Properties

Label 1792.2.a.t.1.1
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
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: \(0\)
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 \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.23607 q^{3} -1.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} +O(q^{10})\) \(q-1.23607 q^{3} -1.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} +4.00000 q^{11} +1.23607 q^{13} +1.52786 q^{15} -2.00000 q^{17} +1.23607 q^{19} +1.23607 q^{21} -6.47214 q^{23} -3.47214 q^{25} +5.52786 q^{27} -1.52786 q^{29} -4.94427 q^{33} +1.23607 q^{35} -6.47214 q^{37} -1.52786 q^{39} -2.00000 q^{41} +8.94427 q^{43} +1.81966 q^{45} +12.9443 q^{47} +1.00000 q^{49} +2.47214 q^{51} +8.94427 q^{53} -4.94427 q^{55} -1.52786 q^{57} +9.23607 q^{59} -1.23607 q^{61} +1.47214 q^{63} -1.52786 q^{65} +1.52786 q^{67} +8.00000 q^{69} +4.94427 q^{71} +14.9443 q^{73} +4.29180 q^{75} -4.00000 q^{77} -4.94427 q^{79} -2.41641 q^{81} +9.23607 q^{83} +2.47214 q^{85} +1.88854 q^{87} +2.00000 q^{89} -1.23607 q^{91} -1.52786 q^{95} +10.9443 q^{97} -5.88854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} - 2q^{7} + 6q^{9} + 8q^{11} - 2q^{13} + 12q^{15} - 4q^{17} - 2q^{19} - 2q^{21} - 4q^{23} + 2q^{25} + 20q^{27} - 12q^{29} + 8q^{33} - 2q^{35} - 4q^{37} - 12q^{39} - 4q^{41} + 26q^{45} + 8q^{47} + 2q^{49} - 4q^{51} + 8q^{55} - 12q^{57} + 14q^{59} + 2q^{61} - 6q^{63} - 12q^{65} + 12q^{67} + 16q^{69} - 8q^{71} + 12q^{73} + 22q^{75} - 8q^{77} + 8q^{79} + 22q^{81} + 14q^{83} - 4q^{85} - 32q^{87} + 4q^{89} + 2q^{91} - 12q^{95} + 4q^{97} + 24q^{99} + O(q^{100}) \)

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
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 0 0
\(15\) 1.52786 0.394493
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 1.23607 0.283573 0.141787 0.989897i \(-0.454715\pi\)
0.141787 + 0.989897i \(0.454715\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) −6.47214 −1.34953 −0.674767 0.738031i \(-0.735756\pi\)
−0.674767 + 0.738031i \(0.735756\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) −1.52786 −0.283717 −0.141859 0.989887i \(-0.545308\pi\)
−0.141859 + 0.989887i \(0.545308\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.94427 −0.860687
\(34\) 0 0
\(35\) 1.23607 0.208934
\(36\) 0 0
\(37\) −6.47214 −1.06401 −0.532006 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) 0 0
\(39\) −1.52786 −0.244654
\(40\) 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
\(45\) 1.81966 0.271259
\(46\) 0 0
\(47\) 12.9443 1.88812 0.944058 0.329779i \(-0.106974\pi\)
0.944058 + 0.329779i \(0.106974\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.47214 0.346168
\(52\) 0 0
\(53\) 8.94427 1.22859 0.614295 0.789076i \(-0.289440\pi\)
0.614295 + 0.789076i \(0.289440\pi\)
\(54\) 0 0
\(55\) −4.94427 −0.666685
\(56\) 0 0
\(57\) −1.52786 −0.202371
\(58\) 0 0
\(59\) 9.23607 1.20243 0.601217 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) −1.23607 −0.158262 −0.0791311 0.996864i \(-0.525215\pi\)
−0.0791311 + 0.996864i \(0.525215\pi\)
\(62\) 0 0
\(63\) 1.47214 0.185472
\(64\) 0 0
\(65\) −1.52786 −0.189508
\(66\) 0 0
\(67\) 1.52786 0.186658 0.0933292 0.995635i \(-0.470249\pi\)
0.0933292 + 0.995635i \(0.470249\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 0 0
\(73\) 14.9443 1.74909 0.874547 0.484940i \(-0.161159\pi\)
0.874547 + 0.484940i \(0.161159\pi\)
\(74\) 0 0
\(75\) 4.29180 0.495574
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) 9.23607 1.01379 0.506895 0.862008i \(-0.330793\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) 0 0
\(87\) 1.88854 0.202473
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.52786 −0.156756
\(96\) 0 0
\(97\) 10.9443 1.11122 0.555611 0.831442i \(-0.312484\pi\)
0.555611 + 0.831442i \(0.312484\pi\)
\(98\) 0 0
\(99\) −5.88854 −0.591821
\(100\) 0 0
\(101\) −14.1803 −1.41100 −0.705498 0.708712i \(-0.749277\pi\)
−0.705498 + 0.708712i \(0.749277\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −1.52786 −0.149104
\(106\) 0 0
\(107\) 9.52786 0.921093 0.460547 0.887635i \(-0.347653\pi\)
0.460547 + 0.887635i \(0.347653\pi\)
\(108\) 0 0
\(109\) −1.52786 −0.146343 −0.0731714 0.997319i \(-0.523312\pi\)
−0.0731714 + 0.997319i \(0.523312\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 13.4164 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) −1.81966 −0.168228
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 2.47214 0.222905
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −19.4164 −1.72293 −0.861464 0.507819i \(-0.830452\pi\)
−0.861464 + 0.507819i \(0.830452\pi\)
\(128\) 0 0
\(129\) −11.0557 −0.973403
\(130\) 0 0
\(131\) 14.1803 1.23894 0.619471 0.785020i \(-0.287347\pi\)
0.619471 + 0.785020i \(0.287347\pi\)
\(132\) 0 0
\(133\) −1.23607 −0.107181
\(134\) 0 0
\(135\) −6.83282 −0.588075
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −17.2361 −1.46194 −0.730972 0.682407i \(-0.760933\pi\)
−0.730972 + 0.682407i \(0.760933\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) 4.94427 0.413461
\(144\) 0 0
\(145\) 1.88854 0.156835
\(146\) 0 0
\(147\) −1.23607 −0.101949
\(148\) 0 0
\(149\) −16.9443 −1.38813 −0.694064 0.719913i \(-0.744182\pi\)
−0.694064 + 0.719913i \(0.744182\pi\)
\(150\) 0 0
\(151\) 6.47214 0.526695 0.263347 0.964701i \(-0.415173\pi\)
0.263347 + 0.964701i \(0.415173\pi\)
\(152\) 0 0
\(153\) 2.94427 0.238030
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.1803 1.77018 0.885092 0.465416i \(-0.154095\pi\)
0.885092 + 0.465416i \(0.154095\pi\)
\(158\) 0 0
\(159\) −11.0557 −0.876776
\(160\) 0 0
\(161\) 6.47214 0.510076
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 6.11146 0.475776
\(166\) 0 0
\(167\) −4.94427 −0.382599 −0.191300 0.981532i \(-0.561270\pi\)
−0.191300 + 0.981532i \(0.561270\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) −1.81966 −0.139153
\(172\) 0 0
\(173\) −3.70820 −0.281930 −0.140965 0.990015i \(-0.545020\pi\)
−0.140965 + 0.990015i \(0.545020\pi\)
\(174\) 0 0
\(175\) 3.47214 0.262469
\(176\) 0 0
\(177\) −11.4164 −0.858110
\(178\) 0 0
\(179\) 6.47214 0.483750 0.241875 0.970307i \(-0.422238\pi\)
0.241875 + 0.970307i \(0.422238\pi\)
\(180\) 0 0
\(181\) 25.2361 1.87578 0.937891 0.346930i \(-0.112776\pi\)
0.937891 + 0.346930i \(0.112776\pi\)
\(182\) 0 0
\(183\) 1.52786 0.112943
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −5.52786 −0.402093
\(190\) 0 0
\(191\) −17.8885 −1.29437 −0.647185 0.762333i \(-0.724054\pi\)
−0.647185 + 0.762333i \(0.724054\pi\)
\(192\) 0 0
\(193\) 12.4721 0.897764 0.448882 0.893591i \(-0.351822\pi\)
0.448882 + 0.893591i \(0.351822\pi\)
\(194\) 0 0
\(195\) 1.88854 0.135241
\(196\) 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 \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −1.88854 −0.133208
\(202\) 0 0
\(203\) 1.52786 0.107235
\(204\) 0 0
\(205\) 2.47214 0.172661
\(206\) 0 0
\(207\) 9.52786 0.662232
\(208\) 0 0
\(209\) 4.94427 0.342002
\(210\) 0 0
\(211\) 11.4164 0.785938 0.392969 0.919552i \(-0.371448\pi\)
0.392969 + 0.919552i \(0.371448\pi\)
\(212\) 0 0
\(213\) −6.11146 −0.418750
\(214\) 0 0
\(215\) −11.0557 −0.753994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −18.4721 −1.24823
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 0 0
\(223\) −28.9443 −1.93825 −0.969126 0.246566i \(-0.920698\pi\)
−0.969126 + 0.246566i \(0.920698\pi\)
\(224\) 0 0
\(225\) 5.11146 0.340764
\(226\) 0 0
\(227\) −3.70820 −0.246122 −0.123061 0.992399i \(-0.539271\pi\)
−0.123061 + 0.992399i \(0.539271\pi\)
\(228\) 0 0
\(229\) −24.6525 −1.62908 −0.814541 0.580106i \(-0.803011\pi\)
−0.814541 + 0.580106i \(0.803011\pi\)
\(230\) 0 0
\(231\) 4.94427 0.325309
\(232\) 0 0
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 6.11146 0.396982
\(238\) 0 0
\(239\) 9.52786 0.616306 0.308153 0.951337i \(-0.400289\pi\)
0.308153 + 0.951337i \(0.400289\pi\)
\(240\) 0 0
\(241\) −14.9443 −0.962645 −0.481323 0.876544i \(-0.659843\pi\)
−0.481323 + 0.876544i \(0.659843\pi\)
\(242\) 0 0
\(243\) −13.5967 −0.872232
\(244\) 0 0
\(245\) −1.23607 −0.0789695
\(246\) 0 0
\(247\) 1.52786 0.0972157
\(248\) 0 0
\(249\) −11.4164 −0.723485
\(250\) 0 0
\(251\) 16.6525 1.05109 0.525547 0.850764i \(-0.323861\pi\)
0.525547 + 0.850764i \(0.323861\pi\)
\(252\) 0 0
\(253\) −25.8885 −1.62760
\(254\) 0 0
\(255\) −3.05573 −0.191357
\(256\) 0 0
\(257\) −7.88854 −0.492074 −0.246037 0.969260i \(-0.579128\pi\)
−0.246037 + 0.969260i \(0.579128\pi\)
\(258\) 0 0
\(259\) 6.47214 0.402159
\(260\) 0 0
\(261\) 2.24922 0.139223
\(262\) 0 0
\(263\) 20.9443 1.29148 0.645740 0.763558i \(-0.276549\pi\)
0.645740 + 0.763558i \(0.276549\pi\)
\(264\) 0 0
\(265\) −11.0557 −0.679148
\(266\) 0 0
\(267\) −2.47214 −0.151292
\(268\) 0 0
\(269\) 27.1246 1.65382 0.826908 0.562337i \(-0.190097\pi\)
0.826908 + 0.562337i \(0.190097\pi\)
\(270\) 0 0
\(271\) −3.05573 −0.185622 −0.0928111 0.995684i \(-0.529585\pi\)
−0.0928111 + 0.995684i \(0.529585\pi\)
\(272\) 0 0
\(273\) 1.52786 0.0924705
\(274\) 0 0
\(275\) −13.8885 −0.837511
\(276\) 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
\(283\) −24.6525 −1.46544 −0.732719 0.680532i \(-0.761749\pi\)
−0.732719 + 0.680532i \(0.761749\pi\)
\(284\) 0 0
\(285\) 1.88854 0.111868
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −13.5279 −0.793017
\(292\) 0 0
\(293\) 27.7082 1.61873 0.809365 0.587306i \(-0.199811\pi\)
0.809365 + 0.587306i \(0.199811\pi\)
\(294\) 0 0
\(295\) −11.4164 −0.664689
\(296\) 0 0
\(297\) 22.1115 1.28304
\(298\) 0 0
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −8.94427 −0.515539
\(302\) 0 0
\(303\) 17.5279 1.00695
\(304\) 0 0
\(305\) 1.52786 0.0874852
\(306\) 0 0
\(307\) −21.5967 −1.23259 −0.616296 0.787515i \(-0.711367\pi\)
−0.616296 + 0.787515i \(0.711367\pi\)
\(308\) 0 0
\(309\) −9.88854 −0.562540
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 7.88854 0.445887 0.222943 0.974831i \(-0.428433\pi\)
0.222943 + 0.974831i \(0.428433\pi\)
\(314\) 0 0
\(315\) −1.81966 −0.102526
\(316\) 0 0
\(317\) −21.8885 −1.22938 −0.614692 0.788768i \(-0.710720\pi\)
−0.614692 + 0.788768i \(0.710720\pi\)
\(318\) 0 0
\(319\) −6.11146 −0.342176
\(320\) 0 0
\(321\) −11.7771 −0.657333
\(322\) 0 0
\(323\) −2.47214 −0.137553
\(324\) 0 0
\(325\) −4.29180 −0.238066
\(326\) 0 0
\(327\) 1.88854 0.104437
\(328\) 0 0
\(329\) −12.9443 −0.713641
\(330\) 0 0
\(331\) −7.05573 −0.387818 −0.193909 0.981020i \(-0.562117\pi\)
−0.193909 + 0.981020i \(0.562117\pi\)
\(332\) 0 0
\(333\) 9.52786 0.522124
\(334\) 0 0
\(335\) −1.88854 −0.103182
\(336\) 0 0
\(337\) −9.41641 −0.512944 −0.256472 0.966552i \(-0.582560\pi\)
−0.256472 + 0.966552i \(0.582560\pi\)
\(338\) 0 0
\(339\) −16.5836 −0.900697
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −9.88854 −0.532381
\(346\) 0 0
\(347\) 15.0557 0.808234 0.404117 0.914707i \(-0.367579\pi\)
0.404117 + 0.914707i \(0.367579\pi\)
\(348\) 0 0
\(349\) 4.29180 0.229735 0.114867 0.993381i \(-0.463356\pi\)
0.114867 + 0.993381i \(0.463356\pi\)
\(350\) 0 0
\(351\) 6.83282 0.364709
\(352\) 0 0
\(353\) −17.0557 −0.907785 −0.453892 0.891056i \(-0.649965\pi\)
−0.453892 + 0.891056i \(0.649965\pi\)
\(354\) 0 0
\(355\) −6.11146 −0.324362
\(356\) 0 0
\(357\) −2.47214 −0.130839
\(358\) 0 0
\(359\) 3.41641 0.180311 0.0901556 0.995928i \(-0.471264\pi\)
0.0901556 + 0.995928i \(0.471264\pi\)
\(360\) 0 0
\(361\) −17.4721 −0.919586
\(362\) 0 0
\(363\) −6.18034 −0.324384
\(364\) 0 0
\(365\) −18.4721 −0.966876
\(366\) 0 0
\(367\) 28.9443 1.51088 0.755439 0.655219i \(-0.227423\pi\)
0.755439 + 0.655219i \(0.227423\pi\)
\(368\) 0 0
\(369\) 2.94427 0.153273
\(370\) 0 0
\(371\) −8.94427 −0.464363
\(372\) 0 0
\(373\) −32.9443 −1.70579 −0.852895 0.522083i \(-0.825155\pi\)
−0.852895 + 0.522083i \(0.825155\pi\)
\(374\) 0 0
\(375\) −12.9443 −0.668439
\(376\) 0 0
\(377\) −1.88854 −0.0972650
\(378\) 0 0
\(379\) −37.8885 −1.94620 −0.973102 0.230375i \(-0.926005\pi\)
−0.973102 + 0.230375i \(0.926005\pi\)
\(380\) 0 0
\(381\) 24.0000 1.22956
\(382\) 0 0
\(383\) 3.05573 0.156140 0.0780702 0.996948i \(-0.475124\pi\)
0.0780702 + 0.996948i \(0.475124\pi\)
\(384\) 0 0
\(385\) 4.94427 0.251983
\(386\) 0 0
\(387\) −13.1672 −0.669326
\(388\) 0 0
\(389\) 19.4164 0.984451 0.492225 0.870468i \(-0.336184\pi\)
0.492225 + 0.870468i \(0.336184\pi\)
\(390\) 0 0
\(391\) 12.9443 0.654620
\(392\) 0 0
\(393\) −17.5279 −0.884164
\(394\) 0 0
\(395\) 6.11146 0.307501
\(396\) 0 0
\(397\) 3.12461 0.156820 0.0784099 0.996921i \(-0.475016\pi\)
0.0784099 + 0.996921i \(0.475016\pi\)
\(398\) 0 0
\(399\) 1.52786 0.0764889
\(400\) 0 0
\(401\) −0.472136 −0.0235773 −0.0117887 0.999931i \(-0.503753\pi\)
−0.0117887 + 0.999931i \(0.503753\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.98684 0.148417
\(406\) 0 0
\(407\) −25.8885 −1.28325
\(408\) 0 0
\(409\) −14.9443 −0.738947 −0.369473 0.929241i \(-0.620462\pi\)
−0.369473 + 0.929241i \(0.620462\pi\)
\(410\) 0 0
\(411\) 12.3607 0.609707
\(412\) 0 0
\(413\) −9.23607 −0.454477
\(414\) 0 0
\(415\) −11.4164 −0.560409
\(416\) 0 0
\(417\) 21.3050 1.04331
\(418\) 0 0
\(419\) −6.18034 −0.301929 −0.150965 0.988539i \(-0.548238\pi\)
−0.150965 + 0.988539i \(0.548238\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 0 0
\(423\) −19.0557 −0.926521
\(424\) 0 0
\(425\) 6.94427 0.336847
\(426\) 0 0
\(427\) 1.23607 0.0598175
\(428\) 0 0
\(429\) −6.11146 −0.295064
\(430\) 0 0
\(431\) 32.3607 1.55876 0.779380 0.626552i \(-0.215534\pi\)
0.779380 + 0.626552i \(0.215534\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −2.33437 −0.111924
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 20.9443 0.999616 0.499808 0.866136i \(-0.333404\pi\)
0.499808 + 0.866136i \(0.333404\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 0 0
\(443\) 24.3607 1.15741 0.578705 0.815537i \(-0.303558\pi\)
0.578705 + 0.815537i \(0.303558\pi\)
\(444\) 0 0
\(445\) −2.47214 −0.117190
\(446\) 0 0
\(447\) 20.9443 0.990630
\(448\) 0 0
\(449\) 23.8885 1.12737 0.563685 0.825990i \(-0.309383\pi\)
0.563685 + 0.825990i \(0.309383\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 1.52786 0.0716274
\(456\) 0 0
\(457\) 20.4721 0.957646 0.478823 0.877911i \(-0.341064\pi\)
0.478823 + 0.877911i \(0.341064\pi\)
\(458\) 0 0
\(459\) −11.0557 −0.516037
\(460\) 0 0
\(461\) 40.0689 1.86619 0.933097 0.359625i \(-0.117095\pi\)
0.933097 + 0.359625i \(0.117095\pi\)
\(462\) 0 0
\(463\) 1.88854 0.0877681 0.0438840 0.999037i \(-0.486027\pi\)
0.0438840 + 0.999037i \(0.486027\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.6525 −1.14078 −0.570390 0.821374i \(-0.693208\pi\)
−0.570390 + 0.821374i \(0.693208\pi\)
\(468\) 0 0
\(469\) −1.52786 −0.0705502
\(470\) 0 0
\(471\) −27.4164 −1.26328
\(472\) 0 0
\(473\) 35.7771 1.64503
\(474\) 0 0
\(475\) −4.29180 −0.196921
\(476\) 0 0
\(477\) −13.1672 −0.602884
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −13.5279 −0.614269
\(486\) 0 0
\(487\) −22.4721 −1.01831 −0.509155 0.860675i \(-0.670042\pi\)
−0.509155 + 0.860675i \(0.670042\pi\)
\(488\) 0 0
\(489\) −14.8328 −0.670763
\(490\) 0 0
\(491\) 14.4721 0.653118 0.326559 0.945177i \(-0.394111\pi\)
0.326559 + 0.945177i \(0.394111\pi\)
\(492\) 0 0
\(493\) 3.05573 0.137623
\(494\) 0 0
\(495\) 7.27864 0.327151
\(496\) 0 0
\(497\) −4.94427 −0.221781
\(498\) 0 0
\(499\) −24.3607 −1.09053 −0.545267 0.838262i \(-0.683572\pi\)
−0.545267 + 0.838262i \(0.683572\pi\)
\(500\) 0 0
\(501\) 6.11146 0.273040
\(502\) 0 0
\(503\) −11.0557 −0.492951 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(504\) 0 0
\(505\) 17.5279 0.779980
\(506\) 0 0
\(507\) 14.1803 0.629771
\(508\) 0 0
\(509\) −0.652476 −0.0289205 −0.0144602 0.999895i \(-0.504603\pi\)
−0.0144602 + 0.999895i \(0.504603\pi\)
\(510\) 0 0
\(511\) −14.9443 −0.661096
\(512\) 0 0
\(513\) 6.83282 0.301676
\(514\) 0 0
\(515\) −9.88854 −0.435741
\(516\) 0 0
\(517\) 51.7771 2.27715
\(518\) 0 0
\(519\) 4.58359 0.201197
\(520\) 0 0
\(521\) 26.9443 1.18045 0.590225 0.807239i \(-0.299039\pi\)
0.590225 + 0.807239i \(0.299039\pi\)
\(522\) 0 0
\(523\) −1.81966 −0.0795682 −0.0397841 0.999208i \(-0.512667\pi\)
−0.0397841 + 0.999208i \(0.512667\pi\)
\(524\) 0 0
\(525\) −4.29180 −0.187309
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 0 0
\(531\) −13.5967 −0.590049
\(532\) 0 0
\(533\) −2.47214 −0.107080
\(534\) 0 0
\(535\) −11.7771 −0.509168
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −0.944272 −0.0405974 −0.0202987 0.999794i \(-0.506462\pi\)
−0.0202987 + 0.999794i \(0.506462\pi\)
\(542\) 0 0
\(543\) −31.1935 −1.33864
\(544\) 0 0
\(545\) 1.88854 0.0808963
\(546\) 0 0
\(547\) −34.8328 −1.48934 −0.744672 0.667431i \(-0.767394\pi\)
−0.744672 + 0.667431i \(0.767394\pi\)
\(548\) 0 0
\(549\) 1.81966 0.0776612
\(550\) 0 0
\(551\) −1.88854 −0.0804547
\(552\) 0 0
\(553\) 4.94427 0.210252
\(554\) 0 0
\(555\) −9.88854 −0.419745
\(556\) 0 0
\(557\) −5.88854 −0.249506 −0.124753 0.992188i \(-0.539814\pi\)
−0.124753 + 0.992188i \(0.539814\pi\)
\(558\) 0 0
\(559\) 11.0557 0.467607
\(560\) 0 0
\(561\) 9.88854 0.417495
\(562\) 0 0
\(563\) 42.5410 1.79289 0.896445 0.443155i \(-0.146141\pi\)
0.896445 + 0.443155i \(0.146141\pi\)
\(564\) 0 0
\(565\) −16.5836 −0.697677
\(566\) 0 0
\(567\) 2.41641 0.101480
\(568\) 0 0
\(569\) −14.5836 −0.611376 −0.305688 0.952132i \(-0.598886\pi\)
−0.305688 + 0.952132i \(0.598886\pi\)
\(570\) 0 0
\(571\) −31.7771 −1.32983 −0.664915 0.746919i \(-0.731532\pi\)
−0.664915 + 0.746919i \(0.731532\pi\)
\(572\) 0 0
\(573\) 22.1115 0.923719
\(574\) 0 0
\(575\) 22.4721 0.937153
\(576\) 0 0
\(577\) −20.8328 −0.867281 −0.433641 0.901086i \(-0.642771\pi\)
−0.433641 + 0.901086i \(0.642771\pi\)
\(578\) 0 0
\(579\) −15.4164 −0.640684
\(580\) 0 0
\(581\) −9.23607 −0.383177
\(582\) 0 0
\(583\) 35.7771 1.48174
\(584\) 0 0
\(585\) 2.24922 0.0929940
\(586\) 0 0
\(587\) 14.7639 0.609373 0.304686 0.952453i \(-0.401448\pi\)
0.304686 + 0.952453i \(0.401448\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −4.94427 −0.203380
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −2.47214 −0.101348
\(596\) 0 0
\(597\) −29.6656 −1.21413
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −4.11146 −0.167710 −0.0838549 0.996478i \(-0.526723\pi\)
−0.0838549 + 0.996478i \(0.526723\pi\)
\(602\) 0 0
\(603\) −2.24922 −0.0915955
\(604\) 0 0
\(605\) −6.18034 −0.251267
\(606\) 0 0
\(607\) 9.88854 0.401364 0.200682 0.979656i \(-0.435684\pi\)
0.200682 + 0.979656i \(0.435684\pi\)
\(608\) 0 0
\(609\) −1.88854 −0.0765277
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 14.4721 0.584524 0.292262 0.956338i \(-0.405592\pi\)
0.292262 + 0.956338i \(0.405592\pi\)
\(614\) 0 0
\(615\) −3.05573 −0.123219
\(616\) 0 0
\(617\) −23.5279 −0.947196 −0.473598 0.880741i \(-0.657045\pi\)
−0.473598 + 0.880741i \(0.657045\pi\)
\(618\) 0 0
\(619\) −9.81966 −0.394685 −0.197343 0.980335i \(-0.563231\pi\)
−0.197343 + 0.980335i \(0.563231\pi\)
\(620\) 0 0
\(621\) −35.7771 −1.43569
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) −6.11146 −0.244068
\(628\) 0 0
\(629\) 12.9443 0.516122
\(630\) 0 0
\(631\) 17.8885 0.712132 0.356066 0.934461i \(-0.384118\pi\)
0.356066 + 0.934461i \(0.384118\pi\)
\(632\) 0 0
\(633\) −14.1115 −0.560880
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) 0 0
\(639\) −7.27864 −0.287939
\(640\) 0 0
\(641\) −10.3607 −0.409222 −0.204611 0.978843i \(-0.565593\pi\)
−0.204611 + 0.978843i \(0.565593\pi\)
\(642\) 0 0
\(643\) 26.5410 1.04668 0.523338 0.852125i \(-0.324687\pi\)
0.523338 + 0.852125i \(0.324687\pi\)
\(644\) 0 0
\(645\) 13.6656 0.538084
\(646\) 0 0
\(647\) −17.8885 −0.703271 −0.351636 0.936137i \(-0.614374\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(648\) 0 0
\(649\) 36.9443 1.45019
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12.5836 −0.492434 −0.246217 0.969215i \(-0.579188\pi\)
−0.246217 + 0.969215i \(0.579188\pi\)
\(654\) 0 0
\(655\) −17.5279 −0.684870
\(656\) 0 0
\(657\) −22.0000 −0.858302
\(658\) 0 0
\(659\) 42.8328 1.66853 0.834265 0.551364i \(-0.185892\pi\)
0.834265 + 0.551364i \(0.185892\pi\)
\(660\) 0 0
\(661\) 22.7639 0.885414 0.442707 0.896666i \(-0.354018\pi\)
0.442707 + 0.896666i \(0.354018\pi\)
\(662\) 0 0
\(663\) 3.05573 0.118675
\(664\) 0 0
\(665\) 1.52786 0.0592480
\(666\) 0 0
\(667\) 9.88854 0.382886
\(668\) 0 0
\(669\) 35.7771 1.38322
\(670\) 0 0
\(671\) −4.94427 −0.190872
\(672\) 0 0
\(673\) −11.8885 −0.458270 −0.229135 0.973395i \(-0.573590\pi\)
−0.229135 + 0.973395i \(0.573590\pi\)
\(674\) 0 0
\(675\) −19.1935 −0.738758
\(676\) 0 0
\(677\) 19.1246 0.735019 0.367509 0.930020i \(-0.380211\pi\)
0.367509 + 0.930020i \(0.380211\pi\)
\(678\) 0 0
\(679\) −10.9443 −0.420003
\(680\) 0 0
\(681\) 4.58359 0.175644
\(682\) 0 0
\(683\) 30.4721 1.16598 0.582992 0.812478i \(-0.301882\pi\)
0.582992 + 0.812478i \(0.301882\pi\)
\(684\) 0 0
\(685\) 12.3607 0.472277
\(686\) 0 0
\(687\) 30.4721 1.16258
\(688\) 0 0
\(689\) 11.0557 0.421190
\(690\) 0 0
\(691\) 20.2918 0.771936 0.385968 0.922512i \(-0.373867\pi\)
0.385968 + 0.922512i \(0.373867\pi\)
\(692\) 0 0
\(693\) 5.88854 0.223687
\(694\) 0 0
\(695\) 21.3050 0.808143
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −24.5836 −0.929837
\(700\) 0 0
\(701\) −16.3607 −0.617934 −0.308967 0.951073i \(-0.599983\pi\)
−0.308967 + 0.951073i \(0.599983\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 19.7771 0.744848
\(706\) 0 0
\(707\) 14.1803 0.533307
\(708\) 0 0
\(709\) −6.47214 −0.243066 −0.121533 0.992587i \(-0.538781\pi\)
−0.121533 + 0.992587i \(0.538781\pi\)
\(710\) 0 0
\(711\) 7.27864 0.272970
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −6.11146 −0.228556
\(716\) 0 0
\(717\) −11.7771 −0.439823
\(718\) 0 0
\(719\) −19.0557 −0.710659 −0.355329 0.934741i \(-0.615631\pi\)
−0.355329 + 0.934741i \(0.615631\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 18.4721 0.686986
\(724\) 0 0
\(725\) 5.30495 0.197021
\(726\) 0 0
\(727\) 14.8328 0.550119 0.275059 0.961427i \(-0.411302\pi\)
0.275059 + 0.961427i \(0.411302\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) −17.8885 −0.661632
\(732\) 0 0
\(733\) −32.0689 −1.18449 −0.592246 0.805757i \(-0.701758\pi\)
−0.592246 + 0.805757i \(0.701758\pi\)
\(734\) 0 0
\(735\) 1.52786 0.0563561
\(736\) 0 0
\(737\) 6.11146 0.225118
\(738\) 0 0
\(739\) 21.8885 0.805183 0.402592 0.915380i \(-0.368110\pi\)
0.402592 + 0.915380i \(0.368110\pi\)
\(740\) 0 0
\(741\) −1.88854 −0.0693774
\(742\) 0 0
\(743\) 25.5279 0.936527 0.468263 0.883589i \(-0.344880\pi\)
0.468263 + 0.883589i \(0.344880\pi\)
\(744\) 0 0
\(745\) 20.9443 0.767339
\(746\) 0 0
\(747\) −13.5967 −0.497479
\(748\) 0 0
\(749\) −9.52786 −0.348141
\(750\) 0 0
\(751\) 16.3607 0.597010 0.298505 0.954408i \(-0.403512\pi\)
0.298505 + 0.954408i \(0.403512\pi\)
\(752\) 0 0
\(753\) −20.5836 −0.750108
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −43.4164 −1.57800 −0.788998 0.614396i \(-0.789400\pi\)
−0.788998 + 0.614396i \(0.789400\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −27.8885 −1.01096 −0.505479 0.862839i \(-0.668684\pi\)
−0.505479 + 0.862839i \(0.668684\pi\)
\(762\) 0 0
\(763\) 1.52786 0.0553124
\(764\) 0 0
\(765\) −3.63932 −0.131580
\(766\) 0 0
\(767\) 11.4164 0.412223
\(768\) 0 0
\(769\) 17.0557 0.615045 0.307523 0.951541i \(-0.400500\pi\)
0.307523 + 0.951541i \(0.400500\pi\)
\(770\) 0 0
\(771\) 9.75078 0.351166
\(772\) 0 0
\(773\) −11.1246 −0.400124 −0.200062 0.979783i \(-0.564114\pi\)
−0.200062 + 0.979783i \(0.564114\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) −2.47214 −0.0885735
\(780\) 0 0
\(781\) 19.7771 0.707680
\(782\) 0 0
\(783\) −8.44582 −0.301829
\(784\) 0 0
\(785\) −27.4164 −0.978534
\(786\) 0 0
\(787\) −4.87539 −0.173789 −0.0868944 0.996218i \(-0.527694\pi\)
−0.0868944 + 0.996218i \(0.527694\pi\)
\(788\) 0 0
\(789\) −25.8885 −0.921657
\(790\) 0 0
\(791\) −13.4164 −0.477033
\(792\) 0 0
\(793\) −1.52786 −0.0542560
\(794\) 0 0
\(795\) 13.6656 0.484670
\(796\) 0 0
\(797\) 33.2361 1.17728 0.588641 0.808395i \(-0.299663\pi\)
0.588641 + 0.808395i \(0.299663\pi\)
\(798\) 0 0
\(799\) −25.8885 −0.915871
\(800\) 0 0
\(801\) −2.94427 −0.104031
\(802\) 0 0
\(803\) 59.7771 2.10949
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −33.5279 −1.18024
\(808\) 0 0
\(809\) −5.41641 −0.190431 −0.0952154 0.995457i \(-0.530354\pi\)
−0.0952154 + 0.995457i \(0.530354\pi\)
\(810\) 0 0
\(811\) 27.1246 0.952474 0.476237 0.879317i \(-0.342000\pi\)
0.476237 + 0.879317i \(0.342000\pi\)
\(812\) 0 0
\(813\) 3.77709 0.132468
\(814\) 0 0
\(815\) −14.8328 −0.519571
\(816\) 0 0
\(817\) 11.0557 0.386791
\(818\) 0 0
\(819\) 1.81966 0.0635841
\(820\) 0 0
\(821\) 4.00000 0.139601 0.0698005 0.997561i \(-0.477764\pi\)
0.0698005 + 0.997561i \(0.477764\pi\)
\(822\) 0 0
\(823\) 30.8328 1.07476 0.537382 0.843339i \(-0.319413\pi\)
0.537382 + 0.843339i \(0.319413\pi\)
\(824\) 0 0
\(825\) 17.1672 0.597685
\(826\) 0 0
\(827\) 35.4164 1.23155 0.615775 0.787922i \(-0.288843\pi\)
0.615775 + 0.787922i \(0.288843\pi\)
\(828\) 0 0
\(829\) −27.1246 −0.942077 −0.471038 0.882113i \(-0.656121\pi\)
−0.471038 + 0.882113i \(0.656121\pi\)
\(830\) 0 0
\(831\) −24.7214 −0.857574
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 6.11146 0.211496
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −26.6656 −0.919505
\(842\) 0 0
\(843\) 7.41641 0.255435
\(844\) 0 0
\(845\) 14.1803 0.487819
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) 30.4721 1.04580
\(850\) 0 0
\(851\) 41.8885 1.43592
\(852\) 0 0
\(853\) 33.2361 1.13798 0.568991 0.822344i \(-0.307334\pi\)
0.568991 + 0.822344i \(0.307334\pi\)
\(854\) 0 0
\(855\) 2.24922 0.0769218
\(856\) 0 0
\(857\) −56.8328 −1.94137 −0.970686 0.240351i \(-0.922737\pi\)
−0.970686 + 0.240351i \(0.922737\pi\)
\(858\) 0 0
\(859\) −30.1803 −1.02974 −0.514870 0.857268i \(-0.672160\pi\)
−0.514870 + 0.857268i \(0.672160\pi\)
\(860\) 0 0
\(861\) −2.47214 −0.0842502
\(862\) 0 0
\(863\) 33.8885 1.15358 0.576790 0.816893i \(-0.304305\pi\)
0.576790 + 0.816893i \(0.304305\pi\)
\(864\) 0 0
\(865\) 4.58359 0.155847
\(866\) 0 0
\(867\) 16.0689 0.545728
\(868\) 0 0
\(869\) −19.7771 −0.670892
\(870\) 0 0
\(871\) 1.88854 0.0639909
\(872\) 0 0
\(873\) −16.1115 −0.545290
\(874\) 0 0
\(875\) −10.4721 −0.354023
\(876\) 0 0
\(877\) −0.360680 −0.0121793 −0.00608965 0.999981i \(-0.501938\pi\)
−0.00608965 + 0.999981i \(0.501938\pi\)
\(878\) 0 0
\(879\) −34.2492 −1.15520
\(880\) 0 0
\(881\) −20.8328 −0.701875 −0.350938 0.936399i \(-0.614137\pi\)
−0.350938 + 0.936399i \(0.614137\pi\)
\(882\) 0 0
\(883\) 38.4721 1.29469 0.647345 0.762197i \(-0.275879\pi\)
0.647345 + 0.762197i \(0.275879\pi\)
\(884\) 0 0
\(885\) 14.1115 0.474351
\(886\) 0 0
\(887\) −14.8328 −0.498037 −0.249019 0.968499i \(-0.580108\pi\)
−0.249019 + 0.968499i \(0.580108\pi\)
\(888\) 0 0
\(889\) 19.4164 0.651205
\(890\) 0 0
\(891\) −9.66563 −0.323811
\(892\) 0 0
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 9.88854 0.330169
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −17.8885 −0.595954
\(902\) 0 0
\(903\) 11.0557 0.367912
\(904\) 0 0
\(905\) −31.1935 −1.03691
\(906\) 0 0
\(907\) 35.4164 1.17598 0.587991 0.808867i \(-0.299919\pi\)
0.587991 + 0.808867i \(0.299919\pi\)
\(908\) 0 0
\(909\) 20.8754 0.692393
\(910\) 0 0
\(911\) −54.4721 −1.80474 −0.902371 0.430960i \(-0.858175\pi\)
−0.902371 + 0.430960i \(0.858175\pi\)
\(912\) 0 0
\(913\) 36.9443 1.22268
\(914\) 0 0
\(915\) −1.88854 −0.0624333
\(916\) 0 0
\(917\) −14.1803 −0.468276
\(918\) 0 0
\(919\) 46.8328 1.54487 0.772436 0.635093i \(-0.219038\pi\)
0.772436 + 0.635093i \(0.219038\pi\)
\(920\) 0 0
\(921\) 26.6950 0.879632
\(922\) 0 0
\(923\) 6.11146 0.201161
\(924\) 0 0
\(925\) 22.4721 0.738879
\(926\) 0 0
\(927\) −11.7771 −0.386810
\(928\) 0 0
\(929\) −27.8885 −0.914993 −0.457497 0.889211i \(-0.651254\pi\)
−0.457497 + 0.889211i \(0.651254\pi\)
\(930\) 0 0
\(931\) 1.23607 0.0405105
\(932\) 0 0
\(933\) 9.88854 0.323736
\(934\) 0 0
\(935\) 9.88854 0.323390
\(936\) 0 0
\(937\) −26.9443 −0.880231 −0.440115 0.897941i \(-0.645063\pi\)
−0.440115 + 0.897941i \(0.645063\pi\)
\(938\) 0 0
\(939\) −9.75078 −0.318205
\(940\) 0 0
\(941\) 0.652476 0.0212701 0.0106351 0.999943i \(-0.496615\pi\)
0.0106351 + 0.999943i \(0.496615\pi\)
\(942\) 0 0
\(943\) 12.9443 0.421523
\(944\) 0 0
\(945\) 6.83282 0.222272
\(946\) 0 0
\(947\) 39.0557 1.26914 0.634570 0.772865i \(-0.281177\pi\)
0.634570 + 0.772865i \(0.281177\pi\)
\(948\) 0 0
\(949\) 18.4721 0.599631
\(950\) 0 0
\(951\) 27.0557 0.877342
\(952\) 0 0
\(953\) 31.8885 1.03297 0.516486 0.856296i \(-0.327240\pi\)
0.516486 + 0.856296i \(0.327240\pi\)
\(954\) 0 0
\(955\) 22.1115 0.715510
\(956\) 0 0
\(957\) 7.55418 0.244192
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −14.0263 −0.451992
\(964\) 0 0
\(965\) −15.4164 −0.496272
\(966\) 0 0
\(967\) −12.5836 −0.404661 −0.202331 0.979317i \(-0.564852\pi\)
−0.202331 + 0.979317i \(0.564852\pi\)
\(968\) 0 0
\(969\) 3.05573 0.0981641
\(970\) 0 0
\(971\) 15.3475 0.492525 0.246263 0.969203i \(-0.420797\pi\)
0.246263 + 0.969203i \(0.420797\pi\)
\(972\) 0 0
\(973\) 17.2361 0.552563
\(974\) 0 0
\(975\) 5.30495 0.169894
\(976\) 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
\(981\) 2.24922 0.0718122
\(982\) 0 0
\(983\) −27.7771 −0.885952 −0.442976 0.896534i \(-0.646077\pi\)
−0.442976 + 0.896534i \(0.646077\pi\)
\(984\) 0 0
\(985\) −4.94427 −0.157538
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) −57.8885 −1.84075
\(990\) 0 0
\(991\) 30.8328 0.979437 0.489718 0.871881i \(-0.337100\pi\)
0.489718 + 0.871881i \(0.337100\pi\)
\(992\) 0 0
\(993\) 8.72136 0.276764
\(994\) 0 0
\(995\) −29.6656 −0.940464
\(996\) 0 0
\(997\) 30.7639 0.974304 0.487152 0.873317i \(-0.338036\pi\)
0.487152 + 0.873317i \(0.338036\pi\)
\(998\) 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.t.1.1 2
4.3 odd 2 1792.2.a.l.1.2 2
8.3 odd 2 1792.2.a.r.1.1 2
8.5 even 2 1792.2.a.j.1.2 2
16.3 odd 4 896.2.b.e.449.3 yes 4
16.5 even 4 896.2.b.g.449.3 yes 4
16.11 odd 4 896.2.b.e.449.2 4
16.13 even 4 896.2.b.g.449.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.e.449.2 4 16.11 odd 4
896.2.b.e.449.3 yes 4 16.3 odd 4
896.2.b.g.449.2 yes 4 16.13 even 4
896.2.b.g.449.3 yes 4 16.5 even 4
1792.2.a.j.1.2 2 8.5 even 2
1792.2.a.l.1.2 2 4.3 odd 2
1792.2.a.r.1.1 2 8.3 odd 2
1792.2.a.t.1.1 2 1.1 even 1 trivial