Properties

Label 1792.2.a.l.1.1
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 896)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.23607 q^{3} +3.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +O(q^{10})\) \(q-3.23607 q^{3} +3.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} -4.00000 q^{11} -3.23607 q^{13} -10.4721 q^{15} -2.00000 q^{17} +3.23607 q^{19} -3.23607 q^{21} -2.47214 q^{23} +5.47214 q^{25} -14.4721 q^{27} -10.4721 q^{29} +12.9443 q^{33} +3.23607 q^{35} +2.47214 q^{37} +10.4721 q^{39} -2.00000 q^{41} +8.94427 q^{43} +24.1803 q^{45} +4.94427 q^{47} +1.00000 q^{49} +6.47214 q^{51} -8.94427 q^{53} -12.9443 q^{55} -10.4721 q^{57} -4.76393 q^{59} +3.23607 q^{61} +7.47214 q^{63} -10.4721 q^{65} -10.4721 q^{67} +8.00000 q^{69} +12.9443 q^{71} -2.94427 q^{73} -17.7082 q^{75} -4.00000 q^{77} -12.9443 q^{79} +24.4164 q^{81} -4.76393 q^{83} -6.47214 q^{85} +33.8885 q^{87} +2.00000 q^{89} -3.23607 q^{91} +10.4721 q^{95} -6.94427 q^{97} -29.8885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 2q^{7} + 6q^{9} - 8q^{11} - 2q^{13} - 12q^{15} - 4q^{17} + 2q^{19} - 2q^{21} + 4q^{23} + 2q^{25} - 20q^{27} - 12q^{29} + 8q^{33} + 2q^{35} - 4q^{37} + 12q^{39} - 4q^{41} + 26q^{45} - 8q^{47} + 2q^{49} + 4q^{51} - 8q^{55} - 12q^{57} - 14q^{59} + 2q^{61} + 6q^{63} - 12q^{65} - 12q^{67} + 16q^{69} + 8q^{71} + 12q^{73} - 22q^{75} - 8q^{77} - 8q^{79} + 22q^{81} - 14q^{83} - 4q^{85} + 32q^{87} + 4q^{89} - 2q^{91} + 12q^{95} + 4q^{97} - 24q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 0 0
\(15\) −10.4721 −2.70389
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 3.23607 0.742405 0.371202 0.928552i \(-0.378946\pi\)
0.371202 + 0.928552i \(0.378946\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) −10.4721 −1.94463 −0.972313 0.233681i \(-0.924923\pi\)
−0.972313 + 0.233681i \(0.924923\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 12.9443 2.25331
\(34\) 0 0
\(35\) 3.23607 0.546995
\(36\) 0 0
\(37\) 2.47214 0.406417 0.203208 0.979136i \(-0.434863\pi\)
0.203208 + 0.979136i \(0.434863\pi\)
\(38\) 0 0
\(39\) 10.4721 1.67688
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0 0
\(45\) 24.1803 3.60459
\(46\) 0 0
\(47\) 4.94427 0.721196 0.360598 0.932721i \(-0.382573\pi\)
0.360598 + 0.932721i \(0.382573\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.47214 0.906280
\(52\) 0 0
\(53\) −8.94427 −1.22859 −0.614295 0.789076i \(-0.710560\pi\)
−0.614295 + 0.789076i \(0.710560\pi\)
\(54\) 0 0
\(55\) −12.9443 −1.74541
\(56\) 0 0
\(57\) −10.4721 −1.38707
\(58\) 0 0
\(59\) −4.76393 −0.620211 −0.310106 0.950702i \(-0.600364\pi\)
−0.310106 + 0.950702i \(0.600364\pi\)
\(60\) 0 0
\(61\) 3.23607 0.414336 0.207168 0.978305i \(-0.433575\pi\)
0.207168 + 0.978305i \(0.433575\pi\)
\(62\) 0 0
\(63\) 7.47214 0.941401
\(64\) 0 0
\(65\) −10.4721 −1.29891
\(66\) 0 0
\(67\) −10.4721 −1.27938 −0.639688 0.768635i \(-0.720936\pi\)
−0.639688 + 0.768635i \(0.720936\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 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
\(75\) −17.7082 −2.04477
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −12.9443 −1.45634 −0.728172 0.685394i \(-0.759630\pi\)
−0.728172 + 0.685394i \(0.759630\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) −4.76393 −0.522909 −0.261455 0.965216i \(-0.584202\pi\)
−0.261455 + 0.965216i \(0.584202\pi\)
\(84\) 0 0
\(85\) −6.47214 −0.702002
\(86\) 0 0
\(87\) 33.8885 3.63323
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.4721 1.07442
\(96\) 0 0
\(97\) −6.94427 −0.705084 −0.352542 0.935796i \(-0.614683\pi\)
−0.352542 + 0.935796i \(0.614683\pi\)
\(98\) 0 0
\(99\) −29.8885 −3.00391
\(100\) 0 0
\(101\) 8.18034 0.813974 0.406987 0.913434i \(-0.366579\pi\)
0.406987 + 0.913434i \(0.366579\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −10.4721 −1.02198
\(106\) 0 0
\(107\) −18.4721 −1.78577 −0.892884 0.450286i \(-0.851322\pi\)
−0.892884 + 0.450286i \(0.851322\pi\)
\(108\) 0 0
\(109\) −10.4721 −1.00305 −0.501524 0.865144i \(-0.667227\pi\)
−0.501524 + 0.865144i \(0.667227\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −13.4164 −1.26211 −0.631055 0.775738i \(-0.717378\pi\)
−0.631055 + 0.775738i \(0.717378\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −24.1803 −2.23547
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 6.47214 0.583573
\(124\) 0 0
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) −7.41641 −0.658100 −0.329050 0.944313i \(-0.606728\pi\)
−0.329050 + 0.944313i \(0.606728\pi\)
\(128\) 0 0
\(129\) −28.9443 −2.54840
\(130\) 0 0
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) 0 0
\(133\) 3.23607 0.280603
\(134\) 0 0
\(135\) −46.8328 −4.03073
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 12.7639 1.08262 0.541311 0.840822i \(-0.317928\pi\)
0.541311 + 0.840822i \(0.317928\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) 12.9443 1.08245
\(144\) 0 0
\(145\) −33.8885 −2.81429
\(146\) 0 0
\(147\) −3.23607 −0.266906
\(148\) 0 0
\(149\) 0.944272 0.0773578 0.0386789 0.999252i \(-0.487685\pi\)
0.0386789 + 0.999252i \(0.487685\pi\)
\(150\) 0 0
\(151\) 2.47214 0.201180 0.100590 0.994928i \(-0.467927\pi\)
0.100590 + 0.994928i \(0.467927\pi\)
\(152\) 0 0
\(153\) −14.9443 −1.20817
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.180340 −0.0143927 −0.00719634 0.999974i \(-0.502291\pi\)
−0.00719634 + 0.999974i \(0.502291\pi\)
\(158\) 0 0
\(159\) 28.9443 2.29543
\(160\) 0 0
\(161\) −2.47214 −0.194832
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 41.8885 3.26102
\(166\) 0 0
\(167\) −12.9443 −1.00166 −0.500829 0.865546i \(-0.666971\pi\)
−0.500829 + 0.865546i \(0.666971\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 24.1803 1.84912
\(172\) 0 0
\(173\) 9.70820 0.738101 0.369051 0.929409i \(-0.379683\pi\)
0.369051 + 0.929409i \(0.379683\pi\)
\(174\) 0 0
\(175\) 5.47214 0.413655
\(176\) 0 0
\(177\) 15.4164 1.15877
\(178\) 0 0
\(179\) 2.47214 0.184776 0.0923881 0.995723i \(-0.470550\pi\)
0.0923881 + 0.995723i \(0.470550\pi\)
\(180\) 0 0
\(181\) 20.7639 1.54337 0.771685 0.636004i \(-0.219414\pi\)
0.771685 + 0.636004i \(0.219414\pi\)
\(182\) 0 0
\(183\) −10.4721 −0.774123
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) −14.4721 −1.05269
\(190\) 0 0
\(191\) −17.8885 −1.29437 −0.647185 0.762333i \(-0.724054\pi\)
−0.647185 + 0.762333i \(0.724054\pi\)
\(192\) 0 0
\(193\) 3.52786 0.253941 0.126971 0.991906i \(-0.459475\pi\)
0.126971 + 0.991906i \(0.459475\pi\)
\(194\) 0 0
\(195\) 33.8885 2.42681
\(196\) 0 0
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 33.8885 2.39031
\(202\) 0 0
\(203\) −10.4721 −0.735000
\(204\) 0 0
\(205\) −6.47214 −0.452034
\(206\) 0 0
\(207\) −18.4721 −1.28390
\(208\) 0 0
\(209\) −12.9443 −0.895374
\(210\) 0 0
\(211\) 15.4164 1.06131 0.530655 0.847588i \(-0.321946\pi\)
0.530655 + 0.847588i \(0.321946\pi\)
\(212\) 0 0
\(213\) −41.8885 −2.87016
\(214\) 0 0
\(215\) 28.9443 1.97398
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 9.52786 0.643833
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 0 0
\(223\) 11.0557 0.740346 0.370173 0.928963i \(-0.379298\pi\)
0.370173 + 0.928963i \(0.379298\pi\)
\(224\) 0 0
\(225\) 40.8885 2.72590
\(226\) 0 0
\(227\) −9.70820 −0.644356 −0.322178 0.946679i \(-0.604415\pi\)
−0.322178 + 0.946679i \(0.604415\pi\)
\(228\) 0 0
\(229\) 6.65248 0.439608 0.219804 0.975544i \(-0.429458\pi\)
0.219804 + 0.975544i \(0.429458\pi\)
\(230\) 0 0
\(231\) 12.9443 0.851671
\(232\) 0 0
\(233\) −15.8885 −1.04089 −0.520447 0.853894i \(-0.674235\pi\)
−0.520447 + 0.853894i \(0.674235\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 41.8885 2.72095
\(238\) 0 0
\(239\) −18.4721 −1.19486 −0.597432 0.801920i \(-0.703812\pi\)
−0.597432 + 0.801920i \(0.703812\pi\)
\(240\) 0 0
\(241\) 2.94427 0.189657 0.0948286 0.995494i \(-0.469770\pi\)
0.0948286 + 0.995494i \(0.469770\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) 0 0
\(245\) 3.23607 0.206745
\(246\) 0 0
\(247\) −10.4721 −0.666326
\(248\) 0 0
\(249\) 15.4164 0.976975
\(250\) 0 0
\(251\) 14.6525 0.924856 0.462428 0.886657i \(-0.346978\pi\)
0.462428 + 0.886657i \(0.346978\pi\)
\(252\) 0 0
\(253\) 9.88854 0.621687
\(254\) 0 0
\(255\) 20.9443 1.31158
\(256\) 0 0
\(257\) 27.8885 1.73964 0.869820 0.493370i \(-0.164235\pi\)
0.869820 + 0.493370i \(0.164235\pi\)
\(258\) 0 0
\(259\) 2.47214 0.153611
\(260\) 0 0
\(261\) −78.2492 −4.84351
\(262\) 0 0
\(263\) −3.05573 −0.188424 −0.0942121 0.995552i \(-0.530033\pi\)
−0.0942121 + 0.995552i \(0.530033\pi\)
\(264\) 0 0
\(265\) −28.9443 −1.77803
\(266\) 0 0
\(267\) −6.47214 −0.396088
\(268\) 0 0
\(269\) −13.1246 −0.800222 −0.400111 0.916467i \(-0.631028\pi\)
−0.400111 + 0.916467i \(0.631028\pi\)
\(270\) 0 0
\(271\) 20.9443 1.27227 0.636137 0.771576i \(-0.280531\pi\)
0.636137 + 0.771576i \(0.280531\pi\)
\(272\) 0 0
\(273\) 10.4721 0.633803
\(274\) 0 0
\(275\) −21.8885 −1.31993
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −6.65248 −0.395449 −0.197724 0.980258i \(-0.563355\pi\)
−0.197724 + 0.980258i \(0.563355\pi\)
\(284\) 0 0
\(285\) −33.8885 −2.00738
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 22.4721 1.31734
\(292\) 0 0
\(293\) 14.2918 0.834936 0.417468 0.908692i \(-0.362918\pi\)
0.417468 + 0.908692i \(0.362918\pi\)
\(294\) 0 0
\(295\) −15.4164 −0.897578
\(296\) 0 0
\(297\) 57.8885 3.35903
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 8.94427 0.515539
\(302\) 0 0
\(303\) −26.4721 −1.52078
\(304\) 0 0
\(305\) 10.4721 0.599633
\(306\) 0 0
\(307\) −27.5967 −1.57503 −0.787515 0.616296i \(-0.788633\pi\)
−0.787515 + 0.616296i \(0.788633\pi\)
\(308\) 0 0
\(309\) 25.8885 1.47275
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −27.8885 −1.57635 −0.788177 0.615449i \(-0.788975\pi\)
−0.788177 + 0.615449i \(0.788975\pi\)
\(314\) 0 0
\(315\) 24.1803 1.36241
\(316\) 0 0
\(317\) 13.8885 0.780058 0.390029 0.920803i \(-0.372465\pi\)
0.390029 + 0.920803i \(0.372465\pi\)
\(318\) 0 0
\(319\) 41.8885 2.34531
\(320\) 0 0
\(321\) 59.7771 3.33643
\(322\) 0 0
\(323\) −6.47214 −0.360119
\(324\) 0 0
\(325\) −17.7082 −0.982274
\(326\) 0 0
\(327\) 33.8885 1.87404
\(328\) 0 0
\(329\) 4.94427 0.272587
\(330\) 0 0
\(331\) 24.9443 1.37106 0.685531 0.728044i \(-0.259570\pi\)
0.685531 + 0.728044i \(0.259570\pi\)
\(332\) 0 0
\(333\) 18.4721 1.01227
\(334\) 0 0
\(335\) −33.8885 −1.85153
\(336\) 0 0
\(337\) 17.4164 0.948732 0.474366 0.880328i \(-0.342677\pi\)
0.474366 + 0.880328i \(0.342677\pi\)
\(338\) 0 0
\(339\) 43.4164 2.35806
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 25.8885 1.39379
\(346\) 0 0
\(347\) −32.9443 −1.76854 −0.884271 0.466975i \(-0.845344\pi\)
−0.884271 + 0.466975i \(0.845344\pi\)
\(348\) 0 0
\(349\) 17.7082 0.947898 0.473949 0.880552i \(-0.342828\pi\)
0.473949 + 0.880552i \(0.342828\pi\)
\(350\) 0 0
\(351\) 46.8328 2.49975
\(352\) 0 0
\(353\) −34.9443 −1.85990 −0.929948 0.367691i \(-0.880148\pi\)
−0.929948 + 0.367691i \(0.880148\pi\)
\(354\) 0 0
\(355\) 41.8885 2.22321
\(356\) 0 0
\(357\) 6.47214 0.342542
\(358\) 0 0
\(359\) 23.4164 1.23587 0.617935 0.786229i \(-0.287969\pi\)
0.617935 + 0.786229i \(0.287969\pi\)
\(360\) 0 0
\(361\) −8.52786 −0.448835
\(362\) 0 0
\(363\) −16.1803 −0.849248
\(364\) 0 0
\(365\) −9.52786 −0.498711
\(366\) 0 0
\(367\) −11.0557 −0.577104 −0.288552 0.957464i \(-0.593174\pi\)
−0.288552 + 0.957464i \(0.593174\pi\)
\(368\) 0 0
\(369\) −14.9443 −0.777968
\(370\) 0 0
\(371\) −8.94427 −0.464363
\(372\) 0 0
\(373\) −15.0557 −0.779556 −0.389778 0.920909i \(-0.627448\pi\)
−0.389778 + 0.920909i \(0.627448\pi\)
\(374\) 0 0
\(375\) −4.94427 −0.255321
\(376\) 0 0
\(377\) 33.8885 1.74535
\(378\) 0 0
\(379\) 2.11146 0.108458 0.0542291 0.998529i \(-0.482730\pi\)
0.0542291 + 0.998529i \(0.482730\pi\)
\(380\) 0 0
\(381\) 24.0000 1.22956
\(382\) 0 0
\(383\) −20.9443 −1.07020 −0.535101 0.844788i \(-0.679727\pi\)
−0.535101 + 0.844788i \(0.679727\pi\)
\(384\) 0 0
\(385\) −12.9443 −0.659701
\(386\) 0 0
\(387\) 66.8328 3.39730
\(388\) 0 0
\(389\) −7.41641 −0.376027 −0.188013 0.982166i \(-0.560205\pi\)
−0.188013 + 0.982166i \(0.560205\pi\)
\(390\) 0 0
\(391\) 4.94427 0.250043
\(392\) 0 0
\(393\) −26.4721 −1.33534
\(394\) 0 0
\(395\) −41.8885 −2.10764
\(396\) 0 0
\(397\) −37.1246 −1.86323 −0.931615 0.363446i \(-0.881600\pi\)
−0.931615 + 0.363446i \(0.881600\pi\)
\(398\) 0 0
\(399\) −10.4721 −0.524263
\(400\) 0 0
\(401\) 8.47214 0.423078 0.211539 0.977370i \(-0.432152\pi\)
0.211539 + 0.977370i \(0.432152\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 79.0132 3.92620
\(406\) 0 0
\(407\) −9.88854 −0.490157
\(408\) 0 0
\(409\) 2.94427 0.145585 0.0727924 0.997347i \(-0.476809\pi\)
0.0727924 + 0.997347i \(0.476809\pi\)
\(410\) 0 0
\(411\) 32.3607 1.59623
\(412\) 0 0
\(413\) −4.76393 −0.234418
\(414\) 0 0
\(415\) −15.4164 −0.756762
\(416\) 0 0
\(417\) −41.3050 −2.02271
\(418\) 0 0
\(419\) −16.1803 −0.790461 −0.395231 0.918582i \(-0.629335\pi\)
−0.395231 + 0.918582i \(0.629335\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 0 0
\(423\) 36.9443 1.79629
\(424\) 0 0
\(425\) −10.9443 −0.530875
\(426\) 0 0
\(427\) 3.23607 0.156604
\(428\) 0 0
\(429\) −41.8885 −2.02240
\(430\) 0 0
\(431\) 12.3607 0.595393 0.297696 0.954661i \(-0.403782\pi\)
0.297696 + 0.954661i \(0.403782\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 109.666 5.25806
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −3.05573 −0.145842 −0.0729210 0.997338i \(-0.523232\pi\)
−0.0729210 + 0.997338i \(0.523232\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) 0 0
\(443\) 20.3607 0.967365 0.483683 0.875244i \(-0.339299\pi\)
0.483683 + 0.875244i \(0.339299\pi\)
\(444\) 0 0
\(445\) 6.47214 0.306809
\(446\) 0 0
\(447\) −3.05573 −0.144531
\(448\) 0 0
\(449\) −11.8885 −0.561055 −0.280528 0.959846i \(-0.590509\pi\)
−0.280528 + 0.959846i \(0.590509\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) −10.4721 −0.490941
\(456\) 0 0
\(457\) 11.5279 0.539251 0.269625 0.962965i \(-0.413100\pi\)
0.269625 + 0.962965i \(0.413100\pi\)
\(458\) 0 0
\(459\) 28.9443 1.35100
\(460\) 0 0
\(461\) −18.0689 −0.841552 −0.420776 0.907165i \(-0.638242\pi\)
−0.420776 + 0.907165i \(0.638242\pi\)
\(462\) 0 0
\(463\) 33.8885 1.57493 0.787467 0.616357i \(-0.211392\pi\)
0.787467 + 0.616357i \(0.211392\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.65248 −0.307840 −0.153920 0.988083i \(-0.549190\pi\)
−0.153920 + 0.988083i \(0.549190\pi\)
\(468\) 0 0
\(469\) −10.4721 −0.483558
\(470\) 0 0
\(471\) 0.583592 0.0268905
\(472\) 0 0
\(473\) −35.7771 −1.64503
\(474\) 0 0
\(475\) 17.7082 0.812508
\(476\) 0 0
\(477\) −66.8328 −3.06006
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) −22.4721 −1.02041
\(486\) 0 0
\(487\) 13.5279 0.613006 0.306503 0.951870i \(-0.400841\pi\)
0.306503 + 0.951870i \(0.400841\pi\)
\(488\) 0 0
\(489\) 38.8328 1.75608
\(490\) 0 0
\(491\) −5.52786 −0.249469 −0.124735 0.992190i \(-0.539808\pi\)
−0.124735 + 0.992190i \(0.539808\pi\)
\(492\) 0 0
\(493\) 20.9443 0.943283
\(494\) 0 0
\(495\) −96.7214 −4.34730
\(496\) 0 0
\(497\) 12.9443 0.580630
\(498\) 0 0
\(499\) −20.3607 −0.911469 −0.455735 0.890116i \(-0.650623\pi\)
−0.455735 + 0.890116i \(0.650623\pi\)
\(500\) 0 0
\(501\) 41.8885 1.87144
\(502\) 0 0
\(503\) 28.9443 1.29056 0.645281 0.763946i \(-0.276740\pi\)
0.645281 + 0.763946i \(0.276740\pi\)
\(504\) 0 0
\(505\) 26.4721 1.17799
\(506\) 0 0
\(507\) 8.18034 0.363302
\(508\) 0 0
\(509\) 30.6525 1.35865 0.679324 0.733839i \(-0.262273\pi\)
0.679324 + 0.733839i \(0.262273\pi\)
\(510\) 0 0
\(511\) −2.94427 −0.130247
\(512\) 0 0
\(513\) −46.8328 −2.06772
\(514\) 0 0
\(515\) −25.8885 −1.14079
\(516\) 0 0
\(517\) −19.7771 −0.869795
\(518\) 0 0
\(519\) −31.4164 −1.37903
\(520\) 0 0
\(521\) 9.05573 0.396739 0.198369 0.980127i \(-0.436435\pi\)
0.198369 + 0.980127i \(0.436435\pi\)
\(522\) 0 0
\(523\) 24.1803 1.05733 0.528666 0.848830i \(-0.322692\pi\)
0.528666 + 0.848830i \(0.322692\pi\)
\(524\) 0 0
\(525\) −17.7082 −0.772849
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) −35.5967 −1.54477
\(532\) 0 0
\(533\) 6.47214 0.280339
\(534\) 0 0
\(535\) −59.7771 −2.58439
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 16.9443 0.728491 0.364246 0.931303i \(-0.381327\pi\)
0.364246 + 0.931303i \(0.381327\pi\)
\(542\) 0 0
\(543\) −67.1935 −2.88355
\(544\) 0 0
\(545\) −33.8885 −1.45163
\(546\) 0 0
\(547\) −18.8328 −0.805233 −0.402617 0.915369i \(-0.631899\pi\)
−0.402617 + 0.915369i \(0.631899\pi\)
\(548\) 0 0
\(549\) 24.1803 1.03199
\(550\) 0 0
\(551\) −33.8885 −1.44370
\(552\) 0 0
\(553\) −12.9443 −0.550446
\(554\) 0 0
\(555\) −25.8885 −1.09891
\(556\) 0 0
\(557\) 29.8885 1.26642 0.633209 0.773981i \(-0.281737\pi\)
0.633209 + 0.773981i \(0.281737\pi\)
\(558\) 0 0
\(559\) −28.9443 −1.22421
\(560\) 0 0
\(561\) −25.8885 −1.09302
\(562\) 0 0
\(563\) 24.5410 1.03428 0.517140 0.855901i \(-0.326997\pi\)
0.517140 + 0.855901i \(0.326997\pi\)
\(564\) 0 0
\(565\) −43.4164 −1.82654
\(566\) 0 0
\(567\) 24.4164 1.02539
\(568\) 0 0
\(569\) −41.4164 −1.73627 −0.868133 0.496332i \(-0.834680\pi\)
−0.868133 + 0.496332i \(0.834680\pi\)
\(570\) 0 0
\(571\) −39.7771 −1.66462 −0.832310 0.554311i \(-0.812982\pi\)
−0.832310 + 0.554311i \(0.812982\pi\)
\(572\) 0 0
\(573\) 57.8885 2.41833
\(574\) 0 0
\(575\) −13.5279 −0.564151
\(576\) 0 0
\(577\) 32.8328 1.36685 0.683424 0.730022i \(-0.260490\pi\)
0.683424 + 0.730022i \(0.260490\pi\)
\(578\) 0 0
\(579\) −11.4164 −0.474450
\(580\) 0 0
\(581\) −4.76393 −0.197641
\(582\) 0 0
\(583\) 35.7771 1.48174
\(584\) 0 0
\(585\) −78.2492 −3.23521
\(586\) 0 0
\(587\) −19.2361 −0.793957 −0.396979 0.917828i \(-0.629941\pi\)
−0.396979 + 0.917828i \(0.629941\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.9443 −0.532456
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −6.47214 −0.265332
\(596\) 0 0
\(597\) 77.6656 3.17864
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −39.8885 −1.62709 −0.813544 0.581504i \(-0.802465\pi\)
−0.813544 + 0.581504i \(0.802465\pi\)
\(602\) 0 0
\(603\) −78.2492 −3.18655
\(604\) 0 0
\(605\) 16.1803 0.657824
\(606\) 0 0
\(607\) 25.8885 1.05078 0.525392 0.850860i \(-0.323919\pi\)
0.525392 + 0.850860i \(0.323919\pi\)
\(608\) 0 0
\(609\) 33.8885 1.37323
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 5.52786 0.223268 0.111634 0.993749i \(-0.464392\pi\)
0.111634 + 0.993749i \(0.464392\pi\)
\(614\) 0 0
\(615\) 20.9443 0.844555
\(616\) 0 0
\(617\) −32.4721 −1.30728 −0.653639 0.756806i \(-0.726759\pi\)
−0.653639 + 0.756806i \(0.726759\pi\)
\(618\) 0 0
\(619\) 32.1803 1.29344 0.646719 0.762729i \(-0.276141\pi\)
0.646719 + 0.762729i \(0.276141\pi\)
\(620\) 0 0
\(621\) 35.7771 1.43569
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 41.8885 1.67287
\(628\) 0 0
\(629\) −4.94427 −0.197141
\(630\) 0 0
\(631\) 17.8885 0.712132 0.356066 0.934461i \(-0.384118\pi\)
0.356066 + 0.934461i \(0.384118\pi\)
\(632\) 0 0
\(633\) −49.8885 −1.98289
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 96.7214 3.82624
\(640\) 0 0
\(641\) 34.3607 1.35717 0.678583 0.734524i \(-0.262595\pi\)
0.678583 + 0.734524i \(0.262595\pi\)
\(642\) 0 0
\(643\) 40.5410 1.59878 0.799391 0.600811i \(-0.205156\pi\)
0.799391 + 0.600811i \(0.205156\pi\)
\(644\) 0 0
\(645\) −93.6656 −3.68808
\(646\) 0 0
\(647\) −17.8885 −0.703271 −0.351636 0.936137i \(-0.614374\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(648\) 0 0
\(649\) 19.0557 0.748003
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −39.4164 −1.54248 −0.771242 0.636543i \(-0.780364\pi\)
−0.771242 + 0.636543i \(0.780364\pi\)
\(654\) 0 0
\(655\) 26.4721 1.03435
\(656\) 0 0
\(657\) −22.0000 −0.858302
\(658\) 0 0
\(659\) 10.8328 0.421987 0.210993 0.977488i \(-0.432330\pi\)
0.210993 + 0.977488i \(0.432330\pi\)
\(660\) 0 0
\(661\) 27.2361 1.05936 0.529680 0.848197i \(-0.322312\pi\)
0.529680 + 0.848197i \(0.322312\pi\)
\(662\) 0 0
\(663\) −20.9443 −0.813408
\(664\) 0 0
\(665\) 10.4721 0.406092
\(666\) 0 0
\(667\) 25.8885 1.00241
\(668\) 0 0
\(669\) −35.7771 −1.38322
\(670\) 0 0
\(671\) −12.9443 −0.499708
\(672\) 0 0
\(673\) 23.8885 0.920836 0.460418 0.887702i \(-0.347700\pi\)
0.460418 + 0.887702i \(0.347700\pi\)
\(674\) 0 0
\(675\) −79.1935 −3.04816
\(676\) 0 0
\(677\) −21.1246 −0.811885 −0.405942 0.913899i \(-0.633057\pi\)
−0.405942 + 0.913899i \(0.633057\pi\)
\(678\) 0 0
\(679\) −6.94427 −0.266497
\(680\) 0 0
\(681\) 31.4164 1.20388
\(682\) 0 0
\(683\) −21.5279 −0.823741 −0.411870 0.911242i \(-0.635124\pi\)
−0.411870 + 0.911242i \(0.635124\pi\)
\(684\) 0 0
\(685\) −32.3607 −1.23644
\(686\) 0 0
\(687\) −21.5279 −0.821339
\(688\) 0 0
\(689\) 28.9443 1.10269
\(690\) 0 0
\(691\) −33.7082 −1.28232 −0.641160 0.767407i \(-0.721547\pi\)
−0.641160 + 0.767407i \(0.721547\pi\)
\(692\) 0 0
\(693\) −29.8885 −1.13537
\(694\) 0 0
\(695\) 41.3050 1.56679
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 51.4164 1.94475
\(700\) 0 0
\(701\) 28.3607 1.07117 0.535584 0.844482i \(-0.320092\pi\)
0.535584 + 0.844482i \(0.320092\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −51.7771 −1.95004
\(706\) 0 0
\(707\) 8.18034 0.307653
\(708\) 0 0
\(709\) 2.47214 0.0928430 0.0464215 0.998922i \(-0.485218\pi\)
0.0464215 + 0.998922i \(0.485218\pi\)
\(710\) 0 0
\(711\) −96.7214 −3.62733
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 41.8885 1.56654
\(716\) 0 0
\(717\) 59.7771 2.23242
\(718\) 0 0
\(719\) 36.9443 1.37779 0.688894 0.724862i \(-0.258096\pi\)
0.688894 + 0.724862i \(0.258096\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −9.52786 −0.354345
\(724\) 0 0
\(725\) −57.3050 −2.12825
\(726\) 0 0
\(727\) 38.8328 1.44023 0.720115 0.693855i \(-0.244089\pi\)
0.720115 + 0.693855i \(0.244089\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) −17.8885 −0.661632
\(732\) 0 0
\(733\) 26.0689 0.962876 0.481438 0.876480i \(-0.340115\pi\)
0.481438 + 0.876480i \(0.340115\pi\)
\(734\) 0 0
\(735\) −10.4721 −0.386271
\(736\) 0 0
\(737\) 41.8885 1.54298
\(738\) 0 0
\(739\) 13.8885 0.510898 0.255449 0.966822i \(-0.417777\pi\)
0.255449 + 0.966822i \(0.417777\pi\)
\(740\) 0 0
\(741\) 33.8885 1.24493
\(742\) 0 0
\(743\) −34.4721 −1.26466 −0.632330 0.774699i \(-0.717901\pi\)
−0.632330 + 0.774699i \(0.717901\pi\)
\(744\) 0 0
\(745\) 3.05573 0.111953
\(746\) 0 0
\(747\) −35.5967 −1.30242
\(748\) 0 0
\(749\) −18.4721 −0.674957
\(750\) 0 0
\(751\) 28.3607 1.03490 0.517448 0.855715i \(-0.326882\pi\)
0.517448 + 0.855715i \(0.326882\pi\)
\(752\) 0 0
\(753\) −47.4164 −1.72795
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −16.5836 −0.602741 −0.301370 0.953507i \(-0.597444\pi\)
−0.301370 + 0.953507i \(0.597444\pi\)
\(758\) 0 0
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 7.88854 0.285959 0.142980 0.989726i \(-0.454332\pi\)
0.142980 + 0.989726i \(0.454332\pi\)
\(762\) 0 0
\(763\) −10.4721 −0.379117
\(764\) 0 0
\(765\) −48.3607 −1.74848
\(766\) 0 0
\(767\) 15.4164 0.556654
\(768\) 0 0
\(769\) 34.9443 1.26012 0.630061 0.776545i \(-0.283030\pi\)
0.630061 + 0.776545i \(0.283030\pi\)
\(770\) 0 0
\(771\) −90.2492 −3.25025
\(772\) 0 0
\(773\) 29.1246 1.04754 0.523770 0.851860i \(-0.324525\pi\)
0.523770 + 0.851860i \(0.324525\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) −6.47214 −0.231888
\(780\) 0 0
\(781\) −51.7771 −1.85273
\(782\) 0 0
\(783\) 151.554 5.41610
\(784\) 0 0
\(785\) −0.583592 −0.0208293
\(786\) 0 0
\(787\) 45.1246 1.60852 0.804259 0.594278i \(-0.202562\pi\)
0.804259 + 0.594278i \(0.202562\pi\)
\(788\) 0 0
\(789\) 9.88854 0.352041
\(790\) 0 0
\(791\) −13.4164 −0.477033
\(792\) 0 0
\(793\) −10.4721 −0.371876
\(794\) 0 0
\(795\) 93.6656 3.32198
\(796\) 0 0
\(797\) 28.7639 1.01887 0.509435 0.860509i \(-0.329854\pi\)
0.509435 + 0.860509i \(0.329854\pi\)
\(798\) 0 0
\(799\) −9.88854 −0.349832
\(800\) 0 0
\(801\) 14.9443 0.528030
\(802\) 0 0
\(803\) 11.7771 0.415604
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 42.4721 1.49509
\(808\) 0 0
\(809\) 21.4164 0.752961 0.376480 0.926425i \(-0.377134\pi\)
0.376480 + 0.926425i \(0.377134\pi\)
\(810\) 0 0
\(811\) 13.1246 0.460867 0.230434 0.973088i \(-0.425986\pi\)
0.230434 + 0.973088i \(0.425986\pi\)
\(812\) 0 0
\(813\) −67.7771 −2.37705
\(814\) 0 0
\(815\) −38.8328 −1.36025
\(816\) 0 0
\(817\) 28.9443 1.01263
\(818\) 0 0
\(819\) −24.1803 −0.844929
\(820\) 0 0
\(821\) 4.00000 0.139601 0.0698005 0.997561i \(-0.477764\pi\)
0.0698005 + 0.997561i \(0.477764\pi\)
\(822\) 0 0
\(823\) 22.8328 0.795902 0.397951 0.917407i \(-0.369721\pi\)
0.397951 + 0.917407i \(0.369721\pi\)
\(824\) 0 0
\(825\) 70.8328 2.46608
\(826\) 0 0
\(827\) −8.58359 −0.298481 −0.149240 0.988801i \(-0.547683\pi\)
−0.149240 + 0.988801i \(0.547683\pi\)
\(828\) 0 0
\(829\) 13.1246 0.455837 0.227918 0.973680i \(-0.426808\pi\)
0.227918 + 0.973680i \(0.426808\pi\)
\(830\) 0 0
\(831\) −64.7214 −2.24516
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −41.8885 −1.44961
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 80.6656 2.78157
\(842\) 0 0
\(843\) 19.4164 0.668737
\(844\) 0 0
\(845\) −8.18034 −0.281412
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) 21.5279 0.738834
\(850\) 0 0
\(851\) −6.11146 −0.209498
\(852\) 0 0
\(853\) 28.7639 0.984858 0.492429 0.870353i \(-0.336109\pi\)
0.492429 + 0.870353i \(0.336109\pi\)
\(854\) 0 0
\(855\) 78.2492 2.67607
\(856\) 0 0
\(857\) −3.16718 −0.108189 −0.0540945 0.998536i \(-0.517227\pi\)
−0.0540945 + 0.998536i \(0.517227\pi\)
\(858\) 0 0
\(859\) 7.81966 0.266803 0.133402 0.991062i \(-0.457410\pi\)
0.133402 + 0.991062i \(0.457410\pi\)
\(860\) 0 0
\(861\) 6.47214 0.220570
\(862\) 0 0
\(863\) 1.88854 0.0642868 0.0321434 0.999483i \(-0.489767\pi\)
0.0321434 + 0.999483i \(0.489767\pi\)
\(864\) 0 0
\(865\) 31.4164 1.06819
\(866\) 0 0
\(867\) 42.0689 1.42873
\(868\) 0 0
\(869\) 51.7771 1.75642
\(870\) 0 0
\(871\) 33.8885 1.14827
\(872\) 0 0
\(873\) −51.8885 −1.75616
\(874\) 0 0
\(875\) 1.52786 0.0516512
\(876\) 0 0
\(877\) 44.3607 1.49795 0.748977 0.662596i \(-0.230545\pi\)
0.748977 + 0.662596i \(0.230545\pi\)
\(878\) 0 0
\(879\) −46.2492 −1.55995
\(880\) 0 0
\(881\) 32.8328 1.10617 0.553083 0.833126i \(-0.313451\pi\)
0.553083 + 0.833126i \(0.313451\pi\)
\(882\) 0 0
\(883\) −29.5279 −0.993692 −0.496846 0.867839i \(-0.665509\pi\)
−0.496846 + 0.867839i \(0.665509\pi\)
\(884\) 0 0
\(885\) 49.8885 1.67699
\(886\) 0 0
\(887\) −38.8328 −1.30388 −0.651939 0.758271i \(-0.726044\pi\)
−0.651939 + 0.758271i \(0.726044\pi\)
\(888\) 0 0
\(889\) −7.41641 −0.248738
\(890\) 0 0
\(891\) −97.6656 −3.27192
\(892\) 0 0
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −25.8885 −0.864393
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 17.8885 0.595954
\(902\) 0 0
\(903\) −28.9443 −0.963205
\(904\) 0 0
\(905\) 67.1935 2.23359
\(906\) 0 0
\(907\) −8.58359 −0.285013 −0.142507 0.989794i \(-0.545516\pi\)
−0.142507 + 0.989794i \(0.545516\pi\)
\(908\) 0 0
\(909\) 61.1246 2.02738
\(910\) 0 0
\(911\) 45.5279 1.50841 0.754203 0.656642i \(-0.228024\pi\)
0.754203 + 0.656642i \(0.228024\pi\)
\(912\) 0 0
\(913\) 19.0557 0.630653
\(914\) 0 0
\(915\) −33.8885 −1.12032
\(916\) 0 0
\(917\) 8.18034 0.270139
\(918\) 0 0
\(919\) 6.83282 0.225394 0.112697 0.993629i \(-0.464051\pi\)
0.112697 + 0.993629i \(0.464051\pi\)
\(920\) 0 0
\(921\) 89.3050 2.94270
\(922\) 0 0
\(923\) −41.8885 −1.37878
\(924\) 0 0
\(925\) 13.5279 0.444793
\(926\) 0 0
\(927\) −59.7771 −1.96334
\(928\) 0 0
\(929\) 7.88854 0.258815 0.129407 0.991592i \(-0.458693\pi\)
0.129407 + 0.991592i \(0.458693\pi\)
\(930\) 0 0
\(931\) 3.23607 0.106058
\(932\) 0 0
\(933\) −25.8885 −0.847553
\(934\) 0 0
\(935\) 25.8885 0.846646
\(936\) 0 0
\(937\) −9.05573 −0.295838 −0.147919 0.988999i \(-0.547257\pi\)
−0.147919 + 0.988999i \(0.547257\pi\)
\(938\) 0 0
\(939\) 90.2492 2.94517
\(940\) 0 0
\(941\) −30.6525 −0.999242 −0.499621 0.866244i \(-0.666527\pi\)
−0.499621 + 0.866244i \(0.666527\pi\)
\(942\) 0 0
\(943\) 4.94427 0.161008
\(944\) 0 0
\(945\) −46.8328 −1.52347
\(946\) 0 0
\(947\) −56.9443 −1.85044 −0.925220 0.379431i \(-0.876120\pi\)
−0.925220 + 0.379431i \(0.876120\pi\)
\(948\) 0 0
\(949\) 9.52786 0.309288
\(950\) 0 0
\(951\) −44.9443 −1.45742
\(952\) 0 0
\(953\) −3.88854 −0.125962 −0.0629811 0.998015i \(-0.520061\pi\)
−0.0629811 + 0.998015i \(0.520061\pi\)
\(954\) 0 0
\(955\) −57.8885 −1.87323
\(956\) 0 0
\(957\) −135.554 −4.38184
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −138.026 −4.44784
\(964\) 0 0
\(965\) 11.4164 0.367507
\(966\) 0 0
\(967\) 39.4164 1.26755 0.633773 0.773519i \(-0.281505\pi\)
0.633773 + 0.773519i \(0.281505\pi\)
\(968\) 0 0
\(969\) 20.9443 0.672827
\(970\) 0 0
\(971\) −46.6525 −1.49715 −0.748575 0.663051i \(-0.769261\pi\)
−0.748575 + 0.663051i \(0.769261\pi\)
\(972\) 0 0
\(973\) 12.7639 0.409193
\(974\) 0 0
\(975\) 57.3050 1.83523
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −78.2492 −2.49831
\(982\) 0 0
\(983\) −43.7771 −1.39627 −0.698136 0.715965i \(-0.745987\pi\)
−0.698136 + 0.715965i \(0.745987\pi\)
\(984\) 0 0
\(985\) 12.9443 0.412439
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) −22.1115 −0.703103
\(990\) 0 0
\(991\) 22.8328 0.725308 0.362654 0.931924i \(-0.381871\pi\)
0.362654 + 0.931924i \(0.381871\pi\)
\(992\) 0 0
\(993\) −80.7214 −2.56161
\(994\) 0 0
\(995\) −77.6656 −2.46217
\(996\) 0 0
\(997\) 35.2361 1.11594 0.557969 0.829862i \(-0.311581\pi\)
0.557969 + 0.829862i \(0.311581\pi\)
\(998\) 0 0
\(999\) −35.7771 −1.13194
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 1792.2.a.l.1.1 2
4.3 odd 2 1792.2.a.t.1.2 2
8.3 odd 2 1792.2.a.j.1.1 2
8.5 even 2 1792.2.a.r.1.2 2
16.3 odd 4 896.2.b.g.449.4 yes 4
16.5 even 4 896.2.b.e.449.4 yes 4
16.11 odd 4 896.2.b.g.449.1 yes 4
16.13 even 4 896.2.b.e.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.e.449.1 4 16.13 even 4
896.2.b.e.449.4 yes 4 16.5 even 4
896.2.b.g.449.1 yes 4 16.11 odd 4
896.2.b.g.449.4 yes 4 16.3 odd 4
1792.2.a.j.1.1 2 8.3 odd 2
1792.2.a.l.1.1 2 1.1 even 1 trivial
1792.2.a.r.1.2 2 8.5 even 2
1792.2.a.t.1.2 2 4.3 odd 2