Properties

Label 1849.2.a.g.1.1
Level $1849$
Weight $2$
Character 1849.1
Self dual yes
Analytic conductor $14.764$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7643393337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{2} +2.44949 q^{3} +4.00000 q^{4} +2.44949 q^{5} -6.00000 q^{6} -2.44949 q^{7} -4.89898 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-2.44949 q^{2} +2.44949 q^{3} +4.00000 q^{4} +2.44949 q^{5} -6.00000 q^{6} -2.44949 q^{7} -4.89898 q^{8} +3.00000 q^{9} -6.00000 q^{10} -1.00000 q^{11} +9.79796 q^{12} -3.00000 q^{13} +6.00000 q^{14} +6.00000 q^{15} +4.00000 q^{16} -7.00000 q^{17} -7.34847 q^{18} -4.89898 q^{19} +9.79796 q^{20} -6.00000 q^{21} +2.44949 q^{22} +1.00000 q^{23} -12.0000 q^{24} +1.00000 q^{25} +7.34847 q^{26} -9.79796 q^{28} +2.44949 q^{29} -14.6969 q^{30} -3.00000 q^{31} -2.44949 q^{33} +17.1464 q^{34} -6.00000 q^{35} +12.0000 q^{36} +4.89898 q^{37} +12.0000 q^{38} -7.34847 q^{39} -12.0000 q^{40} -5.00000 q^{41} +14.6969 q^{42} -4.00000 q^{44} +7.34847 q^{45} -2.44949 q^{46} -10.0000 q^{47} +9.79796 q^{48} -1.00000 q^{49} -2.44949 q^{50} -17.1464 q^{51} -12.0000 q^{52} -1.00000 q^{53} -2.44949 q^{55} +12.0000 q^{56} -12.0000 q^{57} -6.00000 q^{58} -10.0000 q^{59} +24.0000 q^{60} -7.34847 q^{61} +7.34847 q^{62} -7.34847 q^{63} -8.00000 q^{64} -7.34847 q^{65} +6.00000 q^{66} +9.00000 q^{67} -28.0000 q^{68} +2.44949 q^{69} +14.6969 q^{70} -4.89898 q^{71} -14.6969 q^{72} +12.2474 q^{73} -12.0000 q^{74} +2.44949 q^{75} -19.5959 q^{76} +2.44949 q^{77} +18.0000 q^{78} -6.00000 q^{79} +9.79796 q^{80} -9.00000 q^{81} +12.2474 q^{82} +1.00000 q^{83} -24.0000 q^{84} -17.1464 q^{85} +6.00000 q^{87} +4.89898 q^{88} +17.1464 q^{89} -18.0000 q^{90} +7.34847 q^{91} +4.00000 q^{92} -7.34847 q^{93} +24.4949 q^{94} -12.0000 q^{95} +11.0000 q^{97} +2.44949 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} - 12q^{6} + 6q^{9} + O(q^{10}) \) \( 2q + 8q^{4} - 12q^{6} + 6q^{9} - 12q^{10} - 2q^{11} - 6q^{13} + 12q^{14} + 12q^{15} + 8q^{16} - 14q^{17} - 12q^{21} + 2q^{23} - 24q^{24} + 2q^{25} - 6q^{31} - 12q^{35} + 24q^{36} + 24q^{38} - 24q^{40} - 10q^{41} - 8q^{44} - 20q^{47} - 2q^{49} - 24q^{52} - 2q^{53} + 24q^{56} - 24q^{57} - 12q^{58} - 20q^{59} + 48q^{60} - 16q^{64} + 12q^{66} + 18q^{67} - 56q^{68} - 24q^{74} + 36q^{78} - 12q^{79} - 18q^{81} + 2q^{83} - 48q^{84} + 12q^{87} - 36q^{90} + 8q^{92} - 24q^{95} + 22q^{97} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 2.44949 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 4.00000 2.00000
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) −6.00000 −2.44949
\(7\) −2.44949 −0.925820 −0.462910 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) −4.89898 −1.73205
\(9\) 3.00000 1.00000
\(10\) −6.00000 −1.89737
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 9.79796 2.82843
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 6.00000 1.60357
\(15\) 6.00000 1.54919
\(16\) 4.00000 1.00000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −7.34847 −1.73205
\(19\) −4.89898 −1.12390 −0.561951 0.827170i \(-0.689949\pi\)
−0.561951 + 0.827170i \(0.689949\pi\)
\(20\) 9.79796 2.19089
\(21\) −6.00000 −1.30931
\(22\) 2.44949 0.522233
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) −12.0000 −2.44949
\(25\) 1.00000 0.200000
\(26\) 7.34847 1.44115
\(27\) 0 0
\(28\) −9.79796 −1.85164
\(29\) 2.44949 0.454859 0.227429 0.973795i \(-0.426968\pi\)
0.227429 + 0.973795i \(0.426968\pi\)
\(30\) −14.6969 −2.68328
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −2.44949 −0.426401
\(34\) 17.1464 2.94059
\(35\) −6.00000 −1.01419
\(36\) 12.0000 2.00000
\(37\) 4.89898 0.805387 0.402694 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 12.0000 1.94666
\(39\) −7.34847 −1.17670
\(40\) −12.0000 −1.89737
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 14.6969 2.26779
\(43\) 0 0
\(44\) −4.00000 −0.603023
\(45\) 7.34847 1.09545
\(46\) −2.44949 −0.361158
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 9.79796 1.41421
\(49\) −1.00000 −0.142857
\(50\) −2.44949 −0.346410
\(51\) −17.1464 −2.40098
\(52\) −12.0000 −1.66410
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) −2.44949 −0.330289
\(56\) 12.0000 1.60357
\(57\) −12.0000 −1.58944
\(58\) −6.00000 −0.787839
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 24.0000 3.09839
\(61\) −7.34847 −0.940875 −0.470438 0.882433i \(-0.655904\pi\)
−0.470438 + 0.882433i \(0.655904\pi\)
\(62\) 7.34847 0.933257
\(63\) −7.34847 −0.925820
\(64\) −8.00000 −1.00000
\(65\) −7.34847 −0.911465
\(66\) 6.00000 0.738549
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) −28.0000 −3.39550
\(69\) 2.44949 0.294884
\(70\) 14.6969 1.75662
\(71\) −4.89898 −0.581402 −0.290701 0.956814i \(-0.593888\pi\)
−0.290701 + 0.956814i \(0.593888\pi\)
\(72\) −14.6969 −1.73205
\(73\) 12.2474 1.43346 0.716728 0.697353i \(-0.245639\pi\)
0.716728 + 0.697353i \(0.245639\pi\)
\(74\) −12.0000 −1.39497
\(75\) 2.44949 0.282843
\(76\) −19.5959 −2.24781
\(77\) 2.44949 0.279145
\(78\) 18.0000 2.03810
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 9.79796 1.09545
\(81\) −9.00000 −1.00000
\(82\) 12.2474 1.35250
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) −24.0000 −2.61861
\(85\) −17.1464 −1.85979
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 4.89898 0.522233
\(89\) 17.1464 1.81752 0.908759 0.417322i \(-0.137031\pi\)
0.908759 + 0.417322i \(0.137031\pi\)
\(90\) −18.0000 −1.89737
\(91\) 7.34847 0.770329
\(92\) 4.00000 0.417029
\(93\) −7.34847 −0.762001
\(94\) 24.4949 2.52646
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 2.44949 0.247436
\(99\) −3.00000 −0.301511
\(100\) 4.00000 0.400000
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 42.0000 4.15862
\(103\) 17.0000 1.67506 0.837530 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) 14.6969 1.44115
\(105\) −14.6969 −1.43427
\(106\) 2.44949 0.237915
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 6.00000 0.572078
\(111\) 12.0000 1.13899
\(112\) −9.79796 −0.925820
\(113\) −14.6969 −1.38257 −0.691286 0.722581i \(-0.742955\pi\)
−0.691286 + 0.722581i \(0.742955\pi\)
\(114\) 29.3939 2.75299
\(115\) 2.44949 0.228416
\(116\) 9.79796 0.909718
\(117\) −9.00000 −0.832050
\(118\) 24.4949 2.25494
\(119\) 17.1464 1.57181
\(120\) −29.3939 −2.68328
\(121\) −10.0000 −0.909091
\(122\) 18.0000 1.62964
\(123\) −12.2474 −1.10432
\(124\) −12.0000 −1.07763
\(125\) −9.79796 −0.876356
\(126\) 18.0000 1.60357
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 19.5959 1.73205
\(129\) 0 0
\(130\) 18.0000 1.57870
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −9.79796 −0.852803
\(133\) 12.0000 1.04053
\(134\) −22.0454 −1.90443
\(135\) 0 0
\(136\) 34.2929 2.94059
\(137\) 4.89898 0.418548 0.209274 0.977857i \(-0.432890\pi\)
0.209274 + 0.977857i \(0.432890\pi\)
\(138\) −6.00000 −0.510754
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) −24.0000 −2.02837
\(141\) −24.4949 −2.06284
\(142\) 12.0000 1.00702
\(143\) 3.00000 0.250873
\(144\) 12.0000 1.00000
\(145\) 6.00000 0.498273
\(146\) −30.0000 −2.48282
\(147\) −2.44949 −0.202031
\(148\) 19.5959 1.61077
\(149\) −4.89898 −0.401340 −0.200670 0.979659i \(-0.564312\pi\)
−0.200670 + 0.979659i \(0.564312\pi\)
\(150\) −6.00000 −0.489898
\(151\) 17.1464 1.39536 0.697678 0.716411i \(-0.254217\pi\)
0.697678 + 0.716411i \(0.254217\pi\)
\(152\) 24.0000 1.94666
\(153\) −21.0000 −1.69775
\(154\) −6.00000 −0.483494
\(155\) −7.34847 −0.590243
\(156\) −29.3939 −2.35339
\(157\) 9.79796 0.781962 0.390981 0.920399i \(-0.372136\pi\)
0.390981 + 0.920399i \(0.372136\pi\)
\(158\) 14.6969 1.16923
\(159\) −2.44949 −0.194257
\(160\) 0 0
\(161\) −2.44949 −0.193047
\(162\) 22.0454 1.73205
\(163\) −2.44949 −0.191859 −0.0959294 0.995388i \(-0.530582\pi\)
−0.0959294 + 0.995388i \(0.530582\pi\)
\(164\) −20.0000 −1.56174
\(165\) −6.00000 −0.467099
\(166\) −2.44949 −0.190117
\(167\) 5.00000 0.386912 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 29.3939 2.26779
\(169\) −4.00000 −0.307692
\(170\) 42.0000 3.22125
\(171\) −14.6969 −1.12390
\(172\) 0 0
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −14.6969 −1.11417
\(175\) −2.44949 −0.185164
\(176\) −4.00000 −0.301511
\(177\) −24.4949 −1.84115
\(178\) −42.0000 −3.14803
\(179\) 2.44949 0.183083 0.0915417 0.995801i \(-0.470821\pi\)
0.0915417 + 0.995801i \(0.470821\pi\)
\(180\) 29.3939 2.19089
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) −18.0000 −1.33425
\(183\) −18.0000 −1.33060
\(184\) −4.89898 −0.361158
\(185\) 12.0000 0.882258
\(186\) 18.0000 1.31982
\(187\) 7.00000 0.511891
\(188\) −40.0000 −2.91730
\(189\) 0 0
\(190\) 29.3939 2.13246
\(191\) −14.6969 −1.06343 −0.531717 0.846922i \(-0.678453\pi\)
−0.531717 + 0.846922i \(0.678453\pi\)
\(192\) −19.5959 −1.41421
\(193\) 15.0000 1.07972 0.539862 0.841754i \(-0.318476\pi\)
0.539862 + 0.841754i \(0.318476\pi\)
\(194\) −26.9444 −1.93449
\(195\) −18.0000 −1.28901
\(196\) −4.00000 −0.285714
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 7.34847 0.522233
\(199\) 19.5959 1.38912 0.694559 0.719436i \(-0.255600\pi\)
0.694559 + 0.719436i \(0.255600\pi\)
\(200\) −4.89898 −0.346410
\(201\) 22.0454 1.55496
\(202\) 12.2474 0.861727
\(203\) −6.00000 −0.421117
\(204\) −68.5857 −4.80196
\(205\) −12.2474 −0.855399
\(206\) −41.6413 −2.90129
\(207\) 3.00000 0.208514
\(208\) −12.0000 −0.832050
\(209\) 4.89898 0.338869
\(210\) 36.0000 2.48424
\(211\) −14.6969 −1.01178 −0.505889 0.862598i \(-0.668836\pi\)
−0.505889 + 0.862598i \(0.668836\pi\)
\(212\) −4.00000 −0.274721
\(213\) −12.0000 −0.822226
\(214\) −34.2929 −2.34421
\(215\) 0 0
\(216\) 0 0
\(217\) 7.34847 0.498847
\(218\) −31.8434 −2.15670
\(219\) 30.0000 2.02721
\(220\) −9.79796 −0.660578
\(221\) 21.0000 1.41261
\(222\) −29.3939 −1.97279
\(223\) −17.1464 −1.14821 −0.574105 0.818782i \(-0.694650\pi\)
−0.574105 + 0.818782i \(0.694650\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 36.0000 2.39468
\(227\) −9.79796 −0.650313 −0.325157 0.945660i \(-0.605417\pi\)
−0.325157 + 0.945660i \(0.605417\pi\)
\(228\) −48.0000 −3.17888
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) −6.00000 −0.395628
\(231\) 6.00000 0.394771
\(232\) −12.0000 −0.787839
\(233\) −4.89898 −0.320943 −0.160471 0.987040i \(-0.551301\pi\)
−0.160471 + 0.987040i \(0.551301\pi\)
\(234\) 22.0454 1.44115
\(235\) −24.4949 −1.59787
\(236\) −40.0000 −2.60378
\(237\) −14.6969 −0.954669
\(238\) −42.0000 −2.72246
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 24.0000 1.54919
\(241\) −19.5959 −1.26228 −0.631142 0.775667i \(-0.717413\pi\)
−0.631142 + 0.775667i \(0.717413\pi\)
\(242\) 24.4949 1.57459
\(243\) −22.0454 −1.41421
\(244\) −29.3939 −1.88175
\(245\) −2.44949 −0.156492
\(246\) 30.0000 1.91273
\(247\) 14.6969 0.935144
\(248\) 14.6969 0.933257
\(249\) 2.44949 0.155230
\(250\) 24.0000 1.51789
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) −29.3939 −1.85164
\(253\) −1.00000 −0.0628695
\(254\) −2.44949 −0.153695
\(255\) −42.0000 −2.63014
\(256\) −32.0000 −2.00000
\(257\) −2.44949 −0.152795 −0.0763975 0.997077i \(-0.524342\pi\)
−0.0763975 + 0.997077i \(0.524342\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) −29.3939 −1.82293
\(261\) 7.34847 0.454859
\(262\) 0 0
\(263\) −14.6969 −0.906252 −0.453126 0.891446i \(-0.649691\pi\)
−0.453126 + 0.891446i \(0.649691\pi\)
\(264\) 12.0000 0.738549
\(265\) −2.44949 −0.150471
\(266\) −29.3939 −1.80225
\(267\) 42.0000 2.57036
\(268\) 36.0000 2.19905
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) −28.0000 −1.69775
\(273\) 18.0000 1.08941
\(274\) −12.0000 −0.724947
\(275\) −1.00000 −0.0603023
\(276\) 9.79796 0.589768
\(277\) 17.1464 1.03023 0.515115 0.857121i \(-0.327749\pi\)
0.515115 + 0.857121i \(0.327749\pi\)
\(278\) 17.1464 1.02837
\(279\) −9.00000 −0.538816
\(280\) 29.3939 1.75662
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) 60.0000 3.57295
\(283\) −21.0000 −1.24832 −0.624160 0.781296i \(-0.714559\pi\)
−0.624160 + 0.781296i \(0.714559\pi\)
\(284\) −19.5959 −1.16280
\(285\) −29.3939 −1.74114
\(286\) −7.34847 −0.434524
\(287\) 12.2474 0.722944
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) −14.6969 −0.863034
\(291\) 26.9444 1.57951
\(292\) 48.9898 2.86691
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 6.00000 0.349927
\(295\) −24.4949 −1.42615
\(296\) −24.0000 −1.39497
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) −3.00000 −0.173494
\(300\) 9.79796 0.565685
\(301\) 0 0
\(302\) −42.0000 −2.41683
\(303\) −12.2474 −0.703598
\(304\) −19.5959 −1.12390
\(305\) −18.0000 −1.03068
\(306\) 51.4393 2.94059
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 9.79796 0.558291
\(309\) 41.6413 2.36889
\(310\) 18.0000 1.02233
\(311\) −25.0000 −1.41762 −0.708810 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 36.0000 2.03810
\(313\) −7.34847 −0.415360 −0.207680 0.978197i \(-0.566591\pi\)
−0.207680 + 0.978197i \(0.566591\pi\)
\(314\) −24.0000 −1.35440
\(315\) −18.0000 −1.01419
\(316\) −24.0000 −1.35011
\(317\) −17.0000 −0.954815 −0.477408 0.878682i \(-0.658423\pi\)
−0.477408 + 0.878682i \(0.658423\pi\)
\(318\) 6.00000 0.336463
\(319\) −2.44949 −0.137145
\(320\) −19.5959 −1.09545
\(321\) 34.2929 1.91404
\(322\) 6.00000 0.334367
\(323\) 34.2929 1.90811
\(324\) −36.0000 −2.00000
\(325\) −3.00000 −0.166410
\(326\) 6.00000 0.332309
\(327\) 31.8434 1.76094
\(328\) 24.4949 1.35250
\(329\) 24.4949 1.35045
\(330\) 14.6969 0.809040
\(331\) 2.44949 0.134636 0.0673181 0.997732i \(-0.478556\pi\)
0.0673181 + 0.997732i \(0.478556\pi\)
\(332\) 4.00000 0.219529
\(333\) 14.6969 0.805387
\(334\) −12.2474 −0.670151
\(335\) 22.0454 1.20447
\(336\) −24.0000 −1.30931
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 9.79796 0.532939
\(339\) −36.0000 −1.95525
\(340\) −68.5857 −3.71958
\(341\) 3.00000 0.162459
\(342\) 36.0000 1.94666
\(343\) 19.5959 1.05808
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 24.4949 1.31685
\(347\) −17.1464 −0.920468 −0.460234 0.887798i \(-0.652235\pi\)
−0.460234 + 0.887798i \(0.652235\pi\)
\(348\) 24.0000 1.28654
\(349\) 2.44949 0.131118 0.0655591 0.997849i \(-0.479117\pi\)
0.0655591 + 0.997849i \(0.479117\pi\)
\(350\) 6.00000 0.320713
\(351\) 0 0
\(352\) 0 0
\(353\) −29.0000 −1.54351 −0.771757 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(354\) 60.0000 3.18896
\(355\) −12.0000 −0.636894
\(356\) 68.5857 3.63504
\(357\) 42.0000 2.22288
\(358\) −6.00000 −0.317110
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) −36.0000 −1.89737
\(361\) 5.00000 0.263158
\(362\) −29.3939 −1.54491
\(363\) −24.4949 −1.28565
\(364\) 29.3939 1.54066
\(365\) 30.0000 1.57027
\(366\) 44.0908 2.30466
\(367\) 6.00000 0.313197 0.156599 0.987662i \(-0.449947\pi\)
0.156599 + 0.987662i \(0.449947\pi\)
\(368\) 4.00000 0.208514
\(369\) −15.0000 −0.780869
\(370\) −29.3939 −1.52811
\(371\) 2.44949 0.127171
\(372\) −29.3939 −1.52400
\(373\) −34.2929 −1.77562 −0.887808 0.460213i \(-0.847773\pi\)
−0.887808 + 0.460213i \(0.847773\pi\)
\(374\) −17.1464 −0.886621
\(375\) −24.0000 −1.23935
\(376\) 48.9898 2.52646
\(377\) −7.34847 −0.378465
\(378\) 0 0
\(379\) −21.0000 −1.07870 −0.539349 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(380\) −48.0000 −2.46235
\(381\) 2.44949 0.125491
\(382\) 36.0000 1.84192
\(383\) 24.4949 1.25163 0.625815 0.779971i \(-0.284766\pi\)
0.625815 + 0.779971i \(0.284766\pi\)
\(384\) 48.0000 2.44949
\(385\) 6.00000 0.305788
\(386\) −36.7423 −1.87014
\(387\) 0 0
\(388\) 44.0000 2.23376
\(389\) 9.79796 0.496776 0.248388 0.968661i \(-0.420099\pi\)
0.248388 + 0.968661i \(0.420099\pi\)
\(390\) 44.0908 2.23263
\(391\) −7.00000 −0.354005
\(392\) 4.89898 0.247436
\(393\) 0 0
\(394\) −19.5959 −0.987228
\(395\) −14.6969 −0.739483
\(396\) −12.0000 −0.603023
\(397\) −12.0000 −0.602263 −0.301131 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) −48.0000 −2.40602
\(399\) 29.3939 1.47153
\(400\) 4.00000 0.200000
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) −54.0000 −2.69328
\(403\) 9.00000 0.448322
\(404\) −20.0000 −0.995037
\(405\) −22.0454 −1.09545
\(406\) 14.6969 0.729397
\(407\) −4.89898 −0.242833
\(408\) 84.0000 4.15862
\(409\) −36.7423 −1.81679 −0.908396 0.418111i \(-0.862692\pi\)
−0.908396 + 0.418111i \(0.862692\pi\)
\(410\) 30.0000 1.48159
\(411\) 12.0000 0.591916
\(412\) 68.0000 3.35012
\(413\) 24.4949 1.20532
\(414\) −7.34847 −0.361158
\(415\) 2.44949 0.120241
\(416\) 0 0
\(417\) −17.1464 −0.839664
\(418\) −12.0000 −0.586939
\(419\) 2.44949 0.119665 0.0598327 0.998208i \(-0.480943\pi\)
0.0598327 + 0.998208i \(0.480943\pi\)
\(420\) −58.7878 −2.86855
\(421\) 9.79796 0.477523 0.238762 0.971078i \(-0.423259\pi\)
0.238762 + 0.971078i \(0.423259\pi\)
\(422\) 36.0000 1.75245
\(423\) −30.0000 −1.45865
\(424\) 4.89898 0.237915
\(425\) −7.00000 −0.339550
\(426\) 29.3939 1.42414
\(427\) 18.0000 0.871081
\(428\) 56.0000 2.70686
\(429\) 7.34847 0.354787
\(430\) 0 0
\(431\) −17.0000 −0.818861 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(432\) 0 0
\(433\) 12.2474 0.588575 0.294287 0.955717i \(-0.404918\pi\)
0.294287 + 0.955717i \(0.404918\pi\)
\(434\) −18.0000 −0.864028
\(435\) 14.6969 0.704664
\(436\) 52.0000 2.49035
\(437\) −4.89898 −0.234350
\(438\) −73.4847 −3.51123
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 12.0000 0.572078
\(441\) −3.00000 −0.142857
\(442\) −51.4393 −2.44672
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 48.0000 2.27798
\(445\) 42.0000 1.99099
\(446\) 42.0000 1.98876
\(447\) −12.0000 −0.567581
\(448\) 19.5959 0.925820
\(449\) 26.9444 1.27158 0.635792 0.771860i \(-0.280674\pi\)
0.635792 + 0.771860i \(0.280674\pi\)
\(450\) −7.34847 −0.346410
\(451\) 5.00000 0.235441
\(452\) −58.7878 −2.76514
\(453\) 42.0000 1.97333
\(454\) 24.0000 1.12638
\(455\) 18.0000 0.843853
\(456\) 58.7878 2.75299
\(457\) 31.8434 1.48957 0.744785 0.667305i \(-0.232552\pi\)
0.744785 + 0.667305i \(0.232552\pi\)
\(458\) −17.1464 −0.801200
\(459\) 0 0
\(460\) 9.79796 0.456832
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) −14.6969 −0.683763
\(463\) −12.2474 −0.569187 −0.284594 0.958648i \(-0.591859\pi\)
−0.284594 + 0.958648i \(0.591859\pi\)
\(464\) 9.79796 0.454859
\(465\) −18.0000 −0.834730
\(466\) 12.0000 0.555889
\(467\) 19.5959 0.906791 0.453395 0.891309i \(-0.350213\pi\)
0.453395 + 0.891309i \(0.350213\pi\)
\(468\) −36.0000 −1.66410
\(469\) −22.0454 −1.01796
\(470\) 60.0000 2.76759
\(471\) 24.0000 1.10586
\(472\) 48.9898 2.25494
\(473\) 0 0
\(474\) 36.0000 1.65353
\(475\) −4.89898 −0.224781
\(476\) 68.5857 3.14362
\(477\) −3.00000 −0.137361
\(478\) 63.6867 2.91296
\(479\) −1.00000 −0.0456912 −0.0228456 0.999739i \(-0.507273\pi\)
−0.0228456 + 0.999739i \(0.507273\pi\)
\(480\) 0 0
\(481\) −14.6969 −0.670123
\(482\) 48.0000 2.18634
\(483\) −6.00000 −0.273009
\(484\) −40.0000 −1.81818
\(485\) 26.9444 1.22348
\(486\) 54.0000 2.44949
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 36.0000 1.62964
\(489\) −6.00000 −0.271329
\(490\) 6.00000 0.271052
\(491\) 22.0454 0.994895 0.497448 0.867494i \(-0.334271\pi\)
0.497448 + 0.867494i \(0.334271\pi\)
\(492\) −48.9898 −2.20863
\(493\) −17.1464 −0.772236
\(494\) −36.0000 −1.61972
\(495\) −7.34847 −0.330289
\(496\) −12.0000 −0.538816
\(497\) 12.0000 0.538274
\(498\) −6.00000 −0.268866
\(499\) 7.34847 0.328963 0.164481 0.986380i \(-0.447405\pi\)
0.164481 + 0.986380i \(0.447405\pi\)
\(500\) −39.1918 −1.75271
\(501\) 12.2474 0.547176
\(502\) −12.2474 −0.546630
\(503\) −14.6969 −0.655304 −0.327652 0.944798i \(-0.606257\pi\)
−0.327652 + 0.944798i \(0.606257\pi\)
\(504\) 36.0000 1.60357
\(505\) −12.2474 −0.545004
\(506\) 2.44949 0.108893
\(507\) −9.79796 −0.435143
\(508\) 4.00000 0.177471
\(509\) −19.0000 −0.842160 −0.421080 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 102.879 4.55554
\(511\) −30.0000 −1.32712
\(512\) 39.1918 1.73205
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 41.6413 1.83494
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 29.3939 1.29149
\(519\) −24.4949 −1.07521
\(520\) 36.0000 1.57870
\(521\) −22.0454 −0.965827 −0.482913 0.875668i \(-0.660421\pi\)
−0.482913 + 0.875668i \(0.660421\pi\)
\(522\) −18.0000 −0.787839
\(523\) 2.44949 0.107109 0.0535544 0.998565i \(-0.482945\pi\)
0.0535544 + 0.998565i \(0.482945\pi\)
\(524\) 0 0
\(525\) −6.00000 −0.261861
\(526\) 36.0000 1.56967
\(527\) 21.0000 0.914774
\(528\) −9.79796 −0.426401
\(529\) −22.0000 −0.956522
\(530\) 6.00000 0.260623
\(531\) −30.0000 −1.30189
\(532\) 48.0000 2.08106
\(533\) 15.0000 0.649722
\(534\) −102.879 −4.45199
\(535\) 34.2929 1.48261
\(536\) −44.0908 −1.90443
\(537\) 6.00000 0.258919
\(538\) −12.2474 −0.528025
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 33.0000 1.41878 0.709390 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) −7.34847 −0.315644
\(543\) 29.3939 1.26141
\(544\) 0 0
\(545\) 31.8434 1.36402
\(546\) −44.0908 −1.88691
\(547\) −27.0000 −1.15444 −0.577218 0.816590i \(-0.695862\pi\)
−0.577218 + 0.816590i \(0.695862\pi\)
\(548\) 19.5959 0.837096
\(549\) −22.0454 −0.940875
\(550\) 2.44949 0.104447
\(551\) −12.0000 −0.511217
\(552\) −12.0000 −0.510754
\(553\) 14.6969 0.624977
\(554\) −42.0000 −1.78441
\(555\) 29.3939 1.24770
\(556\) −28.0000 −1.18746
\(557\) −41.0000 −1.73723 −0.868613 0.495491i \(-0.834988\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(558\) 22.0454 0.933257
\(559\) 0 0
\(560\) −24.0000 −1.01419
\(561\) 17.1464 0.723923
\(562\) −2.44949 −0.103325
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) −97.9796 −4.12568
\(565\) −36.0000 −1.51453
\(566\) 51.4393 2.16215
\(567\) 22.0454 0.925820
\(568\) 24.0000 1.00702
\(569\) 23.0000 0.964210 0.482105 0.876113i \(-0.339872\pi\)
0.482105 + 0.876113i \(0.339872\pi\)
\(570\) 72.0000 3.01575
\(571\) 7.34847 0.307524 0.153762 0.988108i \(-0.450861\pi\)
0.153762 + 0.988108i \(0.450861\pi\)
\(572\) 12.0000 0.501745
\(573\) −36.0000 −1.50392
\(574\) −30.0000 −1.25218
\(575\) 1.00000 0.0417029
\(576\) −24.0000 −1.00000
\(577\) 12.2474 0.509868 0.254934 0.966958i \(-0.417946\pi\)
0.254934 + 0.966958i \(0.417946\pi\)
\(578\) −78.3837 −3.26033
\(579\) 36.7423 1.52696
\(580\) 24.0000 0.996546
\(581\) −2.44949 −0.101622
\(582\) −66.0000 −2.73579
\(583\) 1.00000 0.0414158
\(584\) −60.0000 −2.48282
\(585\) −22.0454 −0.911465
\(586\) −39.1918 −1.61900
\(587\) −24.4949 −1.01101 −0.505506 0.862823i \(-0.668694\pi\)
−0.505506 + 0.862823i \(0.668694\pi\)
\(588\) −9.79796 −0.404061
\(589\) 14.6969 0.605577
\(590\) 60.0000 2.47016
\(591\) 19.5959 0.806068
\(592\) 19.5959 0.805387
\(593\) 7.34847 0.301765 0.150883 0.988552i \(-0.451788\pi\)
0.150883 + 0.988552i \(0.451788\pi\)
\(594\) 0 0
\(595\) 42.0000 1.72183
\(596\) −19.5959 −0.802680
\(597\) 48.0000 1.96451
\(598\) 7.34847 0.300501
\(599\) −17.0000 −0.694601 −0.347301 0.937754i \(-0.612902\pi\)
−0.347301 + 0.937754i \(0.612902\pi\)
\(600\) −12.0000 −0.489898
\(601\) 19.5959 0.799334 0.399667 0.916660i \(-0.369126\pi\)
0.399667 + 0.916660i \(0.369126\pi\)
\(602\) 0 0
\(603\) 27.0000 1.09952
\(604\) 68.5857 2.79071
\(605\) −24.4949 −0.995859
\(606\) 30.0000 1.21867
\(607\) 39.1918 1.59075 0.795374 0.606119i \(-0.207275\pi\)
0.795374 + 0.606119i \(0.207275\pi\)
\(608\) 0 0
\(609\) −14.6969 −0.595550
\(610\) 44.0908 1.78518
\(611\) 30.0000 1.21367
\(612\) −84.0000 −3.39550
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) −56.3383 −2.27363
\(615\) −30.0000 −1.20972
\(616\) −12.0000 −0.483494
\(617\) 43.0000 1.73111 0.865557 0.500810i \(-0.166964\pi\)
0.865557 + 0.500810i \(0.166964\pi\)
\(618\) −102.000 −4.10304
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) −29.3939 −1.18049
\(621\) 0 0
\(622\) 61.2372 2.45539
\(623\) −42.0000 −1.68269
\(624\) −29.3939 −1.17670
\(625\) −29.0000 −1.16000
\(626\) 18.0000 0.719425
\(627\) 12.0000 0.479234
\(628\) 39.1918 1.56392
\(629\) −34.2929 −1.36735
\(630\) 44.0908 1.75662
\(631\) −14.6969 −0.585076 −0.292538 0.956254i \(-0.594500\pi\)
−0.292538 + 0.956254i \(0.594500\pi\)
\(632\) 29.3939 1.16923
\(633\) −36.0000 −1.43087
\(634\) 41.6413 1.65379
\(635\) 2.44949 0.0972050
\(636\) −9.79796 −0.388514
\(637\) 3.00000 0.118864
\(638\) 6.00000 0.237542
\(639\) −14.6969 −0.581402
\(640\) 48.0000 1.89737
\(641\) 22.0454 0.870741 0.435371 0.900251i \(-0.356617\pi\)
0.435371 + 0.900251i \(0.356617\pi\)
\(642\) −84.0000 −3.31522
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) −9.79796 −0.386094
\(645\) 0 0
\(646\) −84.0000 −3.30494
\(647\) −44.0908 −1.73339 −0.866694 0.498839i \(-0.833760\pi\)
−0.866694 + 0.498839i \(0.833760\pi\)
\(648\) 44.0908 1.73205
\(649\) 10.0000 0.392534
\(650\) 7.34847 0.288231
\(651\) 18.0000 0.705476
\(652\) −9.79796 −0.383718
\(653\) 34.2929 1.34198 0.670992 0.741465i \(-0.265869\pi\)
0.670992 + 0.741465i \(0.265869\pi\)
\(654\) −78.0000 −3.05004
\(655\) 0 0
\(656\) −20.0000 −0.780869
\(657\) 36.7423 1.43346
\(658\) −60.0000 −2.33904
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) −24.0000 −0.934199
\(661\) −45.0000 −1.75030 −0.875149 0.483854i \(-0.839236\pi\)
−0.875149 + 0.483854i \(0.839236\pi\)
\(662\) −6.00000 −0.233197
\(663\) 51.4393 1.99774
\(664\) −4.89898 −0.190117
\(665\) 29.3939 1.13985
\(666\) −36.0000 −1.39497
\(667\) 2.44949 0.0948446
\(668\) 20.0000 0.773823
\(669\) −42.0000 −1.62381
\(670\) −54.0000 −2.08620
\(671\) 7.34847 0.283685
\(672\) 0 0
\(673\) 26.9444 1.03863 0.519315 0.854583i \(-0.326187\pi\)
0.519315 + 0.854583i \(0.326187\pi\)
\(674\) 2.44949 0.0943508
\(675\) 0 0
\(676\) −16.0000 −0.615385
\(677\) −24.4949 −0.941415 −0.470708 0.882289i \(-0.656001\pi\)
−0.470708 + 0.882289i \(0.656001\pi\)
\(678\) 88.1816 3.38660
\(679\) −26.9444 −1.03403
\(680\) 84.0000 3.22125
\(681\) −24.0000 −0.919682
\(682\) −7.34847 −0.281387
\(683\) 23.0000 0.880071 0.440035 0.897980i \(-0.354966\pi\)
0.440035 + 0.897980i \(0.354966\pi\)
\(684\) −58.7878 −2.24781
\(685\) 12.0000 0.458496
\(686\) −48.0000 −1.83265
\(687\) 17.1464 0.654177
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) −14.6969 −0.559503
\(691\) 26.9444 1.02501 0.512506 0.858683i \(-0.328717\pi\)
0.512506 + 0.858683i \(0.328717\pi\)
\(692\) −40.0000 −1.52057
\(693\) 7.34847 0.279145
\(694\) 42.0000 1.59430
\(695\) −17.1464 −0.650401
\(696\) −29.3939 −1.11417
\(697\) 35.0000 1.32572
\(698\) −6.00000 −0.227103
\(699\) −12.0000 −0.453882
\(700\) −9.79796 −0.370328
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 8.00000 0.301511
\(705\) −60.0000 −2.25973
\(706\) 71.0352 2.67345
\(707\) 12.2474 0.460613
\(708\) −97.9796 −3.68230
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 29.3939 1.10313
\(711\) −18.0000 −0.675053
\(712\) −84.0000 −3.14803
\(713\) −3.00000 −0.112351
\(714\) −102.879 −3.85013
\(715\) 7.34847 0.274817
\(716\) 9.79796 0.366167
\(717\) −63.6867 −2.37842
\(718\) 26.9444 1.00556
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 29.3939 1.09545
\(721\) −41.6413 −1.55080
\(722\) −12.2474 −0.455803
\(723\) −48.0000 −1.78514
\(724\) 48.0000 1.78391
\(725\) 2.44949 0.0909718
\(726\) 60.0000 2.22681
\(727\) 29.3939 1.09016 0.545079 0.838385i \(-0.316500\pi\)
0.545079 + 0.838385i \(0.316500\pi\)
\(728\) −36.0000 −1.33425
\(729\) −27.0000 −1.00000
\(730\) −73.4847 −2.71979
\(731\) 0 0
\(732\) −72.0000 −2.66120
\(733\) −48.9898 −1.80948 −0.904740 0.425965i \(-0.859935\pi\)
−0.904740 + 0.425965i \(0.859935\pi\)
\(734\) −14.6969 −0.542474
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) −9.00000 −0.331519
\(738\) 36.7423 1.35250
\(739\) −4.89898 −0.180212 −0.0901059 0.995932i \(-0.528721\pi\)
−0.0901059 + 0.995932i \(0.528721\pi\)
\(740\) 48.0000 1.76452
\(741\) 36.0000 1.32249
\(742\) −6.00000 −0.220267
\(743\) −14.6969 −0.539178 −0.269589 0.962975i \(-0.586888\pi\)
−0.269589 + 0.962975i \(0.586888\pi\)
\(744\) 36.0000 1.31982
\(745\) −12.0000 −0.439646
\(746\) 84.0000 3.07546
\(747\) 3.00000 0.109764
\(748\) 28.0000 1.02378
\(749\) −34.2929 −1.25303
\(750\) 58.7878 2.14663
\(751\) −51.4393 −1.87705 −0.938523 0.345217i \(-0.887805\pi\)
−0.938523 + 0.345217i \(0.887805\pi\)
\(752\) −40.0000 −1.45865
\(753\) 12.2474 0.446322
\(754\) 18.0000 0.655521
\(755\) 42.0000 1.52854
\(756\) 0 0
\(757\) −4.89898 −0.178056 −0.0890282 0.996029i \(-0.528376\pi\)
−0.0890282 + 0.996029i \(0.528376\pi\)
\(758\) 51.4393 1.86836
\(759\) −2.44949 −0.0889108
\(760\) 58.7878 2.13246
\(761\) 4.89898 0.177588 0.0887939 0.996050i \(-0.471699\pi\)
0.0887939 + 0.996050i \(0.471699\pi\)
\(762\) −6.00000 −0.217357
\(763\) −31.8434 −1.15281
\(764\) −58.7878 −2.12687
\(765\) −51.4393 −1.85979
\(766\) −60.0000 −2.16789
\(767\) 30.0000 1.08324
\(768\) −78.3837 −2.82843
\(769\) 24.0000 0.865462 0.432731 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) −14.6969 −0.529641
\(771\) −6.00000 −0.216085
\(772\) 60.0000 2.15945
\(773\) −26.9444 −0.969122 −0.484561 0.874757i \(-0.661021\pi\)
−0.484561 + 0.874757i \(0.661021\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) −53.8888 −1.93449
\(777\) −29.3939 −1.05450
\(778\) −24.0000 −0.860442
\(779\) 24.4949 0.877621
\(780\) −72.0000 −2.57801
\(781\) 4.89898 0.175299
\(782\) 17.1464 0.613155
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) 32.0000 1.13995
\(789\) −36.0000 −1.28163
\(790\) 36.0000 1.28082
\(791\) 36.0000 1.28001
\(792\) 14.6969 0.522233
\(793\) 22.0454 0.782855
\(794\) 29.3939 1.04315
\(795\) −6.00000 −0.212798
\(796\) 78.3837 2.77824
\(797\) −28.0000 −0.991811 −0.495905 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) −72.0000 −2.54877
\(799\) 70.0000 2.47642
\(800\) 0 0
\(801\) 51.4393 1.81752
\(802\) 12.2474 0.432472
\(803\) −12.2474 −0.432203
\(804\) 88.1816 3.10993
\(805\) −6.00000 −0.211472
\(806\) −22.0454 −0.776516
\(807\) 12.2474 0.431131
\(808\) 24.4949 0.861727
\(809\) 28.0000 0.984428 0.492214 0.870474i \(-0.336188\pi\)
0.492214 + 0.870474i \(0.336188\pi\)
\(810\) 54.0000 1.89737
\(811\) 7.34847 0.258040 0.129020 0.991642i \(-0.458817\pi\)
0.129020 + 0.991642i \(0.458817\pi\)
\(812\) −24.0000 −0.842235
\(813\) 7.34847 0.257722
\(814\) 12.0000 0.420600
\(815\) −6.00000 −0.210171
\(816\) −68.5857 −2.40098
\(817\) 0 0
\(818\) 90.0000 3.14678
\(819\) 22.0454 0.770329
\(820\) −48.9898 −1.71080
\(821\) 31.0000 1.08191 0.540954 0.841052i \(-0.318063\pi\)
0.540954 + 0.841052i \(0.318063\pi\)
\(822\) −29.3939 −1.02523
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) −83.2827 −2.90129
\(825\) −2.44949 −0.0852803
\(826\) −60.0000 −2.08767
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 12.0000 0.417029
\(829\) 44.0908 1.53134 0.765669 0.643235i \(-0.222408\pi\)
0.765669 + 0.643235i \(0.222408\pi\)
\(830\) −6.00000 −0.208263
\(831\) 42.0000 1.45696
\(832\) 24.0000 0.832050
\(833\) 7.00000 0.242536
\(834\) 42.0000 1.45434
\(835\) 12.2474 0.423840
\(836\) 19.5959 0.677739
\(837\) 0 0
\(838\) −6.00000 −0.207267
\(839\) −26.9444 −0.930224 −0.465112 0.885252i \(-0.653986\pi\)
−0.465112 + 0.885252i \(0.653986\pi\)
\(840\) 72.0000 2.48424
\(841\) −23.0000 −0.793103
\(842\) −24.0000 −0.827095
\(843\) 2.44949 0.0843649
\(844\) −58.7878 −2.02356
\(845\) −9.79796 −0.337060
\(846\) 73.4847 2.52646
\(847\) 24.4949 0.841655
\(848\) −4.00000 −0.137361
\(849\) −51.4393 −1.76539
\(850\) 17.1464 0.588118
\(851\) 4.89898 0.167935
\(852\) −48.0000 −1.64445
\(853\) −5.00000 −0.171197 −0.0855984 0.996330i \(-0.527280\pi\)
−0.0855984 + 0.996330i \(0.527280\pi\)
\(854\) −44.0908 −1.50876
\(855\) −36.0000 −1.23117
\(856\) −68.5857 −2.34421
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) −18.0000 −0.614510
\(859\) −48.9898 −1.67151 −0.835755 0.549102i \(-0.814970\pi\)
−0.835755 + 0.549102i \(0.814970\pi\)
\(860\) 0 0
\(861\) 30.0000 1.02240
\(862\) 41.6413 1.41831
\(863\) −9.79796 −0.333526 −0.166763 0.985997i \(-0.553332\pi\)
−0.166763 + 0.985997i \(0.553332\pi\)
\(864\) 0 0
\(865\) −24.4949 −0.832851
\(866\) −30.0000 −1.01944
\(867\) 78.3837 2.66205
\(868\) 29.3939 0.997693
\(869\) 6.00000 0.203536
\(870\) −36.0000 −1.22051
\(871\) −27.0000 −0.914860
\(872\) −63.6867 −2.15670
\(873\) 33.0000 1.11688
\(874\) 12.0000 0.405906
\(875\) 24.0000 0.811348
\(876\) 120.000 4.05442
\(877\) −55.0000 −1.85722 −0.928609 0.371060i \(-0.878995\pi\)
−0.928609 + 0.371060i \(0.878995\pi\)
\(878\) 71.0352 2.39732
\(879\) 39.1918 1.32191
\(880\) −9.79796 −0.330289
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 7.34847 0.247436
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 84.0000 2.82523
\(885\) −60.0000 −2.01688
\(886\) 83.2827 2.79794
\(887\) −29.3939 −0.986950 −0.493475 0.869760i \(-0.664274\pi\)
−0.493475 + 0.869760i \(0.664274\pi\)
\(888\) −58.7878 −1.97279
\(889\) −2.44949 −0.0821532
\(890\) −102.879 −3.44850
\(891\) 9.00000 0.301511
\(892\) −68.5857 −2.29642
\(893\) 48.9898 1.63938
\(894\) 29.3939 0.983078
\(895\) 6.00000 0.200558
\(896\) −48.0000 −1.60357
\(897\) −7.34847 −0.245358
\(898\) −66.0000 −2.20245
\(899\) −7.34847 −0.245085
\(900\) 12.0000 0.400000
\(901\) 7.00000 0.233204
\(902\) −12.2474 −0.407795
\(903\) 0 0
\(904\) 72.0000 2.39468
\(905\) 29.3939 0.977086
\(906\) −102.879 −3.41791
\(907\) −27.0000 −0.896520 −0.448260 0.893903i \(-0.647956\pi\)
−0.448260 + 0.893903i \(0.647956\pi\)
\(908\) −39.1918 −1.30063
\(909\) −15.0000 −0.497519
\(910\) −44.0908 −1.46160
\(911\) 22.0454 0.730397 0.365198 0.930930i \(-0.381001\pi\)
0.365198 + 0.930930i \(0.381001\pi\)
\(912\) −48.0000 −1.58944
\(913\) −1.00000 −0.0330952
\(914\) −78.0000 −2.58001
\(915\) −44.0908 −1.45760
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) 0 0
\(919\) 3.00000 0.0989609 0.0494804 0.998775i \(-0.484243\pi\)
0.0494804 + 0.998775i \(0.484243\pi\)
\(920\) −12.0000 −0.395628
\(921\) 56.3383 1.85641
\(922\) −39.1918 −1.29071
\(923\) 14.6969 0.483756
\(924\) 24.0000 0.789542
\(925\) 4.89898 0.161077
\(926\) 30.0000 0.985861
\(927\) 51.0000 1.67506
\(928\) 0 0
\(929\) −53.8888 −1.76803 −0.884017 0.467455i \(-0.845171\pi\)
−0.884017 + 0.467455i \(0.845171\pi\)
\(930\) 44.0908 1.44579
\(931\) 4.89898 0.160558
\(932\) −19.5959 −0.641886
\(933\) −61.2372 −2.00482
\(934\) −48.0000 −1.57061
\(935\) 17.1464 0.560748
\(936\) 44.0908 1.44115
\(937\) −2.44949 −0.0800213 −0.0400107 0.999199i \(-0.512739\pi\)
−0.0400107 + 0.999199i \(0.512739\pi\)
\(938\) 54.0000 1.76316
\(939\) −18.0000 −0.587408
\(940\) −97.9796 −3.19574
\(941\) −29.0000 −0.945373 −0.472686 0.881231i \(-0.656716\pi\)
−0.472686 + 0.881231i \(0.656716\pi\)
\(942\) −58.7878 −1.91541
\(943\) −5.00000 −0.162822
\(944\) −40.0000 −1.30189
\(945\) 0 0
\(946\) 0 0
\(947\) 25.0000 0.812391 0.406195 0.913786i \(-0.366855\pi\)
0.406195 + 0.913786i \(0.366855\pi\)
\(948\) −58.7878 −1.90934
\(949\) −36.7423 −1.19271
\(950\) 12.0000 0.389331
\(951\) −41.6413 −1.35031
\(952\) −84.0000 −2.72246
\(953\) 56.3383 1.82498 0.912488 0.409104i \(-0.134159\pi\)
0.912488 + 0.409104i \(0.134159\pi\)
\(954\) 7.34847 0.237915
\(955\) −36.0000 −1.16493
\(956\) −104.000 −3.36360
\(957\) −6.00000 −0.193952
\(958\) 2.44949 0.0791394
\(959\) −12.0000 −0.387500
\(960\) −48.0000 −1.54919
\(961\) −22.0000 −0.709677
\(962\) 36.0000 1.16069
\(963\) 42.0000 1.35343
\(964\) −78.3837 −2.52457
\(965\) 36.7423 1.18278
\(966\) 14.6969 0.472866
\(967\) 49.0000 1.57573 0.787867 0.615846i \(-0.211185\pi\)
0.787867 + 0.615846i \(0.211185\pi\)
\(968\) 48.9898 1.57459
\(969\) 84.0000 2.69847
\(970\) −66.0000 −2.11913
\(971\) −41.0000 −1.31575 −0.657876 0.753126i \(-0.728545\pi\)
−0.657876 + 0.753126i \(0.728545\pi\)
\(972\) −88.1816 −2.82843
\(973\) 17.1464 0.549689
\(974\) 9.79796 0.313947
\(975\) −7.34847 −0.235339
\(976\) −29.3939 −0.940875
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 14.6969 0.469956
\(979\) −17.1464 −0.548002
\(980\) −9.79796 −0.312984
\(981\) 39.0000 1.24517
\(982\) −54.0000 −1.72321
\(983\) 12.2474 0.390633 0.195316 0.980740i \(-0.437427\pi\)
0.195316 + 0.980740i \(0.437427\pi\)
\(984\) 60.0000 1.91273
\(985\) 19.5959 0.624378
\(986\) 42.0000 1.33755
\(987\) 60.0000 1.90982
\(988\) 58.7878 1.87029
\(989\) 0 0
\(990\) 18.0000 0.572078
\(991\) −31.8434 −1.01154 −0.505769 0.862669i \(-0.668791\pi\)
−0.505769 + 0.862669i \(0.668791\pi\)
\(992\) 0 0
\(993\) 6.00000 0.190404
\(994\) −29.3939 −0.932317
\(995\) 48.0000 1.52170
\(996\) 9.79796 0.310460
\(997\) −31.8434 −1.00849 −0.504245 0.863561i \(-0.668229\pi\)
−0.504245 + 0.863561i \(0.668229\pi\)
\(998\) −18.0000 −0.569780
\(999\) 0 0
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 1849.2.a.g.1.1 2
43.42 odd 2 inner 1849.2.a.g.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.2.a.g.1.1 2 1.1 even 1 trivial
1849.2.a.g.1.2 yes 2 43.42 odd 2 inner