Properties

Label 1840.2.a.t.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.66908 q^{3} -1.00000 q^{5} -3.66908 q^{7} +4.12398 q^{9} +O(q^{10})\) \(q-2.66908 q^{3} -1.00000 q^{5} -3.66908 q^{7} +4.12398 q^{9} +1.21417 q^{11} +2.21417 q^{13} +2.66908 q^{15} +1.21417 q^{17} +2.57889 q^{19} +9.79306 q^{21} -1.00000 q^{23} +1.00000 q^{25} -3.00000 q^{27} +1.45490 q^{29} +6.46214 q^{31} -3.24073 q^{33} +3.66908 q^{35} +4.00000 q^{37} -5.90981 q^{39} -10.9170 q^{41} -6.90981 q^{43} -4.12398 q^{45} -5.45490 q^{47} +6.46214 q^{49} -3.24073 q^{51} +3.81962 q^{53} -1.21417 q^{55} -6.88325 q^{57} +4.24797 q^{59} +6.78583 q^{61} -15.1312 q^{63} -2.21417 q^{65} -12.8567 q^{67} +2.66908 q^{69} +6.91705 q^{71} -15.2214 q^{73} -2.66908 q^{75} -4.45490 q^{77} -4.36471 q^{81} +15.5861 q^{83} -1.21417 q^{85} -3.88325 q^{87} -10.9098 q^{89} -8.12398 q^{91} -17.2480 q^{93} -2.57889 q^{95} +1.69563 q^{97} +5.00724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{5} - 3q^{7} + 3q^{9} - 3q^{11} - 3q^{17} - 3q^{19} + 12q^{21} - 3q^{23} + 3q^{25} - 9q^{27} + 3q^{29} - 6q^{31} - 15q^{33} + 3q^{35} + 12q^{37} - 15q^{39} - 6q^{41} - 18q^{43} - 3q^{45} - 15q^{47} - 6q^{49} - 15q^{51} + 6q^{53} + 3q^{55} - 6q^{57} - 6q^{59} + 27q^{61} - 12q^{63} - 12q^{67} - 6q^{71} - 15q^{73} - 12q^{77} - 9q^{81} + 12q^{83} + 3q^{85} + 3q^{87} - 30q^{89} - 15q^{91} - 33q^{93} + 3q^{95} + 9q^{97} - 9q^{99} + O(q^{100}) \)

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
\(3\) −2.66908 −1.54099 −0.770497 0.637444i \(-0.779992\pi\)
−0.770497 + 0.637444i \(0.779992\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.66908 −1.38678 −0.693391 0.720562i \(-0.743884\pi\)
−0.693391 + 0.720562i \(0.743884\pi\)
\(8\) 0 0
\(9\) 4.12398 1.37466
\(10\) 0 0
\(11\) 1.21417 0.366088 0.183044 0.983105i \(-0.441405\pi\)
0.183044 + 0.983105i \(0.441405\pi\)
\(12\) 0 0
\(13\) 2.21417 0.614102 0.307051 0.951693i \(-0.400658\pi\)
0.307051 + 0.951693i \(0.400658\pi\)
\(14\) 0 0
\(15\) 2.66908 0.689153
\(16\) 0 0
\(17\) 1.21417 0.294481 0.147240 0.989101i \(-0.452961\pi\)
0.147240 + 0.989101i \(0.452961\pi\)
\(18\) 0 0
\(19\) 2.57889 0.591637 0.295819 0.955244i \(-0.404408\pi\)
0.295819 + 0.955244i \(0.404408\pi\)
\(20\) 0 0
\(21\) 9.79306 2.13702
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.00000 −0.577350
\(28\) 0 0
\(29\) 1.45490 0.270169 0.135084 0.990834i \(-0.456869\pi\)
0.135084 + 0.990834i \(0.456869\pi\)
\(30\) 0 0
\(31\) 6.46214 1.16063 0.580317 0.814390i \(-0.302928\pi\)
0.580317 + 0.814390i \(0.302928\pi\)
\(32\) 0 0
\(33\) −3.24073 −0.564139
\(34\) 0 0
\(35\) 3.66908 0.620188
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −5.90981 −0.946327
\(40\) 0 0
\(41\) −10.9170 −1.70496 −0.852478 0.522763i \(-0.824901\pi\)
−0.852478 + 0.522763i \(0.824901\pi\)
\(42\) 0 0
\(43\) −6.90981 −1.05374 −0.526868 0.849947i \(-0.676634\pi\)
−0.526868 + 0.849947i \(0.676634\pi\)
\(44\) 0 0
\(45\) −4.12398 −0.614767
\(46\) 0 0
\(47\) −5.45490 −0.795680 −0.397840 0.917455i \(-0.630240\pi\)
−0.397840 + 0.917455i \(0.630240\pi\)
\(48\) 0 0
\(49\) 6.46214 0.923163
\(50\) 0 0
\(51\) −3.24073 −0.453793
\(52\) 0 0
\(53\) 3.81962 0.524665 0.262332 0.964978i \(-0.415508\pi\)
0.262332 + 0.964978i \(0.415508\pi\)
\(54\) 0 0
\(55\) −1.21417 −0.163719
\(56\) 0 0
\(57\) −6.88325 −0.911709
\(58\) 0 0
\(59\) 4.24797 0.553038 0.276519 0.961008i \(-0.410819\pi\)
0.276519 + 0.961008i \(0.410819\pi\)
\(60\) 0 0
\(61\) 6.78583 0.868836 0.434418 0.900711i \(-0.356954\pi\)
0.434418 + 0.900711i \(0.356954\pi\)
\(62\) 0 0
\(63\) −15.1312 −1.90635
\(64\) 0 0
\(65\) −2.21417 −0.274635
\(66\) 0 0
\(67\) −12.8567 −1.57070 −0.785348 0.619055i \(-0.787516\pi\)
−0.785348 + 0.619055i \(0.787516\pi\)
\(68\) 0 0
\(69\) 2.66908 0.321319
\(70\) 0 0
\(71\) 6.91705 0.820902 0.410451 0.911883i \(-0.365371\pi\)
0.410451 + 0.911883i \(0.365371\pi\)
\(72\) 0 0
\(73\) −15.2214 −1.78153 −0.890766 0.454463i \(-0.849831\pi\)
−0.890766 + 0.454463i \(0.849831\pi\)
\(74\) 0 0
\(75\) −2.66908 −0.308199
\(76\) 0 0
\(77\) −4.45490 −0.507683
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −4.36471 −0.484968
\(82\) 0 0
\(83\) 15.5861 1.71080 0.855400 0.517968i \(-0.173312\pi\)
0.855400 + 0.517968i \(0.173312\pi\)
\(84\) 0 0
\(85\) −1.21417 −0.131696
\(86\) 0 0
\(87\) −3.88325 −0.416329
\(88\) 0 0
\(89\) −10.9098 −1.15644 −0.578219 0.815882i \(-0.696252\pi\)
−0.578219 + 0.815882i \(0.696252\pi\)
\(90\) 0 0
\(91\) −8.12398 −0.851625
\(92\) 0 0
\(93\) −17.2480 −1.78853
\(94\) 0 0
\(95\) −2.57889 −0.264588
\(96\) 0 0
\(97\) 1.69563 0.172165 0.0860827 0.996288i \(-0.472565\pi\)
0.0860827 + 0.996288i \(0.472565\pi\)
\(98\) 0 0
\(99\) 5.00724 0.503246
\(100\) 0 0
\(101\) −2.24797 −0.223681 −0.111841 0.993726i \(-0.535675\pi\)
−0.111841 + 0.993726i \(0.535675\pi\)
\(102\) 0 0
\(103\) 2.78583 0.274495 0.137248 0.990537i \(-0.456174\pi\)
0.137248 + 0.990537i \(0.456174\pi\)
\(104\) 0 0
\(105\) −9.79306 −0.955705
\(106\) 0 0
\(107\) −13.1578 −1.27201 −0.636005 0.771685i \(-0.719414\pi\)
−0.636005 + 0.771685i \(0.719414\pi\)
\(108\) 0 0
\(109\) 3.42111 0.327683 0.163842 0.986487i \(-0.447611\pi\)
0.163842 + 0.986487i \(0.447611\pi\)
\(110\) 0 0
\(111\) −10.6763 −1.01335
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 9.13122 0.844182
\(118\) 0 0
\(119\) −4.45490 −0.408380
\(120\) 0 0
\(121\) −9.52578 −0.865980
\(122\) 0 0
\(123\) 29.1385 2.62733
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.9508 −1.77035 −0.885175 0.465258i \(-0.845962\pi\)
−0.885175 + 0.465258i \(0.845962\pi\)
\(128\) 0 0
\(129\) 18.4428 1.62380
\(130\) 0 0
\(131\) 9.70287 0.847744 0.423872 0.905722i \(-0.360671\pi\)
0.423872 + 0.905722i \(0.360671\pi\)
\(132\) 0 0
\(133\) −9.46214 −0.820472
\(134\) 0 0
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) 2.82685 0.241514 0.120757 0.992682i \(-0.461468\pi\)
0.120757 + 0.992682i \(0.461468\pi\)
\(138\) 0 0
\(139\) −2.36471 −0.200572 −0.100286 0.994959i \(-0.531976\pi\)
−0.100286 + 0.994959i \(0.531976\pi\)
\(140\) 0 0
\(141\) 14.5596 1.22614
\(142\) 0 0
\(143\) 2.68840 0.224815
\(144\) 0 0
\(145\) −1.45490 −0.120823
\(146\) 0 0
\(147\) −17.2480 −1.42259
\(148\) 0 0
\(149\) −11.9170 −0.976282 −0.488141 0.872765i \(-0.662325\pi\)
−0.488141 + 0.872765i \(0.662325\pi\)
\(150\) 0 0
\(151\) −9.57889 −0.779519 −0.389759 0.920917i \(-0.627442\pi\)
−0.389759 + 0.920917i \(0.627442\pi\)
\(152\) 0 0
\(153\) 5.00724 0.404811
\(154\) 0 0
\(155\) −6.46214 −0.519052
\(156\) 0 0
\(157\) −18.7439 −1.49593 −0.747963 0.663740i \(-0.768968\pi\)
−0.747963 + 0.663740i \(0.768968\pi\)
\(158\) 0 0
\(159\) −10.1949 −0.808505
\(160\) 0 0
\(161\) 3.66908 0.289164
\(162\) 0 0
\(163\) 19.3188 1.51317 0.756584 0.653896i \(-0.226867\pi\)
0.756584 + 0.653896i \(0.226867\pi\)
\(164\) 0 0
\(165\) 3.24073 0.252290
\(166\) 0 0
\(167\) 1.81962 0.140806 0.0704031 0.997519i \(-0.477571\pi\)
0.0704031 + 0.997519i \(0.477571\pi\)
\(168\) 0 0
\(169\) −8.09743 −0.622879
\(170\) 0 0
\(171\) 10.6353 0.813301
\(172\) 0 0
\(173\) 15.9581 1.21327 0.606635 0.794981i \(-0.292519\pi\)
0.606635 + 0.794981i \(0.292519\pi\)
\(174\) 0 0
\(175\) −3.66908 −0.277356
\(176\) 0 0
\(177\) −11.3382 −0.852228
\(178\) 0 0
\(179\) −17.9508 −1.34171 −0.670854 0.741589i \(-0.734072\pi\)
−0.670854 + 0.741589i \(0.734072\pi\)
\(180\) 0 0
\(181\) 6.33092 0.470574 0.235287 0.971926i \(-0.424397\pi\)
0.235287 + 0.971926i \(0.424397\pi\)
\(182\) 0 0
\(183\) −18.1119 −1.33887
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 1.47422 0.107806
\(188\) 0 0
\(189\) 11.0072 0.800659
\(190\) 0 0
\(191\) 15.1578 1.09678 0.548389 0.836223i \(-0.315241\pi\)
0.548389 + 0.836223i \(0.315241\pi\)
\(192\) 0 0
\(193\) 18.9879 1.36678 0.683390 0.730053i \(-0.260505\pi\)
0.683390 + 0.730053i \(0.260505\pi\)
\(194\) 0 0
\(195\) 5.90981 0.423210
\(196\) 0 0
\(197\) −19.1650 −1.36545 −0.682725 0.730675i \(-0.739205\pi\)
−0.682725 + 0.730675i \(0.739205\pi\)
\(198\) 0 0
\(199\) 5.76651 0.408777 0.204388 0.978890i \(-0.434479\pi\)
0.204388 + 0.978890i \(0.434479\pi\)
\(200\) 0 0
\(201\) 34.3155 2.42043
\(202\) 0 0
\(203\) −5.33816 −0.374665
\(204\) 0 0
\(205\) 10.9170 0.762480
\(206\) 0 0
\(207\) −4.12398 −0.286637
\(208\) 0 0
\(209\) 3.13122 0.216591
\(210\) 0 0
\(211\) 17.4057 1.19826 0.599130 0.800652i \(-0.295513\pi\)
0.599130 + 0.800652i \(0.295513\pi\)
\(212\) 0 0
\(213\) −18.4621 −1.26501
\(214\) 0 0
\(215\) 6.90981 0.471245
\(216\) 0 0
\(217\) −23.7101 −1.60955
\(218\) 0 0
\(219\) 40.6272 2.74533
\(220\) 0 0
\(221\) 2.68840 0.180841
\(222\) 0 0
\(223\) 14.6763 0.982799 0.491399 0.870934i \(-0.336486\pi\)
0.491399 + 0.870934i \(0.336486\pi\)
\(224\) 0 0
\(225\) 4.12398 0.274932
\(226\) 0 0
\(227\) −16.4283 −1.09039 −0.545194 0.838310i \(-0.683544\pi\)
−0.545194 + 0.838310i \(0.683544\pi\)
\(228\) 0 0
\(229\) 19.3526 1.27886 0.639429 0.768850i \(-0.279171\pi\)
0.639429 + 0.768850i \(0.279171\pi\)
\(230\) 0 0
\(231\) 11.8905 0.782337
\(232\) 0 0
\(233\) −19.4549 −1.27453 −0.637267 0.770643i \(-0.719935\pi\)
−0.637267 + 0.770643i \(0.719935\pi\)
\(234\) 0 0
\(235\) 5.45490 0.355839
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.7029 −1.53321 −0.766606 0.642118i \(-0.778056\pi\)
−0.766606 + 0.642118i \(0.778056\pi\)
\(240\) 0 0
\(241\) 0.233492 0.0150405 0.00752027 0.999972i \(-0.497606\pi\)
0.00752027 + 0.999972i \(0.497606\pi\)
\(242\) 0 0
\(243\) 20.6498 1.32468
\(244\) 0 0
\(245\) −6.46214 −0.412851
\(246\) 0 0
\(247\) 5.71011 0.363325
\(248\) 0 0
\(249\) −41.6006 −2.63633
\(250\) 0 0
\(251\) −21.7511 −1.37292 −0.686460 0.727168i \(-0.740836\pi\)
−0.686460 + 0.727168i \(0.740836\pi\)
\(252\) 0 0
\(253\) −1.21417 −0.0763345
\(254\) 0 0
\(255\) 3.24073 0.202942
\(256\) 0 0
\(257\) −16.3116 −1.01749 −0.508745 0.860917i \(-0.669890\pi\)
−0.508745 + 0.860917i \(0.669890\pi\)
\(258\) 0 0
\(259\) −14.6763 −0.911942
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 2.53786 0.156491 0.0782455 0.996934i \(-0.475068\pi\)
0.0782455 + 0.996934i \(0.475068\pi\)
\(264\) 0 0
\(265\) −3.81962 −0.234637
\(266\) 0 0
\(267\) 29.1191 1.78206
\(268\) 0 0
\(269\) −2.54510 −0.155177 −0.0775886 0.996985i \(-0.524722\pi\)
−0.0775886 + 0.996985i \(0.524722\pi\)
\(270\) 0 0
\(271\) −24.1795 −1.46880 −0.734400 0.678717i \(-0.762536\pi\)
−0.734400 + 0.678717i \(0.762536\pi\)
\(272\) 0 0
\(273\) 21.6836 1.31235
\(274\) 0 0
\(275\) 1.21417 0.0732175
\(276\) 0 0
\(277\) −21.2359 −1.27594 −0.637970 0.770061i \(-0.720226\pi\)
−0.637970 + 0.770061i \(0.720226\pi\)
\(278\) 0 0
\(279\) 26.6498 1.59548
\(280\) 0 0
\(281\) −13.8872 −0.828441 −0.414220 0.910177i \(-0.635946\pi\)
−0.414220 + 0.910177i \(0.635946\pi\)
\(282\) 0 0
\(283\) 29.5330 1.75556 0.877778 0.479068i \(-0.159025\pi\)
0.877778 + 0.479068i \(0.159025\pi\)
\(284\) 0 0
\(285\) 6.88325 0.407729
\(286\) 0 0
\(287\) 40.0555 2.36440
\(288\) 0 0
\(289\) −15.5258 −0.913281
\(290\) 0 0
\(291\) −4.52578 −0.265306
\(292\) 0 0
\(293\) −24.0821 −1.40689 −0.703444 0.710750i \(-0.748356\pi\)
−0.703444 + 0.710750i \(0.748356\pi\)
\(294\) 0 0
\(295\) −4.24797 −0.247326
\(296\) 0 0
\(297\) −3.64252 −0.211361
\(298\) 0 0
\(299\) −2.21417 −0.128049
\(300\) 0 0
\(301\) 25.3526 1.46130
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −6.78583 −0.388555
\(306\) 0 0
\(307\) −19.9436 −1.13824 −0.569121 0.822254i \(-0.692716\pi\)
−0.569121 + 0.822254i \(0.692716\pi\)
\(308\) 0 0
\(309\) −7.43559 −0.422996
\(310\) 0 0
\(311\) 2.97345 0.168609 0.0843043 0.996440i \(-0.473133\pi\)
0.0843043 + 0.996440i \(0.473133\pi\)
\(312\) 0 0
\(313\) 15.4887 0.875473 0.437736 0.899103i \(-0.355780\pi\)
0.437736 + 0.899103i \(0.355780\pi\)
\(314\) 0 0
\(315\) 15.1312 0.852548
\(316\) 0 0
\(317\) 8.82685 0.495765 0.247883 0.968790i \(-0.420265\pi\)
0.247883 + 0.968790i \(0.420265\pi\)
\(318\) 0 0
\(319\) 1.76651 0.0989055
\(320\) 0 0
\(321\) 35.1191 1.96016
\(322\) 0 0
\(323\) 3.13122 0.174226
\(324\) 0 0
\(325\) 2.21417 0.122820
\(326\) 0 0
\(327\) −9.13122 −0.504958
\(328\) 0 0
\(329\) 20.0145 1.10343
\(330\) 0 0
\(331\) −17.4018 −0.956489 −0.478245 0.878227i \(-0.658727\pi\)
−0.478245 + 0.878227i \(0.658727\pi\)
\(332\) 0 0
\(333\) 16.4959 0.903972
\(334\) 0 0
\(335\) 12.8567 0.702437
\(336\) 0 0
\(337\) 26.9541 1.46828 0.734142 0.678995i \(-0.237584\pi\)
0.734142 + 0.678995i \(0.237584\pi\)
\(338\) 0 0
\(339\) 26.6908 1.44964
\(340\) 0 0
\(341\) 7.84617 0.424894
\(342\) 0 0
\(343\) 1.97345 0.106556
\(344\) 0 0
\(345\) −2.66908 −0.143698
\(346\) 0 0
\(347\) 10.1240 0.543484 0.271742 0.962370i \(-0.412400\pi\)
0.271742 + 0.962370i \(0.412400\pi\)
\(348\) 0 0
\(349\) 7.38732 0.395434 0.197717 0.980259i \(-0.436647\pi\)
0.197717 + 0.980259i \(0.436647\pi\)
\(350\) 0 0
\(351\) −6.64252 −0.354552
\(352\) 0 0
\(353\) 13.8833 0.738931 0.369466 0.929244i \(-0.379541\pi\)
0.369466 + 0.929244i \(0.379541\pi\)
\(354\) 0 0
\(355\) −6.91705 −0.367119
\(356\) 0 0
\(357\) 11.8905 0.629312
\(358\) 0 0
\(359\) −15.0902 −0.796430 −0.398215 0.917292i \(-0.630370\pi\)
−0.398215 + 0.917292i \(0.630370\pi\)
\(360\) 0 0
\(361\) −12.3493 −0.649965
\(362\) 0 0
\(363\) 25.4251 1.33447
\(364\) 0 0
\(365\) 15.2214 0.796725
\(366\) 0 0
\(367\) 24.0289 1.25430 0.627150 0.778898i \(-0.284221\pi\)
0.627150 + 0.778898i \(0.284221\pi\)
\(368\) 0 0
\(369\) −45.0217 −2.34374
\(370\) 0 0
\(371\) −14.0145 −0.727595
\(372\) 0 0
\(373\) 30.6908 1.58911 0.794554 0.607193i \(-0.207704\pi\)
0.794554 + 0.607193i \(0.207704\pi\)
\(374\) 0 0
\(375\) 2.66908 0.137831
\(376\) 0 0
\(377\) 3.22141 0.165911
\(378\) 0 0
\(379\) −2.07087 −0.106374 −0.0531869 0.998585i \(-0.516938\pi\)
−0.0531869 + 0.998585i \(0.516938\pi\)
\(380\) 0 0
\(381\) 53.2504 2.72810
\(382\) 0 0
\(383\) 1.58612 0.0810472 0.0405236 0.999179i \(-0.487097\pi\)
0.0405236 + 0.999179i \(0.487097\pi\)
\(384\) 0 0
\(385\) 4.45490 0.227043
\(386\) 0 0
\(387\) −28.4959 −1.44853
\(388\) 0 0
\(389\) 21.2142 1.07560 0.537801 0.843072i \(-0.319255\pi\)
0.537801 + 0.843072i \(0.319255\pi\)
\(390\) 0 0
\(391\) −1.21417 −0.0614035
\(392\) 0 0
\(393\) −25.8977 −1.30637
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.93965 −0.0973485 −0.0486742 0.998815i \(-0.515500\pi\)
−0.0486742 + 0.998815i \(0.515500\pi\)
\(398\) 0 0
\(399\) 25.2552 1.26434
\(400\) 0 0
\(401\) −5.69893 −0.284591 −0.142295 0.989824i \(-0.545448\pi\)
−0.142295 + 0.989824i \(0.545448\pi\)
\(402\) 0 0
\(403\) 14.3083 0.712748
\(404\) 0 0
\(405\) 4.36471 0.216884
\(406\) 0 0
\(407\) 4.85670 0.240738
\(408\) 0 0
\(409\) 3.53786 0.174936 0.0874679 0.996167i \(-0.472122\pi\)
0.0874679 + 0.996167i \(0.472122\pi\)
\(410\) 0 0
\(411\) −7.54510 −0.372172
\(412\) 0 0
\(413\) −15.5861 −0.766943
\(414\) 0 0
\(415\) −15.5861 −0.765093
\(416\) 0 0
\(417\) 6.31160 0.309081
\(418\) 0 0
\(419\) −28.2624 −1.38071 −0.690355 0.723471i \(-0.742546\pi\)
−0.690355 + 0.723471i \(0.742546\pi\)
\(420\) 0 0
\(421\) 34.0974 1.66181 0.830904 0.556417i \(-0.187824\pi\)
0.830904 + 0.556417i \(0.187824\pi\)
\(422\) 0 0
\(423\) −22.4959 −1.09379
\(424\) 0 0
\(425\) 1.21417 0.0588961
\(426\) 0 0
\(427\) −24.8977 −1.20489
\(428\) 0 0
\(429\) −7.17554 −0.346438
\(430\) 0 0
\(431\) 13.2850 0.639918 0.319959 0.947431i \(-0.396331\pi\)
0.319959 + 0.947431i \(0.396331\pi\)
\(432\) 0 0
\(433\) 11.0603 0.531526 0.265763 0.964038i \(-0.414376\pi\)
0.265763 + 0.964038i \(0.414376\pi\)
\(434\) 0 0
\(435\) 3.88325 0.186188
\(436\) 0 0
\(437\) −2.57889 −0.123365
\(438\) 0 0
\(439\) 19.1916 0.915963 0.457982 0.888962i \(-0.348573\pi\)
0.457982 + 0.888962i \(0.348573\pi\)
\(440\) 0 0
\(441\) 26.6498 1.26904
\(442\) 0 0
\(443\) −24.2818 −1.15366 −0.576831 0.816864i \(-0.695711\pi\)
−0.576831 + 0.816864i \(0.695711\pi\)
\(444\) 0 0
\(445\) 10.9098 0.517175
\(446\) 0 0
\(447\) 31.8075 1.50444
\(448\) 0 0
\(449\) 14.8413 0.700406 0.350203 0.936674i \(-0.386113\pi\)
0.350203 + 0.936674i \(0.386113\pi\)
\(450\) 0 0
\(451\) −13.2552 −0.624163
\(452\) 0 0
\(453\) 25.5668 1.20123
\(454\) 0 0
\(455\) 8.12398 0.380858
\(456\) 0 0
\(457\) 15.5861 0.729088 0.364544 0.931186i \(-0.381225\pi\)
0.364544 + 0.931186i \(0.381225\pi\)
\(458\) 0 0
\(459\) −3.64252 −0.170019
\(460\) 0 0
\(461\) −14.1312 −0.658157 −0.329078 0.944303i \(-0.606738\pi\)
−0.329078 + 0.944303i \(0.606738\pi\)
\(462\) 0 0
\(463\) −26.7584 −1.24357 −0.621784 0.783189i \(-0.713592\pi\)
−0.621784 + 0.783189i \(0.713592\pi\)
\(464\) 0 0
\(465\) 17.2480 0.799855
\(466\) 0 0
\(467\) −26.6908 −1.23510 −0.617551 0.786531i \(-0.711875\pi\)
−0.617551 + 0.786531i \(0.711875\pi\)
\(468\) 0 0
\(469\) 47.1722 2.17821
\(470\) 0 0
\(471\) 50.0289 2.30521
\(472\) 0 0
\(473\) −8.38972 −0.385760
\(474\) 0 0
\(475\) 2.57889 0.118327
\(476\) 0 0
\(477\) 15.7520 0.721236
\(478\) 0 0
\(479\) −4.90981 −0.224335 −0.112167 0.993689i \(-0.535779\pi\)
−0.112167 + 0.993689i \(0.535779\pi\)
\(480\) 0 0
\(481\) 8.85670 0.403831
\(482\) 0 0
\(483\) −9.79306 −0.445600
\(484\) 0 0
\(485\) −1.69563 −0.0769947
\(486\) 0 0
\(487\) −16.7786 −0.760310 −0.380155 0.924923i \(-0.624129\pi\)
−0.380155 + 0.924923i \(0.624129\pi\)
\(488\) 0 0
\(489\) −51.5635 −2.33178
\(490\) 0 0
\(491\) −41.4839 −1.87214 −0.936070 0.351814i \(-0.885565\pi\)
−0.936070 + 0.351814i \(0.885565\pi\)
\(492\) 0 0
\(493\) 1.76651 0.0795595
\(494\) 0 0
\(495\) −5.00724 −0.225059
\(496\) 0 0
\(497\) −25.3792 −1.13841
\(498\) 0 0
\(499\) 6.47751 0.289973 0.144987 0.989434i \(-0.453686\pi\)
0.144987 + 0.989434i \(0.453686\pi\)
\(500\) 0 0
\(501\) −4.85670 −0.216981
\(502\) 0 0
\(503\) −37.3825 −1.66680 −0.833401 0.552669i \(-0.813610\pi\)
−0.833401 + 0.552669i \(0.813610\pi\)
\(504\) 0 0
\(505\) 2.24797 0.100033
\(506\) 0 0
\(507\) 21.6127 0.959853
\(508\) 0 0
\(509\) −15.0410 −0.666682 −0.333341 0.942806i \(-0.608176\pi\)
−0.333341 + 0.942806i \(0.608176\pi\)
\(510\) 0 0
\(511\) 55.8486 2.47060
\(512\) 0 0
\(513\) −7.73666 −0.341582
\(514\) 0 0
\(515\) −2.78583 −0.122758
\(516\) 0 0
\(517\) −6.62321 −0.291288
\(518\) 0 0
\(519\) −42.5934 −1.86964
\(520\) 0 0
\(521\) 15.9469 0.698646 0.349323 0.937002i \(-0.386412\pi\)
0.349323 + 0.937002i \(0.386412\pi\)
\(522\) 0 0
\(523\) −10.8567 −0.474730 −0.237365 0.971420i \(-0.576284\pi\)
−0.237365 + 0.971420i \(0.576284\pi\)
\(524\) 0 0
\(525\) 9.79306 0.427404
\(526\) 0 0
\(527\) 7.84617 0.341785
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 17.5185 0.760240
\(532\) 0 0
\(533\) −24.1722 −1.04702
\(534\) 0 0
\(535\) 13.1578 0.568860
\(536\) 0 0
\(537\) 47.9122 2.06756
\(538\) 0 0
\(539\) 7.84617 0.337958
\(540\) 0 0
\(541\) −30.3792 −1.30610 −0.653052 0.757313i \(-0.726512\pi\)
−0.653052 + 0.757313i \(0.726512\pi\)
\(542\) 0 0
\(543\) −16.8977 −0.725151
\(544\) 0 0
\(545\) −3.42111 −0.146544
\(546\) 0 0
\(547\) −28.4090 −1.21468 −0.607341 0.794441i \(-0.707764\pi\)
−0.607341 + 0.794441i \(0.707764\pi\)
\(548\) 0 0
\(549\) 27.9846 1.19435
\(550\) 0 0
\(551\) 3.75203 0.159842
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.6763 0.453184
\(556\) 0 0
\(557\) 36.7584 1.55750 0.778751 0.627333i \(-0.215853\pi\)
0.778751 + 0.627333i \(0.215853\pi\)
\(558\) 0 0
\(559\) −15.2995 −0.647101
\(560\) 0 0
\(561\) −3.93481 −0.166128
\(562\) 0 0
\(563\) 3.03708 0.127998 0.0639989 0.997950i \(-0.479615\pi\)
0.0639989 + 0.997950i \(0.479615\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 16.0145 0.672545
\(568\) 0 0
\(569\) −32.2093 −1.35029 −0.675143 0.737687i \(-0.735918\pi\)
−0.675143 + 0.737687i \(0.735918\pi\)
\(570\) 0 0
\(571\) −27.7777 −1.16246 −0.581230 0.813739i \(-0.697428\pi\)
−0.581230 + 0.813739i \(0.697428\pi\)
\(572\) 0 0
\(573\) −40.4573 −1.69013
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 11.2890 0.469967 0.234984 0.971999i \(-0.424496\pi\)
0.234984 + 0.971999i \(0.424496\pi\)
\(578\) 0 0
\(579\) −50.6803 −2.10620
\(580\) 0 0
\(581\) −57.1867 −2.37251
\(582\) 0 0
\(583\) 4.63768 0.192073
\(584\) 0 0
\(585\) −9.13122 −0.377530
\(586\) 0 0
\(587\) −23.6609 −0.976592 −0.488296 0.872678i \(-0.662381\pi\)
−0.488296 + 0.872678i \(0.662381\pi\)
\(588\) 0 0
\(589\) 16.6651 0.686675
\(590\) 0 0
\(591\) 51.1529 2.10415
\(592\) 0 0
\(593\) −17.8341 −0.732358 −0.366179 0.930544i \(-0.619334\pi\)
−0.366179 + 0.930544i \(0.619334\pi\)
\(594\) 0 0
\(595\) 4.45490 0.182633
\(596\) 0 0
\(597\) −15.3913 −0.629923
\(598\) 0 0
\(599\) 23.8308 0.973700 0.486850 0.873486i \(-0.338146\pi\)
0.486850 + 0.873486i \(0.338146\pi\)
\(600\) 0 0
\(601\) 12.5297 0.511098 0.255549 0.966796i \(-0.417744\pi\)
0.255549 + 0.966796i \(0.417744\pi\)
\(602\) 0 0
\(603\) −53.0208 −2.15917
\(604\) 0 0
\(605\) 9.52578 0.387278
\(606\) 0 0
\(607\) 23.1722 0.940533 0.470266 0.882525i \(-0.344158\pi\)
0.470266 + 0.882525i \(0.344158\pi\)
\(608\) 0 0
\(609\) 14.2480 0.577357
\(610\) 0 0
\(611\) −12.0781 −0.488628
\(612\) 0 0
\(613\) −44.5780 −1.80049 −0.900244 0.435386i \(-0.856612\pi\)
−0.900244 + 0.435386i \(0.856612\pi\)
\(614\) 0 0
\(615\) −29.1385 −1.17498
\(616\) 0 0
\(617\) −32.1385 −1.29385 −0.646923 0.762556i \(-0.723944\pi\)
−0.646923 + 0.762556i \(0.723944\pi\)
\(618\) 0 0
\(619\) −31.5708 −1.26894 −0.634468 0.772949i \(-0.718781\pi\)
−0.634468 + 0.772949i \(0.718781\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 40.0289 1.60373
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.35748 −0.333765
\(628\) 0 0
\(629\) 4.85670 0.193649
\(630\) 0 0
\(631\) −24.0145 −0.956001 −0.478001 0.878360i \(-0.658638\pi\)
−0.478001 + 0.878360i \(0.658638\pi\)
\(632\) 0 0
\(633\) −46.4573 −1.84651
\(634\) 0 0
\(635\) 19.9508 0.791725
\(636\) 0 0
\(637\) 14.3083 0.566916
\(638\) 0 0
\(639\) 28.5258 1.12846
\(640\) 0 0
\(641\) −10.4428 −0.412467 −0.206233 0.978503i \(-0.566121\pi\)
−0.206233 + 0.978503i \(0.566121\pi\)
\(642\) 0 0
\(643\) −4.39940 −0.173495 −0.0867477 0.996230i \(-0.527647\pi\)
−0.0867477 + 0.996230i \(0.527647\pi\)
\(644\) 0 0
\(645\) −18.4428 −0.726186
\(646\) 0 0
\(647\) 23.6498 0.929768 0.464884 0.885372i \(-0.346096\pi\)
0.464884 + 0.885372i \(0.346096\pi\)
\(648\) 0 0
\(649\) 5.15777 0.202460
\(650\) 0 0
\(651\) 63.2842 2.48030
\(652\) 0 0
\(653\) 40.9194 1.60130 0.800651 0.599131i \(-0.204487\pi\)
0.800651 + 0.599131i \(0.204487\pi\)
\(654\) 0 0
\(655\) −9.70287 −0.379123
\(656\) 0 0
\(657\) −62.7728 −2.44900
\(658\) 0 0
\(659\) −48.8567 −1.90319 −0.951593 0.307360i \(-0.900555\pi\)
−0.951593 + 0.307360i \(0.900555\pi\)
\(660\) 0 0
\(661\) −12.1529 −0.472694 −0.236347 0.971669i \(-0.575950\pi\)
−0.236347 + 0.971669i \(0.575950\pi\)
\(662\) 0 0
\(663\) −7.17554 −0.278675
\(664\) 0 0
\(665\) 9.46214 0.366926
\(666\) 0 0
\(667\) −1.45490 −0.0563341
\(668\) 0 0
\(669\) −39.1722 −1.51449
\(670\) 0 0
\(671\) 8.23918 0.318070
\(672\) 0 0
\(673\) 9.20694 0.354901 0.177451 0.984130i \(-0.443215\pi\)
0.177451 + 0.984130i \(0.443215\pi\)
\(674\) 0 0
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) −39.8196 −1.53039 −0.765196 0.643797i \(-0.777358\pi\)
−0.765196 + 0.643797i \(0.777358\pi\)
\(678\) 0 0
\(679\) −6.22141 −0.238756
\(680\) 0 0
\(681\) 43.8486 1.68028
\(682\) 0 0
\(683\) 5.60544 0.214486 0.107243 0.994233i \(-0.465798\pi\)
0.107243 + 0.994233i \(0.465798\pi\)
\(684\) 0 0
\(685\) −2.82685 −0.108009
\(686\) 0 0
\(687\) −51.6537 −1.97071
\(688\) 0 0
\(689\) 8.45730 0.322197
\(690\) 0 0
\(691\) 2.84223 0.108123 0.0540617 0.998538i \(-0.482783\pi\)
0.0540617 + 0.998538i \(0.482783\pi\)
\(692\) 0 0
\(693\) −18.3719 −0.697893
\(694\) 0 0
\(695\) 2.36471 0.0896986
\(696\) 0 0
\(697\) −13.2552 −0.502077
\(698\) 0 0
\(699\) 51.9267 1.96405
\(700\) 0 0
\(701\) −24.4130 −0.922065 −0.461033 0.887383i \(-0.652521\pi\)
−0.461033 + 0.887383i \(0.652521\pi\)
\(702\) 0 0
\(703\) 10.3155 0.389058
\(704\) 0 0
\(705\) −14.5596 −0.548345
\(706\) 0 0
\(707\) 8.24797 0.310197
\(708\) 0 0
\(709\) 0.123983 0.00465629 0.00232814 0.999997i \(-0.499259\pi\)
0.00232814 + 0.999997i \(0.499259\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.46214 −0.242009
\(714\) 0 0
\(715\) −2.68840 −0.100540
\(716\) 0 0
\(717\) 63.2648 2.36267
\(718\) 0 0
\(719\) −47.1569 −1.75865 −0.879327 0.476218i \(-0.842007\pi\)
−0.879327 + 0.476218i \(0.842007\pi\)
\(720\) 0 0
\(721\) −10.2214 −0.380665
\(722\) 0 0
\(723\) −0.623208 −0.0231774
\(724\) 0 0
\(725\) 1.45490 0.0540338
\(726\) 0 0
\(727\) −43.2962 −1.60577 −0.802884 0.596135i \(-0.796702\pi\)
−0.802884 + 0.596135i \(0.796702\pi\)
\(728\) 0 0
\(729\) −42.0217 −1.55636
\(730\) 0 0
\(731\) −8.38972 −0.310305
\(732\) 0 0
\(733\) 11.5861 0.427943 0.213972 0.976840i \(-0.431360\pi\)
0.213972 + 0.976840i \(0.431360\pi\)
\(734\) 0 0
\(735\) 17.2480 0.636201
\(736\) 0 0
\(737\) −15.6103 −0.575012
\(738\) 0 0
\(739\) 30.3792 1.11752 0.558758 0.829331i \(-0.311278\pi\)
0.558758 + 0.829331i \(0.311278\pi\)
\(740\) 0 0
\(741\) −15.2407 −0.559882
\(742\) 0 0
\(743\) −35.5708 −1.30496 −0.652482 0.757804i \(-0.726272\pi\)
−0.652482 + 0.757804i \(0.726272\pi\)
\(744\) 0 0
\(745\) 11.9170 0.436607
\(746\) 0 0
\(747\) 64.2769 2.35177
\(748\) 0 0
\(749\) 48.2769 1.76400
\(750\) 0 0
\(751\) 24.9774 0.911438 0.455719 0.890124i \(-0.349382\pi\)
0.455719 + 0.890124i \(0.349382\pi\)
\(752\) 0 0
\(753\) 58.0555 2.11566
\(754\) 0 0
\(755\) 9.57889 0.348611
\(756\) 0 0
\(757\) 9.35263 0.339927 0.169964 0.985450i \(-0.445635\pi\)
0.169964 + 0.985450i \(0.445635\pi\)
\(758\) 0 0
\(759\) 3.24073 0.117631
\(760\) 0 0
\(761\) −18.1466 −0.657813 −0.328907 0.944362i \(-0.606680\pi\)
−0.328907 + 0.944362i \(0.606680\pi\)
\(762\) 0 0
\(763\) −12.5523 −0.454425
\(764\) 0 0
\(765\) −5.00724 −0.181037
\(766\) 0 0
\(767\) 9.40574 0.339622
\(768\) 0 0
\(769\) 44.3445 1.59910 0.799552 0.600597i \(-0.205070\pi\)
0.799552 + 0.600597i \(0.205070\pi\)
\(770\) 0 0
\(771\) 43.5370 1.56795
\(772\) 0 0
\(773\) −10.8036 −0.388578 −0.194289 0.980944i \(-0.562240\pi\)
−0.194289 + 0.980944i \(0.562240\pi\)
\(774\) 0 0
\(775\) 6.46214 0.232127
\(776\) 0 0
\(777\) 39.1722 1.40530
\(778\) 0 0
\(779\) −28.1538 −1.00872
\(780\) 0 0
\(781\) 8.39850 0.300522
\(782\) 0 0
\(783\) −4.36471 −0.155982
\(784\) 0 0
\(785\) 18.7439 0.668998
\(786\) 0 0
\(787\) 31.7810 1.13287 0.566435 0.824107i \(-0.308322\pi\)
0.566435 + 0.824107i \(0.308322\pi\)
\(788\) 0 0
\(789\) −6.77375 −0.241152
\(790\) 0 0
\(791\) 36.6908 1.30457
\(792\) 0 0
\(793\) 15.0250 0.533554
\(794\) 0 0
\(795\) 10.1949 0.361574
\(796\) 0 0
\(797\) 35.0594 1.24187 0.620935 0.783862i \(-0.286753\pi\)
0.620935 + 0.783862i \(0.286753\pi\)
\(798\) 0 0
\(799\) −6.62321 −0.234312
\(800\) 0 0
\(801\) −44.9919 −1.58971
\(802\) 0 0
\(803\) −18.4815 −0.652196
\(804\) 0 0
\(805\) −3.66908 −0.129318
\(806\) 0 0
\(807\) 6.79306 0.239127
\(808\) 0 0
\(809\) −51.4911 −1.81033 −0.905165 0.425060i \(-0.860253\pi\)
−0.905165 + 0.425060i \(0.860253\pi\)
\(810\) 0 0
\(811\) 19.7705 0.694235 0.347117 0.937822i \(-0.387161\pi\)
0.347117 + 0.937822i \(0.387161\pi\)
\(812\) 0 0
\(813\) 64.5370 2.26341
\(814\) 0 0
\(815\) −19.3188 −0.676709
\(816\) 0 0
\(817\) −17.8196 −0.623429
\(818\) 0 0
\(819\) −33.5032 −1.17070
\(820\) 0 0
\(821\) 31.8486 1.11152 0.555761 0.831342i \(-0.312427\pi\)
0.555761 + 0.831342i \(0.312427\pi\)
\(822\) 0 0
\(823\) −9.34210 −0.325645 −0.162823 0.986655i \(-0.552060\pi\)
−0.162823 + 0.986655i \(0.552060\pi\)
\(824\) 0 0
\(825\) −3.24073 −0.112828
\(826\) 0 0
\(827\) 30.9243 1.07534 0.537671 0.843155i \(-0.319304\pi\)
0.537671 + 0.843155i \(0.319304\pi\)
\(828\) 0 0
\(829\) −32.5701 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(830\) 0 0
\(831\) 56.6803 1.96622
\(832\) 0 0
\(833\) 7.84617 0.271854
\(834\) 0 0
\(835\) −1.81962 −0.0629704
\(836\) 0 0
\(837\) −19.3864 −0.670093
\(838\) 0 0
\(839\) −48.5635 −1.67660 −0.838299 0.545210i \(-0.816450\pi\)
−0.838299 + 0.545210i \(0.816450\pi\)
\(840\) 0 0
\(841\) −26.8833 −0.927009
\(842\) 0 0
\(843\) 37.0660 1.27662
\(844\) 0 0
\(845\) 8.09743 0.278560
\(846\) 0 0
\(847\) 34.9508 1.20092
\(848\) 0 0
\(849\) −78.8260 −2.70530
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −33.6546 −1.15231 −0.576156 0.817340i \(-0.695448\pi\)
−0.576156 + 0.817340i \(0.695448\pi\)
\(854\) 0 0
\(855\) −10.6353 −0.363719
\(856\) 0 0
\(857\) 24.4081 0.833766 0.416883 0.908960i \(-0.363122\pi\)
0.416883 + 0.908960i \(0.363122\pi\)
\(858\) 0 0
\(859\) 11.1394 0.380070 0.190035 0.981777i \(-0.439140\pi\)
0.190035 + 0.981777i \(0.439140\pi\)
\(860\) 0 0
\(861\) −106.911 −3.64353
\(862\) 0 0
\(863\) 15.7173 0.535025 0.267512 0.963554i \(-0.413798\pi\)
0.267512 + 0.963554i \(0.413798\pi\)
\(864\) 0 0
\(865\) −15.9581 −0.542591
\(866\) 0 0
\(867\) 41.4395 1.40736
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −28.4670 −0.964567
\(872\) 0 0
\(873\) 6.99276 0.236669
\(874\) 0 0
\(875\) 3.66908 0.124038
\(876\) 0 0
\(877\) −33.2818 −1.12385 −0.561923 0.827190i \(-0.689938\pi\)
−0.561923 + 0.827190i \(0.689938\pi\)
\(878\) 0 0
\(879\) 64.2769 2.16801
\(880\) 0 0
\(881\) 8.99187 0.302944 0.151472 0.988462i \(-0.451599\pi\)
0.151472 + 0.988462i \(0.451599\pi\)
\(882\) 0 0
\(883\) −58.5258 −1.96955 −0.984775 0.173836i \(-0.944384\pi\)
−0.984775 + 0.173836i \(0.944384\pi\)
\(884\) 0 0
\(885\) 11.3382 0.381128
\(886\) 0 0
\(887\) −32.8220 −1.10206 −0.551028 0.834487i \(-0.685764\pi\)
−0.551028 + 0.834487i \(0.685764\pi\)
\(888\) 0 0
\(889\) 73.2012 2.45509
\(890\) 0 0
\(891\) −5.29952 −0.177541
\(892\) 0 0
\(893\) −14.0676 −0.470754
\(894\) 0 0
\(895\) 17.9508 0.600030
\(896\) 0 0
\(897\) 5.90981 0.197323
\(898\) 0 0
\(899\) 9.40180 0.313567
\(900\) 0 0
\(901\) 4.63768 0.154504
\(902\) 0 0
\(903\) −67.6682 −2.25186
\(904\) 0 0
\(905\) −6.33092 −0.210447
\(906\) 0 0
\(907\) −28.3300 −0.940683 −0.470341 0.882484i \(-0.655869\pi\)
−0.470341 + 0.882484i \(0.655869\pi\)
\(908\) 0 0
\(909\) −9.27058 −0.307486
\(910\) 0 0
\(911\) −15.3382 −0.508176 −0.254088 0.967181i \(-0.581775\pi\)
−0.254088 + 0.967181i \(0.581775\pi\)
\(912\) 0 0
\(913\) 18.9243 0.626302
\(914\) 0 0
\(915\) 18.1119 0.598761
\(916\) 0 0
\(917\) −35.6006 −1.17564
\(918\) 0 0
\(919\) 38.8036 1.28001 0.640006 0.768370i \(-0.278932\pi\)
0.640006 + 0.768370i \(0.278932\pi\)
\(920\) 0 0
\(921\) 53.2310 1.75402
\(922\) 0 0
\(923\) 15.3155 0.504117
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) 11.4887 0.377338
\(928\) 0 0
\(929\) 4.01053 0.131581 0.0657906 0.997833i \(-0.479043\pi\)
0.0657906 + 0.997833i \(0.479043\pi\)
\(930\) 0 0
\(931\) 16.6651 0.546178
\(932\) 0 0
\(933\) −7.93636 −0.259825
\(934\) 0 0
\(935\) −1.47422 −0.0482122
\(936\) 0 0
\(937\) 43.6836 1.42708 0.713540 0.700615i \(-0.247091\pi\)
0.713540 + 0.700615i \(0.247091\pi\)
\(938\) 0 0
\(939\) −41.3406 −1.34910
\(940\) 0 0
\(941\) −24.8534 −0.810198 −0.405099 0.914273i \(-0.632763\pi\)
−0.405099 + 0.914273i \(0.632763\pi\)
\(942\) 0 0
\(943\) 10.9170 0.355508
\(944\) 0 0
\(945\) −11.0072 −0.358065
\(946\) 0 0
\(947\) 22.1690 0.720394 0.360197 0.932876i \(-0.382709\pi\)
0.360197 + 0.932876i \(0.382709\pi\)
\(948\) 0 0
\(949\) −33.7029 −1.09404
\(950\) 0 0
\(951\) −23.5596 −0.763971
\(952\) 0 0
\(953\) 15.9050 0.515212 0.257606 0.966250i \(-0.417066\pi\)
0.257606 + 0.966250i \(0.417066\pi\)
\(954\) 0 0
\(955\) −15.1578 −0.490494
\(956\) 0 0
\(957\) −4.71495 −0.152413
\(958\) 0 0
\(959\) −10.3719 −0.334928
\(960\) 0 0
\(961\) 10.7593 0.347073
\(962\) 0 0
\(963\) −54.2624 −1.74858
\(964\) 0 0
\(965\) −18.9879 −0.611243
\(966\) 0 0
\(967\) −50.6272 −1.62806 −0.814030 0.580823i \(-0.802731\pi\)
−0.814030 + 0.580823i \(0.802731\pi\)
\(968\) 0 0
\(969\) −8.35748 −0.268481
\(970\) 0 0
\(971\) 52.4009 1.68162 0.840812 0.541327i \(-0.182078\pi\)
0.840812 + 0.541327i \(0.182078\pi\)
\(972\) 0 0
\(973\) 8.67632 0.278150
\(974\) 0 0
\(975\) −5.90981 −0.189265
\(976\) 0 0
\(977\) 10.5258 0.336750 0.168375 0.985723i \(-0.446148\pi\)
0.168375 + 0.985723i \(0.446148\pi\)
\(978\) 0 0
\(979\) −13.2464 −0.423357
\(980\) 0 0
\(981\) 14.1086 0.450453
\(982\) 0 0
\(983\) −9.44767 −0.301334 −0.150667 0.988585i \(-0.548142\pi\)
−0.150667 + 0.988585i \(0.548142\pi\)
\(984\) 0 0
\(985\) 19.1650 0.610648
\(986\) 0 0
\(987\) −53.4202 −1.70038
\(988\) 0 0
\(989\) 6.90981 0.219719
\(990\) 0 0
\(991\) −33.3252 −1.05861 −0.529305 0.848432i \(-0.677547\pi\)
−0.529305 + 0.848432i \(0.677547\pi\)
\(992\) 0 0
\(993\) 46.4468 1.47394
\(994\) 0 0
\(995\) −5.76651 −0.182811
\(996\) 0 0
\(997\) 54.0289 1.71111 0.855557 0.517709i \(-0.173215\pi\)
0.855557 + 0.517709i \(0.173215\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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 1840.2.a.t.1.1 3
4.3 odd 2 920.2.a.g.1.3 3
5.4 even 2 9200.2.a.cd.1.3 3
8.3 odd 2 7360.2.a.cb.1.1 3
8.5 even 2 7360.2.a.ca.1.3 3
12.11 even 2 8280.2.a.bo.1.3 3
20.3 even 4 4600.2.e.r.4049.6 6
20.7 even 4 4600.2.e.r.4049.1 6
20.19 odd 2 4600.2.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.3 3 4.3 odd 2
1840.2.a.t.1.1 3 1.1 even 1 trivial
4600.2.a.y.1.1 3 20.19 odd 2
4600.2.e.r.4049.1 6 20.7 even 4
4600.2.e.r.4049.6 6 20.3 even 4
7360.2.a.ca.1.3 3 8.5 even 2
7360.2.a.cb.1.1 3 8.3 odd 2
8280.2.a.bo.1.3 3 12.11 even 2
9200.2.a.cd.1.3 3 5.4 even 2