084\pi\)
\(954\) 0 0
\(955\) 6.24745 0.202163
\(956\) 0 0
\(957\) 10.6515 0.344315
\(958\) 0 0
\(959\) −7.10102 −0.229304
\(960\) 0 0
\(961\) 66.9898 2.16096
\(962\) 0 0
\(963\) 0.550510 0.0177399
\(964\) 0 0
\(965\) −22.6969 −0.730640
\(966\) 0 0
\(967\) −9.95459 −0.320118 −0.160059 0.987107i \(-0.551168\pi\)
−0.160059 + 0.987107i \(0.551168\pi\)
\(968\) 0 0
\(969\) 0.247449 0.00794920
\(970\) 0 0
\(971\) 46.2929 1.48561 0.742804 0.669509i \(-0.233495\pi\)
0.742804 + 0.669509i \(0.233495\pi\)
\(972\) 0 0
\(973\) −14.7980 −0.474401
\(974\) 0 0
\(975\) 0.449490 0.0143952
\(976\) 0 0
\(977\) 34.3485 1.09890 0.549452 0.835525i \(-0.314836\pi\)
0.549452 + 0.835525i \(0.314836\pi\)
\(978\) 0 0
\(979\) −17.3939 −0.555911
\(980\) 0 0
\(981\) 16.4495 0.525192
\(982\) 0 0
\(983\) −39.7423 −1.26758 −0.633792 0.773504i \(-0.718502\pi\)
−0.633792 + 0.773504i \(0.718502\pi\)
\(984\) 0 0
\(985\) 13.1010 0.417433
\(986\) 0 0
\(987\) 3.55051 0.113014
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 59.7878 1.89922 0.949610 0.313433i \(-0.101479\pi\)
0.949610 + 0.313433i \(0.101479\pi\)
\(992\) 0 0
\(993\) −14.5959 −0.463187
\(994\) 0 0
\(995\) 6.89898 0.218712
\(996\) 0 0
\(997\) −48.0908 −1.52305 −0.761526 0.648135i \(-0.775549\pi\)
−0.761526 + 0.648135i \(0.775549\pi\)
\(998\) 0 0
\(999\) 5.89898 0.186635
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 5520.2.a.bi.1.2 2
4.3 odd 2 345.2.a.i.1.2 2
12.11 even 2 1035.2.a.k.1.1 2
20.3 even 4 1725.2.b.m.1174.2 4
20.7 even 4 1725.2.b.m.1174.3 4
20.19 odd 2 1725.2.a.y.1.1 2
60.59 even 2 5175.2.a.bl.1.2 2
92.91 even 2 7935.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.i.1.2 2 4.3 odd 2
1035.2.a.k.1.1 2 12.11 even 2
1725.2.a.y.1.1 2 20.19 odd 2
1725.2.b.m.1174.2 4 20.3 even 4
1725.2.b.m.1174.3 4 20.7 even 4
5175.2.a.bl.1.2 2 60.59 even 2
5520.2.a.bi.1.2 2 1.1 even 1 trivial
7935.2.a.t.1.2 2 92.91 even 2