Properties

Label 3120.2.a.bg.1.1
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +5.56155 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.68466 q^{17} -3.12311 q^{19} -1.56155 q^{21} +5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +7.12311 q^{31} +5.56155 q^{33} -1.56155 q^{35} +9.80776 q^{37} +1.00000 q^{39} +2.68466 q^{41} +10.2462 q^{43} +1.00000 q^{45} -7.12311 q^{47} -4.56155 q^{49} -6.68466 q^{51} +3.56155 q^{53} +5.56155 q^{55} -3.12311 q^{57} +8.00000 q^{59} -10.6847 q^{61} -1.56155 q^{63} +1.00000 q^{65} +14.2462 q^{67} +5.56155 q^{69} -4.68466 q^{71} +16.2462 q^{73} +1.00000 q^{75} -8.68466 q^{77} -11.8078 q^{79} +1.00000 q^{81} +8.00000 q^{83} -6.68466 q^{85} -2.00000 q^{87} -14.6847 q^{89} -1.56155 q^{91} +7.12311 q^{93} -3.12311 q^{95} -17.8078 q^{97} +5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} - q^{37} + 2 q^{39} - 7 q^{41} + 4 q^{43} + 2 q^{45} - 6 q^{47} - 5 q^{49} - q^{51} + 3 q^{53} + 7 q^{55} + 2 q^{57} + 16 q^{59} - 9 q^{61} + q^{63} + 2 q^{65} + 12 q^{67} + 7 q^{69} + 3 q^{71} + 16 q^{73} + 2 q^{75} - 5 q^{77} - 3 q^{79} + 2 q^{81} + 16 q^{83} - q^{85} - 4 q^{87} - 17 q^{89} + q^{91} + 6 q^{93} + 2 q^{95} - 15 q^{97} + 7 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.56155 1.67687 0.838436 0.545001i \(-0.183471\pi\)
0.838436 + 0.545001i \(0.183471\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −6.68466 −1.62127 −0.810634 0.585553i \(-0.800877\pi\)
−0.810634 + 0.585553i \(0.800877\pi\)
\(18\) 0 0
\(19\) −3.12311 −0.716490 −0.358245 0.933628i \(-0.616625\pi\)
−0.358245 + 0.933628i \(0.616625\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 5.56155 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 7.12311 1.27935 0.639674 0.768647i \(-0.279069\pi\)
0.639674 + 0.768647i \(0.279069\pi\)
\(32\) 0 0
\(33\) 5.56155 0.968142
\(34\) 0 0
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) 9.80776 1.61239 0.806193 0.591652i \(-0.201524\pi\)
0.806193 + 0.591652i \(0.201524\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.68466 0.419273 0.209637 0.977779i \(-0.432772\pi\)
0.209637 + 0.977779i \(0.432772\pi\)
\(42\) 0 0
\(43\) 10.2462 1.56253 0.781266 0.624198i \(-0.214574\pi\)
0.781266 + 0.624198i \(0.214574\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −7.12311 −1.03901 −0.519506 0.854467i \(-0.673884\pi\)
−0.519506 + 0.854467i \(0.673884\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) −6.68466 −0.936039
\(52\) 0 0
\(53\) 3.56155 0.489217 0.244608 0.969622i \(-0.421341\pi\)
0.244608 + 0.969622i \(0.421341\pi\)
\(54\) 0 0
\(55\) 5.56155 0.749920
\(56\) 0 0
\(57\) −3.12311 −0.413665
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −10.6847 −1.36803 −0.684015 0.729468i \(-0.739768\pi\)
−0.684015 + 0.729468i \(0.739768\pi\)
\(62\) 0 0
\(63\) −1.56155 −0.196737
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 14.2462 1.74045 0.870226 0.492653i \(-0.163973\pi\)
0.870226 + 0.492653i \(0.163973\pi\)
\(68\) 0 0
\(69\) 5.56155 0.669532
\(70\) 0 0
\(71\) −4.68466 −0.555967 −0.277983 0.960586i \(-0.589666\pi\)
−0.277983 + 0.960586i \(0.589666\pi\)
\(72\) 0 0
\(73\) 16.2462 1.90148 0.950738 0.309997i \(-0.100328\pi\)
0.950738 + 0.309997i \(0.100328\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −8.68466 −0.989709
\(78\) 0 0
\(79\) −11.8078 −1.32848 −0.664239 0.747521i \(-0.731244\pi\)
−0.664239 + 0.747521i \(0.731244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −6.68466 −0.725053
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −14.6847 −1.55657 −0.778285 0.627911i \(-0.783910\pi\)
−0.778285 + 0.627911i \(0.783910\pi\)
\(90\) 0 0
\(91\) −1.56155 −0.163695
\(92\) 0 0
\(93\) 7.12311 0.738632
\(94\) 0 0
\(95\) −3.12311 −0.320424
\(96\) 0 0
\(97\) −17.8078 −1.80810 −0.904052 0.427422i \(-0.859422\pi\)
−0.904052 + 0.427422i \(0.859422\pi\)
\(98\) 0 0
\(99\) 5.56155 0.558957
\(100\) 0 0
\(101\) 1.12311 0.111753 0.0558766 0.998438i \(-0.482205\pi\)
0.0558766 + 0.998438i \(0.482205\pi\)
\(102\) 0 0
\(103\) −14.2462 −1.40372 −0.701860 0.712314i \(-0.747647\pi\)
−0.701860 + 0.712314i \(0.747647\pi\)
\(104\) 0 0
\(105\) −1.56155 −0.152392
\(106\) 0 0
\(107\) 15.8078 1.52819 0.764097 0.645101i \(-0.223185\pi\)
0.764097 + 0.645101i \(0.223185\pi\)
\(108\) 0 0
\(109\) −13.1231 −1.25697 −0.628483 0.777824i \(-0.716324\pi\)
−0.628483 + 0.777824i \(0.716324\pi\)
\(110\) 0 0
\(111\) 9.80776 0.930912
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 5.56155 0.518617
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 10.4384 0.956891
\(120\) 0 0
\(121\) 19.9309 1.81190
\(122\) 0 0
\(123\) 2.68466 0.242067
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.3693 1.54128 0.770639 0.637272i \(-0.219937\pi\)
0.770639 + 0.637272i \(0.219937\pi\)
\(128\) 0 0
\(129\) 10.2462 0.902129
\(130\) 0 0
\(131\) 15.1231 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(132\) 0 0
\(133\) 4.87689 0.422880
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 16.2462 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(138\) 0 0
\(139\) 9.56155 0.811000 0.405500 0.914095i \(-0.367097\pi\)
0.405500 + 0.914095i \(0.367097\pi\)
\(140\) 0 0
\(141\) −7.12311 −0.599874
\(142\) 0 0
\(143\) 5.56155 0.465080
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) −4.56155 −0.376231
\(148\) 0 0
\(149\) −12.4384 −1.01900 −0.509499 0.860471i \(-0.670169\pi\)
−0.509499 + 0.860471i \(0.670169\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −6.68466 −0.540423
\(154\) 0 0
\(155\) 7.12311 0.572142
\(156\) 0 0
\(157\) 10.4924 0.837386 0.418693 0.908128i \(-0.362488\pi\)
0.418693 + 0.908128i \(0.362488\pi\)
\(158\) 0 0
\(159\) 3.56155 0.282450
\(160\) 0 0
\(161\) −8.68466 −0.684447
\(162\) 0 0
\(163\) −13.5616 −1.06222 −0.531111 0.847302i \(-0.678225\pi\)
−0.531111 + 0.847302i \(0.678225\pi\)
\(164\) 0 0
\(165\) 5.56155 0.432966
\(166\) 0 0
\(167\) 5.75379 0.445242 0.222621 0.974905i \(-0.428539\pi\)
0.222621 + 0.974905i \(0.428539\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.12311 −0.238830
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −1.56155 −0.118042
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) 23.1231 1.72830 0.864151 0.503233i \(-0.167856\pi\)
0.864151 + 0.503233i \(0.167856\pi\)
\(180\) 0 0
\(181\) −8.93087 −0.663826 −0.331913 0.943310i \(-0.607694\pi\)
−0.331913 + 0.943310i \(0.607694\pi\)
\(182\) 0 0
\(183\) −10.6847 −0.789833
\(184\) 0 0
\(185\) 9.80776 0.721081
\(186\) 0 0
\(187\) −37.1771 −2.71866
\(188\) 0 0
\(189\) −1.56155 −0.113586
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 9.31534 0.670533 0.335266 0.942123i \(-0.391174\pi\)
0.335266 + 0.942123i \(0.391174\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 7.36932 0.525042 0.262521 0.964926i \(-0.415446\pi\)
0.262521 + 0.964926i \(0.415446\pi\)
\(198\) 0 0
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 0 0
\(201\) 14.2462 1.00485
\(202\) 0 0
\(203\) 3.12311 0.219199
\(204\) 0 0
\(205\) 2.68466 0.187505
\(206\) 0 0
\(207\) 5.56155 0.386555
\(208\) 0 0
\(209\) −17.3693 −1.20146
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −4.68466 −0.320988
\(214\) 0 0
\(215\) 10.2462 0.698786
\(216\) 0 0
\(217\) −11.1231 −0.755086
\(218\) 0 0
\(219\) 16.2462 1.09782
\(220\) 0 0
\(221\) −6.68466 −0.449659
\(222\) 0 0
\(223\) 2.24621 0.150417 0.0752087 0.997168i \(-0.476038\pi\)
0.0752087 + 0.997168i \(0.476038\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −25.3693 −1.68382 −0.841910 0.539617i \(-0.818569\pi\)
−0.841910 + 0.539617i \(0.818569\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −8.68466 −0.571409
\(232\) 0 0
\(233\) 18.6847 1.22407 0.612036 0.790830i \(-0.290351\pi\)
0.612036 + 0.790830i \(0.290351\pi\)
\(234\) 0 0
\(235\) −7.12311 −0.464660
\(236\) 0 0
\(237\) −11.8078 −0.766997
\(238\) 0 0
\(239\) 3.31534 0.214452 0.107226 0.994235i \(-0.465803\pi\)
0.107226 + 0.994235i \(0.465803\pi\)
\(240\) 0 0
\(241\) 11.7538 0.757128 0.378564 0.925575i \(-0.376418\pi\)
0.378564 + 0.925575i \(0.376418\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.56155 −0.291427
\(246\) 0 0
\(247\) −3.12311 −0.198718
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −8.49242 −0.536037 −0.268018 0.963414i \(-0.586369\pi\)
−0.268018 + 0.963414i \(0.586369\pi\)
\(252\) 0 0
\(253\) 30.9309 1.94461
\(254\) 0 0
\(255\) −6.68466 −0.418610
\(256\) 0 0
\(257\) 14.4924 0.904012 0.452006 0.892015i \(-0.350708\pi\)
0.452006 + 0.892015i \(0.350708\pi\)
\(258\) 0 0
\(259\) −15.3153 −0.951649
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −28.4924 −1.75692 −0.878459 0.477818i \(-0.841428\pi\)
−0.878459 + 0.477818i \(0.841428\pi\)
\(264\) 0 0
\(265\) 3.56155 0.218784
\(266\) 0 0
\(267\) −14.6847 −0.898687
\(268\) 0 0
\(269\) −13.1231 −0.800130 −0.400065 0.916487i \(-0.631012\pi\)
−0.400065 + 0.916487i \(0.631012\pi\)
\(270\) 0 0
\(271\) 5.36932 0.326163 0.163081 0.986613i \(-0.447857\pi\)
0.163081 + 0.986613i \(0.447857\pi\)
\(272\) 0 0
\(273\) −1.56155 −0.0945095
\(274\) 0 0
\(275\) 5.56155 0.335374
\(276\) 0 0
\(277\) −22.4924 −1.35144 −0.675719 0.737159i \(-0.736167\pi\)
−0.675719 + 0.737159i \(0.736167\pi\)
\(278\) 0 0
\(279\) 7.12311 0.426449
\(280\) 0 0
\(281\) 3.75379 0.223932 0.111966 0.993712i \(-0.464285\pi\)
0.111966 + 0.993712i \(0.464285\pi\)
\(282\) 0 0
\(283\) −30.7386 −1.82722 −0.913611 0.406589i \(-0.866718\pi\)
−0.913611 + 0.406589i \(0.866718\pi\)
\(284\) 0 0
\(285\) −3.12311 −0.184997
\(286\) 0 0
\(287\) −4.19224 −0.247460
\(288\) 0 0
\(289\) 27.6847 1.62851
\(290\) 0 0
\(291\) −17.8078 −1.04391
\(292\) 0 0
\(293\) −17.6155 −1.02911 −0.514555 0.857457i \(-0.672043\pi\)
−0.514555 + 0.857457i \(0.672043\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) 5.56155 0.321633
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 1.12311 0.0645207
\(304\) 0 0
\(305\) −10.6847 −0.611802
\(306\) 0 0
\(307\) −13.5616 −0.773999 −0.386999 0.922080i \(-0.626488\pi\)
−0.386999 + 0.922080i \(0.626488\pi\)
\(308\) 0 0
\(309\) −14.2462 −0.810439
\(310\) 0 0
\(311\) 26.7386 1.51621 0.758104 0.652133i \(-0.226126\pi\)
0.758104 + 0.652133i \(0.226126\pi\)
\(312\) 0 0
\(313\) −24.7386 −1.39831 −0.699155 0.714970i \(-0.746440\pi\)
−0.699155 + 0.714970i \(0.746440\pi\)
\(314\) 0 0
\(315\) −1.56155 −0.0879835
\(316\) 0 0
\(317\) 2.87689 0.161582 0.0807912 0.996731i \(-0.474255\pi\)
0.0807912 + 0.996731i \(0.474255\pi\)
\(318\) 0 0
\(319\) −11.1231 −0.622774
\(320\) 0 0
\(321\) 15.8078 0.882303
\(322\) 0 0
\(323\) 20.8769 1.16162
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −13.1231 −0.725709
\(328\) 0 0
\(329\) 11.1231 0.613237
\(330\) 0 0
\(331\) −9.75379 −0.536117 −0.268058 0.963403i \(-0.586382\pi\)
−0.268058 + 0.963403i \(0.586382\pi\)
\(332\) 0 0
\(333\) 9.80776 0.537462
\(334\) 0 0
\(335\) 14.2462 0.778354
\(336\) 0 0
\(337\) −2.87689 −0.156714 −0.0783572 0.996925i \(-0.524967\pi\)
−0.0783572 + 0.996925i \(0.524967\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 39.6155 2.14530
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 0 0
\(345\) 5.56155 0.299424
\(346\) 0 0
\(347\) 6.05398 0.324994 0.162497 0.986709i \(-0.448045\pi\)
0.162497 + 0.986709i \(0.448045\pi\)
\(348\) 0 0
\(349\) −0.630683 −0.0337597 −0.0168798 0.999858i \(-0.505373\pi\)
−0.0168798 + 0.999858i \(0.505373\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −4.24621 −0.226003 −0.113002 0.993595i \(-0.536046\pi\)
−0.113002 + 0.993595i \(0.536046\pi\)
\(354\) 0 0
\(355\) −4.68466 −0.248636
\(356\) 0 0
\(357\) 10.4384 0.552461
\(358\) 0 0
\(359\) 2.24621 0.118550 0.0592752 0.998242i \(-0.481121\pi\)
0.0592752 + 0.998242i \(0.481121\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) 0 0
\(363\) 19.9309 1.04610
\(364\) 0 0
\(365\) 16.2462 0.850366
\(366\) 0 0
\(367\) −14.2462 −0.743646 −0.371823 0.928304i \(-0.621267\pi\)
−0.371823 + 0.928304i \(0.621267\pi\)
\(368\) 0 0
\(369\) 2.68466 0.139758
\(370\) 0 0
\(371\) −5.56155 −0.288741
\(372\) 0 0
\(373\) −14.8769 −0.770296 −0.385148 0.922855i \(-0.625850\pi\)
−0.385148 + 0.922855i \(0.625850\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −11.1231 −0.571356 −0.285678 0.958326i \(-0.592219\pi\)
−0.285678 + 0.958326i \(0.592219\pi\)
\(380\) 0 0
\(381\) 17.3693 0.889857
\(382\) 0 0
\(383\) −2.63068 −0.134422 −0.0672108 0.997739i \(-0.521410\pi\)
−0.0672108 + 0.997739i \(0.521410\pi\)
\(384\) 0 0
\(385\) −8.68466 −0.442611
\(386\) 0 0
\(387\) 10.2462 0.520844
\(388\) 0 0
\(389\) 35.8617 1.81826 0.909131 0.416510i \(-0.136747\pi\)
0.909131 + 0.416510i \(0.136747\pi\)
\(390\) 0 0
\(391\) −37.1771 −1.88013
\(392\) 0 0
\(393\) 15.1231 0.762860
\(394\) 0 0
\(395\) −11.8078 −0.594113
\(396\) 0 0
\(397\) 33.8078 1.69676 0.848382 0.529385i \(-0.177577\pi\)
0.848382 + 0.529385i \(0.177577\pi\)
\(398\) 0 0
\(399\) 4.87689 0.244150
\(400\) 0 0
\(401\) −4.24621 −0.212046 −0.106023 0.994364i \(-0.533812\pi\)
−0.106023 + 0.994364i \(0.533812\pi\)
\(402\) 0 0
\(403\) 7.12311 0.354827
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 54.5464 2.70376
\(408\) 0 0
\(409\) −18.4924 −0.914391 −0.457196 0.889366i \(-0.651146\pi\)
−0.457196 + 0.889366i \(0.651146\pi\)
\(410\) 0 0
\(411\) 16.2462 0.801367
\(412\) 0 0
\(413\) −12.4924 −0.614712
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 9.56155 0.468231
\(418\) 0 0
\(419\) −0.876894 −0.0428391 −0.0214195 0.999771i \(-0.506819\pi\)
−0.0214195 + 0.999771i \(0.506819\pi\)
\(420\) 0 0
\(421\) 19.8617 0.968002 0.484001 0.875067i \(-0.339183\pi\)
0.484001 + 0.875067i \(0.339183\pi\)
\(422\) 0 0
\(423\) −7.12311 −0.346337
\(424\) 0 0
\(425\) −6.68466 −0.324254
\(426\) 0 0
\(427\) 16.6847 0.807427
\(428\) 0 0
\(429\) 5.56155 0.268514
\(430\) 0 0
\(431\) −13.7538 −0.662497 −0.331248 0.943544i \(-0.607470\pi\)
−0.331248 + 0.943544i \(0.607470\pi\)
\(432\) 0 0
\(433\) 35.3693 1.69974 0.849870 0.526992i \(-0.176680\pi\)
0.849870 + 0.526992i \(0.176680\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) −17.3693 −0.830887
\(438\) 0 0
\(439\) −6.93087 −0.330792 −0.165396 0.986227i \(-0.552890\pi\)
−0.165396 + 0.986227i \(0.552890\pi\)
\(440\) 0 0
\(441\) −4.56155 −0.217217
\(442\) 0 0
\(443\) 3.31534 0.157517 0.0787583 0.996894i \(-0.474904\pi\)
0.0787583 + 0.996894i \(0.474904\pi\)
\(444\) 0 0
\(445\) −14.6847 −0.696120
\(446\) 0 0
\(447\) −12.4384 −0.588318
\(448\) 0 0
\(449\) 10.6847 0.504240 0.252120 0.967696i \(-0.418872\pi\)
0.252120 + 0.967696i \(0.418872\pi\)
\(450\) 0 0
\(451\) 14.9309 0.703067
\(452\) 0 0
\(453\) −4.00000 −0.187936
\(454\) 0 0
\(455\) −1.56155 −0.0732067
\(456\) 0 0
\(457\) −35.5616 −1.66350 −0.831750 0.555151i \(-0.812660\pi\)
−0.831750 + 0.555151i \(0.812660\pi\)
\(458\) 0 0
\(459\) −6.68466 −0.312013
\(460\) 0 0
\(461\) −15.5616 −0.724774 −0.362387 0.932028i \(-0.618038\pi\)
−0.362387 + 0.932028i \(0.618038\pi\)
\(462\) 0 0
\(463\) 6.43845 0.299220 0.149610 0.988745i \(-0.452198\pi\)
0.149610 + 0.988745i \(0.452198\pi\)
\(464\) 0 0
\(465\) 7.12311 0.330326
\(466\) 0 0
\(467\) −25.1771 −1.16506 −0.582528 0.812811i \(-0.697936\pi\)
−0.582528 + 0.812811i \(0.697936\pi\)
\(468\) 0 0
\(469\) −22.2462 −1.02723
\(470\) 0 0
\(471\) 10.4924 0.483465
\(472\) 0 0
\(473\) 56.9848 2.62017
\(474\) 0 0
\(475\) −3.12311 −0.143298
\(476\) 0 0
\(477\) 3.56155 0.163072
\(478\) 0 0
\(479\) −34.5464 −1.57847 −0.789233 0.614094i \(-0.789521\pi\)
−0.789233 + 0.614094i \(0.789521\pi\)
\(480\) 0 0
\(481\) 9.80776 0.447196
\(482\) 0 0
\(483\) −8.68466 −0.395166
\(484\) 0 0
\(485\) −17.8078 −0.808609
\(486\) 0 0
\(487\) −4.68466 −0.212282 −0.106141 0.994351i \(-0.533850\pi\)
−0.106141 + 0.994351i \(0.533850\pi\)
\(488\) 0 0
\(489\) −13.5616 −0.613275
\(490\) 0 0
\(491\) 7.50758 0.338812 0.169406 0.985546i \(-0.445815\pi\)
0.169406 + 0.985546i \(0.445815\pi\)
\(492\) 0 0
\(493\) 13.3693 0.602124
\(494\) 0 0
\(495\) 5.56155 0.249973
\(496\) 0 0
\(497\) 7.31534 0.328138
\(498\) 0 0
\(499\) 37.8617 1.69492 0.847462 0.530856i \(-0.178129\pi\)
0.847462 + 0.530856i \(0.178129\pi\)
\(500\) 0 0
\(501\) 5.75379 0.257060
\(502\) 0 0
\(503\) −18.7386 −0.835514 −0.417757 0.908559i \(-0.637184\pi\)
−0.417757 + 0.908559i \(0.637184\pi\)
\(504\) 0 0
\(505\) 1.12311 0.0499775
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −42.6847 −1.89196 −0.945982 0.324219i \(-0.894899\pi\)
−0.945982 + 0.324219i \(0.894899\pi\)
\(510\) 0 0
\(511\) −25.3693 −1.12227
\(512\) 0 0
\(513\) −3.12311 −0.137888
\(514\) 0 0
\(515\) −14.2462 −0.627763
\(516\) 0 0
\(517\) −39.6155 −1.74229
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −13.7538 −0.601411 −0.300706 0.953717i \(-0.597222\pi\)
−0.300706 + 0.953717i \(0.597222\pi\)
\(524\) 0 0
\(525\) −1.56155 −0.0681518
\(526\) 0 0
\(527\) −47.6155 −2.07416
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 2.68466 0.116285
\(534\) 0 0
\(535\) 15.8078 0.683429
\(536\) 0 0
\(537\) 23.1231 0.997836
\(538\) 0 0
\(539\) −25.3693 −1.09273
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) −8.93087 −0.383260
\(544\) 0 0
\(545\) −13.1231 −0.562132
\(546\) 0 0
\(547\) 6.73863 0.288123 0.144062 0.989569i \(-0.453984\pi\)
0.144062 + 0.989569i \(0.453984\pi\)
\(548\) 0 0
\(549\) −10.6847 −0.456010
\(550\) 0 0
\(551\) 6.24621 0.266098
\(552\) 0 0
\(553\) 18.4384 0.784083
\(554\) 0 0
\(555\) 9.80776 0.416316
\(556\) 0 0
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 0 0
\(559\) 10.2462 0.433369
\(560\) 0 0
\(561\) −37.1771 −1.56962
\(562\) 0 0
\(563\) 25.5616 1.07729 0.538646 0.842533i \(-0.318936\pi\)
0.538646 + 0.842533i \(0.318936\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) −1.56155 −0.0655791
\(568\) 0 0
\(569\) −1.12311 −0.0470830 −0.0235415 0.999723i \(-0.507494\pi\)
−0.0235415 + 0.999723i \(0.507494\pi\)
\(570\) 0 0
\(571\) −25.5616 −1.06972 −0.534859 0.844941i \(-0.679635\pi\)
−0.534859 + 0.844941i \(0.679635\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 5.56155 0.231933
\(576\) 0 0
\(577\) −36.5464 −1.52145 −0.760723 0.649076i \(-0.775156\pi\)
−0.760723 + 0.649076i \(0.775156\pi\)
\(578\) 0 0
\(579\) 9.31534 0.387132
\(580\) 0 0
\(581\) −12.4924 −0.518273
\(582\) 0 0
\(583\) 19.8078 0.820354
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 17.3693 0.716908 0.358454 0.933547i \(-0.383304\pi\)
0.358454 + 0.933547i \(0.383304\pi\)
\(588\) 0 0
\(589\) −22.2462 −0.916639
\(590\) 0 0
\(591\) 7.36932 0.303133
\(592\) 0 0
\(593\) −5.61553 −0.230602 −0.115301 0.993331i \(-0.536783\pi\)
−0.115301 + 0.993331i \(0.536783\pi\)
\(594\) 0 0
\(595\) 10.4384 0.427935
\(596\) 0 0
\(597\) −1.75379 −0.0717778
\(598\) 0 0
\(599\) 18.7386 0.765640 0.382820 0.923823i \(-0.374953\pi\)
0.382820 + 0.923823i \(0.374953\pi\)
\(600\) 0 0
\(601\) −3.56155 −0.145279 −0.0726394 0.997358i \(-0.523142\pi\)
−0.0726394 + 0.997358i \(0.523142\pi\)
\(602\) 0 0
\(603\) 14.2462 0.580151
\(604\) 0 0
\(605\) 19.9309 0.810305
\(606\) 0 0
\(607\) −26.7386 −1.08529 −0.542644 0.839963i \(-0.682577\pi\)
−0.542644 + 0.839963i \(0.682577\pi\)
\(608\) 0 0
\(609\) 3.12311 0.126555
\(610\) 0 0
\(611\) −7.12311 −0.288170
\(612\) 0 0
\(613\) 4.93087 0.199156 0.0995780 0.995030i \(-0.468251\pi\)
0.0995780 + 0.995030i \(0.468251\pi\)
\(614\) 0 0
\(615\) 2.68466 0.108256
\(616\) 0 0
\(617\) 0.246211 0.00991209 0.00495605 0.999988i \(-0.498422\pi\)
0.00495605 + 0.999988i \(0.498422\pi\)
\(618\) 0 0
\(619\) −6.63068 −0.266510 −0.133255 0.991082i \(-0.542543\pi\)
−0.133255 + 0.991082i \(0.542543\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 22.9309 0.918706
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.3693 −0.693664
\(628\) 0 0
\(629\) −65.5616 −2.61411
\(630\) 0 0
\(631\) −26.2462 −1.04485 −0.522423 0.852687i \(-0.674972\pi\)
−0.522423 + 0.852687i \(0.674972\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 17.3693 0.689280
\(636\) 0 0
\(637\) −4.56155 −0.180735
\(638\) 0 0
\(639\) −4.68466 −0.185322
\(640\) 0 0
\(641\) 0.630683 0.0249105 0.0124552 0.999922i \(-0.496035\pi\)
0.0124552 + 0.999922i \(0.496035\pi\)
\(642\) 0 0
\(643\) −11.8078 −0.465653 −0.232826 0.972518i \(-0.574797\pi\)
−0.232826 + 0.972518i \(0.574797\pi\)
\(644\) 0 0
\(645\) 10.2462 0.403444
\(646\) 0 0
\(647\) 0.684658 0.0269167 0.0134584 0.999909i \(-0.495716\pi\)
0.0134584 + 0.999909i \(0.495716\pi\)
\(648\) 0 0
\(649\) 44.4924 1.74648
\(650\) 0 0
\(651\) −11.1231 −0.435949
\(652\) 0 0
\(653\) 23.7538 0.929558 0.464779 0.885427i \(-0.346134\pi\)
0.464779 + 0.885427i \(0.346134\pi\)
\(654\) 0 0
\(655\) 15.1231 0.590909
\(656\) 0 0
\(657\) 16.2462 0.633825
\(658\) 0 0
\(659\) −8.49242 −0.330818 −0.165409 0.986225i \(-0.552894\pi\)
−0.165409 + 0.986225i \(0.552894\pi\)
\(660\) 0 0
\(661\) −30.8769 −1.20097 −0.600486 0.799635i \(-0.705026\pi\)
−0.600486 + 0.799635i \(0.705026\pi\)
\(662\) 0 0
\(663\) −6.68466 −0.259611
\(664\) 0 0
\(665\) 4.87689 0.189118
\(666\) 0 0
\(667\) −11.1231 −0.430688
\(668\) 0 0
\(669\) 2.24621 0.0868435
\(670\) 0 0
\(671\) −59.4233 −2.29401
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 7.06913 0.271689 0.135844 0.990730i \(-0.456625\pi\)
0.135844 + 0.990730i \(0.456625\pi\)
\(678\) 0 0
\(679\) 27.8078 1.06716
\(680\) 0 0
\(681\) −25.3693 −0.972154
\(682\) 0 0
\(683\) −4.49242 −0.171898 −0.0859489 0.996300i \(-0.527392\pi\)
−0.0859489 + 0.996300i \(0.527392\pi\)
\(684\) 0 0
\(685\) 16.2462 0.620736
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) 3.56155 0.135684
\(690\) 0 0
\(691\) −28.8769 −1.09853 −0.549264 0.835649i \(-0.685092\pi\)
−0.549264 + 0.835649i \(0.685092\pi\)
\(692\) 0 0
\(693\) −8.68466 −0.329903
\(694\) 0 0
\(695\) 9.56155 0.362690
\(696\) 0 0
\(697\) −17.9460 −0.679754
\(698\) 0 0
\(699\) 18.6847 0.706719
\(700\) 0 0
\(701\) −19.3693 −0.731569 −0.365785 0.930700i \(-0.619199\pi\)
−0.365785 + 0.930700i \(0.619199\pi\)
\(702\) 0 0
\(703\) −30.6307 −1.15526
\(704\) 0 0
\(705\) −7.12311 −0.268272
\(706\) 0 0
\(707\) −1.75379 −0.0659580
\(708\) 0 0
\(709\) 17.1231 0.643072 0.321536 0.946897i \(-0.395801\pi\)
0.321536 + 0.946897i \(0.395801\pi\)
\(710\) 0 0
\(711\) −11.8078 −0.442826
\(712\) 0 0
\(713\) 39.6155 1.48361
\(714\) 0 0
\(715\) 5.56155 0.207990
\(716\) 0 0
\(717\) 3.31534 0.123814
\(718\) 0 0
\(719\) −9.75379 −0.363755 −0.181877 0.983321i \(-0.558217\pi\)
−0.181877 + 0.983321i \(0.558217\pi\)
\(720\) 0 0
\(721\) 22.2462 0.828492
\(722\) 0 0
\(723\) 11.7538 0.437128
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 22.6307 0.839326 0.419663 0.907680i \(-0.362148\pi\)
0.419663 + 0.907680i \(0.362148\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −68.4924 −2.53328
\(732\) 0 0
\(733\) −29.4233 −1.08677 −0.543387 0.839482i \(-0.682858\pi\)
−0.543387 + 0.839482i \(0.682858\pi\)
\(734\) 0 0
\(735\) −4.56155 −0.168255
\(736\) 0 0
\(737\) 79.2311 2.91851
\(738\) 0 0
\(739\) −29.8617 −1.09848 −0.549241 0.835664i \(-0.685083\pi\)
−0.549241 + 0.835664i \(0.685083\pi\)
\(740\) 0 0
\(741\) −3.12311 −0.114730
\(742\) 0 0
\(743\) 8.87689 0.325662 0.162831 0.986654i \(-0.447938\pi\)
0.162831 + 0.986654i \(0.447938\pi\)
\(744\) 0 0
\(745\) −12.4384 −0.455709
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −24.6847 −0.901958
\(750\) 0 0
\(751\) 7.31534 0.266941 0.133470 0.991053i \(-0.457388\pi\)
0.133470 + 0.991053i \(0.457388\pi\)
\(752\) 0 0
\(753\) −8.49242 −0.309481
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −14.8769 −0.540710 −0.270355 0.962761i \(-0.587141\pi\)
−0.270355 + 0.962761i \(0.587141\pi\)
\(758\) 0 0
\(759\) 30.9309 1.12272
\(760\) 0 0
\(761\) 22.4924 0.815350 0.407675 0.913127i \(-0.366340\pi\)
0.407675 + 0.913127i \(0.366340\pi\)
\(762\) 0 0
\(763\) 20.4924 0.741876
\(764\) 0 0
\(765\) −6.68466 −0.241684
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 25.6155 0.923720 0.461860 0.886953i \(-0.347182\pi\)
0.461860 + 0.886953i \(0.347182\pi\)
\(770\) 0 0
\(771\) 14.4924 0.521932
\(772\) 0 0
\(773\) −21.1231 −0.759745 −0.379873 0.925039i \(-0.624032\pi\)
−0.379873 + 0.925039i \(0.624032\pi\)
\(774\) 0 0
\(775\) 7.12311 0.255870
\(776\) 0 0
\(777\) −15.3153 −0.549435
\(778\) 0 0
\(779\) −8.38447 −0.300405
\(780\) 0 0
\(781\) −26.0540 −0.932285
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 10.4924 0.374491
\(786\) 0 0
\(787\) 18.7386 0.667960 0.333980 0.942580i \(-0.391608\pi\)
0.333980 + 0.942580i \(0.391608\pi\)
\(788\) 0 0
\(789\) −28.4924 −1.01436
\(790\) 0 0
\(791\) −3.12311 −0.111045
\(792\) 0 0
\(793\) −10.6847 −0.379423
\(794\) 0 0
\(795\) 3.56155 0.126315
\(796\) 0 0
\(797\) 41.4233 1.46729 0.733644 0.679534i \(-0.237818\pi\)
0.733644 + 0.679534i \(0.237818\pi\)
\(798\) 0 0
\(799\) 47.6155 1.68452
\(800\) 0 0
\(801\) −14.6847 −0.518857
\(802\) 0 0
\(803\) 90.3542 3.18853
\(804\) 0 0
\(805\) −8.68466 −0.306094
\(806\) 0 0
\(807\) −13.1231 −0.461955
\(808\) 0 0
\(809\) −18.4924 −0.650159 −0.325079 0.945687i \(-0.605391\pi\)
−0.325079 + 0.945687i \(0.605391\pi\)
\(810\) 0 0
\(811\) −40.9848 −1.43917 −0.719586 0.694403i \(-0.755669\pi\)
−0.719586 + 0.694403i \(0.755669\pi\)
\(812\) 0 0
\(813\) 5.36932 0.188310
\(814\) 0 0
\(815\) −13.5616 −0.475040
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) −1.56155 −0.0545651
\(820\) 0 0
\(821\) −33.3153 −1.16271 −0.581357 0.813649i \(-0.697478\pi\)
−0.581357 + 0.813649i \(0.697478\pi\)
\(822\) 0 0
\(823\) 11.5076 0.401129 0.200564 0.979681i \(-0.435722\pi\)
0.200564 + 0.979681i \(0.435722\pi\)
\(824\) 0 0
\(825\) 5.56155 0.193628
\(826\) 0 0
\(827\) −22.6307 −0.786946 −0.393473 0.919336i \(-0.628727\pi\)
−0.393473 + 0.919336i \(0.628727\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −22.4924 −0.780253
\(832\) 0 0
\(833\) 30.4924 1.05650
\(834\) 0 0
\(835\) 5.75379 0.199118
\(836\) 0 0
\(837\) 7.12311 0.246211
\(838\) 0 0
\(839\) 31.8078 1.09813 0.549063 0.835781i \(-0.314985\pi\)
0.549063 + 0.835781i \(0.314985\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 3.75379 0.129287
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −31.1231 −1.06940
\(848\) 0 0
\(849\) −30.7386 −1.05495
\(850\) 0 0
\(851\) 54.5464 1.86983
\(852\) 0 0
\(853\) −39.5616 −1.35456 −0.677281 0.735725i \(-0.736842\pi\)
−0.677281 + 0.735725i \(0.736842\pi\)
\(854\) 0 0
\(855\) −3.12311 −0.106808
\(856\) 0 0
\(857\) −34.7926 −1.18849 −0.594246 0.804283i \(-0.702550\pi\)
−0.594246 + 0.804283i \(0.702550\pi\)
\(858\) 0 0
\(859\) −32.1922 −1.09838 −0.549192 0.835696i \(-0.685065\pi\)
−0.549192 + 0.835696i \(0.685065\pi\)
\(860\) 0 0
\(861\) −4.19224 −0.142871
\(862\) 0 0
\(863\) −1.86174 −0.0633743 −0.0316872 0.999498i \(-0.510088\pi\)
−0.0316872 + 0.999498i \(0.510088\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 27.6847 0.940220
\(868\) 0 0
\(869\) −65.6695 −2.22769
\(870\) 0 0
\(871\) 14.2462 0.482714
\(872\) 0 0
\(873\) −17.8078 −0.602701
\(874\) 0 0
\(875\) −1.56155 −0.0527901
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) −17.6155 −0.594157
\(880\) 0 0
\(881\) 8.24621 0.277822 0.138911 0.990305i \(-0.455640\pi\)
0.138911 + 0.990305i \(0.455640\pi\)
\(882\) 0 0
\(883\) −32.4924 −1.09346 −0.546729 0.837310i \(-0.684127\pi\)
−0.546729 + 0.837310i \(0.684127\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) −26.0540 −0.874807 −0.437403 0.899265i \(-0.644102\pi\)
−0.437403 + 0.899265i \(0.644102\pi\)
\(888\) 0 0
\(889\) −27.1231 −0.909680
\(890\) 0 0
\(891\) 5.56155 0.186319
\(892\) 0 0
\(893\) 22.2462 0.744441
\(894\) 0 0
\(895\) 23.1231 0.772920
\(896\) 0 0
\(897\) 5.56155 0.185695
\(898\) 0 0
\(899\) −14.2462 −0.475138
\(900\) 0 0
\(901\) −23.8078 −0.793152
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 0 0
\(905\) −8.93087 −0.296872
\(906\) 0 0
\(907\) 47.1231 1.56470 0.782349 0.622841i \(-0.214022\pi\)
0.782349 + 0.622841i \(0.214022\pi\)
\(908\) 0 0
\(909\) 1.12311 0.0372511
\(910\) 0 0
\(911\) −17.7538 −0.588209 −0.294105 0.955773i \(-0.595021\pi\)
−0.294105 + 0.955773i \(0.595021\pi\)
\(912\) 0 0
\(913\) 44.4924 1.47248
\(914\) 0 0
\(915\) −10.6847 −0.353224
\(916\) 0 0
\(917\) −23.6155 −0.779853
\(918\) 0 0
\(919\) 12.1922 0.402185 0.201092 0.979572i \(-0.435551\pi\)
0.201092 + 0.979572i \(0.435551\pi\)
\(920\) 0 0
\(921\) −13.5616 −0.446868
\(922\) 0 0
\(923\) −4.68466 −0.154197
\(924\) 0 0
\(925\) 9.80776 0.322477
\(926\) 0 0
\(927\) −14.2462 −0.467907
\(928\) 0 0
\(929\) 18.3002 0.600410 0.300205 0.953875i \(-0.402945\pi\)
0.300205 + 0.953875i \(0.402945\pi\)
\(930\) 0 0
\(931\) 14.2462 0.466901
\(932\) 0 0
\(933\) 26.7386 0.875384
\(934\) 0 0
\(935\) −37.1771 −1.21582
\(936\) 0 0
\(937\) −31.7538 −1.03735 −0.518676 0.854971i \(-0.673575\pi\)
−0.518676 + 0.854971i \(0.673575\pi\)
\(938\) 0 0
\(939\) −24.7386 −0.807315
\(940\) 0 0
\(941\) 20.5464 0.669793 0.334897 0.942255i \(-0.391299\pi\)
0.334897 + 0.942255i \(0.391299\pi\)
\(942\) 0 0
\(943\) 14.9309 0.486216
\(944\) 0 0
\(945\) −1.56155 −0.0507973
\(946\) 0 0
\(947\) 50.7386 1.64878 0.824392 0.566019i \(-0.191517\pi\)
0.824392 + 0.566019i \(0.191517\pi\)
\(948\) 0 0
\(949\) 16.2462 0.527374
\(950\) 0 0
\(951\) 2.87689 0.0932897
\(952\) 0 0
\(953\) 29.8078 0.965568 0.482784 0.875739i \(-0.339626\pi\)
0.482784 + 0.875739i \(0.339626\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −11.1231 −0.359559
\(958\) 0 0
\(959\) −25.3693 −0.819218
\(960\) 0 0
\(961\) 19.7386 0.636730
\(962\) 0 0
\(963\) 15.8078 0.509398
\(964\) 0 0
\(965\) 9.31534 0.299871
\(966\) 0 0
\(967\) 42.2462 1.35855 0.679273 0.733885i \(-0.262295\pi\)
0.679273 + 0.733885i \(0.262295\pi\)
\(968\) 0 0
\(969\) 20.8769 0.670662
\(970\) 0 0
\(971\) 8.49242 0.272535 0.136267 0.990672i \(-0.456489\pi\)
0.136267 + 0.990672i \(0.456489\pi\)
\(972\) 0 0
\(973\) −14.9309 −0.478662
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −43.8617 −1.40326 −0.701631 0.712541i \(-0.747544\pi\)
−0.701631 + 0.712541i \(0.747544\pi\)
\(978\) 0 0
\(979\) −81.6695 −2.61017
\(980\) 0 0
\(981\) −13.1231 −0.418989
\(982\) 0 0
\(983\) 52.9848 1.68995 0.844977 0.534803i \(-0.179614\pi\)
0.844977 + 0.534803i \(0.179614\pi\)
\(984\) 0 0
\(985\) 7.36932 0.234806
\(986\) 0 0
\(987\) 11.1231 0.354052
\(988\) 0 0
\(989\) 56.9848 1.81201
\(990\) 0 0
\(991\) 34.0540 1.08176 0.540880 0.841100i \(-0.318091\pi\)
0.540880 + 0.841100i \(0.318091\pi\)
\(992\) 0 0
\(993\) −9.75379 −0.309527
\(994\) 0 0
\(995\) −1.75379 −0.0555988
\(996\) 0 0
\(997\) 10.8769 0.344475 0.172237 0.985055i \(-0.444900\pi\)
0.172237 + 0.985055i \(0.444900\pi\)
\(998\) 0 0
\(999\) 9.80776 0.310304
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 3120.2.a.bg.1.1 2
3.2 odd 2 9360.2.a.ci.1.1 2
4.3 odd 2 1560.2.a.o.1.2 2
12.11 even 2 4680.2.a.y.1.2 2
20.19 odd 2 7800.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.o.1.2 2 4.3 odd 2
3120.2.a.bg.1.1 2 1.1 even 1 trivial
4680.2.a.y.1.2 2 12.11 even 2
7800.2.a.bd.1.1 2 20.19 odd 2
9360.2.a.ci.1.1 2 3.2 odd 2