Properties

Label 7728.2.a.bd.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} -2.00000 q^{15} -4.47214 q^{17} +6.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.47214 q^{31} -2.00000 q^{35} -10.9443 q^{37} -4.47214 q^{39} -6.00000 q^{41} -12.9443 q^{43} +2.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} +6.94427 q^{53} -6.47214 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +8.94427 q^{65} +12.9443 q^{67} -1.00000 q^{69} +12.9443 q^{71} -2.94427 q^{73} +1.00000 q^{75} -12.9443 q^{79} +1.00000 q^{81} +6.47214 q^{83} -8.94427 q^{85} +2.00000 q^{87} -17.4164 q^{89} -4.47214 q^{91} +6.47214 q^{93} +12.9443 q^{95} +8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97} + O(q^{100}) \)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 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
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −12.9443 −1.97398 −0.986991 0.160773i \(-0.948601\pi\)
−0.986991 + 0.160773i \(0.948601\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −6.47214 −0.944058 −0.472029 0.881583i \(-0.656478\pi\)
−0.472029 + 0.881583i \(0.656478\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) 6.94427 0.953869 0.476935 0.878939i \(-0.341748\pi\)
0.476935 + 0.878939i \(0.341748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.47214 −0.857255
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 8.94427 1.10940
\(66\) 0 0
\(67\) 12.9443 1.58139 0.790697 0.612207i \(-0.209718\pi\)
0.790697 + 0.612207i \(0.209718\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.9443 1.53620 0.768101 0.640328i \(-0.221202\pi\)
0.768101 + 0.640328i \(0.221202\pi\)
\(72\) 0 0
\(73\) −2.94427 −0.344601 −0.172300 0.985044i \(-0.555120\pi\)
−0.172300 + 0.985044i \(0.555120\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.9443 −1.45634 −0.728172 0.685394i \(-0.759630\pi\)
−0.728172 + 0.685394i \(0.759630\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.47214 0.710409 0.355205 0.934789i \(-0.384411\pi\)
0.355205 + 0.934789i \(0.384411\pi\)
\(84\) 0 0
\(85\) −8.94427 −0.970143
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −17.4164 −1.84614 −0.923068 0.384637i \(-0.874327\pi\)
−0.923068 + 0.384637i \(0.874327\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) 0 0
\(93\) 6.47214 0.671129
\(94\) 0 0
\(95\) 12.9443 1.32805
\(96\) 0 0
\(97\) 8.47214 0.860215 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.4164 1.73300 0.866499 0.499179i \(-0.166365\pi\)
0.866499 + 0.499179i \(0.166365\pi\)
\(102\) 0 0
\(103\) −12.9443 −1.27544 −0.637719 0.770270i \(-0.720122\pi\)
−0.637719 + 0.770270i \(0.720122\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 14.9443 1.43140 0.715701 0.698407i \(-0.246107\pi\)
0.715701 + 0.698407i \(0.246107\pi\)
\(110\) 0 0
\(111\) 10.9443 1.03878
\(112\) 0 0
\(113\) −1.05573 −0.0993145 −0.0496573 0.998766i \(-0.515813\pi\)
−0.0496573 + 0.998766i \(0.515813\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 4.47214 0.413449
\(118\) 0 0
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −3.05573 −0.271152 −0.135576 0.990767i \(-0.543288\pi\)
−0.135576 + 0.990767i \(0.543288\pi\)
\(128\) 0 0
\(129\) 12.9443 1.13968
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −6.47214 −0.561205
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −18.9443 −1.55198 −0.775988 0.630748i \(-0.782748\pi\)
−0.775988 + 0.630748i \(0.782748\pi\)
\(150\) 0 0
\(151\) 12.9443 1.05339 0.526695 0.850054i \(-0.323431\pi\)
0.526695 + 0.850054i \(0.323431\pi\)
\(152\) 0 0
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) −12.9443 −1.03971
\(156\) 0 0
\(157\) 14.9443 1.19268 0.596341 0.802731i \(-0.296620\pi\)
0.596341 + 0.802731i \(0.296620\pi\)
\(158\) 0 0
\(159\) −6.94427 −0.550717
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −8.94427 −0.700569 −0.350285 0.936643i \(-0.613915\pi\)
−0.350285 + 0.936643i \(0.613915\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.4164 −0.883428 −0.441714 0.897156i \(-0.645629\pi\)
−0.441714 + 0.897156i \(0.645629\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 6.47214 0.494937
\(172\) 0 0
\(173\) −8.47214 −0.644125 −0.322062 0.946718i \(-0.604376\pi\)
−0.322062 + 0.946718i \(0.604376\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) −21.8885 −1.60928
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 20.9443 1.51547 0.757737 0.652560i \(-0.226305\pi\)
0.757737 + 0.652560i \(0.226305\pi\)
\(192\) 0 0
\(193\) −23.8885 −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(194\) 0 0
\(195\) −8.94427 −0.640513
\(196\) 0 0
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) 0 0
\(199\) −20.9443 −1.48470 −0.742350 0.670012i \(-0.766289\pi\)
−0.742350 + 0.670012i \(0.766289\pi\)
\(200\) 0 0
\(201\) −12.9443 −0.913019
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.9443 −1.16649 −0.583246 0.812296i \(-0.698218\pi\)
−0.583246 + 0.812296i \(0.698218\pi\)
\(212\) 0 0
\(213\) −12.9443 −0.886927
\(214\) 0 0
\(215\) −25.8885 −1.76558
\(216\) 0 0
\(217\) 6.47214 0.439357
\(218\) 0 0
\(219\) 2.94427 0.198955
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) −9.52786 −0.638033 −0.319016 0.947749i \(-0.603353\pi\)
−0.319016 + 0.947749i \(0.603353\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −14.4721 −0.960549 −0.480275 0.877118i \(-0.659463\pi\)
−0.480275 + 0.877118i \(0.659463\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −12.9443 −0.844391
\(236\) 0 0
\(237\) 12.9443 0.840821
\(238\) 0 0
\(239\) −3.05573 −0.197659 −0.0988293 0.995104i \(-0.531510\pi\)
−0.0988293 + 0.995104i \(0.531510\pi\)
\(240\) 0 0
\(241\) 11.5279 0.742575 0.371288 0.928518i \(-0.378916\pi\)
0.371288 + 0.928518i \(0.378916\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 28.9443 1.84168
\(248\) 0 0
\(249\) −6.47214 −0.410155
\(250\) 0 0
\(251\) 14.4721 0.913473 0.456737 0.889602i \(-0.349018\pi\)
0.456737 + 0.889602i \(0.349018\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 8.94427 0.560112
\(256\) 0 0
\(257\) 14.9443 0.932198 0.466099 0.884733i \(-0.345659\pi\)
0.466099 + 0.884733i \(0.345659\pi\)
\(258\) 0 0
\(259\) 10.9443 0.680044
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 13.8885 0.853166
\(266\) 0 0
\(267\) 17.4164 1.06587
\(268\) 0 0
\(269\) 7.52786 0.458982 0.229491 0.973311i \(-0.426294\pi\)
0.229491 + 0.973311i \(0.426294\pi\)
\(270\) 0 0
\(271\) 11.4164 0.693497 0.346749 0.937958i \(-0.387286\pi\)
0.346749 + 0.937958i \(0.387286\pi\)
\(272\) 0 0
\(273\) 4.47214 0.270666
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.8885 1.43532 0.717662 0.696392i \(-0.245212\pi\)
0.717662 + 0.696392i \(0.245212\pi\)
\(278\) 0 0
\(279\) −6.47214 −0.387477
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −9.52786 −0.566373 −0.283186 0.959065i \(-0.591391\pi\)
−0.283186 + 0.959065i \(0.591391\pi\)
\(284\) 0 0
\(285\) −12.9443 −0.766752
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) −8.47214 −0.496645
\(292\) 0 0
\(293\) −7.88854 −0.460854 −0.230427 0.973090i \(-0.574012\pi\)
−0.230427 + 0.973090i \(0.574012\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) 12.9443 0.746095
\(302\) 0 0
\(303\) −17.4164 −1.00055
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −7.05573 −0.402692 −0.201346 0.979520i \(-0.564532\pi\)
−0.201346 + 0.979520i \(0.564532\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) 0 0
\(311\) −6.47214 −0.367001 −0.183501 0.983020i \(-0.558743\pi\)
−0.183501 + 0.983020i \(0.558743\pi\)
\(312\) 0 0
\(313\) 13.4164 0.758340 0.379170 0.925327i \(-0.376210\pi\)
0.379170 + 0.925327i \(0.376210\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −30.9443 −1.73800 −0.869002 0.494809i \(-0.835238\pi\)
−0.869002 + 0.494809i \(0.835238\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −28.9443 −1.61050
\(324\) 0 0
\(325\) −4.47214 −0.248069
\(326\) 0 0
\(327\) −14.9443 −0.826420
\(328\) 0 0
\(329\) 6.47214 0.356820
\(330\) 0 0
\(331\) −21.8885 −1.20310 −0.601552 0.798834i \(-0.705451\pi\)
−0.601552 + 0.798834i \(0.705451\pi\)
\(332\) 0 0
\(333\) −10.9443 −0.599742
\(334\) 0 0
\(335\) 25.8885 1.41444
\(336\) 0 0
\(337\) 14.9443 0.814066 0.407033 0.913413i \(-0.366563\pi\)
0.407033 + 0.913413i \(0.366563\pi\)
\(338\) 0 0
\(339\) 1.05573 0.0573393
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) −16.9443 −0.909616 −0.454808 0.890589i \(-0.650292\pi\)
−0.454808 + 0.890589i \(0.650292\pi\)
\(348\) 0 0
\(349\) 15.5279 0.831188 0.415594 0.909550i \(-0.363574\pi\)
0.415594 + 0.909550i \(0.363574\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 25.8885 1.37402
\(356\) 0 0
\(357\) −4.47214 −0.236691
\(358\) 0 0
\(359\) −22.8328 −1.20507 −0.602535 0.798092i \(-0.705843\pi\)
−0.602535 + 0.798092i \(0.705843\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −5.88854 −0.308220
\(366\) 0 0
\(367\) 22.8328 1.19186 0.595932 0.803035i \(-0.296783\pi\)
0.595932 + 0.803035i \(0.296783\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −6.94427 −0.360529
\(372\) 0 0
\(373\) −10.9443 −0.566673 −0.283336 0.959021i \(-0.591441\pi\)
−0.283336 + 0.959021i \(0.591441\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 3.05573 0.156550
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.9443 −0.657994
\(388\) 0 0
\(389\) −1.05573 −0.0535275 −0.0267638 0.999642i \(-0.508520\pi\)
−0.0267638 + 0.999642i \(0.508520\pi\)
\(390\) 0 0
\(391\) −4.47214 −0.226166
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) −25.8885 −1.30259
\(396\) 0 0
\(397\) −27.5279 −1.38158 −0.690792 0.723054i \(-0.742738\pi\)
−0.690792 + 0.723054i \(0.742738\pi\)
\(398\) 0 0
\(399\) 6.47214 0.324012
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.8885 0.983425 0.491713 0.870758i \(-0.336371\pi\)
0.491713 + 0.870758i \(0.336371\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 12.9443 0.635409
\(416\) 0 0
\(417\) −8.94427 −0.438003
\(418\) 0 0
\(419\) −19.4164 −0.948554 −0.474277 0.880376i \(-0.657290\pi\)
−0.474277 + 0.880376i \(0.657290\pi\)
\(420\) 0 0
\(421\) −20.8328 −1.01533 −0.507665 0.861555i \(-0.669491\pi\)
−0.507665 + 0.861555i \(0.669491\pi\)
\(422\) 0 0
\(423\) −6.47214 −0.314686
\(424\) 0 0
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.8885 0.861661 0.430830 0.902433i \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(432\) 0 0
\(433\) −1.41641 −0.0680682 −0.0340341 0.999421i \(-0.510835\pi\)
−0.0340341 + 0.999421i \(0.510835\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 0 0
\(437\) 6.47214 0.309604
\(438\) 0 0
\(439\) 16.3607 0.780853 0.390426 0.920634i \(-0.372328\pi\)
0.390426 + 0.920634i \(0.372328\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.8328 0.894774 0.447387 0.894340i \(-0.352355\pi\)
0.447387 + 0.894340i \(0.352355\pi\)
\(444\) 0 0
\(445\) −34.8328 −1.65123
\(446\) 0 0
\(447\) 18.9443 0.896033
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −12.9443 −0.608175
\(454\) 0 0
\(455\) −8.94427 −0.419314
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) 33.4164 1.55636 0.778179 0.628043i \(-0.216144\pi\)
0.778179 + 0.628043i \(0.216144\pi\)
\(462\) 0 0
\(463\) −41.8885 −1.94673 −0.973363 0.229270i \(-0.926366\pi\)
−0.973363 + 0.229270i \(0.926366\pi\)
\(464\) 0 0
\(465\) 12.9443 0.600276
\(466\) 0 0
\(467\) 29.3050 1.35607 0.678036 0.735029i \(-0.262831\pi\)
0.678036 + 0.735029i \(0.262831\pi\)
\(468\) 0 0
\(469\) −12.9443 −0.597711
\(470\) 0 0
\(471\) −14.9443 −0.688596
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.47214 −0.296962
\(476\) 0 0
\(477\) 6.94427 0.317956
\(478\) 0 0
\(479\) −12.9443 −0.591439 −0.295719 0.955275i \(-0.595559\pi\)
−0.295719 + 0.955275i \(0.595559\pi\)
\(480\) 0 0
\(481\) −48.9443 −2.23167
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 16.9443 0.769400
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) 8.94427 0.404474
\(490\) 0 0
\(491\) −23.0557 −1.04049 −0.520245 0.854017i \(-0.674159\pi\)
−0.520245 + 0.854017i \(0.674159\pi\)
\(492\) 0 0
\(493\) 8.94427 0.402830
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.9443 −0.580630
\(498\) 0 0
\(499\) −15.0557 −0.673987 −0.336993 0.941507i \(-0.609410\pi\)
−0.336993 + 0.941507i \(0.609410\pi\)
\(500\) 0 0
\(501\) 11.4164 0.510047
\(502\) 0 0
\(503\) −25.8885 −1.15431 −0.577157 0.816634i \(-0.695838\pi\)
−0.577157 + 0.816634i \(0.695838\pi\)
\(504\) 0 0
\(505\) 34.8328 1.55004
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 0 0
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) 0 0
\(511\) 2.94427 0.130247
\(512\) 0 0
\(513\) −6.47214 −0.285752
\(514\) 0 0
\(515\) −25.8885 −1.14079
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 8.47214 0.371885
\(520\) 0 0
\(521\) 6.58359 0.288432 0.144216 0.989546i \(-0.453934\pi\)
0.144216 + 0.989546i \(0.453934\pi\)
\(522\) 0 0
\(523\) −37.3050 −1.63123 −0.815616 0.578594i \(-0.803602\pi\)
−0.815616 + 0.578594i \(0.803602\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 28.9443 1.26083
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −26.8328 −1.16226
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 29.8885 1.28028
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −12.9443 −0.551445
\(552\) 0 0
\(553\) 12.9443 0.550446
\(554\) 0 0
\(555\) 21.8885 0.929117
\(556\) 0 0
\(557\) −34.9443 −1.48064 −0.740318 0.672257i \(-0.765325\pi\)
−0.740318 + 0.672257i \(0.765325\pi\)
\(558\) 0 0
\(559\) −57.8885 −2.44842
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.5279 −0.738711 −0.369356 0.929288i \(-0.620422\pi\)
−0.369356 + 0.929288i \(0.620422\pi\)
\(564\) 0 0
\(565\) −2.11146 −0.0888296
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 27.8885 1.16915 0.584574 0.811340i \(-0.301262\pi\)
0.584574 + 0.811340i \(0.301262\pi\)
\(570\) 0 0
\(571\) −4.94427 −0.206911 −0.103456 0.994634i \(-0.532990\pi\)
−0.103456 + 0.994634i \(0.532990\pi\)
\(572\) 0 0
\(573\) −20.9443 −0.874960
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 14.9443 0.622138 0.311069 0.950387i \(-0.399313\pi\)
0.311069 + 0.950387i \(0.399313\pi\)
\(578\) 0 0
\(579\) 23.8885 0.992774
\(580\) 0 0
\(581\) −6.47214 −0.268509
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.94427 0.369800
\(586\) 0 0
\(587\) −16.9443 −0.699365 −0.349682 0.936868i \(-0.613711\pi\)
−0.349682 + 0.936868i \(0.613711\pi\)
\(588\) 0 0
\(589\) −41.8885 −1.72599
\(590\) 0 0
\(591\) −2.94427 −0.121111
\(592\) 0 0
\(593\) 24.8328 1.01976 0.509881 0.860245i \(-0.329690\pi\)
0.509881 + 0.860245i \(0.329690\pi\)
\(594\) 0 0
\(595\) 8.94427 0.366679
\(596\) 0 0
\(597\) 20.9443 0.857192
\(598\) 0 0
\(599\) 1.88854 0.0771638 0.0385819 0.999255i \(-0.487716\pi\)
0.0385819 + 0.999255i \(0.487716\pi\)
\(600\) 0 0
\(601\) −12.8328 −0.523461 −0.261731 0.965141i \(-0.584293\pi\)
−0.261731 + 0.965141i \(0.584293\pi\)
\(602\) 0 0
\(603\) 12.9443 0.527132
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) −16.3607 −0.664060 −0.332030 0.943269i \(-0.607733\pi\)
−0.332030 + 0.943269i \(0.607733\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −28.9443 −1.17096
\(612\) 0 0
\(613\) −12.8328 −0.518313 −0.259156 0.965835i \(-0.583444\pi\)
−0.259156 + 0.965835i \(0.583444\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 25.5279 1.02605 0.513026 0.858373i \(-0.328525\pi\)
0.513026 + 0.858373i \(0.328525\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 17.4164 0.697774
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.9443 1.95154
\(630\) 0 0
\(631\) 28.9443 1.15225 0.576127 0.817360i \(-0.304564\pi\)
0.576127 + 0.817360i \(0.304564\pi\)
\(632\) 0 0
\(633\) 16.9443 0.673474
\(634\) 0 0
\(635\) −6.11146 −0.242526
\(636\) 0 0
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) 12.9443 0.512067
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 1.52786 0.0602531 0.0301265 0.999546i \(-0.490409\pi\)
0.0301265 + 0.999546i \(0.490409\pi\)
\(644\) 0 0
\(645\) 25.8885 1.01936
\(646\) 0 0
\(647\) 32.3607 1.27223 0.636115 0.771594i \(-0.280540\pi\)
0.636115 + 0.771594i \(0.280540\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.47214 −0.253663
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) −2.94427 −0.114867
\(658\) 0 0
\(659\) 6.83282 0.266169 0.133084 0.991105i \(-0.457512\pi\)
0.133084 + 0.991105i \(0.457512\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) −12.9443 −0.501957
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 9.52786 0.368369
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −39.8885 −1.53304 −0.766521 0.642220i \(-0.778014\pi\)
−0.766521 + 0.642220i \(0.778014\pi\)
\(678\) 0 0
\(679\) −8.47214 −0.325131
\(680\) 0 0
\(681\) 14.4721 0.554573
\(682\) 0 0
\(683\) 0.944272 0.0361316 0.0180658 0.999837i \(-0.494249\pi\)
0.0180658 + 0.999837i \(0.494249\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) 31.0557 1.18313
\(690\) 0 0
\(691\) 32.9443 1.25326 0.626630 0.779317i \(-0.284434\pi\)
0.626630 + 0.779317i \(0.284434\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.8885 0.678551
\(696\) 0 0
\(697\) 26.8328 1.01637
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 32.8328 1.24008 0.620039 0.784571i \(-0.287117\pi\)
0.620039 + 0.784571i \(0.287117\pi\)
\(702\) 0 0
\(703\) −70.8328 −2.67151
\(704\) 0 0
\(705\) 12.9443 0.487509
\(706\) 0 0
\(707\) −17.4164 −0.655011
\(708\) 0 0
\(709\) −10.9443 −0.411021 −0.205510 0.978655i \(-0.565885\pi\)
−0.205510 + 0.978655i \(0.565885\pi\)
\(710\) 0 0
\(711\) −12.9443 −0.485448
\(712\) 0 0
\(713\) −6.47214 −0.242383
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.05573 0.114118
\(718\) 0 0
\(719\) 14.4721 0.539720 0.269860 0.962900i \(-0.413023\pi\)
0.269860 + 0.962900i \(0.413023\pi\)
\(720\) 0 0
\(721\) 12.9443 0.482070
\(722\) 0 0
\(723\) −11.5279 −0.428726
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −20.9443 −0.776780 −0.388390 0.921495i \(-0.626969\pi\)
−0.388390 + 0.921495i \(0.626969\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 57.8885 2.14109
\(732\) 0 0
\(733\) −33.0557 −1.22094 −0.610471 0.792039i \(-0.709020\pi\)
−0.610471 + 0.792039i \(0.709020\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) 0 0
\(741\) −28.9443 −1.06329
\(742\) 0 0
\(743\) −30.8328 −1.13115 −0.565573 0.824698i \(-0.691345\pi\)
−0.565573 + 0.824698i \(0.691345\pi\)
\(744\) 0 0
\(745\) −37.8885 −1.38813
\(746\) 0 0
\(747\) 6.47214 0.236803
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 17.8885 0.652762 0.326381 0.945238i \(-0.394171\pi\)
0.326381 + 0.945238i \(0.394171\pi\)
\(752\) 0 0
\(753\) −14.4721 −0.527394
\(754\) 0 0
\(755\) 25.8885 0.942181
\(756\) 0 0
\(757\) −25.0557 −0.910666 −0.455333 0.890321i \(-0.650480\pi\)
−0.455333 + 0.890321i \(0.650480\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.9443 1.41173 0.705864 0.708347i \(-0.250559\pi\)
0.705864 + 0.708347i \(0.250559\pi\)
\(762\) 0 0
\(763\) −14.9443 −0.541019
\(764\) 0 0
\(765\) −8.94427 −0.323381
\(766\) 0 0
\(767\) −17.8885 −0.645918
\(768\) 0 0
\(769\) −14.3607 −0.517859 −0.258930 0.965896i \(-0.583370\pi\)
−0.258930 + 0.965896i \(0.583370\pi\)
\(770\) 0 0
\(771\) −14.9443 −0.538205
\(772\) 0 0
\(773\) 22.9443 0.825248 0.412624 0.910901i \(-0.364612\pi\)
0.412624 + 0.910901i \(0.364612\pi\)
\(774\) 0 0
\(775\) 6.47214 0.232486
\(776\) 0 0
\(777\) −10.9443 −0.392624
\(778\) 0 0
\(779\) −38.8328 −1.39133
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 29.8885 1.06677
\(786\) 0 0
\(787\) 27.4164 0.977289 0.488645 0.872483i \(-0.337491\pi\)
0.488645 + 0.872483i \(0.337491\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 1.05573 0.0375374
\(792\) 0 0
\(793\) −26.8328 −0.952861
\(794\) 0 0
\(795\) −13.8885 −0.492576
\(796\) 0 0
\(797\) 24.8328 0.879623 0.439812 0.898090i \(-0.355045\pi\)
0.439812 + 0.898090i \(0.355045\pi\)
\(798\) 0 0
\(799\) 28.9443 1.02397
\(800\) 0 0
\(801\) −17.4164 −0.615379
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) −7.52786 −0.264993
\(808\) 0 0
\(809\) 0.111456 0.00391859 0.00195930 0.999998i \(-0.499376\pi\)
0.00195930 + 0.999998i \(0.499376\pi\)
\(810\) 0 0
\(811\) 2.11146 0.0741433 0.0370716 0.999313i \(-0.488197\pi\)
0.0370716 + 0.999313i \(0.488197\pi\)
\(812\) 0 0
\(813\) −11.4164 −0.400391
\(814\) 0 0
\(815\) −17.8885 −0.626608
\(816\) 0 0
\(817\) −83.7771 −2.93099
\(818\) 0 0
\(819\) −4.47214 −0.156269
\(820\) 0 0
\(821\) 36.8328 1.28547 0.642737 0.766087i \(-0.277799\pi\)
0.642737 + 0.766087i \(0.277799\pi\)
\(822\) 0 0
\(823\) −3.05573 −0.106516 −0.0532580 0.998581i \(-0.516961\pi\)
−0.0532580 + 0.998581i \(0.516961\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.88854 0.0656711 0.0328356 0.999461i \(-0.489546\pi\)
0.0328356 + 0.999461i \(0.489546\pi\)
\(828\) 0 0
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 0 0
\(831\) −23.8885 −0.828684
\(832\) 0 0
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) −22.8328 −0.790162
\(836\) 0 0
\(837\) 6.47214 0.223710
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 14.0000 0.481615
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 9.52786 0.326995
\(850\) 0 0
\(851\) −10.9443 −0.375165
\(852\) 0 0
\(853\) 38.3607 1.31344 0.656722 0.754132i \(-0.271942\pi\)
0.656722 + 0.754132i \(0.271942\pi\)
\(854\) 0 0
\(855\) 12.9443 0.442685
\(856\) 0 0
\(857\) −12.1115 −0.413719 −0.206860 0.978371i \(-0.566324\pi\)
−0.206860 + 0.978371i \(0.566324\pi\)
\(858\) 0 0
\(859\) −15.0557 −0.513695 −0.256847 0.966452i \(-0.582684\pi\)
−0.256847 + 0.966452i \(0.582684\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −3.05573 −0.104018 −0.0520091 0.998647i \(-0.516562\pi\)
−0.0520091 + 0.998647i \(0.516562\pi\)
\(864\) 0 0
\(865\) −16.9443 −0.576123
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 57.8885 1.96148
\(872\) 0 0
\(873\) 8.47214 0.286738
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 54.7214 1.84781 0.923905 0.382623i \(-0.124979\pi\)
0.923905 + 0.382623i \(0.124979\pi\)
\(878\) 0 0
\(879\) 7.88854 0.266074
\(880\) 0 0
\(881\) −22.3607 −0.753350 −0.376675 0.926345i \(-0.622933\pi\)
−0.376675 + 0.926345i \(0.622933\pi\)
\(882\) 0 0
\(883\) 50.8328 1.71066 0.855330 0.518083i \(-0.173354\pi\)
0.855330 + 0.518083i \(0.173354\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 14.4721 0.485927 0.242963 0.970035i \(-0.421881\pi\)
0.242963 + 0.970035i \(0.421881\pi\)
\(888\) 0 0
\(889\) 3.05573 0.102486
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −41.8885 −1.40175
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) −4.47214 −0.149320
\(898\) 0 0
\(899\) 12.9443 0.431716
\(900\) 0 0
\(901\) −31.0557 −1.03462
\(902\) 0 0
\(903\) −12.9443 −0.430758
\(904\) 0 0
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 6.11146 0.202928 0.101464 0.994839i \(-0.467647\pi\)
0.101464 + 0.994839i \(0.467647\pi\)
\(908\) 0 0
\(909\) 17.4164 0.577666
\(910\) 0 0
\(911\) −40.7214 −1.34916 −0.674579 0.738202i \(-0.735675\pi\)
−0.674579 + 0.738202i \(0.735675\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 12.0000 0.396708
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −52.9443 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(920\) 0 0
\(921\) 7.05573 0.232494
\(922\) 0 0
\(923\) 57.8885 1.90542
\(924\) 0 0
\(925\) 10.9443 0.359845
\(926\) 0 0
\(927\) −12.9443 −0.425146
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 6.47214 0.212116
\(932\) 0 0
\(933\) 6.47214 0.211888
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40.2492 −1.31488 −0.657442 0.753505i \(-0.728362\pi\)
−0.657442 + 0.753505i \(0.728362\pi\)
\(938\) 0 0
\(939\) −13.4164 −0.437828
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 16.9443 0.550615 0.275307 0.961356i \(-0.411220\pi\)
0.275307 + 0.961356i \(0.411220\pi\)
\(948\) 0 0
\(949\) −13.1672 −0.427425
\(950\) 0 0
\(951\) 30.9443 1.00344
\(952\) 0 0
\(953\) −26.9443 −0.872811 −0.436405 0.899750i \(-0.643749\pi\)
−0.436405 + 0.899750i \(0.643749\pi\)
\(954\) 0 0
\(955\) 41.8885 1.35548
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) −47.7771 −1.53800
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) 28.9443 0.929824
\(970\) 0 0
\(971\) −3.41641 −0.109638 −0.0548189 0.998496i \(-0.517458\pi\)
−0.0548189 + 0.998496i \(0.517458\pi\)
\(972\) 0 0
\(973\) −8.94427 −0.286740
\(974\) 0 0
\(975\) 4.47214 0.143223
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.9443 0.477134
\(982\) 0 0
\(983\) 57.8885 1.84636 0.923179 0.384371i \(-0.125581\pi\)
0.923179 + 0.384371i \(0.125581\pi\)
\(984\) 0 0
\(985\) 5.88854 0.187625
\(986\) 0 0
\(987\) −6.47214 −0.206010
\(988\) 0 0
\(989\) −12.9443 −0.411604
\(990\) 0 0
\(991\) −43.0557 −1.36771 −0.683855 0.729618i \(-0.739698\pi\)
−0.683855 + 0.729618i \(0.739698\pi\)
\(992\) 0 0
\(993\) 21.8885 0.694612
\(994\) 0 0
\(995\) −41.8885 −1.32796
\(996\) 0 0
\(997\) −1.63932 −0.0519178 −0.0259589 0.999663i \(-0.508264\pi\)
−0.0259589 + 0.999663i \(0.508264\pi\)
\(998\) 0 0
\(999\) 10.9443 0.346261
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.bd.1.2 2
4.3 odd 2 966.2.a.p.1.2 2
12.11 even 2 2898.2.a.v.1.2 2
28.27 even 2 6762.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.2 2 4.3 odd 2
2898.2.a.v.1.2 2 12.11 even 2
6762.2.a.bz.1.1 2 28.27 even 2
7728.2.a.bd.1.2 2 1.1 even 1 trivial