Properties

Label 1856.2.a.y.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.12489 q^{3} -1.48486 q^{5} +1.51514 q^{9} +O(q^{10})\) \(q-2.12489 q^{3} -1.48486 q^{5} +1.51514 q^{9} +3.15516 q^{11} -6.76491 q^{13} +3.15516 q^{15} +2.00000 q^{17} -1.03028 q^{19} -4.24977 q^{23} -2.79518 q^{25} +3.15516 q^{27} -1.00000 q^{29} +1.87511 q^{31} -6.70436 q^{33} +0.969724 q^{37} +14.3747 q^{39} -7.52982 q^{41} +1.09461 q^{43} -2.24977 q^{45} +9.34438 q^{47} -7.00000 q^{49} -4.24977 q^{51} -5.73463 q^{53} -4.68498 q^{55} +2.18922 q^{57} +8.24977 q^{59} +10.4995 q^{61} +10.0450 q^{65} -4.49954 q^{67} +9.03028 q^{69} -10.1892 q^{71} +11.5298 q^{73} +5.93945 q^{75} +13.0946 q^{79} -11.2498 q^{81} +16.2498 q^{83} -2.96972 q^{85} +2.12489 q^{87} +13.5904 q^{89} -3.98440 q^{93} +1.52982 q^{95} -4.96972 q^{97} +4.78051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43} + 10 q^{45} + 2 q^{47} - 21 q^{49} + 4 q^{51} + 26 q^{55} - 12 q^{57} + 8 q^{59} - 2 q^{61} - 2 q^{65} + 20 q^{67} + 28 q^{69} - 12 q^{71} + 2 q^{73} + 16 q^{75} + 30 q^{79} - 17 q^{81} + 32 q^{83} - 8 q^{85} - 2 q^{87} + 10 q^{89} + 22 q^{93} - 28 q^{95} - 14 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.12489 −1.22680 −0.613402 0.789771i \(-0.710199\pi\)
−0.613402 + 0.789771i \(0.710199\pi\)
\(4\) 0 0
\(5\) −1.48486 −0.664050 −0.332025 0.943271i \(-0.607732\pi\)
−0.332025 + 0.943271i \(0.607732\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.51514 0.505046
\(10\) 0 0
\(11\) 3.15516 0.951317 0.475658 0.879630i \(-0.342210\pi\)
0.475658 + 0.879630i \(0.342210\pi\)
\(12\) 0 0
\(13\) −6.76491 −1.87625 −0.938124 0.346299i \(-0.887438\pi\)
−0.938124 + 0.346299i \(0.887438\pi\)
\(14\) 0 0
\(15\) 3.15516 0.814659
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.03028 −0.236362 −0.118181 0.992992i \(-0.537706\pi\)
−0.118181 + 0.992992i \(0.537706\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.24977 −0.886138 −0.443069 0.896487i \(-0.646110\pi\)
−0.443069 + 0.896487i \(0.646110\pi\)
\(24\) 0 0
\(25\) −2.79518 −0.559037
\(26\) 0 0
\(27\) 3.15516 0.607211
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.87511 0.336781 0.168390 0.985720i \(-0.446143\pi\)
0.168390 + 0.985720i \(0.446143\pi\)
\(32\) 0 0
\(33\) −6.70436 −1.16708
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.969724 0.159422 0.0797108 0.996818i \(-0.474600\pi\)
0.0797108 + 0.996818i \(0.474600\pi\)
\(38\) 0 0
\(39\) 14.3747 2.30179
\(40\) 0 0
\(41\) −7.52982 −1.17596 −0.587980 0.808875i \(-0.700077\pi\)
−0.587980 + 0.808875i \(0.700077\pi\)
\(42\) 0 0
\(43\) 1.09461 0.166926 0.0834632 0.996511i \(-0.473402\pi\)
0.0834632 + 0.996511i \(0.473402\pi\)
\(44\) 0 0
\(45\) −2.24977 −0.335376
\(46\) 0 0
\(47\) 9.34438 1.36302 0.681509 0.731810i \(-0.261324\pi\)
0.681509 + 0.731810i \(0.261324\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −4.24977 −0.595087
\(52\) 0 0
\(53\) −5.73463 −0.787712 −0.393856 0.919172i \(-0.628859\pi\)
−0.393856 + 0.919172i \(0.628859\pi\)
\(54\) 0 0
\(55\) −4.68498 −0.631722
\(56\) 0 0
\(57\) 2.18922 0.289969
\(58\) 0 0
\(59\) 8.24977 1.07403 0.537014 0.843573i \(-0.319552\pi\)
0.537014 + 0.843573i \(0.319552\pi\)
\(60\) 0 0
\(61\) 10.4995 1.34433 0.672164 0.740402i \(-0.265365\pi\)
0.672164 + 0.740402i \(0.265365\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.0450 1.24592
\(66\) 0 0
\(67\) −4.49954 −0.549707 −0.274853 0.961486i \(-0.588629\pi\)
−0.274853 + 0.961486i \(0.588629\pi\)
\(68\) 0 0
\(69\) 9.03028 1.08712
\(70\) 0 0
\(71\) −10.1892 −1.20924 −0.604619 0.796515i \(-0.706675\pi\)
−0.604619 + 0.796515i \(0.706675\pi\)
\(72\) 0 0
\(73\) 11.5298 1.34946 0.674732 0.738063i \(-0.264259\pi\)
0.674732 + 0.738063i \(0.264259\pi\)
\(74\) 0 0
\(75\) 5.93945 0.685828
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0946 1.47326 0.736629 0.676297i \(-0.236416\pi\)
0.736629 + 0.676297i \(0.236416\pi\)
\(80\) 0 0
\(81\) −11.2498 −1.24997
\(82\) 0 0
\(83\) 16.2498 1.78364 0.891822 0.452386i \(-0.149427\pi\)
0.891822 + 0.452386i \(0.149427\pi\)
\(84\) 0 0
\(85\) −2.96972 −0.322112
\(86\) 0 0
\(87\) 2.12489 0.227812
\(88\) 0 0
\(89\) 13.5904 1.44058 0.720288 0.693675i \(-0.244010\pi\)
0.720288 + 0.693675i \(0.244010\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.98440 −0.413163
\(94\) 0 0
\(95\) 1.52982 0.156956
\(96\) 0 0
\(97\) −4.96972 −0.504599 −0.252300 0.967649i \(-0.581187\pi\)
−0.252300 + 0.967649i \(0.581187\pi\)
\(98\) 0 0
\(99\) 4.78051 0.480459
\(100\) 0 0
\(101\) 9.46927 0.942227 0.471114 0.882073i \(-0.343852\pi\)
0.471114 + 0.882073i \(0.343852\pi\)
\(102\) 0 0
\(103\) −4.24977 −0.418742 −0.209371 0.977836i \(-0.567142\pi\)
−0.209371 + 0.977836i \(0.567142\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.56009 −0.634188 −0.317094 0.948394i \(-0.602707\pi\)
−0.317094 + 0.948394i \(0.602707\pi\)
\(108\) 0 0
\(109\) 12.9541 1.24078 0.620390 0.784293i \(-0.286974\pi\)
0.620390 + 0.784293i \(0.286974\pi\)
\(110\) 0 0
\(111\) −2.06055 −0.195579
\(112\) 0 0
\(113\) −0.0605522 −0.00569627 −0.00284814 0.999996i \(-0.500907\pi\)
−0.00284814 + 0.999996i \(0.500907\pi\)
\(114\) 0 0
\(115\) 6.31032 0.588441
\(116\) 0 0
\(117\) −10.2498 −0.947592
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.04496 −0.0949960
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) 11.5748 1.03528
\(126\) 0 0
\(127\) 13.5298 1.20058 0.600289 0.799783i \(-0.295052\pi\)
0.600289 + 0.799783i \(0.295052\pi\)
\(128\) 0 0
\(129\) −2.32592 −0.204786
\(130\) 0 0
\(131\) 9.03028 0.788979 0.394489 0.918900i \(-0.370921\pi\)
0.394489 + 0.918900i \(0.370921\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.68498 −0.403219
\(136\) 0 0
\(137\) 0.969724 0.0828491 0.0414246 0.999142i \(-0.486810\pi\)
0.0414246 + 0.999142i \(0.486810\pi\)
\(138\) 0 0
\(139\) 16.7493 1.42066 0.710329 0.703870i \(-0.248546\pi\)
0.710329 + 0.703870i \(0.248546\pi\)
\(140\) 0 0
\(141\) −19.8557 −1.67215
\(142\) 0 0
\(143\) −21.3444 −1.78491
\(144\) 0 0
\(145\) 1.48486 0.123311
\(146\) 0 0
\(147\) 14.8742 1.22680
\(148\) 0 0
\(149\) 20.2947 1.66261 0.831304 0.555817i \(-0.187595\pi\)
0.831304 + 0.555817i \(0.187595\pi\)
\(150\) 0 0
\(151\) 18.5601 1.51040 0.755200 0.655495i \(-0.227540\pi\)
0.755200 + 0.655495i \(0.227540\pi\)
\(152\) 0 0
\(153\) 3.03028 0.244983
\(154\) 0 0
\(155\) −2.78429 −0.223639
\(156\) 0 0
\(157\) −7.03028 −0.561077 −0.280539 0.959843i \(-0.590513\pi\)
−0.280539 + 0.959843i \(0.590513\pi\)
\(158\) 0 0
\(159\) 12.1854 0.966368
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −24.9348 −1.95304 −0.976520 0.215426i \(-0.930886\pi\)
−0.976520 + 0.215426i \(0.930886\pi\)
\(164\) 0 0
\(165\) 9.95504 0.774999
\(166\) 0 0
\(167\) −10.1892 −0.788465 −0.394233 0.919011i \(-0.628990\pi\)
−0.394233 + 0.919011i \(0.628990\pi\)
\(168\) 0 0
\(169\) 32.7640 2.52031
\(170\) 0 0
\(171\) −1.56101 −0.119373
\(172\) 0 0
\(173\) −0.0605522 −0.00460370 −0.00230185 0.999997i \(-0.500733\pi\)
−0.00230185 + 0.999997i \(0.500733\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −17.5298 −1.31762
\(178\) 0 0
\(179\) −14.1892 −1.06055 −0.530276 0.847825i \(-0.677912\pi\)
−0.530276 + 0.847825i \(0.677912\pi\)
\(180\) 0 0
\(181\) −16.9541 −1.26019 −0.630095 0.776518i \(-0.716984\pi\)
−0.630095 + 0.776518i \(0.716984\pi\)
\(182\) 0 0
\(183\) −22.3103 −1.64923
\(184\) 0 0
\(185\) −1.43991 −0.105864
\(186\) 0 0
\(187\) 6.31032 0.461457
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.59037 −0.549220 −0.274610 0.961556i \(-0.588549\pi\)
−0.274610 + 0.961556i \(0.588549\pi\)
\(192\) 0 0
\(193\) −13.4693 −0.969539 −0.484769 0.874642i \(-0.661097\pi\)
−0.484769 + 0.874642i \(0.661097\pi\)
\(194\) 0 0
\(195\) −21.3444 −1.52850
\(196\) 0 0
\(197\) 7.93945 0.565662 0.282831 0.959170i \(-0.408726\pi\)
0.282831 + 0.959170i \(0.408726\pi\)
\(198\) 0 0
\(199\) 26.5601 1.88280 0.941398 0.337299i \(-0.109513\pi\)
0.941398 + 0.337299i \(0.109513\pi\)
\(200\) 0 0
\(201\) 9.56101 0.674382
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 11.1807 0.780897
\(206\) 0 0
\(207\) −6.43899 −0.447541
\(208\) 0 0
\(209\) −3.25069 −0.224855
\(210\) 0 0
\(211\) −0.435208 −0.0299610 −0.0149805 0.999888i \(-0.504769\pi\)
−0.0149805 + 0.999888i \(0.504769\pi\)
\(212\) 0 0
\(213\) 21.6509 1.48350
\(214\) 0 0
\(215\) −1.62534 −0.110848
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −24.4995 −1.65553
\(220\) 0 0
\(221\) −13.5298 −0.910114
\(222\) 0 0
\(223\) 8.12867 0.544336 0.272168 0.962250i \(-0.412259\pi\)
0.272168 + 0.962250i \(0.412259\pi\)
\(224\) 0 0
\(225\) −4.23509 −0.282339
\(226\) 0 0
\(227\) −29.3700 −1.94935 −0.974676 0.223620i \(-0.928212\pi\)
−0.974676 + 0.223620i \(0.928212\pi\)
\(228\) 0 0
\(229\) 9.46927 0.625747 0.312873 0.949795i \(-0.398708\pi\)
0.312873 + 0.949795i \(0.398708\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.20482 0.537515 0.268758 0.963208i \(-0.413387\pi\)
0.268758 + 0.963208i \(0.413387\pi\)
\(234\) 0 0
\(235\) −13.8751 −0.905113
\(236\) 0 0
\(237\) −27.8245 −1.80740
\(238\) 0 0
\(239\) −3.75023 −0.242582 −0.121291 0.992617i \(-0.538703\pi\)
−0.121291 + 0.992617i \(0.538703\pi\)
\(240\) 0 0
\(241\) 28.2947 1.82262 0.911312 0.411717i \(-0.135071\pi\)
0.911312 + 0.411717i \(0.135071\pi\)
\(242\) 0 0
\(243\) 14.4390 0.926262
\(244\) 0 0
\(245\) 10.3940 0.664050
\(246\) 0 0
\(247\) 6.96972 0.443473
\(248\) 0 0
\(249\) −34.5289 −2.18818
\(250\) 0 0
\(251\) −12.1854 −0.769138 −0.384569 0.923096i \(-0.625650\pi\)
−0.384569 + 0.923096i \(0.625650\pi\)
\(252\) 0 0
\(253\) −13.4087 −0.842999
\(254\) 0 0
\(255\) 6.31032 0.395168
\(256\) 0 0
\(257\) 4.98532 0.310976 0.155488 0.987838i \(-0.450305\pi\)
0.155488 + 0.987838i \(0.450305\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.51514 −0.0937847
\(262\) 0 0
\(263\) 2.37466 0.146428 0.0732138 0.997316i \(-0.476674\pi\)
0.0732138 + 0.997316i \(0.476674\pi\)
\(264\) 0 0
\(265\) 8.51514 0.523081
\(266\) 0 0
\(267\) −28.8780 −1.76730
\(268\) 0 0
\(269\) −28.9385 −1.76441 −0.882207 0.470862i \(-0.843943\pi\)
−0.882207 + 0.470862i \(0.843943\pi\)
\(270\) 0 0
\(271\) 16.1854 0.983195 0.491598 0.870822i \(-0.336413\pi\)
0.491598 + 0.870822i \(0.336413\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.81926 −0.531821
\(276\) 0 0
\(277\) −25.0596 −1.50569 −0.752844 0.658199i \(-0.771318\pi\)
−0.752844 + 0.658199i \(0.771318\pi\)
\(278\) 0 0
\(279\) 2.84106 0.170090
\(280\) 0 0
\(281\) −13.2351 −0.789539 −0.394770 0.918780i \(-0.629176\pi\)
−0.394770 + 0.918780i \(0.629176\pi\)
\(282\) 0 0
\(283\) 18.3103 1.08844 0.544218 0.838944i \(-0.316826\pi\)
0.544218 + 0.838944i \(0.316826\pi\)
\(284\) 0 0
\(285\) −3.25069 −0.192554
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 10.5601 0.619044
\(292\) 0 0
\(293\) 7.93945 0.463827 0.231914 0.972736i \(-0.425501\pi\)
0.231914 + 0.972736i \(0.425501\pi\)
\(294\) 0 0
\(295\) −12.2498 −0.713209
\(296\) 0 0
\(297\) 9.95504 0.577650
\(298\) 0 0
\(299\) 28.7493 1.66262
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −20.1211 −1.15593
\(304\) 0 0
\(305\) −15.5904 −0.892702
\(306\) 0 0
\(307\) 28.1542 1.60685 0.803424 0.595408i \(-0.203009\pi\)
0.803424 + 0.595408i \(0.203009\pi\)
\(308\) 0 0
\(309\) 9.03028 0.513714
\(310\) 0 0
\(311\) 4.53073 0.256914 0.128457 0.991715i \(-0.458998\pi\)
0.128457 + 0.991715i \(0.458998\pi\)
\(312\) 0 0
\(313\) 5.48486 0.310023 0.155011 0.987913i \(-0.450459\pi\)
0.155011 + 0.987913i \(0.450459\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.4390 −1.14797 −0.573984 0.818867i \(-0.694603\pi\)
−0.573984 + 0.818867i \(0.694603\pi\)
\(318\) 0 0
\(319\) −3.15516 −0.176655
\(320\) 0 0
\(321\) 13.9394 0.778024
\(322\) 0 0
\(323\) −2.06055 −0.114652
\(324\) 0 0
\(325\) 18.9092 1.04889
\(326\) 0 0
\(327\) −27.5260 −1.52219
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.5639 1.35015 0.675076 0.737748i \(-0.264111\pi\)
0.675076 + 0.737748i \(0.264111\pi\)
\(332\) 0 0
\(333\) 1.46927 0.0805153
\(334\) 0 0
\(335\) 6.68120 0.365033
\(336\) 0 0
\(337\) 26.4995 1.44352 0.721761 0.692142i \(-0.243333\pi\)
0.721761 + 0.692142i \(0.243333\pi\)
\(338\) 0 0
\(339\) 0.128666 0.00698820
\(340\) 0 0
\(341\) 5.91629 0.320385
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −13.4087 −0.721901
\(346\) 0 0
\(347\) −20.8704 −1.12038 −0.560191 0.828363i \(-0.689272\pi\)
−0.560191 + 0.828363i \(0.689272\pi\)
\(348\) 0 0
\(349\) 3.26445 0.174742 0.0873710 0.996176i \(-0.472153\pi\)
0.0873710 + 0.996176i \(0.472153\pi\)
\(350\) 0 0
\(351\) −21.3444 −1.13928
\(352\) 0 0
\(353\) −30.9991 −1.64991 −0.824957 0.565195i \(-0.808801\pi\)
−0.824957 + 0.565195i \(0.808801\pi\)
\(354\) 0 0
\(355\) 15.1296 0.802995
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.03406 0.160131 0.0800657 0.996790i \(-0.474487\pi\)
0.0800657 + 0.996790i \(0.474487\pi\)
\(360\) 0 0
\(361\) −17.9385 −0.944133
\(362\) 0 0
\(363\) 2.22041 0.116541
\(364\) 0 0
\(365\) −17.1202 −0.896112
\(366\) 0 0
\(367\) 16.9092 0.882652 0.441326 0.897347i \(-0.354508\pi\)
0.441326 + 0.897347i \(0.354508\pi\)
\(368\) 0 0
\(369\) −11.4087 −0.593914
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12.5757 −0.651145 −0.325572 0.945517i \(-0.605557\pi\)
−0.325572 + 0.945517i \(0.605557\pi\)
\(374\) 0 0
\(375\) −24.5951 −1.27008
\(376\) 0 0
\(377\) 6.76491 0.348411
\(378\) 0 0
\(379\) −13.4087 −0.688759 −0.344380 0.938830i \(-0.611911\pi\)
−0.344380 + 0.938830i \(0.611911\pi\)
\(380\) 0 0
\(381\) −28.7493 −1.47287
\(382\) 0 0
\(383\) −18.0606 −0.922851 −0.461426 0.887179i \(-0.652662\pi\)
−0.461426 + 0.887179i \(0.652662\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.65848 0.0843055
\(388\) 0 0
\(389\) −0.0293595 −0.00148858 −0.000744292 1.00000i \(-0.500237\pi\)
−0.000744292 1.00000i \(0.500237\pi\)
\(390\) 0 0
\(391\) −8.49954 −0.429840
\(392\) 0 0
\(393\) −19.1883 −0.967922
\(394\) 0 0
\(395\) −19.4437 −0.978318
\(396\) 0 0
\(397\) 26.1055 1.31020 0.655099 0.755543i \(-0.272627\pi\)
0.655099 + 0.755543i \(0.272627\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 19.2645 0.962021 0.481010 0.876715i \(-0.340270\pi\)
0.481010 + 0.876715i \(0.340270\pi\)
\(402\) 0 0
\(403\) −12.6850 −0.631884
\(404\) 0 0
\(405\) 16.7044 0.830046
\(406\) 0 0
\(407\) 3.05964 0.151661
\(408\) 0 0
\(409\) 19.4986 0.964145 0.482072 0.876131i \(-0.339884\pi\)
0.482072 + 0.876131i \(0.339884\pi\)
\(410\) 0 0
\(411\) −2.06055 −0.101640
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.1287 −1.18443
\(416\) 0 0
\(417\) −35.5904 −1.74287
\(418\) 0 0
\(419\) −33.2489 −1.62431 −0.812156 0.583440i \(-0.801706\pi\)
−0.812156 + 0.583440i \(0.801706\pi\)
\(420\) 0 0
\(421\) −3.93945 −0.191997 −0.0959985 0.995381i \(-0.530604\pi\)
−0.0959985 + 0.995381i \(0.530604\pi\)
\(422\) 0 0
\(423\) 14.1580 0.688387
\(424\) 0 0
\(425\) −5.59037 −0.271173
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 45.3544 2.18973
\(430\) 0 0
\(431\) −17.3700 −0.836681 −0.418341 0.908290i \(-0.637388\pi\)
−0.418341 + 0.908290i \(0.637388\pi\)
\(432\) 0 0
\(433\) −15.5298 −0.746315 −0.373158 0.927768i \(-0.621725\pi\)
−0.373158 + 0.927768i \(0.621725\pi\)
\(434\) 0 0
\(435\) −3.15516 −0.151278
\(436\) 0 0
\(437\) 4.37844 0.209449
\(438\) 0 0
\(439\) −8.87042 −0.423362 −0.211681 0.977339i \(-0.567894\pi\)
−0.211681 + 0.977339i \(0.567894\pi\)
\(440\) 0 0
\(441\) −10.6060 −0.505046
\(442\) 0 0
\(443\) −3.09083 −0.146850 −0.0734248 0.997301i \(-0.523393\pi\)
−0.0734248 + 0.997301i \(0.523393\pi\)
\(444\) 0 0
\(445\) −20.1798 −0.956615
\(446\) 0 0
\(447\) −43.1240 −2.03969
\(448\) 0 0
\(449\) 16.9385 0.799379 0.399689 0.916651i \(-0.369118\pi\)
0.399689 + 0.916651i \(0.369118\pi\)
\(450\) 0 0
\(451\) −23.7578 −1.11871
\(452\) 0 0
\(453\) −39.4381 −1.85296
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.9991 0.514515 0.257258 0.966343i \(-0.417181\pi\)
0.257258 + 0.966343i \(0.417181\pi\)
\(458\) 0 0
\(459\) 6.31032 0.294541
\(460\) 0 0
\(461\) 23.1202 1.07681 0.538407 0.842685i \(-0.319026\pi\)
0.538407 + 0.842685i \(0.319026\pi\)
\(462\) 0 0
\(463\) 0.870417 0.0404517 0.0202259 0.999795i \(-0.493561\pi\)
0.0202259 + 0.999795i \(0.493561\pi\)
\(464\) 0 0
\(465\) 5.91629 0.274361
\(466\) 0 0
\(467\) 14.2148 0.657782 0.328891 0.944368i \(-0.393325\pi\)
0.328891 + 0.944368i \(0.393325\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.9385 0.688331
\(472\) 0 0
\(473\) 3.45367 0.158800
\(474\) 0 0
\(475\) 2.87981 0.132135
\(476\) 0 0
\(477\) −8.68876 −0.397831
\(478\) 0 0
\(479\) 3.56479 0.162879 0.0814397 0.996678i \(-0.474048\pi\)
0.0814397 + 0.996678i \(0.474048\pi\)
\(480\) 0 0
\(481\) −6.56009 −0.299115
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.37935 0.335079
\(486\) 0 0
\(487\) 8.49954 0.385151 0.192575 0.981282i \(-0.438316\pi\)
0.192575 + 0.981282i \(0.438316\pi\)
\(488\) 0 0
\(489\) 52.9835 2.39600
\(490\) 0 0
\(491\) −0.595068 −0.0268550 −0.0134275 0.999910i \(-0.504274\pi\)
−0.0134275 + 0.999910i \(0.504274\pi\)
\(492\) 0 0
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −7.09839 −0.319049
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.24885 0.414036 0.207018 0.978337i \(-0.433624\pi\)
0.207018 + 0.978337i \(0.433624\pi\)
\(500\) 0 0
\(501\) 21.6509 0.967292
\(502\) 0 0
\(503\) −0.314104 −0.0140052 −0.00700260 0.999975i \(-0.502229\pi\)
−0.00700260 + 0.999975i \(0.502229\pi\)
\(504\) 0 0
\(505\) −14.0606 −0.625686
\(506\) 0 0
\(507\) −69.6197 −3.09192
\(508\) 0 0
\(509\) −1.11399 −0.0493766 −0.0246883 0.999695i \(-0.507859\pi\)
−0.0246883 + 0.999695i \(0.507859\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.25069 −0.143521
\(514\) 0 0
\(515\) 6.31032 0.278066
\(516\) 0 0
\(517\) 29.4830 1.29666
\(518\) 0 0
\(519\) 0.128666 0.00564783
\(520\) 0 0
\(521\) 11.1358 0.487868 0.243934 0.969792i \(-0.421562\pi\)
0.243934 + 0.969792i \(0.421562\pi\)
\(522\) 0 0
\(523\) −4.49954 −0.196751 −0.0983756 0.995149i \(-0.531365\pi\)
−0.0983756 + 0.995149i \(0.531365\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.75023 0.163363
\(528\) 0 0
\(529\) −4.93945 −0.214759
\(530\) 0 0
\(531\) 12.4995 0.542434
\(532\) 0 0
\(533\) 50.9385 2.20639
\(534\) 0 0
\(535\) 9.74083 0.421133
\(536\) 0 0
\(537\) 30.1505 1.30109
\(538\) 0 0
\(539\) −22.0861 −0.951317
\(540\) 0 0
\(541\) −19.6509 −0.844859 −0.422430 0.906396i \(-0.638823\pi\)
−0.422430 + 0.906396i \(0.638823\pi\)
\(542\) 0 0
\(543\) 36.0256 1.54601
\(544\) 0 0
\(545\) −19.2351 −0.823941
\(546\) 0 0
\(547\) −1.93945 −0.0829248 −0.0414624 0.999140i \(-0.513202\pi\)
−0.0414624 + 0.999140i \(0.513202\pi\)
\(548\) 0 0
\(549\) 15.9083 0.678948
\(550\) 0 0
\(551\) 1.03028 0.0438912
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3.05964 0.129874
\(556\) 0 0
\(557\) 38.0587 1.61260 0.806300 0.591507i \(-0.201467\pi\)
0.806300 + 0.591507i \(0.201467\pi\)
\(558\) 0 0
\(559\) −7.40493 −0.313195
\(560\) 0 0
\(561\) −13.4087 −0.566116
\(562\) 0 0
\(563\) 45.1845 1.90430 0.952150 0.305630i \(-0.0988672\pi\)
0.952150 + 0.305630i \(0.0988672\pi\)
\(564\) 0 0
\(565\) 0.0899116 0.00378261
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.4986 −1.32049 −0.660246 0.751050i \(-0.729548\pi\)
−0.660246 + 0.751050i \(0.729548\pi\)
\(570\) 0 0
\(571\) 16.6206 0.695552 0.347776 0.937578i \(-0.386937\pi\)
0.347776 + 0.937578i \(0.386937\pi\)
\(572\) 0 0
\(573\) 16.1287 0.673785
\(574\) 0 0
\(575\) 11.8789 0.495384
\(576\) 0 0
\(577\) −3.43991 −0.143205 −0.0716026 0.997433i \(-0.522811\pi\)
−0.0716026 + 0.997433i \(0.522811\pi\)
\(578\) 0 0
\(579\) 28.6206 1.18943
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.0937 −0.749364
\(584\) 0 0
\(585\) 15.2195 0.629249
\(586\) 0 0
\(587\) 6.93097 0.286072 0.143036 0.989718i \(-0.454314\pi\)
0.143036 + 0.989718i \(0.454314\pi\)
\(588\) 0 0
\(589\) −1.93189 −0.0796020
\(590\) 0 0
\(591\) −16.8704 −0.693956
\(592\) 0 0
\(593\) −31.6041 −1.29783 −0.648913 0.760863i \(-0.724776\pi\)
−0.648913 + 0.760863i \(0.724776\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −56.4372 −2.30982
\(598\) 0 0
\(599\) 31.6547 1.29338 0.646688 0.762755i \(-0.276154\pi\)
0.646688 + 0.762755i \(0.276154\pi\)
\(600\) 0 0
\(601\) 19.0303 0.776261 0.388131 0.921604i \(-0.373121\pi\)
0.388131 + 0.921604i \(0.373121\pi\)
\(602\) 0 0
\(603\) −6.81743 −0.277627
\(604\) 0 0
\(605\) 1.55162 0.0630821
\(606\) 0 0
\(607\) −24.3141 −0.986879 −0.493440 0.869780i \(-0.664261\pi\)
−0.493440 + 0.869780i \(0.664261\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −63.2139 −2.55736
\(612\) 0 0
\(613\) 8.07615 0.326193 0.163096 0.986610i \(-0.447852\pi\)
0.163096 + 0.986610i \(0.447852\pi\)
\(614\) 0 0
\(615\) −23.7578 −0.958007
\(616\) 0 0
\(617\) 16.9697 0.683175 0.341588 0.939850i \(-0.389035\pi\)
0.341588 + 0.939850i \(0.389035\pi\)
\(618\) 0 0
\(619\) −23.3737 −0.939470 −0.469735 0.882807i \(-0.655651\pi\)
−0.469735 + 0.882807i \(0.655651\pi\)
\(620\) 0 0
\(621\) −13.4087 −0.538073
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.21102 −0.128441
\(626\) 0 0
\(627\) 6.90734 0.275853
\(628\) 0 0
\(629\) 1.93945 0.0773308
\(630\) 0 0
\(631\) 3.75023 0.149294 0.0746471 0.997210i \(-0.476217\pi\)
0.0746471 + 0.997210i \(0.476217\pi\)
\(632\) 0 0
\(633\) 0.924768 0.0367562
\(634\) 0 0
\(635\) −20.0899 −0.797244
\(636\) 0 0
\(637\) 47.3544 1.87625
\(638\) 0 0
\(639\) −15.4381 −0.610721
\(640\) 0 0
\(641\) 29.5592 1.16752 0.583759 0.811927i \(-0.301581\pi\)
0.583759 + 0.811927i \(0.301581\pi\)
\(642\) 0 0
\(643\) 44.4995 1.75489 0.877445 0.479677i \(-0.159246\pi\)
0.877445 + 0.479677i \(0.159246\pi\)
\(644\) 0 0
\(645\) 3.45367 0.135988
\(646\) 0 0
\(647\) 6.80986 0.267723 0.133862 0.991000i \(-0.457262\pi\)
0.133862 + 0.991000i \(0.457262\pi\)
\(648\) 0 0
\(649\) 26.0294 1.02174
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.1514 −0.436387 −0.218194 0.975905i \(-0.570016\pi\)
−0.218194 + 0.975905i \(0.570016\pi\)
\(654\) 0 0
\(655\) −13.4087 −0.523922
\(656\) 0 0
\(657\) 17.4693 0.681541
\(658\) 0 0
\(659\) 35.2526 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(660\) 0 0
\(661\) −38.9991 −1.51689 −0.758444 0.651738i \(-0.774040\pi\)
−0.758444 + 0.651738i \(0.774040\pi\)
\(662\) 0 0
\(663\) 28.7493 1.11653
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.24977 0.164552
\(668\) 0 0
\(669\) −17.2725 −0.667793
\(670\) 0 0
\(671\) 33.1277 1.27888
\(672\) 0 0
\(673\) −10.1443 −0.391033 −0.195516 0.980700i \(-0.562638\pi\)
−0.195516 + 0.980700i \(0.562638\pi\)
\(674\) 0 0
\(675\) −8.81926 −0.339453
\(676\) 0 0
\(677\) −47.4986 −1.82552 −0.912760 0.408496i \(-0.866053\pi\)
−0.912760 + 0.408496i \(0.866053\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 62.4078 2.39147
\(682\) 0 0
\(683\) −1.43991 −0.0550965 −0.0275482 0.999620i \(-0.508770\pi\)
−0.0275482 + 0.999620i \(0.508770\pi\)
\(684\) 0 0
\(685\) −1.43991 −0.0550160
\(686\) 0 0
\(687\) −20.1211 −0.767668
\(688\) 0 0
\(689\) 38.7943 1.47794
\(690\) 0 0
\(691\) 26.9385 1.02479 0.512395 0.858750i \(-0.328758\pi\)
0.512395 + 0.858750i \(0.328758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.8704 −0.943389
\(696\) 0 0
\(697\) −15.0596 −0.570424
\(698\) 0 0
\(699\) −17.4343 −0.659425
\(700\) 0 0
\(701\) −17.3250 −0.654356 −0.327178 0.944963i \(-0.606098\pi\)
−0.327178 + 0.944963i \(0.606098\pi\)
\(702\) 0 0
\(703\) −0.999083 −0.0376811
\(704\) 0 0
\(705\) 29.4830 1.11040
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.0837106 −0.00314382 −0.00157191 0.999999i \(-0.500500\pi\)
−0.00157191 + 0.999999i \(0.500500\pi\)
\(710\) 0 0
\(711\) 19.8401 0.744063
\(712\) 0 0
\(713\) −7.96881 −0.298434
\(714\) 0 0
\(715\) 31.6935 1.18527
\(716\) 0 0
\(717\) 7.96881 0.297601
\(718\) 0 0
\(719\) −19.4305 −0.724636 −0.362318 0.932055i \(-0.618015\pi\)
−0.362318 + 0.932055i \(0.618015\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −60.1231 −2.23600
\(724\) 0 0
\(725\) 2.79518 0.103811
\(726\) 0 0
\(727\) 29.2876 1.08622 0.543109 0.839662i \(-0.317247\pi\)
0.543109 + 0.839662i \(0.317247\pi\)
\(728\) 0 0
\(729\) 3.06811 0.113634
\(730\) 0 0
\(731\) 2.18922 0.0809712
\(732\) 0 0
\(733\) −21.7115 −0.801932 −0.400966 0.916093i \(-0.631325\pi\)
−0.400966 + 0.916093i \(0.631325\pi\)
\(734\) 0 0
\(735\) −22.0861 −0.814659
\(736\) 0 0
\(737\) −14.1968 −0.522945
\(738\) 0 0
\(739\) 6.37466 0.234496 0.117248 0.993103i \(-0.462593\pi\)
0.117248 + 0.993103i \(0.462593\pi\)
\(740\) 0 0
\(741\) −14.8099 −0.544054
\(742\) 0 0
\(743\) 27.4693 1.00775 0.503875 0.863777i \(-0.331907\pi\)
0.503875 + 0.863777i \(0.331907\pi\)
\(744\) 0 0
\(745\) −30.1349 −1.10406
\(746\) 0 0
\(747\) 24.6206 0.900822
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 49.0890 1.79128 0.895641 0.444777i \(-0.146717\pi\)
0.895641 + 0.444777i \(0.146717\pi\)
\(752\) 0 0
\(753\) 25.8927 0.943581
\(754\) 0 0
\(755\) −27.5592 −1.00298
\(756\) 0 0
\(757\) 35.2413 1.28087 0.640433 0.768014i \(-0.278755\pi\)
0.640433 + 0.768014i \(0.278755\pi\)
\(758\) 0 0
\(759\) 28.4920 1.03419
\(760\) 0 0
\(761\) −25.0596 −0.908411 −0.454206 0.890897i \(-0.650077\pi\)
−0.454206 + 0.890897i \(0.650077\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.49954 −0.162681
\(766\) 0 0
\(767\) −55.8089 −2.01514
\(768\) 0 0
\(769\) −29.6803 −1.07030 −0.535149 0.844758i \(-0.679745\pi\)
−0.535149 + 0.844758i \(0.679745\pi\)
\(770\) 0 0
\(771\) −10.5932 −0.381506
\(772\) 0 0
\(773\) −23.5298 −0.846309 −0.423154 0.906058i \(-0.639077\pi\)
−0.423154 + 0.906058i \(0.639077\pi\)
\(774\) 0 0
\(775\) −5.24129 −0.188273
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.75779 0.277952
\(780\) 0 0
\(781\) −32.1486 −1.15037
\(782\) 0 0
\(783\) −3.15516 −0.112756
\(784\) 0 0
\(785\) 10.4390 0.372584
\(786\) 0 0
\(787\) −41.7484 −1.48817 −0.744085 0.668085i \(-0.767114\pi\)
−0.744085 + 0.668085i \(0.767114\pi\)
\(788\) 0 0
\(789\) −5.04587 −0.179638
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −71.0284 −2.52229
\(794\) 0 0
\(795\) −18.0937 −0.641717
\(796\) 0 0
\(797\) −5.50046 −0.194836 −0.0974181 0.995244i \(-0.531058\pi\)
−0.0974181 + 0.995244i \(0.531058\pi\)
\(798\) 0 0
\(799\) 18.6888 0.661161
\(800\) 0 0
\(801\) 20.5913 0.727557
\(802\) 0 0
\(803\) 36.3784 1.28377
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 61.4911 2.16459
\(808\) 0 0
\(809\) 40.1505 1.41162 0.705808 0.708404i \(-0.250584\pi\)
0.705808 + 0.708404i \(0.250584\pi\)
\(810\) 0 0
\(811\) 14.0606 0.493733 0.246866 0.969050i \(-0.420599\pi\)
0.246866 + 0.969050i \(0.420599\pi\)
\(812\) 0 0
\(813\) −34.3922 −1.20619
\(814\) 0 0
\(815\) 37.0247 1.29692
\(816\) 0 0
\(817\) −1.12775 −0.0394550
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.1131 1.05095 0.525477 0.850808i \(-0.323887\pi\)
0.525477 + 0.850808i \(0.323887\pi\)
\(822\) 0 0
\(823\) −48.5895 −1.69372 −0.846861 0.531814i \(-0.821510\pi\)
−0.846861 + 0.531814i \(0.821510\pi\)
\(824\) 0 0
\(825\) 18.7399 0.652440
\(826\) 0 0
\(827\) 31.0029 1.07808 0.539038 0.842282i \(-0.318788\pi\)
0.539038 + 0.842282i \(0.318788\pi\)
\(828\) 0 0
\(829\) 46.6206 1.61920 0.809601 0.586981i \(-0.199684\pi\)
0.809601 + 0.586981i \(0.199684\pi\)
\(830\) 0 0
\(831\) 53.2489 1.84718
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 15.1296 0.523581
\(836\) 0 0
\(837\) 5.91629 0.204497
\(838\) 0 0
\(839\) −4.09553 −0.141393 −0.0706966 0.997498i \(-0.522522\pi\)
−0.0706966 + 0.997498i \(0.522522\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 28.1231 0.968609
\(844\) 0 0
\(845\) −48.6500 −1.67361
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −38.9073 −1.33530
\(850\) 0 0
\(851\) −4.12110 −0.141270
\(852\) 0 0
\(853\) 28.0606 0.960775 0.480388 0.877056i \(-0.340496\pi\)
0.480388 + 0.877056i \(0.340496\pi\)
\(854\) 0 0
\(855\) 2.31789 0.0792700
\(856\) 0 0
\(857\) 9.60597 0.328134 0.164067 0.986449i \(-0.447539\pi\)
0.164067 + 0.986449i \(0.447539\pi\)
\(858\) 0 0
\(859\) 8.96594 0.305914 0.152957 0.988233i \(-0.451120\pi\)
0.152957 + 0.988233i \(0.451120\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 45.4911 1.54853 0.774267 0.632859i \(-0.218119\pi\)
0.774267 + 0.632859i \(0.218119\pi\)
\(864\) 0 0
\(865\) 0.0899116 0.00305709
\(866\) 0 0
\(867\) 27.6235 0.938144
\(868\) 0 0
\(869\) 41.3156 1.40154
\(870\) 0 0
\(871\) 30.4390 1.03139
\(872\) 0 0
\(873\) −7.52982 −0.254846
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.2654 1.15706 0.578530 0.815661i \(-0.303627\pi\)
0.578530 + 0.815661i \(0.303627\pi\)
\(878\) 0 0
\(879\) −16.8704 −0.569025
\(880\) 0 0
\(881\) −22.4995 −0.758029 −0.379014 0.925391i \(-0.623737\pi\)
−0.379014 + 0.925391i \(0.623737\pi\)
\(882\) 0 0
\(883\) 31.5592 1.06205 0.531025 0.847356i \(-0.321807\pi\)
0.531025 + 0.847356i \(0.321807\pi\)
\(884\) 0 0
\(885\) 26.0294 0.874967
\(886\) 0 0
\(887\) −17.6841 −0.593773 −0.296886 0.954913i \(-0.595948\pi\)
−0.296886 + 0.954913i \(0.595948\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −35.4948 −1.18912
\(892\) 0 0
\(893\) −9.62729 −0.322165
\(894\) 0 0
\(895\) 21.0690 0.704260
\(896\) 0 0
\(897\) −61.0890 −2.03970
\(898\) 0 0
\(899\) −1.87511 −0.0625386
\(900\) 0 0
\(901\) −11.4693 −0.382097
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 25.1745 0.836830
\(906\) 0 0
\(907\) 1.03028 0.0342098 0.0171049 0.999854i \(-0.494555\pi\)
0.0171049 + 0.999854i \(0.494555\pi\)
\(908\) 0 0
\(909\) 14.3472 0.475868
\(910\) 0 0
\(911\) 21.3056 0.705887 0.352943 0.935645i \(-0.385181\pi\)
0.352943 + 0.935645i \(0.385181\pi\)
\(912\) 0 0
\(913\) 51.2707 1.69681
\(914\) 0 0
\(915\) 33.1277 1.09517
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −19.1883 −0.632964 −0.316482 0.948599i \(-0.602502\pi\)
−0.316482 + 0.948599i \(0.602502\pi\)
\(920\) 0 0
\(921\) −59.8245 −1.97129
\(922\) 0 0
\(923\) 68.9291 2.26883
\(924\) 0 0
\(925\) −2.71056 −0.0891226
\(926\) 0 0
\(927\) −6.43899 −0.211484
\(928\) 0 0
\(929\) −4.18166 −0.137196 −0.0685979 0.997644i \(-0.521853\pi\)
−0.0685979 + 0.997644i \(0.521853\pi\)
\(930\) 0 0
\(931\) 7.21193 0.236362
\(932\) 0 0
\(933\) −9.62729 −0.315183
\(934\) 0 0
\(935\) −9.36996 −0.306430
\(936\) 0 0
\(937\) −45.1807 −1.47599 −0.737995 0.674806i \(-0.764227\pi\)
−0.737995 + 0.674806i \(0.764227\pi\)
\(938\) 0 0
\(939\) −11.6547 −0.380337
\(940\) 0 0
\(941\) 17.0752 0.556637 0.278318 0.960489i \(-0.410223\pi\)
0.278318 + 0.960489i \(0.410223\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50.3359 1.63570 0.817849 0.575434i \(-0.195167\pi\)
0.817849 + 0.575434i \(0.195167\pi\)
\(948\) 0 0
\(949\) −77.9982 −2.53193
\(950\) 0 0
\(951\) 43.4305 1.40833
\(952\) 0 0
\(953\) 8.94657 0.289808 0.144904 0.989446i \(-0.453713\pi\)
0.144904 + 0.989446i \(0.453713\pi\)
\(954\) 0 0
\(955\) 11.2707 0.364710
\(956\) 0 0
\(957\) 6.70436 0.216721
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.4839 −0.886579
\(962\) 0 0
\(963\) −9.93945 −0.320294
\(964\) 0 0
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −44.5327 −1.43207 −0.716037 0.698062i \(-0.754046\pi\)
−0.716037 + 0.698062i \(0.754046\pi\)
\(968\) 0 0
\(969\) 4.37844 0.140656
\(970\) 0 0
\(971\) −42.0294 −1.34879 −0.674393 0.738372i \(-0.735595\pi\)
−0.674393 + 0.738372i \(0.735595\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −40.1798 −1.28678
\(976\) 0 0
\(977\) −15.3931 −0.492469 −0.246235 0.969210i \(-0.579193\pi\)
−0.246235 + 0.969210i \(0.579193\pi\)
\(978\) 0 0
\(979\) 42.8798 1.37044
\(980\) 0 0
\(981\) 19.6273 0.626651
\(982\) 0 0
\(983\) −33.7153 −1.07535 −0.537675 0.843152i \(-0.680697\pi\)
−0.537675 + 0.843152i \(0.680697\pi\)
\(984\) 0 0
\(985\) −11.7890 −0.375628
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.65184 −0.147920
\(990\) 0 0
\(991\) 23.3094 0.740448 0.370224 0.928943i \(-0.379281\pi\)
0.370224 + 0.928943i \(0.379281\pi\)
\(992\) 0 0
\(993\) −52.1954 −1.65637
\(994\) 0 0
\(995\) −39.4381 −1.25027
\(996\) 0 0
\(997\) 4.02936 0.127611 0.0638055 0.997962i \(-0.479676\pi\)
0.0638055 + 0.997962i \(0.479676\pi\)
\(998\) 0 0
\(999\) 3.05964 0.0968026
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 1856.2.a.y.1.1 3
4.3 odd 2 1856.2.a.x.1.3 3
8.3 odd 2 232.2.a.d.1.1 3
8.5 even 2 464.2.a.j.1.3 3
24.5 odd 2 4176.2.a.bu.1.2 3
24.11 even 2 2088.2.a.s.1.2 3
40.19 odd 2 5800.2.a.p.1.3 3
232.115 odd 2 6728.2.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.a.d.1.1 3 8.3 odd 2
464.2.a.j.1.3 3 8.5 even 2
1856.2.a.x.1.3 3 4.3 odd 2
1856.2.a.y.1.1 3 1.1 even 1 trivial
2088.2.a.s.1.2 3 24.11 even 2
4176.2.a.bu.1.2 3 24.5 odd 2
5800.2.a.p.1.3 3 40.19 odd 2
6728.2.a.j.1.3 3 232.115 odd 2