Properties

Label 4560.2.a.bf.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +2.24264 q^{11} -3.41421 q^{13} +1.00000 q^{15} +1.17157 q^{17} +1.00000 q^{19} -1.41421 q^{21} -7.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.41421 q^{29} +3.17157 q^{31} -2.24264 q^{33} -1.41421 q^{35} -3.41421 q^{37} +3.41421 q^{39} -0.242641 q^{41} -12.2426 q^{43} -1.00000 q^{45} +7.65685 q^{47} -5.00000 q^{49} -1.17157 q^{51} +8.00000 q^{53} -2.24264 q^{55} -1.00000 q^{57} -12.4853 q^{59} +7.31371 q^{61} +1.41421 q^{63} +3.41421 q^{65} +9.65685 q^{67} +7.65685 q^{69} +10.8284 q^{71} -7.65685 q^{73} -1.00000 q^{75} +3.17157 q^{77} +1.00000 q^{81} -12.8284 q^{83} -1.17157 q^{85} -1.41421 q^{87} +5.89949 q^{89} -4.82843 q^{91} -3.17157 q^{93} -1.00000 q^{95} -9.75736 q^{97} +2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} - 4q^{11} - 4q^{13} + 2q^{15} + 8q^{17} + 2q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 12q^{31} + 4q^{33} - 4q^{37} + 4q^{39} + 8q^{41} - 16q^{43} - 2q^{45} + 4q^{47} - 10q^{49} - 8q^{51} + 16q^{53} + 4q^{55} - 2q^{57} - 8q^{59} - 8q^{61} + 4q^{65} + 8q^{67} + 4q^{69} + 16q^{71} - 4q^{73} - 2q^{75} + 12q^{77} + 2q^{81} - 20q^{83} - 8q^{85} - 8q^{89} - 4q^{91} - 12q^{93} - 2q^{95} - 28q^{97} - 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.24264 0.676182 0.338091 0.941113i \(-0.390219\pi\)
0.338091 + 0.941113i \(0.390219\pi\)
\(12\) 0 0
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) 3.17157 0.569631 0.284816 0.958582i \(-0.408068\pi\)
0.284816 + 0.958582i \(0.408068\pi\)
\(32\) 0 0
\(33\) −2.24264 −0.390394
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) 0 0
\(39\) 3.41421 0.546712
\(40\) 0 0
\(41\) −0.242641 −0.0378941 −0.0189471 0.999820i \(-0.506031\pi\)
−0.0189471 + 0.999820i \(0.506031\pi\)
\(42\) 0 0
\(43\) −12.2426 −1.86699 −0.933493 0.358597i \(-0.883255\pi\)
−0.933493 + 0.358597i \(0.883255\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 7.65685 1.11687 0.558433 0.829549i \(-0.311403\pi\)
0.558433 + 0.829549i \(0.311403\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −1.17157 −0.164053
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) −2.24264 −0.302398
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −12.4853 −1.62545 −0.812723 0.582651i \(-0.802016\pi\)
−0.812723 + 0.582651i \(0.802016\pi\)
\(60\) 0 0
\(61\) 7.31371 0.936424 0.468212 0.883616i \(-0.344898\pi\)
0.468212 + 0.883616i \(0.344898\pi\)
\(62\) 0 0
\(63\) 1.41421 0.178174
\(64\) 0 0
\(65\) 3.41421 0.423481
\(66\) 0 0
\(67\) 9.65685 1.17977 0.589886 0.807486i \(-0.299173\pi\)
0.589886 + 0.807486i \(0.299173\pi\)
\(68\) 0 0
\(69\) 7.65685 0.921777
\(70\) 0 0
\(71\) 10.8284 1.28510 0.642549 0.766245i \(-0.277877\pi\)
0.642549 + 0.766245i \(0.277877\pi\)
\(72\) 0 0
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.17157 0.361434
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.8284 −1.40810 −0.704051 0.710149i \(-0.748628\pi\)
−0.704051 + 0.710149i \(0.748628\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) 0 0
\(87\) −1.41421 −0.151620
\(88\) 0 0
\(89\) 5.89949 0.625345 0.312673 0.949861i \(-0.398776\pi\)
0.312673 + 0.949861i \(0.398776\pi\)
\(90\) 0 0
\(91\) −4.82843 −0.506157
\(92\) 0 0
\(93\) −3.17157 −0.328877
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −9.75736 −0.990710 −0.495355 0.868691i \(-0.664962\pi\)
−0.495355 + 0.868691i \(0.664962\pi\)
\(98\) 0 0
\(99\) 2.24264 0.225394
\(100\) 0 0
\(101\) −3.17157 −0.315583 −0.157792 0.987472i \(-0.550437\pi\)
−0.157792 + 0.987472i \(0.550437\pi\)
\(102\) 0 0
\(103\) 7.31371 0.720641 0.360321 0.932829i \(-0.382667\pi\)
0.360321 + 0.932829i \(0.382667\pi\)
\(104\) 0 0
\(105\) 1.41421 0.138013
\(106\) 0 0
\(107\) −19.3137 −1.86713 −0.933563 0.358412i \(-0.883318\pi\)
−0.933563 + 0.358412i \(0.883318\pi\)
\(108\) 0 0
\(109\) −6.48528 −0.621177 −0.310589 0.950544i \(-0.600526\pi\)
−0.310589 + 0.950544i \(0.600526\pi\)
\(110\) 0 0
\(111\) 3.41421 0.324063
\(112\) 0 0
\(113\) 10.1421 0.954092 0.477046 0.878878i \(-0.341708\pi\)
0.477046 + 0.878878i \(0.341708\pi\)
\(114\) 0 0
\(115\) 7.65685 0.714005
\(116\) 0 0
\(117\) −3.41421 −0.315644
\(118\) 0 0
\(119\) 1.65685 0.151884
\(120\) 0 0
\(121\) −5.97056 −0.542778
\(122\) 0 0
\(123\) 0.242641 0.0218782
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.3137 −1.71381 −0.856907 0.515471i \(-0.827617\pi\)
−0.856907 + 0.515471i \(0.827617\pi\)
\(128\) 0 0
\(129\) 12.2426 1.07790
\(130\) 0 0
\(131\) 8.58579 0.750144 0.375072 0.926996i \(-0.377618\pi\)
0.375072 + 0.926996i \(0.377618\pi\)
\(132\) 0 0
\(133\) 1.41421 0.122628
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 14.1421 1.19952 0.599760 0.800180i \(-0.295263\pi\)
0.599760 + 0.800180i \(0.295263\pi\)
\(140\) 0 0
\(141\) −7.65685 −0.644823
\(142\) 0 0
\(143\) −7.65685 −0.640298
\(144\) 0 0
\(145\) −1.41421 −0.117444
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 6.48528 0.527765 0.263882 0.964555i \(-0.414997\pi\)
0.263882 + 0.964555i \(0.414997\pi\)
\(152\) 0 0
\(153\) 1.17157 0.0947161
\(154\) 0 0
\(155\) −3.17157 −0.254747
\(156\) 0 0
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) −10.8284 −0.853400
\(162\) 0 0
\(163\) −21.8995 −1.71530 −0.857650 0.514233i \(-0.828077\pi\)
−0.857650 + 0.514233i \(0.828077\pi\)
\(164\) 0 0
\(165\) 2.24264 0.174589
\(166\) 0 0
\(167\) 17.3137 1.33977 0.669887 0.742463i \(-0.266342\pi\)
0.669887 + 0.742463i \(0.266342\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0 0
\(175\) 1.41421 0.106904
\(176\) 0 0
\(177\) 12.4853 0.938451
\(178\) 0 0
\(179\) −16.4853 −1.23217 −0.616084 0.787681i \(-0.711282\pi\)
−0.616084 + 0.787681i \(0.711282\pi\)
\(180\) 0 0
\(181\) −20.8284 −1.54816 −0.774082 0.633085i \(-0.781788\pi\)
−0.774082 + 0.633085i \(0.781788\pi\)
\(182\) 0 0
\(183\) −7.31371 −0.540645
\(184\) 0 0
\(185\) 3.41421 0.251018
\(186\) 0 0
\(187\) 2.62742 0.192136
\(188\) 0 0
\(189\) −1.41421 −0.102869
\(190\) 0 0
\(191\) −10.2426 −0.741131 −0.370566 0.928806i \(-0.620836\pi\)
−0.370566 + 0.928806i \(0.620836\pi\)
\(192\) 0 0
\(193\) 5.07107 0.365023 0.182512 0.983204i \(-0.441577\pi\)
0.182512 + 0.983204i \(0.441577\pi\)
\(194\) 0 0
\(195\) −3.41421 −0.244497
\(196\) 0 0
\(197\) 6.82843 0.486505 0.243253 0.969963i \(-0.421786\pi\)
0.243253 + 0.969963i \(0.421786\pi\)
\(198\) 0 0
\(199\) −18.1421 −1.28606 −0.643031 0.765840i \(-0.722323\pi\)
−0.643031 + 0.765840i \(0.722323\pi\)
\(200\) 0 0
\(201\) −9.65685 −0.681142
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0.242641 0.0169468
\(206\) 0 0
\(207\) −7.65685 −0.532188
\(208\) 0 0
\(209\) 2.24264 0.155127
\(210\) 0 0
\(211\) −7.31371 −0.503496 −0.251748 0.967793i \(-0.581005\pi\)
−0.251748 + 0.967793i \(0.581005\pi\)
\(212\) 0 0
\(213\) −10.8284 −0.741952
\(214\) 0 0
\(215\) 12.2426 0.834941
\(216\) 0 0
\(217\) 4.48528 0.304481
\(218\) 0 0
\(219\) 7.65685 0.517402
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −18.6274 −1.24738 −0.623692 0.781670i \(-0.714368\pi\)
−0.623692 + 0.781670i \(0.714368\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −14.9706 −0.993631 −0.496816 0.867856i \(-0.665497\pi\)
−0.496816 + 0.867856i \(0.665497\pi\)
\(228\) 0 0
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) 0 0
\(231\) −3.17157 −0.208674
\(232\) 0 0
\(233\) −0.343146 −0.0224802 −0.0112401 0.999937i \(-0.503578\pi\)
−0.0112401 + 0.999937i \(0.503578\pi\)
\(234\) 0 0
\(235\) −7.65685 −0.499478
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.7279 1.72889 0.864443 0.502731i \(-0.167671\pi\)
0.864443 + 0.502731i \(0.167671\pi\)
\(240\) 0 0
\(241\) −19.6569 −1.26621 −0.633105 0.774066i \(-0.718220\pi\)
−0.633105 + 0.774066i \(0.718220\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.00000 0.319438
\(246\) 0 0
\(247\) −3.41421 −0.217241
\(248\) 0 0
\(249\) 12.8284 0.812969
\(250\) 0 0
\(251\) 1.75736 0.110924 0.0554618 0.998461i \(-0.482337\pi\)
0.0554618 + 0.998461i \(0.482337\pi\)
\(252\) 0 0
\(253\) −17.1716 −1.07957
\(254\) 0 0
\(255\) 1.17157 0.0733667
\(256\) 0 0
\(257\) −4.48528 −0.279784 −0.139892 0.990167i \(-0.544676\pi\)
−0.139892 + 0.990167i \(0.544676\pi\)
\(258\) 0 0
\(259\) −4.82843 −0.300024
\(260\) 0 0
\(261\) 1.41421 0.0875376
\(262\) 0 0
\(263\) 23.4558 1.44635 0.723175 0.690665i \(-0.242682\pi\)
0.723175 + 0.690665i \(0.242682\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) −5.89949 −0.361043
\(268\) 0 0
\(269\) −3.07107 −0.187246 −0.0936232 0.995608i \(-0.529845\pi\)
−0.0936232 + 0.995608i \(0.529845\pi\)
\(270\) 0 0
\(271\) 10.8284 0.657780 0.328890 0.944368i \(-0.393325\pi\)
0.328890 + 0.944368i \(0.393325\pi\)
\(272\) 0 0
\(273\) 4.82843 0.292230
\(274\) 0 0
\(275\) 2.24264 0.135236
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 3.17157 0.189877
\(280\) 0 0
\(281\) −14.5858 −0.870115 −0.435058 0.900403i \(-0.643272\pi\)
−0.435058 + 0.900403i \(0.643272\pi\)
\(282\) 0 0
\(283\) 10.3848 0.617311 0.308655 0.951174i \(-0.400121\pi\)
0.308655 + 0.951174i \(0.400121\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −0.343146 −0.0202553
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 9.75736 0.571987
\(292\) 0 0
\(293\) −11.5147 −0.672697 −0.336349 0.941738i \(-0.609192\pi\)
−0.336349 + 0.941738i \(0.609192\pi\)
\(294\) 0 0
\(295\) 12.4853 0.726921
\(296\) 0 0
\(297\) −2.24264 −0.130131
\(298\) 0 0
\(299\) 26.1421 1.51184
\(300\) 0 0
\(301\) −17.3137 −0.997946
\(302\) 0 0
\(303\) 3.17157 0.182202
\(304\) 0 0
\(305\) −7.31371 −0.418782
\(306\) 0 0
\(307\) −21.1716 −1.20833 −0.604163 0.796861i \(-0.706492\pi\)
−0.604163 + 0.796861i \(0.706492\pi\)
\(308\) 0 0
\(309\) −7.31371 −0.416062
\(310\) 0 0
\(311\) 6.24264 0.353988 0.176994 0.984212i \(-0.443363\pi\)
0.176994 + 0.984212i \(0.443363\pi\)
\(312\) 0 0
\(313\) 5.79899 0.327778 0.163889 0.986479i \(-0.447596\pi\)
0.163889 + 0.986479i \(0.447596\pi\)
\(314\) 0 0
\(315\) −1.41421 −0.0796819
\(316\) 0 0
\(317\) −26.6274 −1.49554 −0.747772 0.663955i \(-0.768877\pi\)
−0.747772 + 0.663955i \(0.768877\pi\)
\(318\) 0 0
\(319\) 3.17157 0.177574
\(320\) 0 0
\(321\) 19.3137 1.07799
\(322\) 0 0
\(323\) 1.17157 0.0651881
\(324\) 0 0
\(325\) −3.41421 −0.189386
\(326\) 0 0
\(327\) 6.48528 0.358637
\(328\) 0 0
\(329\) 10.8284 0.596991
\(330\) 0 0
\(331\) −28.1421 −1.54683 −0.773416 0.633899i \(-0.781453\pi\)
−0.773416 + 0.633899i \(0.781453\pi\)
\(332\) 0 0
\(333\) −3.41421 −0.187098
\(334\) 0 0
\(335\) −9.65685 −0.527610
\(336\) 0 0
\(337\) −24.5858 −1.33927 −0.669637 0.742689i \(-0.733550\pi\)
−0.669637 + 0.742689i \(0.733550\pi\)
\(338\) 0 0
\(339\) −10.1421 −0.550845
\(340\) 0 0
\(341\) 7.11270 0.385174
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −7.65685 −0.412231
\(346\) 0 0
\(347\) 27.4558 1.47391 0.736953 0.675943i \(-0.236264\pi\)
0.736953 + 0.675943i \(0.236264\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 3.41421 0.182237
\(352\) 0 0
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) 0 0
\(355\) −10.8284 −0.574713
\(356\) 0 0
\(357\) −1.65685 −0.0876900
\(358\) 0 0
\(359\) −24.8701 −1.31259 −0.656296 0.754504i \(-0.727878\pi\)
−0.656296 + 0.754504i \(0.727878\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.97056 0.313373
\(364\) 0 0
\(365\) 7.65685 0.400778
\(366\) 0 0
\(367\) 14.3848 0.750879 0.375440 0.926847i \(-0.377492\pi\)
0.375440 + 0.926847i \(0.377492\pi\)
\(368\) 0 0
\(369\) −0.242641 −0.0126314
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) 0.585786 0.0303309 0.0151654 0.999885i \(-0.495173\pi\)
0.0151654 + 0.999885i \(0.495173\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −4.82843 −0.248677
\(378\) 0 0
\(379\) −3.17157 −0.162913 −0.0814564 0.996677i \(-0.525957\pi\)
−0.0814564 + 0.996677i \(0.525957\pi\)
\(380\) 0 0
\(381\) 19.3137 0.989471
\(382\) 0 0
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) −3.17157 −0.161638
\(386\) 0 0
\(387\) −12.2426 −0.622328
\(388\) 0 0
\(389\) 30.9706 1.57027 0.785135 0.619325i \(-0.212594\pi\)
0.785135 + 0.619325i \(0.212594\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) 0 0
\(393\) −8.58579 −0.433096
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.6274 1.43677 0.718384 0.695646i \(-0.244882\pi\)
0.718384 + 0.695646i \(0.244882\pi\)
\(398\) 0 0
\(399\) −1.41421 −0.0707992
\(400\) 0 0
\(401\) −7.75736 −0.387384 −0.193692 0.981062i \(-0.562046\pi\)
−0.193692 + 0.981062i \(0.562046\pi\)
\(402\) 0 0
\(403\) −10.8284 −0.539402
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −7.65685 −0.379536
\(408\) 0 0
\(409\) −12.8284 −0.634325 −0.317162 0.948371i \(-0.602730\pi\)
−0.317162 + 0.948371i \(0.602730\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) −17.6569 −0.868837
\(414\) 0 0
\(415\) 12.8284 0.629723
\(416\) 0 0
\(417\) −14.1421 −0.692543
\(418\) 0 0
\(419\) −3.41421 −0.166795 −0.0833976 0.996516i \(-0.526577\pi\)
−0.0833976 + 0.996516i \(0.526577\pi\)
\(420\) 0 0
\(421\) 9.31371 0.453922 0.226961 0.973904i \(-0.427121\pi\)
0.226961 + 0.973904i \(0.427121\pi\)
\(422\) 0 0
\(423\) 7.65685 0.372289
\(424\) 0 0
\(425\) 1.17157 0.0568296
\(426\) 0 0
\(427\) 10.3431 0.500540
\(428\) 0 0
\(429\) 7.65685 0.369676
\(430\) 0 0
\(431\) 31.1127 1.49865 0.749323 0.662205i \(-0.230379\pi\)
0.749323 + 0.662205i \(0.230379\pi\)
\(432\) 0 0
\(433\) −10.9289 −0.525211 −0.262605 0.964903i \(-0.584582\pi\)
−0.262605 + 0.964903i \(0.584582\pi\)
\(434\) 0 0
\(435\) 1.41421 0.0678064
\(436\) 0 0
\(437\) −7.65685 −0.366277
\(438\) 0 0
\(439\) 21.6569 1.03363 0.516813 0.856099i \(-0.327118\pi\)
0.516813 + 0.856099i \(0.327118\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) −5.89949 −0.279663
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −26.8701 −1.26808 −0.634038 0.773302i \(-0.718604\pi\)
−0.634038 + 0.773302i \(0.718604\pi\)
\(450\) 0 0
\(451\) −0.544156 −0.0256233
\(452\) 0 0
\(453\) −6.48528 −0.304705
\(454\) 0 0
\(455\) 4.82843 0.226360
\(456\) 0 0
\(457\) 32.8284 1.53565 0.767825 0.640660i \(-0.221339\pi\)
0.767825 + 0.640660i \(0.221339\pi\)
\(458\) 0 0
\(459\) −1.17157 −0.0546843
\(460\) 0 0
\(461\) 19.6569 0.915511 0.457755 0.889078i \(-0.348654\pi\)
0.457755 + 0.889078i \(0.348654\pi\)
\(462\) 0 0
\(463\) −24.2426 −1.12665 −0.563326 0.826235i \(-0.690478\pi\)
−0.563326 + 0.826235i \(0.690478\pi\)
\(464\) 0 0
\(465\) 3.17157 0.147078
\(466\) 0 0
\(467\) −35.6569 −1.65000 −0.825001 0.565131i \(-0.808826\pi\)
−0.825001 + 0.565131i \(0.808826\pi\)
\(468\) 0 0
\(469\) 13.6569 0.630615
\(470\) 0 0
\(471\) −10.4853 −0.483136
\(472\) 0 0
\(473\) −27.4558 −1.26242
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 0 0
\(479\) −17.0711 −0.779997 −0.389998 0.920815i \(-0.627524\pi\)
−0.389998 + 0.920815i \(0.627524\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) 0 0
\(483\) 10.8284 0.492710
\(484\) 0 0
\(485\) 9.75736 0.443059
\(486\) 0 0
\(487\) −20.4853 −0.928277 −0.464138 0.885763i \(-0.653636\pi\)
−0.464138 + 0.885763i \(0.653636\pi\)
\(488\) 0 0
\(489\) 21.8995 0.990329
\(490\) 0 0
\(491\) 1.75736 0.0793085 0.0396543 0.999213i \(-0.487374\pi\)
0.0396543 + 0.999213i \(0.487374\pi\)
\(492\) 0 0
\(493\) 1.65685 0.0746210
\(494\) 0 0
\(495\) −2.24264 −0.100799
\(496\) 0 0
\(497\) 15.3137 0.686914
\(498\) 0 0
\(499\) 5.17157 0.231511 0.115756 0.993278i \(-0.463071\pi\)
0.115756 + 0.993278i \(0.463071\pi\)
\(500\) 0 0
\(501\) −17.3137 −0.773519
\(502\) 0 0
\(503\) −4.82843 −0.215289 −0.107644 0.994189i \(-0.534331\pi\)
−0.107644 + 0.994189i \(0.534331\pi\)
\(504\) 0 0
\(505\) 3.17157 0.141133
\(506\) 0 0
\(507\) 1.34315 0.0596512
\(508\) 0 0
\(509\) −0.727922 −0.0322646 −0.0161323 0.999870i \(-0.505135\pi\)
−0.0161323 + 0.999870i \(0.505135\pi\)
\(510\) 0 0
\(511\) −10.8284 −0.479021
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −7.31371 −0.322281
\(516\) 0 0
\(517\) 17.1716 0.755205
\(518\) 0 0
\(519\) 19.7990 0.869079
\(520\) 0 0
\(521\) 7.75736 0.339856 0.169928 0.985456i \(-0.445646\pi\)
0.169928 + 0.985456i \(0.445646\pi\)
\(522\) 0 0
\(523\) 15.5147 0.678411 0.339206 0.940712i \(-0.389842\pi\)
0.339206 + 0.940712i \(0.389842\pi\)
\(524\) 0 0
\(525\) −1.41421 −0.0617213
\(526\) 0 0
\(527\) 3.71573 0.161860
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −12.4853 −0.541815
\(532\) 0 0
\(533\) 0.828427 0.0358832
\(534\) 0 0
\(535\) 19.3137 0.835004
\(536\) 0 0
\(537\) 16.4853 0.711392
\(538\) 0 0
\(539\) −11.2132 −0.482987
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 20.8284 0.893833
\(544\) 0 0
\(545\) 6.48528 0.277799
\(546\) 0 0
\(547\) −24.4853 −1.04692 −0.523458 0.852052i \(-0.675358\pi\)
−0.523458 + 0.852052i \(0.675358\pi\)
\(548\) 0 0
\(549\) 7.31371 0.312141
\(550\) 0 0
\(551\) 1.41421 0.0602475
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.41421 −0.144925
\(556\) 0 0
\(557\) −25.3137 −1.07258 −0.536288 0.844035i \(-0.680174\pi\)
−0.536288 + 0.844035i \(0.680174\pi\)
\(558\) 0 0
\(559\) 41.7990 1.76791
\(560\) 0 0
\(561\) −2.62742 −0.110930
\(562\) 0 0
\(563\) 30.2843 1.27633 0.638165 0.769900i \(-0.279694\pi\)
0.638165 + 0.769900i \(0.279694\pi\)
\(564\) 0 0
\(565\) −10.1421 −0.426683
\(566\) 0 0
\(567\) 1.41421 0.0593914
\(568\) 0 0
\(569\) 31.0711 1.30257 0.651283 0.758835i \(-0.274231\pi\)
0.651283 + 0.758835i \(0.274231\pi\)
\(570\) 0 0
\(571\) −9.17157 −0.383818 −0.191909 0.981413i \(-0.561468\pi\)
−0.191909 + 0.981413i \(0.561468\pi\)
\(572\) 0 0
\(573\) 10.2426 0.427892
\(574\) 0 0
\(575\) −7.65685 −0.319313
\(576\) 0 0
\(577\) 19.4558 0.809957 0.404979 0.914326i \(-0.367279\pi\)
0.404979 + 0.914326i \(0.367279\pi\)
\(578\) 0 0
\(579\) −5.07107 −0.210746
\(580\) 0 0
\(581\) −18.1421 −0.752663
\(582\) 0 0
\(583\) 17.9411 0.743045
\(584\) 0 0
\(585\) 3.41421 0.141160
\(586\) 0 0
\(587\) 12.3431 0.509456 0.254728 0.967013i \(-0.418014\pi\)
0.254728 + 0.967013i \(0.418014\pi\)
\(588\) 0 0
\(589\) 3.17157 0.130682
\(590\) 0 0
\(591\) −6.82843 −0.280884
\(592\) 0 0
\(593\) 36.6274 1.50411 0.752054 0.659102i \(-0.229063\pi\)
0.752054 + 0.659102i \(0.229063\pi\)
\(594\) 0 0
\(595\) −1.65685 −0.0679244
\(596\) 0 0
\(597\) 18.1421 0.742508
\(598\) 0 0
\(599\) 3.02944 0.123779 0.0618897 0.998083i \(-0.480287\pi\)
0.0618897 + 0.998083i \(0.480287\pi\)
\(600\) 0 0
\(601\) 8.14214 0.332125 0.166062 0.986115i \(-0.446895\pi\)
0.166062 + 0.986115i \(0.446895\pi\)
\(602\) 0 0
\(603\) 9.65685 0.393258
\(604\) 0 0
\(605\) 5.97056 0.242738
\(606\) 0 0
\(607\) −16.4853 −0.669117 −0.334558 0.942375i \(-0.608587\pi\)
−0.334558 + 0.942375i \(0.608587\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −26.1421 −1.05760
\(612\) 0 0
\(613\) −10.4853 −0.423497 −0.211748 0.977324i \(-0.567916\pi\)
−0.211748 + 0.977324i \(0.567916\pi\)
\(614\) 0 0
\(615\) −0.242641 −0.00978422
\(616\) 0 0
\(617\) 28.4853 1.14677 0.573387 0.819285i \(-0.305629\pi\)
0.573387 + 0.819285i \(0.305629\pi\)
\(618\) 0 0
\(619\) 20.4853 0.823373 0.411686 0.911326i \(-0.364940\pi\)
0.411686 + 0.911326i \(0.364940\pi\)
\(620\) 0 0
\(621\) 7.65685 0.307259
\(622\) 0 0
\(623\) 8.34315 0.334261
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.24264 −0.0895624
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 7.31371 0.290694
\(634\) 0 0
\(635\) 19.3137 0.766441
\(636\) 0 0
\(637\) 17.0711 0.676380
\(638\) 0 0
\(639\) 10.8284 0.428366
\(640\) 0 0
\(641\) −31.3553 −1.23846 −0.619231 0.785209i \(-0.712555\pi\)
−0.619231 + 0.785209i \(0.712555\pi\)
\(642\) 0 0
\(643\) −42.3848 −1.67149 −0.835746 0.549116i \(-0.814965\pi\)
−0.835746 + 0.549116i \(0.814965\pi\)
\(644\) 0 0
\(645\) −12.2426 −0.482054
\(646\) 0 0
\(647\) 36.1421 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) −4.48528 −0.175792
\(652\) 0 0
\(653\) 42.8284 1.67601 0.838003 0.545666i \(-0.183723\pi\)
0.838003 + 0.545666i \(0.183723\pi\)
\(654\) 0 0
\(655\) −8.58579 −0.335474
\(656\) 0 0
\(657\) −7.65685 −0.298722
\(658\) 0 0
\(659\) 18.6274 0.725621 0.362811 0.931863i \(-0.381817\pi\)
0.362811 + 0.931863i \(0.381817\pi\)
\(660\) 0 0
\(661\) 7.45584 0.289999 0.144999 0.989432i \(-0.453682\pi\)
0.144999 + 0.989432i \(0.453682\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) −1.41421 −0.0548408
\(666\) 0 0
\(667\) −10.8284 −0.419278
\(668\) 0 0
\(669\) 18.6274 0.720178
\(670\) 0 0
\(671\) 16.4020 0.633193
\(672\) 0 0
\(673\) −44.1838 −1.70316 −0.851580 0.524225i \(-0.824355\pi\)
−0.851580 + 0.524225i \(0.824355\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) 0 0
\(679\) −13.7990 −0.529557
\(680\) 0 0
\(681\) 14.9706 0.573673
\(682\) 0 0
\(683\) −18.3431 −0.701881 −0.350940 0.936398i \(-0.614138\pi\)
−0.350940 + 0.936398i \(0.614138\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 22.6274 0.863290
\(688\) 0 0
\(689\) −27.3137 −1.04057
\(690\) 0 0
\(691\) −46.8284 −1.78144 −0.890719 0.454555i \(-0.849798\pi\)
−0.890719 + 0.454555i \(0.849798\pi\)
\(692\) 0 0
\(693\) 3.17157 0.120478
\(694\) 0 0
\(695\) −14.1421 −0.536442
\(696\) 0 0
\(697\) −0.284271 −0.0107675
\(698\) 0 0
\(699\) 0.343146 0.0129790
\(700\) 0 0
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) 0 0
\(703\) −3.41421 −0.128770
\(704\) 0 0
\(705\) 7.65685 0.288374
\(706\) 0 0
\(707\) −4.48528 −0.168686
\(708\) 0 0
\(709\) 28.9706 1.08801 0.544006 0.839081i \(-0.316907\pi\)
0.544006 + 0.839081i \(0.316907\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.2843 −0.909453
\(714\) 0 0
\(715\) 7.65685 0.286350
\(716\) 0 0
\(717\) −26.7279 −0.998173
\(718\) 0 0
\(719\) 1.07107 0.0399441 0.0199720 0.999801i \(-0.493642\pi\)
0.0199720 + 0.999801i \(0.493642\pi\)
\(720\) 0 0
\(721\) 10.3431 0.385199
\(722\) 0 0
\(723\) 19.6569 0.731046
\(724\) 0 0
\(725\) 1.41421 0.0525226
\(726\) 0 0
\(727\) −47.3553 −1.75631 −0.878156 0.478374i \(-0.841226\pi\)
−0.878156 + 0.478374i \(0.841226\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.3431 −0.530500
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) −5.00000 −0.184428
\(736\) 0 0
\(737\) 21.6569 0.797740
\(738\) 0 0
\(739\) −14.3431 −0.527621 −0.263811 0.964575i \(-0.584979\pi\)
−0.263811 + 0.964575i \(0.584979\pi\)
\(740\) 0 0
\(741\) 3.41421 0.125424
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −12.8284 −0.469368
\(748\) 0 0
\(749\) −27.3137 −0.998021
\(750\) 0 0
\(751\) 24.1421 0.880959 0.440480 0.897763i \(-0.354808\pi\)
0.440480 + 0.897763i \(0.354808\pi\)
\(752\) 0 0
\(753\) −1.75736 −0.0640417
\(754\) 0 0
\(755\) −6.48528 −0.236024
\(756\) 0 0
\(757\) −32.4264 −1.17856 −0.589279 0.807930i \(-0.700588\pi\)
−0.589279 + 0.807930i \(0.700588\pi\)
\(758\) 0 0
\(759\) 17.1716 0.623289
\(760\) 0 0
\(761\) 43.9411 1.59286 0.796432 0.604728i \(-0.206718\pi\)
0.796432 + 0.604728i \(0.206718\pi\)
\(762\) 0 0
\(763\) −9.17157 −0.332033
\(764\) 0 0
\(765\) −1.17157 −0.0423583
\(766\) 0 0
\(767\) 42.6274 1.53919
\(768\) 0 0
\(769\) 42.9706 1.54956 0.774779 0.632232i \(-0.217861\pi\)
0.774779 + 0.632232i \(0.217861\pi\)
\(770\) 0 0
\(771\) 4.48528 0.161533
\(772\) 0 0
\(773\) −25.6569 −0.922813 −0.461406 0.887189i \(-0.652655\pi\)
−0.461406 + 0.887189i \(0.652655\pi\)
\(774\) 0 0
\(775\) 3.17157 0.113926
\(776\) 0 0
\(777\) 4.82843 0.173219
\(778\) 0 0
\(779\) −0.242641 −0.00869350
\(780\) 0 0
\(781\) 24.2843 0.868960
\(782\) 0 0
\(783\) −1.41421 −0.0505399
\(784\) 0 0
\(785\) −10.4853 −0.374236
\(786\) 0 0
\(787\) 42.8284 1.52667 0.763334 0.646004i \(-0.223561\pi\)
0.763334 + 0.646004i \(0.223561\pi\)
\(788\) 0 0
\(789\) −23.4558 −0.835050
\(790\) 0 0
\(791\) 14.3431 0.509984
\(792\) 0 0
\(793\) −24.9706 −0.886731
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) −14.1421 −0.500940 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(798\) 0 0
\(799\) 8.97056 0.317356
\(800\) 0 0
\(801\) 5.89949 0.208448
\(802\) 0 0
\(803\) −17.1716 −0.605972
\(804\) 0 0
\(805\) 10.8284 0.381652
\(806\) 0 0
\(807\) 3.07107 0.108107
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −8.68629 −0.305017 −0.152508 0.988302i \(-0.548735\pi\)
−0.152508 + 0.988302i \(0.548735\pi\)
\(812\) 0 0
\(813\) −10.8284 −0.379770
\(814\) 0 0
\(815\) 21.8995 0.767106
\(816\) 0 0
\(817\) −12.2426 −0.428316
\(818\) 0 0
\(819\) −4.82843 −0.168719
\(820\) 0 0
\(821\) 14.4853 0.505540 0.252770 0.967526i \(-0.418658\pi\)
0.252770 + 0.967526i \(0.418658\pi\)
\(822\) 0 0
\(823\) 25.0122 0.871870 0.435935 0.899978i \(-0.356418\pi\)
0.435935 + 0.899978i \(0.356418\pi\)
\(824\) 0 0
\(825\) −2.24264 −0.0780787
\(826\) 0 0
\(827\) 2.68629 0.0934115 0.0467058 0.998909i \(-0.485128\pi\)
0.0467058 + 0.998909i \(0.485128\pi\)
\(828\) 0 0
\(829\) 46.4853 1.61450 0.807250 0.590209i \(-0.200955\pi\)
0.807250 + 0.590209i \(0.200955\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −5.85786 −0.202963
\(834\) 0 0
\(835\) −17.3137 −0.599166
\(836\) 0 0
\(837\) −3.17157 −0.109626
\(838\) 0 0
\(839\) 9.85786 0.340331 0.170166 0.985415i \(-0.445570\pi\)
0.170166 + 0.985415i \(0.445570\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) 14.5858 0.502361
\(844\) 0 0
\(845\) 1.34315 0.0462056
\(846\) 0 0
\(847\) −8.44365 −0.290127
\(848\) 0 0
\(849\) −10.3848 −0.356405
\(850\) 0 0
\(851\) 26.1421 0.896141
\(852\) 0 0
\(853\) −16.8284 −0.576194 −0.288097 0.957601i \(-0.593023\pi\)
−0.288097 + 0.957601i \(0.593023\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 12.6863 0.433355 0.216678 0.976243i \(-0.430478\pi\)
0.216678 + 0.976243i \(0.430478\pi\)
\(858\) 0 0
\(859\) 49.9411 1.70397 0.851985 0.523567i \(-0.175399\pi\)
0.851985 + 0.523567i \(0.175399\pi\)
\(860\) 0 0
\(861\) 0.343146 0.0116944
\(862\) 0 0
\(863\) −8.68629 −0.295685 −0.147842 0.989011i \(-0.547233\pi\)
−0.147842 + 0.989011i \(0.547233\pi\)
\(864\) 0 0
\(865\) 19.7990 0.673186
\(866\) 0 0
\(867\) 15.6274 0.530735
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −32.9706 −1.11716
\(872\) 0 0
\(873\) −9.75736 −0.330237
\(874\) 0 0
\(875\) −1.41421 −0.0478091
\(876\) 0 0
\(877\) −34.9289 −1.17947 −0.589733 0.807598i \(-0.700767\pi\)
−0.589733 + 0.807598i \(0.700767\pi\)
\(878\) 0 0
\(879\) 11.5147 0.388382
\(880\) 0 0
\(881\) −5.79899 −0.195373 −0.0976865 0.995217i \(-0.531144\pi\)
−0.0976865 + 0.995217i \(0.531144\pi\)
\(882\) 0 0
\(883\) 12.0416 0.405233 0.202617 0.979258i \(-0.435055\pi\)
0.202617 + 0.979258i \(0.435055\pi\)
\(884\) 0 0
\(885\) −12.4853 −0.419688
\(886\) 0 0
\(887\) −25.9411 −0.871018 −0.435509 0.900184i \(-0.643432\pi\)
−0.435509 + 0.900184i \(0.643432\pi\)
\(888\) 0 0
\(889\) −27.3137 −0.916072
\(890\) 0 0
\(891\) 2.24264 0.0751313
\(892\) 0 0
\(893\) 7.65685 0.256227
\(894\) 0 0
\(895\) 16.4853 0.551042
\(896\) 0 0
\(897\) −26.1421 −0.872861
\(898\) 0 0
\(899\) 4.48528 0.149593
\(900\) 0 0
\(901\) 9.37258 0.312246
\(902\) 0 0
\(903\) 17.3137 0.576164
\(904\) 0 0
\(905\) 20.8284 0.692360
\(906\) 0 0
\(907\) 18.1421 0.602400 0.301200 0.953561i \(-0.402613\pi\)
0.301200 + 0.953561i \(0.402613\pi\)
\(908\) 0 0
\(909\) −3.17157 −0.105194
\(910\) 0 0
\(911\) −8.68629 −0.287790 −0.143895 0.989593i \(-0.545963\pi\)
−0.143895 + 0.989593i \(0.545963\pi\)
\(912\) 0 0
\(913\) −28.7696 −0.952133
\(914\) 0 0
\(915\) 7.31371 0.241784
\(916\) 0 0
\(917\) 12.1421 0.400969
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 21.1716 0.697627
\(922\) 0 0
\(923\) −36.9706 −1.21690
\(924\) 0 0
\(925\) −3.41421 −0.112259
\(926\) 0 0
\(927\) 7.31371 0.240214
\(928\) 0 0
\(929\) 33.1127 1.08639 0.543196 0.839606i \(-0.317214\pi\)
0.543196 + 0.839606i \(0.317214\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) −6.24264 −0.204375
\(934\) 0 0
\(935\) −2.62742 −0.0859257
\(936\) 0 0
\(937\) −29.7990 −0.973491 −0.486745 0.873544i \(-0.661816\pi\)
−0.486745 + 0.873544i \(0.661816\pi\)
\(938\) 0 0
\(939\) −5.79899 −0.189243
\(940\) 0 0
\(941\) 22.8701 0.745543 0.372771 0.927923i \(-0.378408\pi\)
0.372771 + 0.927923i \(0.378408\pi\)
\(942\) 0 0
\(943\) 1.85786 0.0605004
\(944\) 0 0
\(945\) 1.41421 0.0460044
\(946\) 0 0
\(947\) 47.1716 1.53287 0.766435 0.642322i \(-0.222029\pi\)
0.766435 + 0.642322i \(0.222029\pi\)
\(948\) 0 0
\(949\) 26.1421 0.848610
\(950\) 0 0
\(951\) 26.6274 0.863453
\(952\) 0 0
\(953\) −53.4558 −1.73160 −0.865802 0.500386i \(-0.833191\pi\)
−0.865802 + 0.500386i \(0.833191\pi\)
\(954\) 0 0
\(955\) 10.2426 0.331444
\(956\) 0 0
\(957\) −3.17157 −0.102522
\(958\) 0 0
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) −20.9411 −0.675520
\(962\) 0 0
\(963\) −19.3137 −0.622376
\(964\) 0 0
\(965\) −5.07107 −0.163243
\(966\) 0 0
\(967\) −8.04163 −0.258601 −0.129301 0.991605i \(-0.541273\pi\)
−0.129301 + 0.991605i \(0.541273\pi\)
\(968\) 0 0
\(969\) −1.17157 −0.0376363
\(970\) 0 0
\(971\) −22.3431 −0.717026 −0.358513 0.933525i \(-0.616716\pi\)
−0.358513 + 0.933525i \(0.616716\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 0 0
\(975\) 3.41421 0.109342
\(976\) 0 0
\(977\) 32.2843 1.03287 0.516433 0.856328i \(-0.327260\pi\)
0.516433 + 0.856328i \(0.327260\pi\)
\(978\) 0 0
\(979\) 13.2304 0.422847
\(980\) 0 0
\(981\) −6.48528 −0.207059
\(982\) 0 0
\(983\) 24.6274 0.785493 0.392746 0.919647i \(-0.371525\pi\)
0.392746 + 0.919647i \(0.371525\pi\)
\(984\) 0 0
\(985\) −6.82843 −0.217572
\(986\) 0 0
\(987\) −10.8284 −0.344673
\(988\) 0 0
\(989\) 93.7401 2.98076
\(990\) 0 0
\(991\) −2.34315 −0.0744325 −0.0372162 0.999307i \(-0.511849\pi\)
−0.0372162 + 0.999307i \(0.511849\pi\)
\(992\) 0 0
\(993\) 28.1421 0.893064
\(994\) 0 0
\(995\) 18.1421 0.575144
\(996\) 0 0
\(997\) −38.0833 −1.20611 −0.603054 0.797700i \(-0.706050\pi\)
−0.603054 + 0.797700i \(0.706050\pi\)
\(998\) 0 0
\(999\) 3.41421 0.108021
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 4560.2.a.bf.1.2 2
4.3 odd 2 285.2.a.g.1.2 2
12.11 even 2 855.2.a.d.1.1 2
20.3 even 4 1425.2.c.l.799.1 4
20.7 even 4 1425.2.c.l.799.4 4
20.19 odd 2 1425.2.a.k.1.1 2
60.59 even 2 4275.2.a.y.1.2 2
76.75 even 2 5415.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.2 2 4.3 odd 2
855.2.a.d.1.1 2 12.11 even 2
1425.2.a.k.1.1 2 20.19 odd 2
1425.2.c.l.799.1 4 20.3 even 4
1425.2.c.l.799.4 4 20.7 even 4
4275.2.a.y.1.2 2 60.59 even 2
4560.2.a.bf.1.2 2 1.1 even 1 trivial
5415.2.a.n.1.1 2 76.75 even 2