Properties

Label 2028.2.b.a.337.2
Level $2028$
Weight $2$
Character 2028.337
Analytic conductor $16.194$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1936615299\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2028.337
Dual form 2028.2.b.a.337.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} -4.00000i q^{15} -2.00000 q^{17} +2.00000i q^{19} +2.00000i q^{21} -11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +10.0000i q^{31} +4.00000i q^{33} +8.00000 q^{35} +10.0000i q^{37} -8.00000i q^{41} -4.00000 q^{43} +4.00000i q^{45} -4.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} +16.0000 q^{55} -2.00000i q^{57} -8.00000i q^{59} -14.0000 q^{61} -2.00000i q^{63} -2.00000i q^{67} -16.0000i q^{71} -10.0000i q^{73} +11.0000 q^{75} -8.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -8.00000i q^{85} +6.00000 q^{87} -4.00000i q^{89} -10.0000i q^{93} -8.00000 q^{95} +2.00000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} - 4q^{17} - 22q^{25} - 2q^{27} - 12q^{29} + 16q^{35} - 8q^{43} + 6q^{49} + 4q^{51} - 20q^{53} + 32q^{55} - 28q^{61} + 22q^{75} - 16q^{77} - 32q^{79} + 2q^{81} + 12q^{87} - 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 4.00000i − 1.03280i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.00000i − 1.24939i −0.780869 0.624695i \(-0.785223\pi\)
0.780869 0.624695i \(-0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 4.00000i 0.596285i
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) − 8.00000i − 1.04151i −0.853706 0.520756i \(-0.825650\pi\)
0.853706 0.520756i \(-0.174350\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 16.0000i − 1.89885i −0.313993 0.949425i \(-0.601667\pi\)
0.313993 0.949425i \(-0.398333\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) − 8.00000i − 0.867722i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) − 4.00000i − 0.423999i −0.977270 0.212000i \(-0.932002\pi\)
0.977270 0.212000i \(-0.0679975\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 10.0000i − 1.03695i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) − 10.0000i − 0.949158i
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000i 0.366679i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) − 24.0000i − 2.14663i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) − 4.00000i − 0.344265i
\(136\) 0 0
\(137\) 8.00000i 0.683486i 0.939793 + 0.341743i \(0.111017\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 24.0000i − 1.99309i
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 18.0000i 1.46482i 0.680864 + 0.732410i \(0.261604\pi\)
−0.680864 + 0.732410i \(0.738396\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −40.0000 −3.21288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000i 0.156652i 0.996928 + 0.0783260i \(0.0249575\pi\)
−0.996928 + 0.0783260i \(0.975042\pi\)
\(164\) 0 0
\(165\) −16.0000 −1.24560
\(166\) 0 0
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 22.0000i 1.66304i
\(176\) 0 0
\(177\) 8.00000i 0.601317i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) −40.0000 −2.94086
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 2.00000i 0.141069i
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 32.0000 2.23498
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) − 16.0000i − 1.09119i
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) 0 0
\(219\) 10.0000i 0.675737i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) −11.0000 −0.733333
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) − 2.00000i − 0.132164i −0.997814 0.0660819i \(-0.978950\pi\)
0.997814 0.0660819i \(-0.0210498\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 24.0000i 1.55243i 0.630468 + 0.776215i \(0.282863\pi\)
−0.630468 + 0.776215i \(0.717137\pi\)
\(240\) 0 0
\(241\) − 2.00000i − 0.128831i −0.997923 0.0644157i \(-0.979482\pi\)
0.997923 0.0644157i \(-0.0205183\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 12.0000i 0.766652i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 8.00000i 0.500979i
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) − 40.0000i − 2.45718i
\(266\) 0 0
\(267\) 4.00000i 0.244796i
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) − 10.0000i − 0.607457i −0.952759 0.303728i \(-0.901768\pi\)
0.952759 0.303728i \(-0.0982315\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 44.0000i 2.65330i
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 10.0000i 0.598684i
\(280\) 0 0
\(281\) − 8.00000i − 0.477240i −0.971113 0.238620i \(-0.923305\pi\)
0.971113 0.238620i \(-0.0766950\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −16.0000 −0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 2.00000i − 0.117242i
\(292\) 0 0
\(293\) 8.00000i 0.467365i 0.972313 + 0.233682i \(0.0750776\pi\)
−0.972313 + 0.233682i \(0.924922\pi\)
\(294\) 0 0
\(295\) 32.0000 1.86311
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) − 56.0000i − 3.20655i
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 24.0000i 1.34374i
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.00000i − 0.110600i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) − 2.00000i − 0.109930i −0.998488 0.0549650i \(-0.982495\pi\)
0.998488 0.0549650i \(-0.0175047\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 40.0000 2.16612
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 18.0000i 0.963518i 0.876304 + 0.481759i \(0.160002\pi\)
−0.876304 + 0.481759i \(0.839998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0000i 0.851594i 0.904819 + 0.425797i \(0.140006\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 64.0000 3.39677
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) − 12.0000i − 0.633336i −0.948536 0.316668i \(-0.897436\pi\)
0.948536 0.316668i \(-0.102564\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 40.0000 2.09370
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) − 8.00000i − 0.416463i
\(370\) 0 0
\(371\) 20.0000i 1.03835i
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) 24.0000i 1.23935i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 6.00000i − 0.308199i −0.988055 0.154100i \(-0.950752\pi\)
0.988055 0.154100i \(-0.0492477\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) − 32.0000i − 1.63087i
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 64.0000i − 3.22019i
\(396\) 0 0
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) − 12.0000i − 0.599251i −0.954057 0.299626i \(-0.903138\pi\)
0.954057 0.299626i \(-0.0968618\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.00000i 0.198762i
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) − 14.0000i − 0.692255i −0.938187 0.346128i \(-0.887496\pi\)
0.938187 0.346128i \(-0.112504\pi\)
\(410\) 0 0
\(411\) − 8.00000i − 0.394611i
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 2.00000i 0.0974740i 0.998812 + 0.0487370i \(0.0155196\pi\)
−0.998812 + 0.0487370i \(0.984480\pi\)
\(422\) 0 0
\(423\) − 4.00000i − 0.194487i
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000i 0.963366i 0.876346 + 0.481683i \(0.159974\pi\)
−0.876346 + 0.481683i \(0.840026\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 24.0000i 1.15071i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.0000i 0.566315i 0.959073 + 0.283158i \(0.0913819\pi\)
−0.959073 + 0.283158i \(0.908618\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) − 18.0000i − 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) − 12.0000i − 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i 0.859454 + 0.511213i \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) 0 0
\(465\) 40.0000 1.85496
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) − 22.0000i − 1.00943i
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) − 16.0000i − 0.731059i −0.930800 0.365529i \(-0.880888\pi\)
0.930800 0.365529i \(-0.119112\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) − 2.00000i − 0.0904431i
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) 6.00000i 0.268597i 0.990941 + 0.134298i \(0.0428781\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) − 40.0000i − 1.77998i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 12.0000i − 0.531891i −0.963988 0.265945i \(-0.914316\pi\)
0.963988 0.265945i \(-0.0856841\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) − 2.00000i − 0.0883022i
\(514\) 0 0
\(515\) 32.0000i 1.41009i
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) − 22.0000i − 0.960159i
\(526\) 0 0
\(527\) − 20.0000i − 0.871214i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 8.00000i − 0.347170i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 48.0000i 2.07522i
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) − 12.0000i − 0.516877i
\(540\) 0 0
\(541\) 38.0000i 1.63375i 0.576816 + 0.816874i \(0.304295\pi\)
−0.576816 + 0.816874i \(0.695705\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) − 12.0000i − 0.511217i
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 40.0000 1.69791
\(556\) 0 0
\(557\) 24.0000i 1.01691i 0.861088 + 0.508456i \(0.169784\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 8.00000i − 0.337760i
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 24.0000i 1.00969i
\(566\) 0 0
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) 0 0
\(579\) 14.0000i 0.581820i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000i 1.65663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) − 12.0000i − 0.493614i
\(592\) 0 0
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 0 0
\(605\) − 20.0000i − 0.813116i
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) − 12.0000i − 0.486265i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) −32.0000 −1.29036
\(616\) 0 0
\(617\) 48.0000i 1.93241i 0.257780 + 0.966204i \(0.417009\pi\)
−0.257780 + 0.966204i \(0.582991\pi\)
\(618\) 0 0
\(619\) 14.0000i 0.562708i 0.959604 + 0.281354i \(0.0907834\pi\)
−0.959604 + 0.281354i \(0.909217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) − 20.0000i − 0.797452i
\(630\) 0 0
\(631\) − 30.0000i − 1.19428i −0.802137 0.597141i \(-0.796303\pi\)
0.802137 0.597141i \(-0.203697\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) − 48.0000i − 1.90482i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 16.0000i − 0.632950i
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) − 26.0000i − 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 0 0
\(645\) 16.0000i 0.629999i
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) 16.0000i 0.625172i
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 10.0000i 0.388955i 0.980907 + 0.194477i \(0.0623011\pi\)
−0.980907 + 0.194477i \(0.937699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000i 0.620453i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.0000i 0.541271i
\(670\) 0 0
\(671\) 56.0000i 2.16186i
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 11.0000 0.423390
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) − 4.00000i − 0.153280i
\(682\) 0 0
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 10.0000i − 0.380418i −0.981744 0.190209i \(-0.939083\pi\)
0.981744 0.190209i \(-0.0609166\pi\)
\(692\) 0 0
\(693\) −8.00000 −0.303895
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) 20.0000i 0.752177i
\(708\) 0 0
\(709\) 6.00000i 0.225335i 0.993633 + 0.112667i \(0.0359394\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) − 16.0000i − 0.595871i
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) − 22.0000i − 0.812589i −0.913742 0.406294i \(-0.866821\pi\)
0.913742 0.406294i \(-0.133179\pi\)
\(734\) 0 0
\(735\) − 12.0000i − 0.442627i
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 22.0000i 0.809283i 0.914475 + 0.404642i \(0.132604\pi\)
−0.914475 + 0.404642i \(0.867396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 24.0000i − 0.876941i
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 0 0
\(755\) −72.0000 −2.62035
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000i 1.45000i 0.688749 + 0.724999i \(0.258160\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) − 8.00000i − 0.289241i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000i 0.793340i 0.917961 + 0.396670i \(0.129834\pi\)
−0.917961 + 0.396670i \(0.870166\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 0 0
\(775\) − 110.000i − 3.95132i
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 8.00000i 0.285532i
\(786\) 0 0
\(787\) − 50.0000i − 1.78231i −0.453701 0.891154i \(-0.649897\pi\)
0.453701 0.891154i \(-0.350103\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) − 12.0000i − 0.426671i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 40.0000i 1.41865i
\(796\) 0 0
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 8.00000i 0.283020i
\(800\) 0 0
\(801\) − 4.00000i − 0.141333i
\(802\) 0 0
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) − 6.00000i − 0.210688i −0.994436 0.105344i \(-0.966406\pi\)
0.994436 0.105344i \(-0.0335944\pi\)
\(812\) 0 0
\(813\) 10.0000i 0.350715i
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) − 8.00000i − 0.279885i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 24.0000i − 0.837606i −0.908077 0.418803i \(-0.862450\pi\)
0.908077 0.418803i \(-0.137550\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) − 44.0000i − 1.53188i
\(826\) 0 0
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 48.0000 1.66111
\(836\) 0 0
\(837\) − 10.0000i − 0.345651i
\(838\) 0 0
\(839\) − 16.0000i − 0.552381i −0.961103 0.276191i \(-0.910928\pi\)
0.961103 0.276191i \(-0.0890721\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 8.00000i 0.275535i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 0 0
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) − 8.00000i − 0.272008i
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 64.0000i 2.17105i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 0 0
\(879\) − 8.00000i − 0.269833i
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) −32.0000 −1.07567
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 24.0000i 0.804934i
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) − 48.0000i − 1.60446i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 60.0000i − 2.00111i
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) 0 0
\(905\) − 8.00000i − 0.265929i
\(906\) 0 0
\(907\) 56.0000 1.85945 0.929725 0.368255i \(-0.120045\pi\)
0.929725 + 0.368255i \(0.120045\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 56.0000i 1.85130i
\(916\) 0 0
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) − 22.0000i − 0.724925i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 110.000i − 3.61678i
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) − 16.0000i − 0.524943i −0.964940 0.262471i \(-0.915462\pi\)
0.964940 0.262471i \(-0.0845376\pi\)
\(930\) 0 0
\(931\) 6.00000i 0.196642i
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −32.0000 −1.04651
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) − 8.00000i − 0.260793i −0.991462 0.130396i \(-0.958375\pi\)
0.991462 0.130396i \(-0.0416250\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 12.0000i − 0.389127i
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) − 32.0000i − 1.03550i
\(956\) 0 0
\(957\) − 24.0000i − 0.775810i
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 56.0000 1.80270
\(966\) 0 0
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 0 0
\(969\) 4.00000i 0.128499i
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) 0 0
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 0 0
\(993\) 2.00000i 0.0634681i
\(994\) 0 0
\(995\) − 16.0000i − 0.507234i
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 0 0
\(999\) − 10.0000i − 0.316386i
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 2028.2.b.a.337.2 2
3.2 odd 2 6084.2.b.j.4393.1 2
13.2 odd 12 2028.2.i.g.529.1 2
13.3 even 3 2028.2.q.h.1837.2 4
13.4 even 6 2028.2.q.h.361.1 4
13.5 odd 4 2028.2.a.c.1.1 1
13.6 odd 12 2028.2.i.g.2005.1 2
13.7 odd 12 2028.2.i.e.2005.1 2
13.8 odd 4 156.2.a.a.1.1 1
13.9 even 3 2028.2.q.h.361.2 4
13.10 even 6 2028.2.q.h.1837.1 4
13.11 odd 12 2028.2.i.e.529.1 2
13.12 even 2 inner 2028.2.b.a.337.1 2
39.5 even 4 6084.2.a.b.1.1 1
39.8 even 4 468.2.a.d.1.1 1
39.38 odd 2 6084.2.b.j.4393.2 2
52.31 even 4 8112.2.a.bi.1.1 1
52.47 even 4 624.2.a.e.1.1 1
65.8 even 4 3900.2.h.b.1249.1 2
65.34 odd 4 3900.2.a.m.1.1 1
65.47 even 4 3900.2.h.b.1249.2 2
91.34 even 4 7644.2.a.k.1.1 1
104.21 odd 4 2496.2.a.bc.1.1 1
104.99 even 4 2496.2.a.o.1.1 1
117.34 odd 12 4212.2.i.l.1405.1 2
117.47 even 12 4212.2.i.b.1405.1 2
117.86 even 12 4212.2.i.b.2809.1 2
117.112 odd 12 4212.2.i.l.2809.1 2
156.47 odd 4 1872.2.a.s.1.1 1
312.125 even 4 7488.2.a.c.1.1 1
312.203 odd 4 7488.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.a.1.1 1 13.8 odd 4
468.2.a.d.1.1 1 39.8 even 4
624.2.a.e.1.1 1 52.47 even 4
1872.2.a.s.1.1 1 156.47 odd 4
2028.2.a.c.1.1 1 13.5 odd 4
2028.2.b.a.337.1 2 13.12 even 2 inner
2028.2.b.a.337.2 2 1.1 even 1 trivial
2028.2.i.e.529.1 2 13.11 odd 12
2028.2.i.e.2005.1 2 13.7 odd 12
2028.2.i.g.529.1 2 13.2 odd 12
2028.2.i.g.2005.1 2 13.6 odd 12
2028.2.q.h.361.1 4 13.4 even 6
2028.2.q.h.361.2 4 13.9 even 3
2028.2.q.h.1837.1 4 13.10 even 6
2028.2.q.h.1837.2 4 13.3 even 3
2496.2.a.o.1.1 1 104.99 even 4
2496.2.a.bc.1.1 1 104.21 odd 4
3900.2.a.m.1.1 1 65.34 odd 4
3900.2.h.b.1249.1 2 65.8 even 4
3900.2.h.b.1249.2 2 65.47 even 4
4212.2.i.b.1405.1 2 117.47 even 12
4212.2.i.b.2809.1 2 117.86 even 12
4212.2.i.l.1405.1 2 117.34 odd 12
4212.2.i.l.2809.1 2 117.112 odd 12
6084.2.a.b.1.1 1 39.5 even 4
6084.2.b.j.4393.1 2 3.2 odd 2
6084.2.b.j.4393.2 2 39.38 odd 2
7488.2.a.c.1.1 1 312.125 even 4
7488.2.a.d.1.1 1 312.203 odd 4
7644.2.a.k.1.1 1 91.34 even 4
8112.2.a.bi.1.1 1 52.31 even 4