Properties

Label 1856.2.a.y.1.3
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 1.3
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +O(q^{10})\) \(q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +4.49084 q^{11} -0.103084 q^{13} +4.49084 q^{15} +2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} +4.49084 q^{27} -1.00000 q^{29} +6.76156 q^{31} +12.4017 q^{33} -5.25240 q^{37} -0.284672 q^{39} +5.79383 q^{41} -10.0140 q^{43} +7.52311 q^{45} -11.5371 q^{47} -7.00000 q^{49} +5.52311 q^{51} +7.14931 q^{53} +7.30299 q^{55} -20.0279 q^{57} -1.52311 q^{59} -9.04623 q^{61} -0.167635 q^{65} +15.0462 q^{67} +15.2524 q^{69} +12.0279 q^{71} -1.79383 q^{73} -6.50479 q^{75} +1.98605 q^{79} -1.47689 q^{81} +6.47689 q^{83} +3.25240 q^{85} -2.76156 q^{87} +12.7110 q^{89} +18.6724 q^{93} -11.7938 q^{95} +1.25240 q^{97} +20.7755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9}+O(q^{10}) \) 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
\(3\) 2.76156 1.59439 0.797193 0.603725i \(-0.206317\pi\)
0.797193 + 0.603725i \(0.206317\pi\)
\(4\) 0 0
\(5\) 1.62620 0.727258 0.363629 0.931544i \(-0.381538\pi\)
0.363629 + 0.931544i \(0.381538\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 4.62620 1.54207
\(10\) 0 0
\(11\) 4.49084 1.35404 0.677019 0.735965i \(-0.263271\pi\)
0.677019 + 0.735965i \(0.263271\pi\)
\(12\) 0 0
\(13\) −0.103084 −0.0285903 −0.0142951 0.999898i \(-0.504550\pi\)
−0.0142951 + 0.999898i \(0.504550\pi\)
\(14\) 0 0
\(15\) 4.49084 1.15953
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −7.25240 −1.66381 −0.831907 0.554915i \(-0.812751\pi\)
−0.831907 + 0.554915i \(0.812751\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.52311 1.15165 0.575824 0.817573i \(-0.304681\pi\)
0.575824 + 0.817573i \(0.304681\pi\)
\(24\) 0 0
\(25\) −2.35548 −0.471096
\(26\) 0 0
\(27\) 4.49084 0.864262
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.76156 1.21441 0.607206 0.794545i \(-0.292290\pi\)
0.607206 + 0.794545i \(0.292290\pi\)
\(32\) 0 0
\(33\) 12.4017 2.15886
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.25240 −0.863489 −0.431744 0.901996i \(-0.642102\pi\)
−0.431744 + 0.901996i \(0.642102\pi\)
\(38\) 0 0
\(39\) −0.284672 −0.0455839
\(40\) 0 0
\(41\) 5.79383 0.904845 0.452422 0.891804i \(-0.350560\pi\)
0.452422 + 0.891804i \(0.350560\pi\)
\(42\) 0 0
\(43\) −10.0140 −1.52711 −0.763557 0.645741i \(-0.776549\pi\)
−0.763557 + 0.645741i \(0.776549\pi\)
\(44\) 0 0
\(45\) 7.52311 1.12148
\(46\) 0 0
\(47\) −11.5371 −1.68285 −0.841427 0.540371i \(-0.818284\pi\)
−0.841427 + 0.540371i \(0.818284\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 5.52311 0.773391
\(52\) 0 0
\(53\) 7.14931 0.982034 0.491017 0.871150i \(-0.336625\pi\)
0.491017 + 0.871150i \(0.336625\pi\)
\(54\) 0 0
\(55\) 7.30299 0.984735
\(56\) 0 0
\(57\) −20.0279 −2.65276
\(58\) 0 0
\(59\) −1.52311 −0.198293 −0.0991463 0.995073i \(-0.531611\pi\)
−0.0991463 + 0.995073i \(0.531611\pi\)
\(60\) 0 0
\(61\) −9.04623 −1.15825 −0.579125 0.815238i \(-0.696606\pi\)
−0.579125 + 0.815238i \(0.696606\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.167635 −0.0207925
\(66\) 0 0
\(67\) 15.0462 1.83819 0.919095 0.394037i \(-0.128922\pi\)
0.919095 + 0.394037i \(0.128922\pi\)
\(68\) 0 0
\(69\) 15.2524 1.83617
\(70\) 0 0
\(71\) 12.0279 1.42745 0.713725 0.700426i \(-0.247007\pi\)
0.713725 + 0.700426i \(0.247007\pi\)
\(72\) 0 0
\(73\) −1.79383 −0.209952 −0.104976 0.994475i \(-0.533477\pi\)
−0.104976 + 0.994475i \(0.533477\pi\)
\(74\) 0 0
\(75\) −6.50479 −0.751109
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.98605 0.223448 0.111724 0.993739i \(-0.464363\pi\)
0.111724 + 0.993739i \(0.464363\pi\)
\(80\) 0 0
\(81\) −1.47689 −0.164098
\(82\) 0 0
\(83\) 6.47689 0.710931 0.355465 0.934689i \(-0.384322\pi\)
0.355465 + 0.934689i \(0.384322\pi\)
\(84\) 0 0
\(85\) 3.25240 0.352772
\(86\) 0 0
\(87\) −2.76156 −0.296070
\(88\) 0 0
\(89\) 12.7110 1.34736 0.673680 0.739024i \(-0.264713\pi\)
0.673680 + 0.739024i \(0.264713\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 18.6724 1.93624
\(94\) 0 0
\(95\) −11.7938 −1.21002
\(96\) 0 0
\(97\) 1.25240 0.127162 0.0635808 0.997977i \(-0.479748\pi\)
0.0635808 + 0.997977i \(0.479748\pi\)
\(98\) 0 0
\(99\) 20.7755 2.08802
\(100\) 0 0
\(101\) −16.2986 −1.62177 −0.810887 0.585203i \(-0.801015\pi\)
−0.810887 + 0.585203i \(0.801015\pi\)
\(102\) 0 0
\(103\) 5.52311 0.544209 0.272104 0.962268i \(-0.412280\pi\)
0.272104 + 0.962268i \(0.412280\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.541436 0.0523426 0.0261713 0.999657i \(-0.491668\pi\)
0.0261713 + 0.999657i \(0.491668\pi\)
\(108\) 0 0
\(109\) −15.9248 −1.52532 −0.762661 0.646799i \(-0.776107\pi\)
−0.762661 + 0.646799i \(0.776107\pi\)
\(110\) 0 0
\(111\) −14.5048 −1.37673
\(112\) 0 0
\(113\) −12.5048 −1.17635 −0.588176 0.808733i \(-0.700154\pi\)
−0.588176 + 0.808733i \(0.700154\pi\)
\(114\) 0 0
\(115\) 8.98168 0.837546
\(116\) 0 0
\(117\) −0.476886 −0.0440881
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.16763 0.833421
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) −11.9615 −1.06987
\(126\) 0 0
\(127\) 0.206167 0.0182944 0.00914720 0.999958i \(-0.497088\pi\)
0.00914720 + 0.999958i \(0.497088\pi\)
\(128\) 0 0
\(129\) −27.6541 −2.43481
\(130\) 0 0
\(131\) 15.2524 1.33261 0.666304 0.745680i \(-0.267875\pi\)
0.666304 + 0.745680i \(0.267875\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 7.30299 0.628542
\(136\) 0 0
\(137\) −5.25240 −0.448742 −0.224371 0.974504i \(-0.572033\pi\)
−0.224371 + 0.974504i \(0.572033\pi\)
\(138\) 0 0
\(139\) −12.5693 −1.06612 −0.533059 0.846078i \(-0.678958\pi\)
−0.533059 + 0.846078i \(0.678958\pi\)
\(140\) 0 0
\(141\) −31.8603 −2.68312
\(142\) 0 0
\(143\) −0.462932 −0.0387123
\(144\) 0 0
\(145\) −1.62620 −0.135048
\(146\) 0 0
\(147\) −19.3309 −1.59439
\(148\) 0 0
\(149\) 0.309251 0.0253348 0.0126674 0.999920i \(-0.495968\pi\)
0.0126674 + 0.999920i \(0.495968\pi\)
\(150\) 0 0
\(151\) 11.4586 0.932485 0.466242 0.884657i \(-0.345607\pi\)
0.466242 + 0.884657i \(0.345607\pi\)
\(152\) 0 0
\(153\) 9.25240 0.748012
\(154\) 0 0
\(155\) 10.9956 0.883190
\(156\) 0 0
\(157\) −13.2524 −1.05766 −0.528828 0.848729i \(-0.677368\pi\)
−0.528828 + 0.848729i \(0.677368\pi\)
\(158\) 0 0
\(159\) 19.7432 1.56574
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.17389 −0.248598 −0.124299 0.992245i \(-0.539668\pi\)
−0.124299 + 0.992245i \(0.539668\pi\)
\(164\) 0 0
\(165\) 20.1676 1.57005
\(166\) 0 0
\(167\) 12.0279 0.930747 0.465374 0.885114i \(-0.345920\pi\)
0.465374 + 0.885114i \(0.345920\pi\)
\(168\) 0 0
\(169\) −12.9894 −0.999183
\(170\) 0 0
\(171\) −33.5510 −2.56571
\(172\) 0 0
\(173\) −12.5048 −0.950722 −0.475361 0.879791i \(-0.657682\pi\)
−0.475361 + 0.879791i \(0.657682\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.20617 −0.316155
\(178\) 0 0
\(179\) 8.02791 0.600034 0.300017 0.953934i \(-0.403008\pi\)
0.300017 + 0.953934i \(0.403008\pi\)
\(180\) 0 0
\(181\) 11.9248 0.886365 0.443183 0.896431i \(-0.353849\pi\)
0.443183 + 0.896431i \(0.353849\pi\)
\(182\) 0 0
\(183\) −24.9817 −1.84670
\(184\) 0 0
\(185\) −8.54144 −0.627979
\(186\) 0 0
\(187\) 8.98168 0.656805
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.71096 −0.485588 −0.242794 0.970078i \(-0.578064\pi\)
−0.242794 + 0.970078i \(0.578064\pi\)
\(192\) 0 0
\(193\) 12.2986 0.885274 0.442637 0.896701i \(-0.354043\pi\)
0.442637 + 0.896701i \(0.354043\pi\)
\(194\) 0 0
\(195\) −0.462932 −0.0331513
\(196\) 0 0
\(197\) −4.50479 −0.320953 −0.160477 0.987040i \(-0.551303\pi\)
−0.160477 + 0.987040i \(0.551303\pi\)
\(198\) 0 0
\(199\) 19.4586 1.37938 0.689690 0.724104i \(-0.257747\pi\)
0.689690 + 0.724104i \(0.257747\pi\)
\(200\) 0 0
\(201\) 41.5510 2.93078
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.42192 0.658055
\(206\) 0 0
\(207\) 25.5510 1.77592
\(208\) 0 0
\(209\) −32.5693 −2.25287
\(210\) 0 0
\(211\) 1.77988 0.122532 0.0612660 0.998121i \(-0.480486\pi\)
0.0612660 + 0.998121i \(0.480486\pi\)
\(212\) 0 0
\(213\) 33.2158 2.27591
\(214\) 0 0
\(215\) −16.2847 −1.11061
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.95377 −0.334745
\(220\) 0 0
\(221\) −0.206167 −0.0138683
\(222\) 0 0
\(223\) −26.5327 −1.77676 −0.888380 0.459108i \(-0.848169\pi\)
−0.888380 + 0.459108i \(0.848169\pi\)
\(224\) 0 0
\(225\) −10.8969 −0.726461
\(226\) 0 0
\(227\) −5.39401 −0.358013 −0.179007 0.983848i \(-0.557288\pi\)
−0.179007 + 0.983848i \(0.557288\pi\)
\(228\) 0 0
\(229\) −16.2986 −1.07704 −0.538522 0.842612i \(-0.681017\pi\)
−0.538522 + 0.842612i \(0.681017\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.64452 0.566321 0.283161 0.959072i \(-0.408617\pi\)
0.283161 + 0.959072i \(0.408617\pi\)
\(234\) 0 0
\(235\) −18.7616 −1.22387
\(236\) 0 0
\(237\) 5.48458 0.356262
\(238\) 0 0
\(239\) −13.5231 −0.874738 −0.437369 0.899282i \(-0.644090\pi\)
−0.437369 + 0.899282i \(0.644090\pi\)
\(240\) 0 0
\(241\) 8.30925 0.535246 0.267623 0.963524i \(-0.413762\pi\)
0.267623 + 0.963524i \(0.413762\pi\)
\(242\) 0 0
\(243\) −17.5510 −1.12590
\(244\) 0 0
\(245\) −11.3834 −0.727258
\(246\) 0 0
\(247\) 0.747604 0.0475689
\(248\) 0 0
\(249\) 17.8863 1.13350
\(250\) 0 0
\(251\) −19.7432 −1.24618 −0.623091 0.782149i \(-0.714123\pi\)
−0.623091 + 0.782149i \(0.714123\pi\)
\(252\) 0 0
\(253\) 24.8034 1.55938
\(254\) 0 0
\(255\) 8.98168 0.562454
\(256\) 0 0
\(257\) 21.4200 1.33614 0.668072 0.744096i \(-0.267120\pi\)
0.668072 + 0.744096i \(0.267120\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.62620 −0.286354
\(262\) 0 0
\(263\) −12.2847 −0.757505 −0.378753 0.925498i \(-0.623647\pi\)
−0.378753 + 0.925498i \(0.623647\pi\)
\(264\) 0 0
\(265\) 11.6262 0.714192
\(266\) 0 0
\(267\) 35.1020 2.14821
\(268\) 0 0
\(269\) 22.5972 1.37778 0.688889 0.724867i \(-0.258099\pi\)
0.688889 + 0.724867i \(0.258099\pi\)
\(270\) 0 0
\(271\) 23.7432 1.44230 0.721149 0.692780i \(-0.243614\pi\)
0.721149 + 0.692780i \(0.243614\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.5781 −0.637882
\(276\) 0 0
\(277\) 1.58767 0.0953936 0.0476968 0.998862i \(-0.484812\pi\)
0.0476968 + 0.998862i \(0.484812\pi\)
\(278\) 0 0
\(279\) 31.2803 1.87270
\(280\) 0 0
\(281\) −19.8969 −1.18695 −0.593475 0.804852i \(-0.702245\pi\)
−0.593475 + 0.804852i \(0.702245\pi\)
\(282\) 0 0
\(283\) 20.9817 1.24723 0.623616 0.781731i \(-0.285663\pi\)
0.623616 + 0.781731i \(0.285663\pi\)
\(284\) 0 0
\(285\) −32.5693 −1.92924
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 3.45856 0.202745
\(292\) 0 0
\(293\) −4.50479 −0.263173 −0.131586 0.991305i \(-0.542007\pi\)
−0.131586 + 0.991305i \(0.542007\pi\)
\(294\) 0 0
\(295\) −2.47689 −0.144210
\(296\) 0 0
\(297\) 20.1676 1.17024
\(298\) 0 0
\(299\) −0.569343 −0.0329260
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −45.0096 −2.58573
\(304\) 0 0
\(305\) −14.7110 −0.842347
\(306\) 0 0
\(307\) −9.60162 −0.547993 −0.273997 0.961731i \(-0.588346\pi\)
−0.273997 + 0.961731i \(0.588346\pi\)
\(308\) 0 0
\(309\) 15.2524 0.867679
\(310\) 0 0
\(311\) 30.2986 1.71808 0.859039 0.511911i \(-0.171062\pi\)
0.859039 + 0.511911i \(0.171062\pi\)
\(312\) 0 0
\(313\) 2.37380 0.134175 0.0670876 0.997747i \(-0.478629\pi\)
0.0670876 + 0.997747i \(0.478629\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.5510 0.648770 0.324385 0.945925i \(-0.394843\pi\)
0.324385 + 0.945925i \(0.394843\pi\)
\(318\) 0 0
\(319\) −4.49084 −0.251439
\(320\) 0 0
\(321\) 1.49521 0.0834544
\(322\) 0 0
\(323\) −14.5048 −0.807068
\(324\) 0 0
\(325\) 0.242812 0.0134688
\(326\) 0 0
\(327\) −43.9773 −2.43195
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.3126 −0.676761 −0.338380 0.941009i \(-0.609879\pi\)
−0.338380 + 0.941009i \(0.609879\pi\)
\(332\) 0 0
\(333\) −24.2986 −1.33156
\(334\) 0 0
\(335\) 24.4681 1.33684
\(336\) 0 0
\(337\) 6.95377 0.378796 0.189398 0.981900i \(-0.439346\pi\)
0.189398 + 0.981900i \(0.439346\pi\)
\(338\) 0 0
\(339\) −34.5327 −1.87556
\(340\) 0 0
\(341\) 30.3651 1.64436
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 24.8034 1.33537
\(346\) 0 0
\(347\) −16.4402 −0.882558 −0.441279 0.897370i \(-0.645475\pi\)
−0.441279 + 0.897370i \(0.645475\pi\)
\(348\) 0 0
\(349\) −22.9431 −1.22812 −0.614059 0.789260i \(-0.710464\pi\)
−0.614059 + 0.789260i \(0.710464\pi\)
\(350\) 0 0
\(351\) −0.462932 −0.0247095
\(352\) 0 0
\(353\) 8.09246 0.430718 0.215359 0.976535i \(-0.430908\pi\)
0.215359 + 0.976535i \(0.430908\pi\)
\(354\) 0 0
\(355\) 19.5598 1.03812
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.5187 −1.08294 −0.541469 0.840721i \(-0.682132\pi\)
−0.541469 + 0.840721i \(0.682132\pi\)
\(360\) 0 0
\(361\) 33.5972 1.76828
\(362\) 0 0
\(363\) 25.3169 1.32880
\(364\) 0 0
\(365\) −2.91713 −0.152689
\(366\) 0 0
\(367\) −1.75719 −0.0917245 −0.0458622 0.998948i \(-0.514604\pi\)
−0.0458622 + 0.998948i \(0.514604\pi\)
\(368\) 0 0
\(369\) 26.8034 1.39533
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −28.1310 −1.45657 −0.728284 0.685276i \(-0.759682\pi\)
−0.728284 + 0.685276i \(0.759682\pi\)
\(374\) 0 0
\(375\) −33.0323 −1.70578
\(376\) 0 0
\(377\) 0.103084 0.00530908
\(378\) 0 0
\(379\) 24.8034 1.27407 0.637033 0.770837i \(-0.280161\pi\)
0.637033 + 0.770837i \(0.280161\pi\)
\(380\) 0 0
\(381\) 0.569343 0.0291683
\(382\) 0 0
\(383\) −30.5048 −1.55872 −0.779361 0.626575i \(-0.784456\pi\)
−0.779361 + 0.626575i \(0.784456\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −46.3265 −2.35491
\(388\) 0 0
\(389\) 32.8401 1.66506 0.832529 0.553982i \(-0.186892\pi\)
0.832529 + 0.553982i \(0.186892\pi\)
\(390\) 0 0
\(391\) 11.0462 0.558632
\(392\) 0 0
\(393\) 42.1204 2.12469
\(394\) 0 0
\(395\) 3.22971 0.162504
\(396\) 0 0
\(397\) 28.3372 1.42220 0.711101 0.703090i \(-0.248197\pi\)
0.711101 + 0.703090i \(0.248197\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.94315 −0.346724 −0.173362 0.984858i \(-0.555463\pi\)
−0.173362 + 0.984858i \(0.555463\pi\)
\(402\) 0 0
\(403\) −0.697006 −0.0347204
\(404\) 0 0
\(405\) −2.40171 −0.119342
\(406\) 0 0
\(407\) −23.5877 −1.16920
\(408\) 0 0
\(409\) −39.1387 −1.93528 −0.967642 0.252328i \(-0.918804\pi\)
−0.967642 + 0.252328i \(0.918804\pi\)
\(410\) 0 0
\(411\) −14.5048 −0.715469
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.5327 0.517030
\(416\) 0 0
\(417\) −34.7110 −1.69980
\(418\) 0 0
\(419\) 15.6156 0.762871 0.381435 0.924396i \(-0.375430\pi\)
0.381435 + 0.924396i \(0.375430\pi\)
\(420\) 0 0
\(421\) 8.50479 0.414498 0.207249 0.978288i \(-0.433549\pi\)
0.207249 + 0.978288i \(0.433549\pi\)
\(422\) 0 0
\(423\) −53.3728 −2.59507
\(424\) 0 0
\(425\) −4.71096 −0.228515
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.27841 −0.0617224
\(430\) 0 0
\(431\) 6.60599 0.318199 0.159100 0.987263i \(-0.449141\pi\)
0.159100 + 0.987263i \(0.449141\pi\)
\(432\) 0 0
\(433\) −2.20617 −0.106022 −0.0530108 0.998594i \(-0.516882\pi\)
−0.0530108 + 0.998594i \(0.516882\pi\)
\(434\) 0 0
\(435\) −4.49084 −0.215319
\(436\) 0 0
\(437\) −40.0558 −1.91613
\(438\) 0 0
\(439\) −4.44024 −0.211921 −0.105961 0.994370i \(-0.533792\pi\)
−0.105961 + 0.994370i \(0.533792\pi\)
\(440\) 0 0
\(441\) −32.3834 −1.54207
\(442\) 0 0
\(443\) −21.7572 −1.03372 −0.516858 0.856071i \(-0.672898\pi\)
−0.516858 + 0.856071i \(0.672898\pi\)
\(444\) 0 0
\(445\) 20.6705 0.979877
\(446\) 0 0
\(447\) 0.854015 0.0403935
\(448\) 0 0
\(449\) −34.5972 −1.63275 −0.816373 0.577526i \(-0.804018\pi\)
−0.816373 + 0.577526i \(0.804018\pi\)
\(450\) 0 0
\(451\) 26.0192 1.22519
\(452\) 0 0
\(453\) 31.6435 1.48674
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −28.0925 −1.31411 −0.657055 0.753843i \(-0.728198\pi\)
−0.657055 + 0.753843i \(0.728198\pi\)
\(458\) 0 0
\(459\) 8.98168 0.419229
\(460\) 0 0
\(461\) 8.91713 0.415312 0.207656 0.978202i \(-0.433417\pi\)
0.207656 + 0.978202i \(0.433417\pi\)
\(462\) 0 0
\(463\) −3.55976 −0.165436 −0.0827180 0.996573i \(-0.526360\pi\)
−0.0827180 + 0.996573i \(0.526360\pi\)
\(464\) 0 0
\(465\) 30.3651 1.40815
\(466\) 0 0
\(467\) −11.0968 −0.513500 −0.256750 0.966478i \(-0.582652\pi\)
−0.256750 + 0.966478i \(0.582652\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −36.5972 −1.68631
\(472\) 0 0
\(473\) −44.9711 −2.06777
\(474\) 0 0
\(475\) 17.0829 0.783816
\(476\) 0 0
\(477\) 33.0741 1.51436
\(478\) 0 0
\(479\) 5.77988 0.264089 0.132045 0.991244i \(-0.457846\pi\)
0.132045 + 0.991244i \(0.457846\pi\)
\(480\) 0 0
\(481\) 0.541436 0.0246874
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.03664 0.0924792
\(486\) 0 0
\(487\) −11.0462 −0.500552 −0.250276 0.968174i \(-0.580521\pi\)
−0.250276 + 0.968174i \(0.580521\pi\)
\(488\) 0 0
\(489\) −8.76488 −0.396362
\(490\) 0 0
\(491\) −9.03228 −0.407621 −0.203810 0.979010i \(-0.565333\pi\)
−0.203810 + 0.979010i \(0.565333\pi\)
\(492\) 0 0
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) 33.7851 1.51853
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −39.6156 −1.77344 −0.886718 0.462310i \(-0.847021\pi\)
−0.886718 + 0.462310i \(0.847021\pi\)
\(500\) 0 0
\(501\) 33.2158 1.48397
\(502\) 0 0
\(503\) 26.7895 1.19448 0.597242 0.802061i \(-0.296263\pi\)
0.597242 + 0.802061i \(0.296263\pi\)
\(504\) 0 0
\(505\) −26.5048 −1.17945
\(506\) 0 0
\(507\) −35.8709 −1.59308
\(508\) 0 0
\(509\) 17.1127 0.758506 0.379253 0.925293i \(-0.376181\pi\)
0.379253 + 0.925293i \(0.376181\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −32.5693 −1.43797
\(514\) 0 0
\(515\) 8.98168 0.395780
\(516\) 0 0
\(517\) −51.8111 −2.27865
\(518\) 0 0
\(519\) −34.5327 −1.51582
\(520\) 0 0
\(521\) 19.5896 0.858234 0.429117 0.903249i \(-0.358825\pi\)
0.429117 + 0.903249i \(0.358825\pi\)
\(522\) 0 0
\(523\) 15.0462 0.657926 0.328963 0.944343i \(-0.393301\pi\)
0.328963 + 0.944343i \(0.393301\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.5231 0.589076
\(528\) 0 0
\(529\) 7.50479 0.326295
\(530\) 0 0
\(531\) −7.04623 −0.305780
\(532\) 0 0
\(533\) −0.597250 −0.0258698
\(534\) 0 0
\(535\) 0.880483 0.0380666
\(536\) 0 0
\(537\) 22.1695 0.956686
\(538\) 0 0
\(539\) −31.4359 −1.35404
\(540\) 0 0
\(541\) −31.2158 −1.34207 −0.671035 0.741426i \(-0.734150\pi\)
−0.671035 + 0.741426i \(0.734150\pi\)
\(542\) 0 0
\(543\) 32.9311 1.41321
\(544\) 0 0
\(545\) −25.8969 −1.10930
\(546\) 0 0
\(547\) 10.5048 0.449152 0.224576 0.974457i \(-0.427900\pi\)
0.224576 + 0.974457i \(0.427900\pi\)
\(548\) 0 0
\(549\) −41.8496 −1.78610
\(550\) 0 0
\(551\) 7.25240 0.308962
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −23.5877 −1.00124
\(556\) 0 0
\(557\) −27.6801 −1.17284 −0.586422 0.810006i \(-0.699464\pi\)
−0.586422 + 0.810006i \(0.699464\pi\)
\(558\) 0 0
\(559\) 1.03228 0.0436606
\(560\) 0 0
\(561\) 24.8034 1.04720
\(562\) 0 0
\(563\) 13.6508 0.575312 0.287656 0.957734i \(-0.407124\pi\)
0.287656 + 0.957734i \(0.407124\pi\)
\(564\) 0 0
\(565\) −20.3353 −0.855511
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.1387 1.13771 0.568856 0.822437i \(-0.307386\pi\)
0.568856 + 0.822437i \(0.307386\pi\)
\(570\) 0 0
\(571\) 21.9634 0.919138 0.459569 0.888142i \(-0.348004\pi\)
0.459569 + 0.888142i \(0.348004\pi\)
\(572\) 0 0
\(573\) −18.5327 −0.774215
\(574\) 0 0
\(575\) −13.0096 −0.542537
\(576\) 0 0
\(577\) −10.5414 −0.438846 −0.219423 0.975630i \(-0.570417\pi\)
−0.219423 + 0.975630i \(0.570417\pi\)
\(578\) 0 0
\(579\) 33.9634 1.41147
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 32.1064 1.32971
\(584\) 0 0
\(585\) −0.775511 −0.0320634
\(586\) 0 0
\(587\) 14.9450 0.616848 0.308424 0.951249i \(-0.400199\pi\)
0.308424 + 0.951249i \(0.400199\pi\)
\(588\) 0 0
\(589\) −49.0375 −2.02055
\(590\) 0 0
\(591\) −12.4402 −0.511723
\(592\) 0 0
\(593\) 24.8015 1.01848 0.509238 0.860626i \(-0.329927\pi\)
0.509238 + 0.860626i \(0.329927\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 53.7359 2.19927
\(598\) 0 0
\(599\) 13.4446 0.549332 0.274666 0.961540i \(-0.411433\pi\)
0.274666 + 0.961540i \(0.411433\pi\)
\(600\) 0 0
\(601\) 25.2524 1.03007 0.515033 0.857170i \(-0.327780\pi\)
0.515033 + 0.857170i \(0.327780\pi\)
\(602\) 0 0
\(603\) 69.6068 2.83461
\(604\) 0 0
\(605\) 14.9084 0.606112
\(606\) 0 0
\(607\) 2.78946 0.113221 0.0566104 0.998396i \(-0.481971\pi\)
0.0566104 + 0.998396i \(0.481971\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.18928 0.0481133
\(612\) 0 0
\(613\) 43.1772 1.74391 0.871956 0.489585i \(-0.162852\pi\)
0.871956 + 0.489585i \(0.162852\pi\)
\(614\) 0 0
\(615\) 26.0192 1.04919
\(616\) 0 0
\(617\) 10.7476 0.432682 0.216341 0.976318i \(-0.430588\pi\)
0.216341 + 0.976318i \(0.430588\pi\)
\(618\) 0 0
\(619\) 30.3771 1.22096 0.610480 0.792032i \(-0.290977\pi\)
0.610480 + 0.792032i \(0.290977\pi\)
\(620\) 0 0
\(621\) 24.8034 0.995327
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.67432 −0.306973
\(626\) 0 0
\(627\) −89.9421 −3.59194
\(628\) 0 0
\(629\) −10.5048 −0.418853
\(630\) 0 0
\(631\) 13.5231 0.538347 0.269173 0.963092i \(-0.413250\pi\)
0.269173 + 0.963092i \(0.413250\pi\)
\(632\) 0 0
\(633\) 4.91524 0.195363
\(634\) 0 0
\(635\) 0.335269 0.0133047
\(636\) 0 0
\(637\) 0.721586 0.0285903
\(638\) 0 0
\(639\) 55.6435 2.20122
\(640\) 0 0
\(641\) −16.6339 −0.656999 −0.328500 0.944504i \(-0.606543\pi\)
−0.328500 + 0.944504i \(0.606543\pi\)
\(642\) 0 0
\(643\) 24.9538 0.984081 0.492040 0.870572i \(-0.336251\pi\)
0.492040 + 0.870572i \(0.336251\pi\)
\(644\) 0 0
\(645\) −44.9711 −1.77073
\(646\) 0 0
\(647\) −10.0646 −0.395678 −0.197839 0.980234i \(-0.563392\pi\)
−0.197839 + 0.980234i \(0.563392\pi\)
\(648\) 0 0
\(649\) −6.84006 −0.268496
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −42.2620 −1.65384 −0.826920 0.562320i \(-0.809909\pi\)
−0.826920 + 0.562320i \(0.809909\pi\)
\(654\) 0 0
\(655\) 24.8034 0.969150
\(656\) 0 0
\(657\) −8.29862 −0.323760
\(658\) 0 0
\(659\) −43.3867 −1.69011 −0.845053 0.534682i \(-0.820431\pi\)
−0.845053 + 0.534682i \(0.820431\pi\)
\(660\) 0 0
\(661\) 0.0924575 0.00359618 0.00179809 0.999998i \(-0.499428\pi\)
0.00179809 + 0.999998i \(0.499428\pi\)
\(662\) 0 0
\(663\) −0.569343 −0.0221115
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.52311 −0.213856
\(668\) 0 0
\(669\) −73.2716 −2.83284
\(670\) 0 0
\(671\) −40.6252 −1.56832
\(672\) 0 0
\(673\) 1.86027 0.0717082 0.0358541 0.999357i \(-0.488585\pi\)
0.0358541 + 0.999357i \(0.488585\pi\)
\(674\) 0 0
\(675\) −10.5781 −0.407151
\(676\) 0 0
\(677\) 11.1387 0.428094 0.214047 0.976823i \(-0.431335\pi\)
0.214047 + 0.976823i \(0.431335\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −14.8959 −0.570811
\(682\) 0 0
\(683\) −8.54144 −0.326829 −0.163414 0.986558i \(-0.552251\pi\)
−0.163414 + 0.986558i \(0.552251\pi\)
\(684\) 0 0
\(685\) −8.54144 −0.326352
\(686\) 0 0
\(687\) −45.0096 −1.71722
\(688\) 0 0
\(689\) −0.736978 −0.0280766
\(690\) 0 0
\(691\) −24.5972 −0.935723 −0.467862 0.883802i \(-0.654975\pi\)
−0.467862 + 0.883802i \(0.654975\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.4402 −0.775343
\(696\) 0 0
\(697\) 11.5877 0.438914
\(698\) 0 0
\(699\) 23.8723 0.902935
\(700\) 0 0
\(701\) −3.56165 −0.134522 −0.0672608 0.997735i \(-0.521426\pi\)
−0.0672608 + 0.997735i \(0.521426\pi\)
\(702\) 0 0
\(703\) 38.0925 1.43668
\(704\) 0 0
\(705\) −51.8111 −1.95132
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 24.3651 0.915049 0.457525 0.889197i \(-0.348736\pi\)
0.457525 + 0.889197i \(0.348736\pi\)
\(710\) 0 0
\(711\) 9.18785 0.344571
\(712\) 0 0
\(713\) 37.3449 1.39858
\(714\) 0 0
\(715\) −0.752820 −0.0281539
\(716\) 0 0
\(717\) −37.3449 −1.39467
\(718\) 0 0
\(719\) −7.89881 −0.294576 −0.147288 0.989094i \(-0.547054\pi\)
−0.147288 + 0.989094i \(0.547054\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 22.9465 0.853388
\(724\) 0 0
\(725\) 2.35548 0.0874803
\(726\) 0 0
\(727\) −33.8130 −1.25405 −0.627027 0.778997i \(-0.715729\pi\)
−0.627027 + 0.778997i \(0.715729\pi\)
\(728\) 0 0
\(729\) −44.0375 −1.63102
\(730\) 0 0
\(731\) −20.0279 −0.740759
\(732\) 0 0
\(733\) −45.7205 −1.68873 −0.844363 0.535771i \(-0.820021\pi\)
−0.844363 + 0.535771i \(0.820021\pi\)
\(734\) 0 0
\(735\) −31.4359 −1.15953
\(736\) 0 0
\(737\) 67.5702 2.48898
\(738\) 0 0
\(739\) −8.28467 −0.304757 −0.152378 0.988322i \(-0.548693\pi\)
−0.152378 + 0.988322i \(0.548693\pi\)
\(740\) 0 0
\(741\) 2.06455 0.0758432
\(742\) 0 0
\(743\) 1.70138 0.0624174 0.0312087 0.999513i \(-0.490064\pi\)
0.0312087 + 0.999513i \(0.490064\pi\)
\(744\) 0 0
\(745\) 0.502904 0.0184250
\(746\) 0 0
\(747\) 29.9634 1.09630
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.4277 −0.380513 −0.190257 0.981734i \(-0.560932\pi\)
−0.190257 + 0.981734i \(0.560932\pi\)
\(752\) 0 0
\(753\) −54.5221 −1.98689
\(754\) 0 0
\(755\) 18.6339 0.678157
\(756\) 0 0
\(757\) 45.9267 1.66923 0.834617 0.550830i \(-0.185689\pi\)
0.834617 + 0.550830i \(0.185689\pi\)
\(758\) 0 0
\(759\) 68.4961 2.48625
\(760\) 0 0
\(761\) 1.58767 0.0575528 0.0287764 0.999586i \(-0.490839\pi\)
0.0287764 + 0.999586i \(0.490839\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 15.0462 0.543998
\(766\) 0 0
\(767\) 0.157008 0.00566924
\(768\) 0 0
\(769\) −8.37569 −0.302035 −0.151018 0.988531i \(-0.548255\pi\)
−0.151018 + 0.988531i \(0.548255\pi\)
\(770\) 0 0
\(771\) 59.1526 2.13033
\(772\) 0 0
\(773\) −10.2062 −0.367090 −0.183545 0.983011i \(-0.558757\pi\)
−0.183545 + 0.983011i \(0.558757\pi\)
\(774\) 0 0
\(775\) −15.9267 −0.572104
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −42.0192 −1.50549
\(780\) 0 0
\(781\) 54.0154 1.93282
\(782\) 0 0
\(783\) −4.49084 −0.160489
\(784\) 0 0
\(785\) −21.5510 −0.769189
\(786\) 0 0
\(787\) 26.6618 0.950391 0.475195 0.879880i \(-0.342377\pi\)
0.475195 + 0.879880i \(0.342377\pi\)
\(788\) 0 0
\(789\) −33.9248 −1.20776
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.932519 0.0331147
\(794\) 0 0
\(795\) 32.1064 1.13870
\(796\) 0 0
\(797\) −25.0462 −0.887183 −0.443591 0.896229i \(-0.646296\pi\)
−0.443591 + 0.896229i \(0.646296\pi\)
\(798\) 0 0
\(799\) −23.0741 −0.816304
\(800\) 0 0
\(801\) 58.8034 2.07772
\(802\) 0 0
\(803\) −8.05581 −0.284283
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 62.4036 2.19671
\(808\) 0 0
\(809\) 32.1695 1.13102 0.565510 0.824741i \(-0.308679\pi\)
0.565510 + 0.824741i \(0.308679\pi\)
\(810\) 0 0
\(811\) 26.5048 0.930709 0.465355 0.885124i \(-0.345927\pi\)
0.465355 + 0.885124i \(0.345927\pi\)
\(812\) 0 0
\(813\) 65.5683 2.29958
\(814\) 0 0
\(815\) −5.16138 −0.180795
\(816\) 0 0
\(817\) 72.6252 2.54083
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.2051 −0.949465 −0.474733 0.880130i \(-0.657455\pi\)
−0.474733 + 0.880130i \(0.657455\pi\)
\(822\) 0 0
\(823\) −8.61850 −0.300422 −0.150211 0.988654i \(-0.547995\pi\)
−0.150211 + 0.988654i \(0.547995\pi\)
\(824\) 0 0
\(825\) −29.2120 −1.01703
\(826\) 0 0
\(827\) −37.8636 −1.31665 −0.658323 0.752735i \(-0.728734\pi\)
−0.658323 + 0.752735i \(0.728734\pi\)
\(828\) 0 0
\(829\) 51.9634 1.80476 0.902381 0.430939i \(-0.141818\pi\)
0.902381 + 0.430939i \(0.141818\pi\)
\(830\) 0 0
\(831\) 4.38443 0.152094
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 19.5598 0.676893
\(836\) 0 0
\(837\) 30.3651 1.04957
\(838\) 0 0
\(839\) −32.0785 −1.10747 −0.553736 0.832692i \(-0.686799\pi\)
−0.553736 + 0.832692i \(0.686799\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −54.9465 −1.89246
\(844\) 0 0
\(845\) −21.1233 −0.726663
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 57.9421 1.98857
\(850\) 0 0
\(851\) −29.0096 −0.994436
\(852\) 0 0
\(853\) 40.5048 1.38686 0.693429 0.720525i \(-0.256099\pi\)
0.693429 + 0.720525i \(0.256099\pi\)
\(854\) 0 0
\(855\) −54.5606 −1.86593
\(856\) 0 0
\(857\) 31.3834 1.07204 0.536018 0.844207i \(-0.319928\pi\)
0.536018 + 0.844207i \(0.319928\pi\)
\(858\) 0 0
\(859\) 32.5187 1.10953 0.554763 0.832009i \(-0.312809\pi\)
0.554763 + 0.832009i \(0.312809\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.4036 1.57960 0.789798 0.613367i \(-0.210185\pi\)
0.789798 + 0.613367i \(0.210185\pi\)
\(864\) 0 0
\(865\) −20.3353 −0.691420
\(866\) 0 0
\(867\) −35.9002 −1.21924
\(868\) 0 0
\(869\) 8.91902 0.302557
\(870\) 0 0
\(871\) −1.55102 −0.0525543
\(872\) 0 0
\(873\) 5.79383 0.196092
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 47.1493 1.59212 0.796060 0.605218i \(-0.206914\pi\)
0.796060 + 0.605218i \(0.206914\pi\)
\(878\) 0 0
\(879\) −12.4402 −0.419599
\(880\) 0 0
\(881\) −2.95377 −0.0995151 −0.0497575 0.998761i \(-0.515845\pi\)
−0.0497575 + 0.998761i \(0.515845\pi\)
\(882\) 0 0
\(883\) −14.6339 −0.492470 −0.246235 0.969210i \(-0.579193\pi\)
−0.246235 + 0.969210i \(0.579193\pi\)
\(884\) 0 0
\(885\) −6.84006 −0.229926
\(886\) 0 0
\(887\) 33.3955 1.12131 0.560655 0.828050i \(-0.310549\pi\)
0.560655 + 0.828050i \(0.310549\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.63246 −0.222196
\(892\) 0 0
\(893\) 83.6714 2.79996
\(894\) 0 0
\(895\) 13.0550 0.436379
\(896\) 0 0
\(897\) −1.57227 −0.0524967
\(898\) 0 0
\(899\) −6.76156 −0.225511
\(900\) 0 0
\(901\) 14.2986 0.476356
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.3921 0.644616
\(906\) 0 0
\(907\) 7.25240 0.240812 0.120406 0.992725i \(-0.461580\pi\)
0.120406 + 0.992725i \(0.461580\pi\)
\(908\) 0 0
\(909\) −75.4007 −2.50088
\(910\) 0 0
\(911\) 14.6604 0.485719 0.242860 0.970061i \(-0.421915\pi\)
0.242860 + 0.970061i \(0.421915\pi\)
\(912\) 0 0
\(913\) 29.0867 0.962628
\(914\) 0 0
\(915\) −40.6252 −1.34303
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 42.1204 1.38942 0.694711 0.719289i \(-0.255532\pi\)
0.694711 + 0.719289i \(0.255532\pi\)
\(920\) 0 0
\(921\) −26.5154 −0.873713
\(922\) 0 0
\(923\) −1.23988 −0.0408112
\(924\) 0 0
\(925\) 12.3719 0.406786
\(926\) 0 0
\(927\) 25.5510 0.839206
\(928\) 0 0
\(929\) −41.5144 −1.36204 −0.681021 0.732264i \(-0.738464\pi\)
−0.681021 + 0.732264i \(0.738464\pi\)
\(930\) 0 0
\(931\) 50.7668 1.66381
\(932\) 0 0
\(933\) 83.6714 2.73928
\(934\) 0 0
\(935\) 14.6060 0.477667
\(936\) 0 0
\(937\) −43.4219 −1.41853 −0.709266 0.704941i \(-0.750974\pi\)
−0.709266 + 0.704941i \(0.750974\pi\)
\(938\) 0 0
\(939\) 6.55539 0.213927
\(940\) 0 0
\(941\) 13.0848 0.426551 0.213276 0.976992i \(-0.431587\pi\)
0.213276 + 0.976992i \(0.431587\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 49.9128 1.62195 0.810973 0.585083i \(-0.198938\pi\)
0.810973 + 0.585083i \(0.198938\pi\)
\(948\) 0 0
\(949\) 0.184915 0.00600259
\(950\) 0 0
\(951\) 31.8988 1.03439
\(952\) 0 0
\(953\) 39.6175 1.28334 0.641668 0.766983i \(-0.278243\pi\)
0.641668 + 0.766983i \(0.278243\pi\)
\(954\) 0 0
\(955\) −10.9133 −0.353148
\(956\) 0 0
\(957\) −12.4017 −0.400890
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 14.7187 0.474795
\(962\) 0 0
\(963\) 2.50479 0.0807158
\(964\) 0 0
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) 37.6574 1.21098 0.605491 0.795852i \(-0.292977\pi\)
0.605491 + 0.795852i \(0.292977\pi\)
\(968\) 0 0
\(969\) −40.0558 −1.28678
\(970\) 0 0
\(971\) −9.15994 −0.293956 −0.146978 0.989140i \(-0.546955\pi\)
−0.146978 + 0.989140i \(0.546955\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0.670538 0.0214744
\(976\) 0 0
\(977\) 45.4758 1.45490 0.727451 0.686160i \(-0.240705\pi\)
0.727451 + 0.686160i \(0.240705\pi\)
\(978\) 0 0
\(979\) 57.0829 1.82438
\(980\) 0 0
\(981\) −73.6714 −2.35215
\(982\) 0 0
\(983\) −27.9494 −0.891447 −0.445724 0.895171i \(-0.647054\pi\)
−0.445724 + 0.895171i \(0.647054\pi\)
\(984\) 0 0
\(985\) −7.32568 −0.233416
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −55.3082 −1.75870
\(990\) 0 0
\(991\) −13.1108 −0.416478 −0.208239 0.978078i \(-0.566773\pi\)
−0.208239 + 0.978078i \(0.566773\pi\)
\(992\) 0 0
\(993\) −34.0019 −1.07902
\(994\) 0 0
\(995\) 31.6435 1.00317
\(996\) 0 0
\(997\) −28.8401 −0.913374 −0.456687 0.889627i \(-0.650964\pi\)
−0.456687 + 0.889627i \(0.650964\pi\)
\(998\) 0 0
\(999\) −23.5877 −0.746281
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.1.3 3
4.3 odd 2 1856.2.a.x.1.1 3
8.3 odd 2 232.2.a.d.1.3 3
8.5 even 2 464.2.a.j.1.1 3
24.5 odd 2 4176.2.a.bu.1.3 3
24.11 even 2 2088.2.a.s.1.3 3
40.19 odd 2 5800.2.a.p.1.1 3
232.115 odd 2 6728.2.a.j.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.a.d.1.3 3 8.3 odd 2
464.2.a.j.1.1 3 8.5 even 2
1856.2.a.x.1.1 3 4.3 odd 2
1856.2.a.y.1.3 3 1.1 even 1 trivial
2088.2.a.s.1.3 3 24.11 even 2
4176.2.a.bu.1.3 3 24.5 odd 2
5800.2.a.p.1.1 3 40.19 odd 2
6728.2.a.j.1.1 3 232.115 odd 2