Properties

Label 4560.2.a.bn.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
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: \(0\)
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 2280)
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

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +3.41421 q^{11} +3.41421 q^{13} -1.00000 q^{15} +2.82843 q^{17} -1.00000 q^{19} +1.41421 q^{21} -0.828427 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.07107 q^{29} -0.828427 q^{31} +3.41421 q^{33} -1.41421 q^{35} -2.24264 q^{37} +3.41421 q^{39} -9.89949 q^{41} +3.07107 q^{43} -1.00000 q^{45} +8.82843 q^{47} -5.00000 q^{49} +2.82843 q^{51} -8.48528 q^{53} -3.41421 q^{55} -1.00000 q^{57} +6.82843 q^{59} +9.65685 q^{61} +1.41421 q^{63} -3.41421 q^{65} -11.3137 q^{67} -0.828427 q^{69} +9.17157 q^{71} -0.343146 q^{73} +1.00000 q^{75} +4.82843 q^{77} +2.34315 q^{79} +1.00000 q^{81} -3.65685 q^{83} -2.82843 q^{85} +7.07107 q^{87} -3.75736 q^{89} +4.82843 q^{91} -0.828427 q^{93} +1.00000 q^{95} -7.89949 q^{97} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} + 4 q^{33} + 4 q^{37} + 4 q^{39} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 10 q^{49} - 4 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} + 24 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{89} + 4 q^{91} + 4 q^{93} + 2 q^{95} + 4 q^{97} + 4 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −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
\(11\) 3.41421 1.02942 0.514712 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 0 0
\(23\) −0.828427 −0.172739 −0.0863695 0.996263i \(-0.527527\pi\)
−0.0863695 + 0.996263i \(0.527527\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.07107 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) −0.828427 −0.148790 −0.0743950 0.997229i \(-0.523703\pi\)
−0.0743950 + 0.997229i \(0.523703\pi\)
\(32\) 0 0
\(33\) 3.41421 0.594338
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −2.24264 −0.368688 −0.184344 0.982862i \(-0.559016\pi\)
−0.184344 + 0.982862i \(0.559016\pi\)
\(38\) 0 0
\(39\) 3.41421 0.546712
\(40\) 0 0
\(41\) −9.89949 −1.54604 −0.773021 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 0 0
\(43\) 3.07107 0.468333 0.234167 0.972196i \(-0.424764\pi\)
0.234167 + 0.972196i \(0.424764\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.82843 1.28776 0.643879 0.765127i \(-0.277324\pi\)
0.643879 + 0.765127i \(0.277324\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 2.82843 0.396059
\(52\) 0 0
\(53\) −8.48528 −1.16554 −0.582772 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 0 0
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 6.82843 0.888985 0.444493 0.895782i \(-0.353384\pi\)
0.444493 + 0.895782i \(0.353384\pi\)
\(60\) 0 0
\(61\) 9.65685 1.23643 0.618217 0.786008i \(-0.287855\pi\)
0.618217 + 0.786008i \(0.287855\pi\)
\(62\) 0 0
\(63\) 1.41421 0.178174
\(64\) 0 0
\(65\) −3.41421 −0.423481
\(66\) 0 0
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) −0.828427 −0.0997309
\(70\) 0 0
\(71\) 9.17157 1.08847 0.544233 0.838934i \(-0.316821\pi\)
0.544233 + 0.838934i \(0.316821\pi\)
\(72\) 0 0
\(73\) −0.343146 −0.0401622 −0.0200811 0.999798i \(-0.506392\pi\)
−0.0200811 + 0.999798i \(0.506392\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.65685 −0.401392 −0.200696 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) 7.07107 0.758098
\(88\) 0 0
\(89\) −3.75736 −0.398279 −0.199140 0.979971i \(-0.563815\pi\)
−0.199140 + 0.979971i \(0.563815\pi\)
\(90\) 0 0
\(91\) 4.82843 0.506157
\(92\) 0 0
\(93\) −0.828427 −0.0859039
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −7.89949 −0.802072 −0.401036 0.916062i \(-0.631350\pi\)
−0.401036 + 0.916062i \(0.631350\pi\)
\(98\) 0 0
\(99\) 3.41421 0.343141
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 0 0
\(103\) 19.3137 1.90304 0.951518 0.307593i \(-0.0995234\pi\)
0.951518 + 0.307593i \(0.0995234\pi\)
\(104\) 0 0
\(105\) −1.41421 −0.138013
\(106\) 0 0
\(107\) 2.34315 0.226520 0.113260 0.993565i \(-0.463871\pi\)
0.113260 + 0.993565i \(0.463871\pi\)
\(108\) 0 0
\(109\) −20.1421 −1.92927 −0.964633 0.263595i \(-0.915092\pi\)
−0.964633 + 0.263595i \(0.915092\pi\)
\(110\) 0 0
\(111\) −2.24264 −0.212862
\(112\) 0 0
\(113\) 7.31371 0.688016 0.344008 0.938967i \(-0.388215\pi\)
0.344008 + 0.938967i \(0.388215\pi\)
\(114\) 0 0
\(115\) 0.828427 0.0772512
\(116\) 0 0
\(117\) 3.41421 0.315644
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 0.656854 0.0597140
\(122\) 0 0
\(123\) −9.89949 −0.892607
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) 3.07107 0.270392
\(130\) 0 0
\(131\) 18.7279 1.63627 0.818133 0.575029i \(-0.195009\pi\)
0.818133 + 0.575029i \(0.195009\pi\)
\(132\) 0 0
\(133\) −1.41421 −0.122628
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 4.34315 0.371060 0.185530 0.982639i \(-0.440600\pi\)
0.185530 + 0.982639i \(0.440600\pi\)
\(138\) 0 0
\(139\) −2.82843 −0.239904 −0.119952 0.992780i \(-0.538274\pi\)
−0.119952 + 0.992780i \(0.538274\pi\)
\(140\) 0 0
\(141\) 8.82843 0.743488
\(142\) 0 0
\(143\) 11.6569 0.974795
\(144\) 0 0
\(145\) −7.07107 −0.587220
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) −15.6569 −1.28266 −0.641330 0.767265i \(-0.721617\pi\)
−0.641330 + 0.767265i \(0.721617\pi\)
\(150\) 0 0
\(151\) 4.82843 0.392932 0.196466 0.980511i \(-0.437053\pi\)
0.196466 + 0.980511i \(0.437053\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0.828427 0.0665409
\(156\) 0 0
\(157\) 24.1421 1.92675 0.963376 0.268154i \(-0.0864136\pi\)
0.963376 + 0.268154i \(0.0864136\pi\)
\(158\) 0 0
\(159\) −8.48528 −0.672927
\(160\) 0 0
\(161\) −1.17157 −0.0923329
\(162\) 0 0
\(163\) −7.55635 −0.591859 −0.295929 0.955210i \(-0.595629\pi\)
−0.295929 + 0.955210i \(0.595629\pi\)
\(164\) 0 0
\(165\) −3.41421 −0.265796
\(166\) 0 0
\(167\) −5.31371 −0.411187 −0.205594 0.978637i \(-0.565912\pi\)
−0.205594 + 0.978637i \(0.565912\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 11.3137 0.860165 0.430083 0.902790i \(-0.358484\pi\)
0.430083 + 0.902790i \(0.358484\pi\)
\(174\) 0 0
\(175\) 1.41421 0.106904
\(176\) 0 0
\(177\) 6.82843 0.513256
\(178\) 0 0
\(179\) 18.8284 1.40730 0.703651 0.710545i \(-0.251552\pi\)
0.703651 + 0.710545i \(0.251552\pi\)
\(180\) 0 0
\(181\) −8.14214 −0.605200 −0.302600 0.953118i \(-0.597855\pi\)
−0.302600 + 0.953118i \(0.597855\pi\)
\(182\) 0 0
\(183\) 9.65685 0.713855
\(184\) 0 0
\(185\) 2.24264 0.164882
\(186\) 0 0
\(187\) 9.65685 0.706179
\(188\) 0 0
\(189\) 1.41421 0.102869
\(190\) 0 0
\(191\) 22.2426 1.60942 0.804710 0.593667i \(-0.202321\pi\)
0.804710 + 0.593667i \(0.202321\pi\)
\(192\) 0 0
\(193\) −1.07107 −0.0770971 −0.0385486 0.999257i \(-0.512273\pi\)
−0.0385486 + 0.999257i \(0.512273\pi\)
\(194\) 0 0
\(195\) −3.41421 −0.244497
\(196\) 0 0
\(197\) −6.14214 −0.437609 −0.218805 0.975769i \(-0.570216\pi\)
−0.218805 + 0.975769i \(0.570216\pi\)
\(198\) 0 0
\(199\) 14.8284 1.05116 0.525580 0.850744i \(-0.323848\pi\)
0.525580 + 0.850744i \(0.323848\pi\)
\(200\) 0 0
\(201\) −11.3137 −0.798007
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 9.89949 0.691411
\(206\) 0 0
\(207\) −0.828427 −0.0575797
\(208\) 0 0
\(209\) −3.41421 −0.236166
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 9.17157 0.628426
\(214\) 0 0
\(215\) −3.07107 −0.209445
\(216\) 0 0
\(217\) −1.17157 −0.0795315
\(218\) 0 0
\(219\) −0.343146 −0.0231876
\(220\) 0 0
\(221\) 9.65685 0.649590
\(222\) 0 0
\(223\) 11.3137 0.757622 0.378811 0.925474i \(-0.376333\pi\)
0.378811 + 0.925474i \(0.376333\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.343146 −0.0227754 −0.0113877 0.999935i \(-0.503625\pi\)
−0.0113877 + 0.999935i \(0.503625\pi\)
\(228\) 0 0
\(229\) −10.3431 −0.683494 −0.341747 0.939792i \(-0.611019\pi\)
−0.341747 + 0.939792i \(0.611019\pi\)
\(230\) 0 0
\(231\) 4.82843 0.317687
\(232\) 0 0
\(233\) 4.34315 0.284529 0.142264 0.989829i \(-0.454562\pi\)
0.142264 + 0.989829i \(0.454562\pi\)
\(234\) 0 0
\(235\) −8.82843 −0.575903
\(236\) 0 0
\(237\) 2.34315 0.152204
\(238\) 0 0
\(239\) 26.2426 1.69750 0.848748 0.528798i \(-0.177357\pi\)
0.848748 + 0.528798i \(0.177357\pi\)
\(240\) 0 0
\(241\) 23.6569 1.52387 0.761936 0.647652i \(-0.224249\pi\)
0.761936 + 0.647652i \(0.224249\pi\)
\(242\) 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
\(249\) −3.65685 −0.231744
\(250\) 0 0
\(251\) 14.2426 0.898988 0.449494 0.893283i \(-0.351604\pi\)
0.449494 + 0.893283i \(0.351604\pi\)
\(252\) 0 0
\(253\) −2.82843 −0.177822
\(254\) 0 0
\(255\) −2.82843 −0.177123
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −3.17157 −0.197072
\(260\) 0 0
\(261\) 7.07107 0.437688
\(262\) 0 0
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) −3.75736 −0.229947
\(268\) 0 0
\(269\) −22.3848 −1.36482 −0.682412 0.730968i \(-0.739069\pi\)
−0.682412 + 0.730968i \(0.739069\pi\)
\(270\) 0 0
\(271\) −2.82843 −0.171815 −0.0859074 0.996303i \(-0.527379\pi\)
−0.0859074 + 0.996303i \(0.527379\pi\)
\(272\) 0 0
\(273\) 4.82843 0.292230
\(274\) 0 0
\(275\) 3.41421 0.205885
\(276\) 0 0
\(277\) 20.6274 1.23938 0.619691 0.784846i \(-0.287258\pi\)
0.619691 + 0.784846i \(0.287258\pi\)
\(278\) 0 0
\(279\) −0.828427 −0.0495966
\(280\) 0 0
\(281\) 6.38478 0.380884 0.190442 0.981698i \(-0.439008\pi\)
0.190442 + 0.981698i \(0.439008\pi\)
\(282\) 0 0
\(283\) −16.2426 −0.965525 −0.482762 0.875751i \(-0.660367\pi\)
−0.482762 + 0.875751i \(0.660367\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −14.0000 −0.826394
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −7.89949 −0.463077
\(292\) 0 0
\(293\) −9.65685 −0.564159 −0.282080 0.959391i \(-0.591024\pi\)
−0.282080 + 0.959391i \(0.591024\pi\)
\(294\) 0 0
\(295\) −6.82843 −0.397566
\(296\) 0 0
\(297\) 3.41421 0.198113
\(298\) 0 0
\(299\) −2.82843 −0.163572
\(300\) 0 0
\(301\) 4.34315 0.250335
\(302\) 0 0
\(303\) 4.82843 0.277386
\(304\) 0 0
\(305\) −9.65685 −0.552950
\(306\) 0 0
\(307\) −1.85786 −0.106034 −0.0530170 0.998594i \(-0.516884\pi\)
−0.0530170 + 0.998594i \(0.516884\pi\)
\(308\) 0 0
\(309\) 19.3137 1.09872
\(310\) 0 0
\(311\) −19.2132 −1.08948 −0.544740 0.838605i \(-0.683372\pi\)
−0.544740 + 0.838605i \(0.683372\pi\)
\(312\) 0 0
\(313\) −13.7990 −0.779965 −0.389983 0.920822i \(-0.627519\pi\)
−0.389983 + 0.920822i \(0.627519\pi\)
\(314\) 0 0
\(315\) −1.41421 −0.0796819
\(316\) 0 0
\(317\) −2.14214 −0.120314 −0.0601572 0.998189i \(-0.519160\pi\)
−0.0601572 + 0.998189i \(0.519160\pi\)
\(318\) 0 0
\(319\) 24.1421 1.35170
\(320\) 0 0
\(321\) 2.34315 0.130782
\(322\) 0 0
\(323\) −2.82843 −0.157378
\(324\) 0 0
\(325\) 3.41421 0.189386
\(326\) 0 0
\(327\) −20.1421 −1.11386
\(328\) 0 0
\(329\) 12.4853 0.688336
\(330\) 0 0
\(331\) −16.1421 −0.887252 −0.443626 0.896212i \(-0.646308\pi\)
−0.443626 + 0.896212i \(0.646308\pi\)
\(332\) 0 0
\(333\) −2.24264 −0.122896
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) 26.2426 1.42953 0.714764 0.699366i \(-0.246534\pi\)
0.714764 + 0.699366i \(0.246534\pi\)
\(338\) 0 0
\(339\) 7.31371 0.397226
\(340\) 0 0
\(341\) −2.82843 −0.153168
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0.828427 0.0446010
\(346\) 0 0
\(347\) 1.31371 0.0705236 0.0352618 0.999378i \(-0.488773\pi\)
0.0352618 + 0.999378i \(0.488773\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 3.41421 0.182237
\(352\) 0 0
\(353\) −29.3137 −1.56021 −0.780106 0.625648i \(-0.784835\pi\)
−0.780106 + 0.625648i \(0.784835\pi\)
\(354\) 0 0
\(355\) −9.17157 −0.486777
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 17.5563 0.926589 0.463294 0.886204i \(-0.346667\pi\)
0.463294 + 0.886204i \(0.346667\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.656854 0.0344759
\(364\) 0 0
\(365\) 0.343146 0.0179611
\(366\) 0 0
\(367\) 0.727922 0.0379972 0.0189986 0.999820i \(-0.493952\pi\)
0.0189986 + 0.999820i \(0.493952\pi\)
\(368\) 0 0
\(369\) −9.89949 −0.515347
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −2.92893 −0.151654 −0.0758272 0.997121i \(-0.524160\pi\)
−0.0758272 + 0.997121i \(0.524160\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 24.1421 1.24338
\(378\) 0 0
\(379\) −26.4853 −1.36046 −0.680229 0.733000i \(-0.738120\pi\)
−0.680229 + 0.733000i \(0.738120\pi\)
\(380\) 0 0
\(381\) −5.65685 −0.289809
\(382\) 0 0
\(383\) −10.6274 −0.543036 −0.271518 0.962433i \(-0.587526\pi\)
−0.271518 + 0.962433i \(0.587526\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) 3.07107 0.156111
\(388\) 0 0
\(389\) −1.02944 −0.0521945 −0.0260973 0.999659i \(-0.508308\pi\)
−0.0260973 + 0.999659i \(0.508308\pi\)
\(390\) 0 0
\(391\) −2.34315 −0.118498
\(392\) 0 0
\(393\) 18.7279 0.944699
\(394\) 0 0
\(395\) −2.34315 −0.117896
\(396\) 0 0
\(397\) −28.6274 −1.43677 −0.718384 0.695646i \(-0.755118\pi\)
−0.718384 + 0.695646i \(0.755118\pi\)
\(398\) 0 0
\(399\) −1.41421 −0.0707992
\(400\) 0 0
\(401\) −18.3848 −0.918092 −0.459046 0.888413i \(-0.651809\pi\)
−0.459046 + 0.888413i \(0.651809\pi\)
\(402\) 0 0
\(403\) −2.82843 −0.140894
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −7.65685 −0.379536
\(408\) 0 0
\(409\) 0.828427 0.0409631 0.0204815 0.999790i \(-0.493480\pi\)
0.0204815 + 0.999790i \(0.493480\pi\)
\(410\) 0 0
\(411\) 4.34315 0.214232
\(412\) 0 0
\(413\) 9.65685 0.475183
\(414\) 0 0
\(415\) 3.65685 0.179508
\(416\) 0 0
\(417\) −2.82843 −0.138509
\(418\) 0 0
\(419\) −29.5563 −1.44392 −0.721961 0.691934i \(-0.756759\pi\)
−0.721961 + 0.691934i \(0.756759\pi\)
\(420\) 0 0
\(421\) 28.6274 1.39521 0.697607 0.716480i \(-0.254248\pi\)
0.697607 + 0.716480i \(0.254248\pi\)
\(422\) 0 0
\(423\) 8.82843 0.429253
\(424\) 0 0
\(425\) 2.82843 0.137199
\(426\) 0 0
\(427\) 13.6569 0.660901
\(428\) 0 0
\(429\) 11.6569 0.562798
\(430\) 0 0
\(431\) 22.1421 1.06655 0.533275 0.845942i \(-0.320961\pi\)
0.533275 + 0.845942i \(0.320961\pi\)
\(432\) 0 0
\(433\) −5.75736 −0.276681 −0.138341 0.990385i \(-0.544177\pi\)
−0.138341 + 0.990385i \(0.544177\pi\)
\(434\) 0 0
\(435\) −7.07107 −0.339032
\(436\) 0 0
\(437\) 0.828427 0.0396290
\(438\) 0 0
\(439\) −27.3137 −1.30361 −0.651806 0.758386i \(-0.725988\pi\)
−0.651806 + 0.758386i \(0.725988\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) 27.4558 1.30447 0.652233 0.758018i \(-0.273832\pi\)
0.652233 + 0.758018i \(0.273832\pi\)
\(444\) 0 0
\(445\) 3.75736 0.178116
\(446\) 0 0
\(447\) −15.6569 −0.740544
\(448\) 0 0
\(449\) 22.3848 1.05640 0.528201 0.849119i \(-0.322867\pi\)
0.528201 + 0.849119i \(0.322867\pi\)
\(450\) 0 0
\(451\) −33.7990 −1.59153
\(452\) 0 0
\(453\) 4.82843 0.226859
\(454\) 0 0
\(455\) −4.82843 −0.226360
\(456\) 0 0
\(457\) −12.1421 −0.567985 −0.283993 0.958826i \(-0.591659\pi\)
−0.283993 + 0.958826i \(0.591659\pi\)
\(458\) 0 0
\(459\) 2.82843 0.132020
\(460\) 0 0
\(461\) 29.3137 1.36528 0.682638 0.730757i \(-0.260833\pi\)
0.682638 + 0.730757i \(0.260833\pi\)
\(462\) 0 0
\(463\) 0.727922 0.0338294 0.0169147 0.999857i \(-0.494616\pi\)
0.0169147 + 0.999857i \(0.494616\pi\)
\(464\) 0 0
\(465\) 0.828427 0.0384174
\(466\) 0 0
\(467\) 11.1716 0.516959 0.258479 0.966017i \(-0.416779\pi\)
0.258479 + 0.966017i \(0.416779\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 24.1421 1.11241
\(472\) 0 0
\(473\) 10.4853 0.482114
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −8.48528 −0.388514
\(478\) 0 0
\(479\) −14.2426 −0.650763 −0.325381 0.945583i \(-0.605493\pi\)
−0.325381 + 0.945583i \(0.605493\pi\)
\(480\) 0 0
\(481\) −7.65685 −0.349123
\(482\) 0 0
\(483\) −1.17157 −0.0533084
\(484\) 0 0
\(485\) 7.89949 0.358698
\(486\) 0 0
\(487\) 15.7990 0.715921 0.357960 0.933737i \(-0.383472\pi\)
0.357960 + 0.933737i \(0.383472\pi\)
\(488\) 0 0
\(489\) −7.55635 −0.341710
\(490\) 0 0
\(491\) −24.3848 −1.10047 −0.550235 0.835010i \(-0.685462\pi\)
−0.550235 + 0.835010i \(0.685462\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −3.41421 −0.153457
\(496\) 0 0
\(497\) 12.9706 0.581809
\(498\) 0 0
\(499\) −30.1421 −1.34935 −0.674674 0.738116i \(-0.735716\pi\)
−0.674674 + 0.738116i \(0.735716\pi\)
\(500\) 0 0
\(501\) −5.31371 −0.237399
\(502\) 0 0
\(503\) 29.3137 1.30703 0.653517 0.756912i \(-0.273293\pi\)
0.653517 + 0.756912i \(0.273293\pi\)
\(504\) 0 0
\(505\) −4.82843 −0.214862
\(506\) 0 0
\(507\) −1.34315 −0.0596512
\(508\) 0 0
\(509\) −19.0711 −0.845310 −0.422655 0.906291i \(-0.638902\pi\)
−0.422655 + 0.906291i \(0.638902\pi\)
\(510\) 0 0
\(511\) −0.485281 −0.0214676
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −19.3137 −0.851064
\(516\) 0 0
\(517\) 30.1421 1.32565
\(518\) 0 0
\(519\) 11.3137 0.496617
\(520\) 0 0
\(521\) −23.5563 −1.03202 −0.516011 0.856582i \(-0.672584\pi\)
−0.516011 + 0.856582i \(0.672584\pi\)
\(522\) 0 0
\(523\) 15.7990 0.690842 0.345421 0.938448i \(-0.387736\pi\)
0.345421 + 0.938448i \(0.387736\pi\)
\(524\) 0 0
\(525\) 1.41421 0.0617213
\(526\) 0 0
\(527\) −2.34315 −0.102069
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) 6.82843 0.296328
\(532\) 0 0
\(533\) −33.7990 −1.46400
\(534\) 0 0
\(535\) −2.34315 −0.101303
\(536\) 0 0
\(537\) 18.8284 0.812507
\(538\) 0 0
\(539\) −17.0711 −0.735303
\(540\) 0 0
\(541\) 9.31371 0.400428 0.200214 0.979752i \(-0.435836\pi\)
0.200214 + 0.979752i \(0.435836\pi\)
\(542\) 0 0
\(543\) −8.14214 −0.349412
\(544\) 0 0
\(545\) 20.1421 0.862794
\(546\) 0 0
\(547\) −8.48528 −0.362804 −0.181402 0.983409i \(-0.558064\pi\)
−0.181402 + 0.983409i \(0.558064\pi\)
\(548\) 0 0
\(549\) 9.65685 0.412144
\(550\) 0 0
\(551\) −7.07107 −0.301238
\(552\) 0 0
\(553\) 3.31371 0.140913
\(554\) 0 0
\(555\) 2.24264 0.0951948
\(556\) 0 0
\(557\) 23.6569 1.00237 0.501187 0.865339i \(-0.332897\pi\)
0.501187 + 0.865339i \(0.332897\pi\)
\(558\) 0 0
\(559\) 10.4853 0.443480
\(560\) 0 0
\(561\) 9.65685 0.407713
\(562\) 0 0
\(563\) −0.343146 −0.0144619 −0.00723093 0.999974i \(-0.502302\pi\)
−0.00723093 + 0.999974i \(0.502302\pi\)
\(564\) 0 0
\(565\) −7.31371 −0.307690
\(566\) 0 0
\(567\) 1.41421 0.0593914
\(568\) 0 0
\(569\) −42.1838 −1.76844 −0.884218 0.467075i \(-0.845308\pi\)
−0.884218 + 0.467075i \(0.845308\pi\)
\(570\) 0 0
\(571\) 2.14214 0.0896456 0.0448228 0.998995i \(-0.485728\pi\)
0.0448228 + 0.998995i \(0.485728\pi\)
\(572\) 0 0
\(573\) 22.2426 0.929200
\(574\) 0 0
\(575\) −0.828427 −0.0345478
\(576\) 0 0
\(577\) −9.51472 −0.396103 −0.198051 0.980192i \(-0.563461\pi\)
−0.198051 + 0.980192i \(0.563461\pi\)
\(578\) 0 0
\(579\) −1.07107 −0.0445121
\(580\) 0 0
\(581\) −5.17157 −0.214553
\(582\) 0 0
\(583\) −28.9706 −1.19984
\(584\) 0 0
\(585\) −3.41421 −0.141160
\(586\) 0 0
\(587\) −27.4558 −1.13322 −0.566612 0.823985i \(-0.691746\pi\)
−0.566612 + 0.823985i \(0.691746\pi\)
\(588\) 0 0
\(589\) 0.828427 0.0341347
\(590\) 0 0
\(591\) −6.14214 −0.252654
\(592\) 0 0
\(593\) 44.6274 1.83263 0.916314 0.400460i \(-0.131150\pi\)
0.916314 + 0.400460i \(0.131150\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 14.8284 0.606887
\(598\) 0 0
\(599\) −16.9706 −0.693398 −0.346699 0.937976i \(-0.612698\pi\)
−0.346699 + 0.937976i \(0.612698\pi\)
\(600\) 0 0
\(601\) 6.20101 0.252944 0.126472 0.991970i \(-0.459635\pi\)
0.126472 + 0.991970i \(0.459635\pi\)
\(602\) 0 0
\(603\) −11.3137 −0.460730
\(604\) 0 0
\(605\) −0.656854 −0.0267049
\(606\) 0 0
\(607\) −33.4558 −1.35793 −0.678965 0.734170i \(-0.737571\pi\)
−0.678965 + 0.734170i \(0.737571\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 30.1421 1.21942
\(612\) 0 0
\(613\) −12.8284 −0.518135 −0.259068 0.965859i \(-0.583415\pi\)
−0.259068 + 0.965859i \(0.583415\pi\)
\(614\) 0 0
\(615\) 9.89949 0.399186
\(616\) 0 0
\(617\) −15.1127 −0.608414 −0.304207 0.952606i \(-0.598392\pi\)
−0.304207 + 0.952606i \(0.598392\pi\)
\(618\) 0 0
\(619\) −3.51472 −0.141268 −0.0706342 0.997502i \(-0.522502\pi\)
−0.0706342 + 0.997502i \(0.522502\pi\)
\(620\) 0 0
\(621\) −0.828427 −0.0332436
\(622\) 0 0
\(623\) −5.31371 −0.212889
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.41421 −0.136351
\(628\) 0 0
\(629\) −6.34315 −0.252918
\(630\) 0 0
\(631\) 36.2843 1.44445 0.722227 0.691656i \(-0.243119\pi\)
0.722227 + 0.691656i \(0.243119\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 5.65685 0.224485
\(636\) 0 0
\(637\) −17.0711 −0.676380
\(638\) 0 0
\(639\) 9.17157 0.362822
\(640\) 0 0
\(641\) 16.9289 0.668653 0.334326 0.942457i \(-0.391491\pi\)
0.334326 + 0.942457i \(0.391491\pi\)
\(642\) 0 0
\(643\) 28.9289 1.14085 0.570423 0.821351i \(-0.306779\pi\)
0.570423 + 0.821351i \(0.306779\pi\)
\(644\) 0 0
\(645\) −3.07107 −0.120923
\(646\) 0 0
\(647\) −25.3137 −0.995185 −0.497592 0.867411i \(-0.665782\pi\)
−0.497592 + 0.867411i \(0.665782\pi\)
\(648\) 0 0
\(649\) 23.3137 0.915143
\(650\) 0 0
\(651\) −1.17157 −0.0459176
\(652\) 0 0
\(653\) 10.1421 0.396892 0.198446 0.980112i \(-0.436410\pi\)
0.198446 + 0.980112i \(0.436410\pi\)
\(654\) 0 0
\(655\) −18.7279 −0.731760
\(656\) 0 0
\(657\) −0.343146 −0.0133874
\(658\) 0 0
\(659\) −28.9706 −1.12853 −0.564266 0.825593i \(-0.690841\pi\)
−0.564266 + 0.825593i \(0.690841\pi\)
\(660\) 0 0
\(661\) −49.1127 −1.91026 −0.955131 0.296183i \(-0.904286\pi\)
−0.955131 + 0.296183i \(0.904286\pi\)
\(662\) 0 0
\(663\) 9.65685 0.375041
\(664\) 0 0
\(665\) 1.41421 0.0548408
\(666\) 0 0
\(667\) −5.85786 −0.226817
\(668\) 0 0
\(669\) 11.3137 0.437413
\(670\) 0 0
\(671\) 32.9706 1.27281
\(672\) 0 0
\(673\) 14.2426 0.549013 0.274507 0.961585i \(-0.411485\pi\)
0.274507 + 0.961585i \(0.411485\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 29.1716 1.12115 0.560577 0.828102i \(-0.310579\pi\)
0.560577 + 0.828102i \(0.310579\pi\)
\(678\) 0 0
\(679\) −11.1716 −0.428726
\(680\) 0 0
\(681\) −0.343146 −0.0131494
\(682\) 0 0
\(683\) −35.3137 −1.35124 −0.675621 0.737249i \(-0.736124\pi\)
−0.675621 + 0.737249i \(0.736124\pi\)
\(684\) 0 0
\(685\) −4.34315 −0.165943
\(686\) 0 0
\(687\) −10.3431 −0.394616
\(688\) 0 0
\(689\) −28.9706 −1.10369
\(690\) 0 0
\(691\) −8.20101 −0.311981 −0.155991 0.987759i \(-0.549857\pi\)
−0.155991 + 0.987759i \(0.549857\pi\)
\(692\) 0 0
\(693\) 4.82843 0.183417
\(694\) 0 0
\(695\) 2.82843 0.107288
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 4.34315 0.164273
\(700\) 0 0
\(701\) 22.9706 0.867586 0.433793 0.901013i \(-0.357175\pi\)
0.433793 + 0.901013i \(0.357175\pi\)
\(702\) 0 0
\(703\) 2.24264 0.0845828
\(704\) 0 0
\(705\) −8.82843 −0.332498
\(706\) 0 0
\(707\) 6.82843 0.256809
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) 0.686292 0.0257018
\(714\) 0 0
\(715\) −11.6569 −0.435942
\(716\) 0 0
\(717\) 26.2426 0.980049
\(718\) 0 0
\(719\) −1.75736 −0.0655384 −0.0327692 0.999463i \(-0.510433\pi\)
−0.0327692 + 0.999463i \(0.510433\pi\)
\(720\) 0 0
\(721\) 27.3137 1.01722
\(722\) 0 0
\(723\) 23.6569 0.879808
\(724\) 0 0
\(725\) 7.07107 0.262613
\(726\) 0 0
\(727\) −15.7574 −0.584408 −0.292204 0.956356i \(-0.594389\pi\)
−0.292204 + 0.956356i \(0.594389\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.68629 0.321274
\(732\) 0 0
\(733\) −33.3137 −1.23047 −0.615235 0.788344i \(-0.710939\pi\)
−0.615235 + 0.788344i \(0.710939\pi\)
\(734\) 0 0
\(735\) 5.00000 0.184428
\(736\) 0 0
\(737\) −38.6274 −1.42286
\(738\) 0 0
\(739\) 15.3137 0.563324 0.281662 0.959514i \(-0.409114\pi\)
0.281662 + 0.959514i \(0.409114\pi\)
\(740\) 0 0
\(741\) −3.41421 −0.125424
\(742\) 0 0
\(743\) 49.2548 1.80698 0.903492 0.428604i \(-0.140995\pi\)
0.903492 + 0.428604i \(0.140995\pi\)
\(744\) 0 0
\(745\) 15.6569 0.573623
\(746\) 0 0
\(747\) −3.65685 −0.133797
\(748\) 0 0
\(749\) 3.31371 0.121080
\(750\) 0 0
\(751\) 15.8579 0.578662 0.289331 0.957229i \(-0.406567\pi\)
0.289331 + 0.957229i \(0.406567\pi\)
\(752\) 0 0
\(753\) 14.2426 0.519031
\(754\) 0 0
\(755\) −4.82843 −0.175724
\(756\) 0 0
\(757\) −9.51472 −0.345818 −0.172909 0.984938i \(-0.555317\pi\)
−0.172909 + 0.984938i \(0.555317\pi\)
\(758\) 0 0
\(759\) −2.82843 −0.102665
\(760\) 0 0
\(761\) −28.3431 −1.02744 −0.513719 0.857958i \(-0.671733\pi\)
−0.513719 + 0.857958i \(0.671733\pi\)
\(762\) 0 0
\(763\) −28.4853 −1.03124
\(764\) 0 0
\(765\) −2.82843 −0.102262
\(766\) 0 0
\(767\) 23.3137 0.841809
\(768\) 0 0
\(769\) 22.2843 0.803591 0.401796 0.915729i \(-0.368386\pi\)
0.401796 + 0.915729i \(0.368386\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) −45.4558 −1.63493 −0.817467 0.575976i \(-0.804622\pi\)
−0.817467 + 0.575976i \(0.804622\pi\)
\(774\) 0 0
\(775\) −0.828427 −0.0297580
\(776\) 0 0
\(777\) −3.17157 −0.113780
\(778\) 0 0
\(779\) 9.89949 0.354686
\(780\) 0 0
\(781\) 31.3137 1.12049
\(782\) 0 0
\(783\) 7.07107 0.252699
\(784\) 0 0
\(785\) −24.1421 −0.861670
\(786\) 0 0
\(787\) 1.17157 0.0417621 0.0208810 0.999782i \(-0.493353\pi\)
0.0208810 + 0.999782i \(0.493353\pi\)
\(788\) 0 0
\(789\) 2.00000 0.0712019
\(790\) 0 0
\(791\) 10.3431 0.367760
\(792\) 0 0
\(793\) 32.9706 1.17082
\(794\) 0 0
\(795\) 8.48528 0.300942
\(796\) 0 0
\(797\) 27.3137 0.967501 0.483751 0.875206i \(-0.339274\pi\)
0.483751 + 0.875206i \(0.339274\pi\)
\(798\) 0 0
\(799\) 24.9706 0.883395
\(800\) 0 0
\(801\) −3.75736 −0.132760
\(802\) 0 0
\(803\) −1.17157 −0.0413439
\(804\) 0 0
\(805\) 1.17157 0.0412925
\(806\) 0 0
\(807\) −22.3848 −0.787981
\(808\) 0 0
\(809\) −30.6863 −1.07887 −0.539436 0.842026i \(-0.681363\pi\)
−0.539436 + 0.842026i \(0.681363\pi\)
\(810\) 0 0
\(811\) −19.0294 −0.668214 −0.334107 0.942535i \(-0.608435\pi\)
−0.334107 + 0.942535i \(0.608435\pi\)
\(812\) 0 0
\(813\) −2.82843 −0.0991973
\(814\) 0 0
\(815\) 7.55635 0.264687
\(816\) 0 0
\(817\) −3.07107 −0.107443
\(818\) 0 0
\(819\) 4.82843 0.168719
\(820\) 0 0
\(821\) 17.7990 0.621189 0.310595 0.950543i \(-0.399472\pi\)
0.310595 + 0.950543i \(0.399472\pi\)
\(822\) 0 0
\(823\) −2.87006 −0.100044 −0.0500220 0.998748i \(-0.515929\pi\)
−0.0500220 + 0.998748i \(0.515929\pi\)
\(824\) 0 0
\(825\) 3.41421 0.118868
\(826\) 0 0
\(827\) 3.65685 0.127161 0.0635806 0.997977i \(-0.479748\pi\)
0.0635806 + 0.997977i \(0.479748\pi\)
\(828\) 0 0
\(829\) −51.4558 −1.78714 −0.893568 0.448929i \(-0.851806\pi\)
−0.893568 + 0.448929i \(0.851806\pi\)
\(830\) 0 0
\(831\) 20.6274 0.715557
\(832\) 0 0
\(833\) −14.1421 −0.489996
\(834\) 0 0
\(835\) 5.31371 0.183888
\(836\) 0 0
\(837\) −0.828427 −0.0286346
\(838\) 0 0
\(839\) −5.85786 −0.202236 −0.101118 0.994874i \(-0.532242\pi\)
−0.101118 + 0.994874i \(0.532242\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 6.38478 0.219903
\(844\) 0 0
\(845\) 1.34315 0.0462056
\(846\) 0 0
\(847\) 0.928932 0.0319185
\(848\) 0 0
\(849\) −16.2426 −0.557446
\(850\) 0 0
\(851\) 1.85786 0.0636868
\(852\) 0 0
\(853\) 8.82843 0.302280 0.151140 0.988512i \(-0.451706\pi\)
0.151140 + 0.988512i \(0.451706\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 25.4558 0.869555 0.434778 0.900538i \(-0.356827\pi\)
0.434778 + 0.900538i \(0.356827\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) −14.0000 −0.477119
\(862\) 0 0
\(863\) −33.2548 −1.13201 −0.566004 0.824403i \(-0.691511\pi\)
−0.566004 + 0.824403i \(0.691511\pi\)
\(864\) 0 0
\(865\) −11.3137 −0.384678
\(866\) 0 0
\(867\) −9.00000 −0.305656
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −38.6274 −1.30884
\(872\) 0 0
\(873\) −7.89949 −0.267357
\(874\) 0 0
\(875\) −1.41421 −0.0478091
\(876\) 0 0
\(877\) −46.0416 −1.55472 −0.777358 0.629059i \(-0.783440\pi\)
−0.777358 + 0.629059i \(0.783440\pi\)
\(878\) 0 0
\(879\) −9.65685 −0.325718
\(880\) 0 0
\(881\) −34.4853 −1.16184 −0.580919 0.813961i \(-0.697307\pi\)
−0.580919 + 0.813961i \(0.697307\pi\)
\(882\) 0 0
\(883\) 35.7574 1.20333 0.601665 0.798748i \(-0.294504\pi\)
0.601665 + 0.798748i \(0.294504\pi\)
\(884\) 0 0
\(885\) −6.82843 −0.229535
\(886\) 0 0
\(887\) 21.6569 0.727166 0.363583 0.931562i \(-0.381553\pi\)
0.363583 + 0.931562i \(0.381553\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 3.41421 0.114380
\(892\) 0 0
\(893\) −8.82843 −0.295432
\(894\) 0 0
\(895\) −18.8284 −0.629365
\(896\) 0 0
\(897\) −2.82843 −0.0944384
\(898\) 0 0
\(899\) −5.85786 −0.195371
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 4.34315 0.144531
\(904\) 0 0
\(905\) 8.14214 0.270654
\(906\) 0 0
\(907\) −52.7696 −1.75218 −0.876092 0.482144i \(-0.839858\pi\)
−0.876092 + 0.482144i \(0.839858\pi\)
\(908\) 0 0
\(909\) 4.82843 0.160149
\(910\) 0 0
\(911\) 10.6274 0.352102 0.176051 0.984381i \(-0.443668\pi\)
0.176051 + 0.984381i \(0.443668\pi\)
\(912\) 0 0
\(913\) −12.4853 −0.413203
\(914\) 0 0
\(915\) −9.65685 −0.319246
\(916\) 0 0
\(917\) 26.4853 0.874621
\(918\) 0 0
\(919\) −27.5980 −0.910373 −0.455187 0.890396i \(-0.650427\pi\)
−0.455187 + 0.890396i \(0.650427\pi\)
\(920\) 0 0
\(921\) −1.85786 −0.0612187
\(922\) 0 0
\(923\) 31.3137 1.03070
\(924\) 0 0
\(925\) −2.24264 −0.0737376
\(926\) 0 0
\(927\) 19.3137 0.634345
\(928\) 0 0
\(929\) −53.7990 −1.76509 −0.882544 0.470230i \(-0.844171\pi\)
−0.882544 + 0.470230i \(0.844171\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 0 0
\(933\) −19.2132 −0.629012
\(934\) 0 0
\(935\) −9.65685 −0.315813
\(936\) 0 0
\(937\) 9.79899 0.320119 0.160060 0.987107i \(-0.448831\pi\)
0.160060 + 0.987107i \(0.448831\pi\)
\(938\) 0 0
\(939\) −13.7990 −0.450313
\(940\) 0 0
\(941\) 16.2426 0.529495 0.264747 0.964318i \(-0.414711\pi\)
0.264747 + 0.964318i \(0.414711\pi\)
\(942\) 0 0
\(943\) 8.20101 0.267062
\(944\) 0 0
\(945\) −1.41421 −0.0460044
\(946\) 0 0
\(947\) 18.2843 0.594159 0.297079 0.954853i \(-0.403987\pi\)
0.297079 + 0.954853i \(0.403987\pi\)
\(948\) 0 0
\(949\) −1.17157 −0.0380309
\(950\) 0 0
\(951\) −2.14214 −0.0694635
\(952\) 0 0
\(953\) 3.02944 0.0981331 0.0490665 0.998796i \(-0.484375\pi\)
0.0490665 + 0.998796i \(0.484375\pi\)
\(954\) 0 0
\(955\) −22.2426 −0.719755
\(956\) 0 0
\(957\) 24.1421 0.780404
\(958\) 0 0
\(959\) 6.14214 0.198340
\(960\) 0 0
\(961\) −30.3137 −0.977862
\(962\) 0 0
\(963\) 2.34315 0.0755068
\(964\) 0 0
\(965\) 1.07107 0.0344789
\(966\) 0 0
\(967\) 14.5858 0.469047 0.234524 0.972110i \(-0.424647\pi\)
0.234524 + 0.972110i \(0.424647\pi\)
\(968\) 0 0
\(969\) −2.82843 −0.0908622
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) 3.41421 0.109342
\(976\) 0 0
\(977\) 20.4853 0.655382 0.327691 0.944785i \(-0.393729\pi\)
0.327691 + 0.944785i \(0.393729\pi\)
\(978\) 0 0
\(979\) −12.8284 −0.409998
\(980\) 0 0
\(981\) −20.1421 −0.643089
\(982\) 0 0
\(983\) −15.9411 −0.508443 −0.254221 0.967146i \(-0.581819\pi\)
−0.254221 + 0.967146i \(0.581819\pi\)
\(984\) 0 0
\(985\) 6.14214 0.195705
\(986\) 0 0
\(987\) 12.4853 0.397411
\(988\) 0 0
\(989\) −2.54416 −0.0808995
\(990\) 0 0
\(991\) 10.3431 0.328561 0.164280 0.986414i \(-0.447470\pi\)
0.164280 + 0.986414i \(0.447470\pi\)
\(992\) 0 0
\(993\) −16.1421 −0.512255
\(994\) 0 0
\(995\) −14.8284 −0.470093
\(996\) 0 0
\(997\) 27.4558 0.869535 0.434768 0.900543i \(-0.356830\pi\)
0.434768 + 0.900543i \(0.356830\pi\)
\(998\) 0 0
\(999\) −2.24264 −0.0709540
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.bn.1.2 2
4.3 odd 2 2280.2.a.k.1.1 2
12.11 even 2 6840.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.k.1.1 2 4.3 odd 2
4560.2.a.bn.1.2 2 1.1 even 1 trivial
6840.2.a.bb.1.1 2 12.11 even 2