Properties

Label 7728.2.a.n.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{13} -3.00000 q^{15} -4.00000 q^{17} +1.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +5.00000 q^{29} +10.0000 q^{31} -3.00000 q^{35} -3.00000 q^{37} -1.00000 q^{39} -11.0000 q^{41} +9.00000 q^{43} -3.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +12.0000 q^{53} -10.0000 q^{59} -10.0000 q^{61} +1.00000 q^{63} +3.00000 q^{65} +12.0000 q^{67} -1.00000 q^{69} -6.00000 q^{71} -4.00000 q^{73} +4.00000 q^{75} -12.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +12.0000 q^{85} +5.00000 q^{87} +6.00000 q^{89} -1.00000 q^{91} +10.0000 q^{93} +9.00000 q^{97} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −11.0000 −0.991837
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −15.0000 −1.33103 −0.665517 0.746382i \(-0.731789\pi\)
−0.665517 + 0.746382i \(0.731789\pi\)
\(128\) 0 0
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −11.0000 −0.939793 −0.469897 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −15.0000 −1.24568
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −30.0000 −2.40966
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) −13.0000 −0.971666 −0.485833 0.874052i \(-0.661484\pi\)
−0.485833 + 0.874052i \(0.661484\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −25.0000 −1.79954 −0.899770 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 33.0000 2.30482
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −27.0000 −1.84138
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 27.0000 1.73922 0.869611 0.493737i \(-0.164369\pi\)
0.869611 + 0.493737i \(0.164369\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) 31.0000 1.91154 0.955771 0.294112i \(-0.0950239\pi\)
0.955771 + 0.294112i \(0.0950239\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.0000 −0.649309
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 9.00000 0.527589
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 30.0000 1.74667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 9.00000 0.518751
\(302\) 0 0
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 19.0000 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 1.00000 0.0553001
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) 0 0
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −33.0000 −1.77153 −0.885766 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −33.0000 −1.75641 −0.878206 0.478282i \(-0.841260\pi\)
−0.878206 + 0.478282i \(0.841260\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 0 0
\(369\) −11.0000 −0.572637
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −21.0000 −1.07870 −0.539349 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(380\) 0 0
\(381\) −15.0000 −0.768473
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.00000 0.457496
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) 36.0000 1.81136
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) −11.0000 −0.542590
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) −3.00000 −0.146911
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.0000 −0.818861 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) −15.0000 −0.719195
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −38.0000 −1.79333 −0.896665 0.442709i \(-0.854018\pi\)
−0.896665 + 0.442709i \(0.854018\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 9.00000 0.422857
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −37.0000 −1.71954 −0.859768 0.510685i \(-0.829392\pi\)
−0.859768 + 0.510685i \(0.829392\pi\)
\(464\) 0 0
\(465\) −30.0000 −1.39122
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) 3.00000 0.136788
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −27.0000 −1.22601
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 39.0000 1.71855
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 11.0000 0.476463
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −13.0000 −0.560991
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 0 0
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) −25.0000 −1.03896
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 15.0000 0.617018
\(592\) 0 0
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) −11.0000 −0.450200
\(598\) 0 0
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) 36.0000 1.46847 0.734235 0.678895i \(-0.237541\pi\)
0.734235 + 0.678895i \(0.237541\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) 0 0
\(609\) 5.00000 0.202610
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) −5.00000 −0.201948 −0.100974 0.994889i \(-0.532196\pi\)
−0.100974 + 0.994889i \(0.532196\pi\)
\(614\) 0 0
\(615\) 33.0000 1.33069
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 45.0000 1.78577
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 35.0000 1.38242 0.691208 0.722655i \(-0.257079\pi\)
0.691208 + 0.722655i \(0.257079\pi\)
\(642\) 0 0
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) 0 0
\(645\) −27.0000 −1.06312
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) 25.0000 0.978326 0.489163 0.872192i \(-0.337302\pi\)
0.489163 + 0.872192i \(0.337302\pi\)
\(654\) 0 0
\(655\) 54.0000 2.10995
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 9.00000 0.345388
\(680\) 0 0
\(681\) −13.0000 −0.498161
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 33.0000 1.26087
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.00000 0.341389
\(696\) 0 0
\(697\) 44.0000 1.66662
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) 19.0000 0.708580 0.354290 0.935136i \(-0.384723\pi\)
0.354290 + 0.935136i \(0.384723\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) 27.0000 1.00414
\(724\) 0 0
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) −5.00000 −0.182210
\(754\) 0 0
\(755\) −27.0000 −0.982631
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 1.00000 0.0362024
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) 10.0000 0.361079
\(768\) 0 0
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 5.00000 0.179838 0.0899188 0.995949i \(-0.471339\pi\)
0.0899188 + 0.995949i \(0.471339\pi\)
\(774\) 0 0
\(775\) 40.0000 1.43684
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) 31.0000 1.10363
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) −36.0000 −1.27679
\(796\) 0 0
\(797\) −23.0000 −0.814702 −0.407351 0.913272i \(-0.633547\pi\)
−0.407351 + 0.913272i \(0.633547\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 52.0000 1.82822 0.914111 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 33.0000 1.15031 0.575154 0.818045i \(-0.304942\pi\)
0.575154 + 0.818045i \(0.304942\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 16.0000 0.555034
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −13.0000 −0.447744
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −27.0000 −0.924462 −0.462231 0.886759i \(-0.652951\pi\)
−0.462231 + 0.886759i \(0.652951\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43.0000 −1.46885 −0.734426 0.678689i \(-0.762549\pi\)
−0.734426 + 0.678689i \(0.762549\pi\)
\(858\) 0 0
\(859\) 49.0000 1.67186 0.835929 0.548837i \(-0.184929\pi\)
0.835929 + 0.548837i \(0.184929\pi\)
\(860\) 0 0
\(861\) −11.0000 −0.374879
\(862\) 0 0
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 9.00000 0.304604
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 6.00000 0.201916 0.100958 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(884\) 0 0
\(885\) 30.0000 1.00844
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −15.0000 −0.503084
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 39.0000 1.30363
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) 0 0
\(899\) 50.0000 1.66759
\(900\) 0 0
\(901\) −48.0000 −1.59911
\(902\) 0 0
\(903\) 9.00000 0.299501
\(904\) 0 0
\(905\) −66.0000 −2.19391
\(906\) 0 0
\(907\) 23.0000 0.763702 0.381851 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 30.0000 0.991769
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 19.0000 0.626071
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) −13.0000 −0.426976
\(928\) 0 0
\(929\) −11.0000 −0.360898 −0.180449 0.983584i \(-0.557755\pi\)
−0.180449 + 0.983584i \(0.557755\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.0000 1.07806 0.539032 0.842286i \(-0.318790\pi\)
0.539032 + 0.842286i \(0.318790\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 13.0000 0.423788 0.211894 0.977293i \(-0.432037\pi\)
0.211894 + 0.977293i \(0.432037\pi\)
\(942\) 0 0
\(943\) 11.0000 0.358209
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) 1.00000 0.0324956 0.0162478 0.999868i \(-0.494828\pi\)
0.0162478 + 0.999868i \(0.494828\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 36.0000 1.16493
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.0000 −0.355209
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 75.0000 2.41434
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) −3.00000 −0.0961756
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 0 0
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) −45.0000 −1.43382
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 33.0000 1.04617
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 0 0
\(999\) −3.00000 −0.0949158
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7728.2.a.n.1.1 1
4.3 odd 2 3864.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.a.1.1 1 4.3 odd 2
7728.2.a.n.1.1 1 1.1 even 1 trivial