Properties

Label 2850.2.a.bc.1.2
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.449490 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.449490 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.44949 q^{11} -1.00000 q^{12} -2.44949 q^{13} -0.449490 q^{14} +1.00000 q^{16} +0.449490 q^{17} -1.00000 q^{18} +1.00000 q^{19} -0.449490 q^{21} +1.44949 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.44949 q^{26} -1.00000 q^{27} +0.449490 q^{28} +10.3485 q^{29} -3.00000 q^{31} -1.00000 q^{32} +1.44949 q^{33} -0.449490 q^{34} +1.00000 q^{36} -11.7980 q^{37} -1.00000 q^{38} +2.44949 q^{39} +8.89898 q^{41} +0.449490 q^{42} -2.44949 q^{43} -1.44949 q^{44} -1.00000 q^{46} -11.7980 q^{47} -1.00000 q^{48} -6.79796 q^{49} -0.449490 q^{51} -2.44949 q^{52} +2.55051 q^{53} +1.00000 q^{54} -0.449490 q^{56} -1.00000 q^{57} -10.3485 q^{58} +1.55051 q^{59} -4.55051 q^{61} +3.00000 q^{62} +0.449490 q^{63} +1.00000 q^{64} -1.44949 q^{66} +9.24745 q^{67} +0.449490 q^{68} -1.00000 q^{69} -6.44949 q^{71} -1.00000 q^{72} +1.00000 q^{73} +11.7980 q^{74} +1.00000 q^{76} -0.651531 q^{77} -2.44949 q^{78} -5.00000 q^{79} +1.00000 q^{81} -8.89898 q^{82} +6.34847 q^{83} -0.449490 q^{84} +2.44949 q^{86} -10.3485 q^{87} +1.44949 q^{88} +11.8990 q^{89} -1.10102 q^{91} +1.00000 q^{92} +3.00000 q^{93} +11.7980 q^{94} +1.00000 q^{96} -6.44949 q^{97} +6.79796 q^{98} -1.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 4q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 4q^{7} - 2q^{8} + 2q^{9} + 2q^{11} - 2q^{12} + 4q^{14} + 2q^{16} - 4q^{17} - 2q^{18} + 2q^{19} + 4q^{21} - 2q^{22} + 2q^{23} + 2q^{24} - 2q^{27} - 4q^{28} + 6q^{29} - 6q^{31} - 2q^{32} - 2q^{33} + 4q^{34} + 2q^{36} - 4q^{37} - 2q^{38} + 8q^{41} - 4q^{42} + 2q^{44} - 2q^{46} - 4q^{47} - 2q^{48} + 6q^{49} + 4q^{51} + 10q^{53} + 2q^{54} + 4q^{56} - 2q^{57} - 6q^{58} + 8q^{59} - 14q^{61} + 6q^{62} - 4q^{63} + 2q^{64} + 2q^{66} - 6q^{67} - 4q^{68} - 2q^{69} - 8q^{71} - 2q^{72} + 2q^{73} + 4q^{74} + 2q^{76} - 16q^{77} - 10q^{79} + 2q^{81} - 8q^{82} - 2q^{83} + 4q^{84} - 6q^{87} - 2q^{88} + 14q^{89} - 12q^{91} + 2q^{92} + 6q^{93} + 4q^{94} + 2q^{96} - 8q^{97} - 6q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0.449490 0.169891 0.0849456 0.996386i \(-0.472928\pi\)
0.0849456 + 0.996386i \(0.472928\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.44949 −0.437038 −0.218519 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.44949 −0.679366 −0.339683 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) −0.449490 −0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.449490 0.109017 0.0545086 0.998513i \(-0.482641\pi\)
0.0545086 + 0.998513i \(0.482641\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.449490 −0.0980867
\(22\) 1.44949 0.309032
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.44949 0.480384
\(27\) −1.00000 −0.192450
\(28\) 0.449490 0.0849456
\(29\) 10.3485 1.92166 0.960831 0.277134i \(-0.0893846\pi\)
0.960831 + 0.277134i \(0.0893846\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.44949 0.252324
\(34\) −0.449490 −0.0770869
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.7980 −1.93957 −0.969786 0.243956i \(-0.921555\pi\)
−0.969786 + 0.243956i \(0.921555\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.44949 0.392232
\(40\) 0 0
\(41\) 8.89898 1.38979 0.694894 0.719113i \(-0.255451\pi\)
0.694894 + 0.719113i \(0.255451\pi\)
\(42\) 0.449490 0.0693578
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) −1.44949 −0.218519
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −11.7980 −1.72091 −0.860455 0.509527i \(-0.829820\pi\)
−0.860455 + 0.509527i \(0.829820\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.79796 −0.971137
\(50\) 0 0
\(51\) −0.449490 −0.0629412
\(52\) −2.44949 −0.339683
\(53\) 2.55051 0.350340 0.175170 0.984538i \(-0.443953\pi\)
0.175170 + 0.984538i \(0.443953\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.449490 −0.0600656
\(57\) −1.00000 −0.132453
\(58\) −10.3485 −1.35882
\(59\) 1.55051 0.201859 0.100930 0.994894i \(-0.467818\pi\)
0.100930 + 0.994894i \(0.467818\pi\)
\(60\) 0 0
\(61\) −4.55051 −0.582633 −0.291317 0.956627i \(-0.594093\pi\)
−0.291317 + 0.956627i \(0.594093\pi\)
\(62\) 3.00000 0.381000
\(63\) 0.449490 0.0566304
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.44949 −0.178420
\(67\) 9.24745 1.12976 0.564878 0.825175i \(-0.308923\pi\)
0.564878 + 0.825175i \(0.308923\pi\)
\(68\) 0.449490 0.0545086
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.44949 −0.765414 −0.382707 0.923870i \(-0.625008\pi\)
−0.382707 + 0.923870i \(0.625008\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 11.7980 1.37148
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.651531 −0.0742488
\(78\) −2.44949 −0.277350
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −8.89898 −0.982728
\(83\) 6.34847 0.696835 0.348418 0.937339i \(-0.386719\pi\)
0.348418 + 0.937339i \(0.386719\pi\)
\(84\) −0.449490 −0.0490434
\(85\) 0 0
\(86\) 2.44949 0.264135
\(87\) −10.3485 −1.10947
\(88\) 1.44949 0.154516
\(89\) 11.8990 1.26129 0.630645 0.776072i \(-0.282791\pi\)
0.630645 + 0.776072i \(0.282791\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) 1.00000 0.104257
\(93\) 3.00000 0.311086
\(94\) 11.7980 1.21687
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.44949 −0.654846 −0.327423 0.944878i \(-0.606180\pi\)
−0.327423 + 0.944878i \(0.606180\pi\)
\(98\) 6.79796 0.686698
\(99\) −1.44949 −0.145679
\(100\) 0 0
\(101\) −6.44949 −0.641748 −0.320874 0.947122i \(-0.603977\pi\)
−0.320874 + 0.947122i \(0.603977\pi\)
\(102\) 0.449490 0.0445061
\(103\) 7.89898 0.778310 0.389155 0.921172i \(-0.372767\pi\)
0.389155 + 0.921172i \(0.372767\pi\)
\(104\) 2.44949 0.240192
\(105\) 0 0
\(106\) −2.55051 −0.247727
\(107\) −2.65153 −0.256333 −0.128167 0.991753i \(-0.540909\pi\)
−0.128167 + 0.991753i \(0.540909\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.89898 0.660802 0.330401 0.943841i \(-0.392816\pi\)
0.330401 + 0.943841i \(0.392816\pi\)
\(110\) 0 0
\(111\) 11.7980 1.11981
\(112\) 0.449490 0.0424728
\(113\) −2.79796 −0.263210 −0.131605 0.991302i \(-0.542013\pi\)
−0.131605 + 0.991302i \(0.542013\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 10.3485 0.960831
\(117\) −2.44949 −0.226455
\(118\) −1.55051 −0.142736
\(119\) 0.202041 0.0185211
\(120\) 0 0
\(121\) −8.89898 −0.808998
\(122\) 4.55051 0.411984
\(123\) −8.89898 −0.802394
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −0.449490 −0.0400437
\(127\) −9.89898 −0.878392 −0.439196 0.898391i \(-0.644737\pi\)
−0.439196 + 0.898391i \(0.644737\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.44949 0.215666
\(130\) 0 0
\(131\) −15.2474 −1.33218 −0.666088 0.745873i \(-0.732032\pi\)
−0.666088 + 0.745873i \(0.732032\pi\)
\(132\) 1.44949 0.126162
\(133\) 0.449490 0.0389757
\(134\) −9.24745 −0.798858
\(135\) 0 0
\(136\) −0.449490 −0.0385434
\(137\) −4.89898 −0.418548 −0.209274 0.977857i \(-0.567110\pi\)
−0.209274 + 0.977857i \(0.567110\pi\)
\(138\) 1.00000 0.0851257
\(139\) 2.24745 0.190626 0.0953131 0.995447i \(-0.469615\pi\)
0.0953131 + 0.995447i \(0.469615\pi\)
\(140\) 0 0
\(141\) 11.7980 0.993567
\(142\) 6.44949 0.541229
\(143\) 3.55051 0.296909
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 6.79796 0.560686
\(148\) −11.7980 −0.969786
\(149\) −13.7980 −1.13037 −0.565186 0.824963i \(-0.691196\pi\)
−0.565186 + 0.824963i \(0.691196\pi\)
\(150\) 0 0
\(151\) −4.20204 −0.341957 −0.170979 0.985275i \(-0.554693\pi\)
−0.170979 + 0.985275i \(0.554693\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0.449490 0.0363391
\(154\) 0.651531 0.0525018
\(155\) 0 0
\(156\) 2.44949 0.196116
\(157\) −4.89898 −0.390981 −0.195491 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(158\) 5.00000 0.397779
\(159\) −2.55051 −0.202269
\(160\) 0 0
\(161\) 0.449490 0.0354248
\(162\) −1.00000 −0.0785674
\(163\) −0.202041 −0.0158251 −0.00791254 0.999969i \(-0.502519\pi\)
−0.00791254 + 0.999969i \(0.502519\pi\)
\(164\) 8.89898 0.694894
\(165\) 0 0
\(166\) −6.34847 −0.492737
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0.449490 0.0346789
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −2.44949 −0.186772
\(173\) 16.3485 1.24295 0.621476 0.783434i \(-0.286534\pi\)
0.621476 + 0.783434i \(0.286534\pi\)
\(174\) 10.3485 0.784515
\(175\) 0 0
\(176\) −1.44949 −0.109259
\(177\) −1.55051 −0.116543
\(178\) −11.8990 −0.891866
\(179\) 6.20204 0.463562 0.231781 0.972768i \(-0.425545\pi\)
0.231781 + 0.972768i \(0.425545\pi\)
\(180\) 0 0
\(181\) −9.55051 −0.709884 −0.354942 0.934888i \(-0.615499\pi\)
−0.354942 + 0.934888i \(0.615499\pi\)
\(182\) 1.10102 0.0816131
\(183\) 4.55051 0.336383
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) −0.651531 −0.0476446
\(188\) −11.7980 −0.860455
\(189\) −0.449490 −0.0326956
\(190\) 0 0
\(191\) −9.89898 −0.716265 −0.358133 0.933671i \(-0.616586\pi\)
−0.358133 + 0.933671i \(0.616586\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0.651531 0.0468982 0.0234491 0.999725i \(-0.492535\pi\)
0.0234491 + 0.999725i \(0.492535\pi\)
\(194\) 6.44949 0.463046
\(195\) 0 0
\(196\) −6.79796 −0.485568
\(197\) −12.6515 −0.901384 −0.450692 0.892679i \(-0.648823\pi\)
−0.450692 + 0.892679i \(0.648823\pi\)
\(198\) 1.44949 0.103011
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −9.24745 −0.652265
\(202\) 6.44949 0.453785
\(203\) 4.65153 0.326473
\(204\) −0.449490 −0.0314706
\(205\) 0 0
\(206\) −7.89898 −0.550348
\(207\) 1.00000 0.0695048
\(208\) −2.44949 −0.169842
\(209\) −1.44949 −0.100263
\(210\) 0 0
\(211\) −18.3485 −1.26316 −0.631580 0.775310i \(-0.717593\pi\)
−0.631580 + 0.775310i \(0.717593\pi\)
\(212\) 2.55051 0.175170
\(213\) 6.44949 0.441912
\(214\) 2.65153 0.181255
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −1.34847 −0.0915401
\(218\) −6.89898 −0.467258
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) −1.10102 −0.0740627
\(222\) −11.7980 −0.791827
\(223\) −25.8990 −1.73432 −0.867162 0.498026i \(-0.834058\pi\)
−0.867162 + 0.498026i \(0.834058\pi\)
\(224\) −0.449490 −0.0300328
\(225\) 0 0
\(226\) 2.79796 0.186117
\(227\) 9.59592 0.636903 0.318452 0.947939i \(-0.396837\pi\)
0.318452 + 0.947939i \(0.396837\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −17.2474 −1.13974 −0.569872 0.821734i \(-0.693007\pi\)
−0.569872 + 0.821734i \(0.693007\pi\)
\(230\) 0 0
\(231\) 0.651531 0.0428676
\(232\) −10.3485 −0.679410
\(233\) 18.2474 1.19543 0.597715 0.801709i \(-0.296075\pi\)
0.597715 + 0.801709i \(0.296075\pi\)
\(234\) 2.44949 0.160128
\(235\) 0 0
\(236\) 1.55051 0.100930
\(237\) 5.00000 0.324785
\(238\) −0.202041 −0.0130964
\(239\) 20.6969 1.33877 0.669387 0.742914i \(-0.266557\pi\)
0.669387 + 0.742914i \(0.266557\pi\)
\(240\) 0 0
\(241\) 0.449490 0.0289542 0.0144771 0.999895i \(-0.495392\pi\)
0.0144771 + 0.999895i \(0.495392\pi\)
\(242\) 8.89898 0.572048
\(243\) −1.00000 −0.0641500
\(244\) −4.55051 −0.291317
\(245\) 0 0
\(246\) 8.89898 0.567378
\(247\) −2.44949 −0.155857
\(248\) 3.00000 0.190500
\(249\) −6.34847 −0.402318
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0.449490 0.0283152
\(253\) −1.44949 −0.0911286
\(254\) 9.89898 0.621117
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.4949 −1.71508 −0.857542 0.514413i \(-0.828010\pi\)
−0.857542 + 0.514413i \(0.828010\pi\)
\(258\) −2.44949 −0.152499
\(259\) −5.30306 −0.329516
\(260\) 0 0
\(261\) 10.3485 0.640554
\(262\) 15.2474 0.941991
\(263\) −12.7980 −0.789156 −0.394578 0.918862i \(-0.629109\pi\)
−0.394578 + 0.918862i \(0.629109\pi\)
\(264\) −1.44949 −0.0892099
\(265\) 0 0
\(266\) −0.449490 −0.0275600
\(267\) −11.8990 −0.728206
\(268\) 9.24745 0.564878
\(269\) 6.89898 0.420638 0.210319 0.977633i \(-0.432550\pi\)
0.210319 + 0.977633i \(0.432550\pi\)
\(270\) 0 0
\(271\) −30.9444 −1.87974 −0.939869 0.341536i \(-0.889053\pi\)
−0.939869 + 0.341536i \(0.889053\pi\)
\(272\) 0.449490 0.0272543
\(273\) 1.10102 0.0666368
\(274\) 4.89898 0.295958
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 23.0454 1.38466 0.692332 0.721579i \(-0.256583\pi\)
0.692332 + 0.721579i \(0.256583\pi\)
\(278\) −2.24745 −0.134793
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) −11.7980 −0.702558
\(283\) −10.2020 −0.606448 −0.303224 0.952919i \(-0.598063\pi\)
−0.303224 + 0.952919i \(0.598063\pi\)
\(284\) −6.44949 −0.382707
\(285\) 0 0
\(286\) −3.55051 −0.209946
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) −16.7980 −0.988115
\(290\) 0 0
\(291\) 6.44949 0.378076
\(292\) 1.00000 0.0585206
\(293\) −1.24745 −0.0728767 −0.0364384 0.999336i \(-0.511601\pi\)
−0.0364384 + 0.999336i \(0.511601\pi\)
\(294\) −6.79796 −0.396465
\(295\) 0 0
\(296\) 11.7980 0.685742
\(297\) 1.44949 0.0841079
\(298\) 13.7980 0.799294
\(299\) −2.44949 −0.141658
\(300\) 0 0
\(301\) −1.10102 −0.0634618
\(302\) 4.20204 0.241800
\(303\) 6.44949 0.370514
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −0.449490 −0.0256956
\(307\) 1.65153 0.0942578 0.0471289 0.998889i \(-0.484993\pi\)
0.0471289 + 0.998889i \(0.484993\pi\)
\(308\) −0.651531 −0.0371244
\(309\) −7.89898 −0.449357
\(310\) 0 0
\(311\) −4.20204 −0.238276 −0.119138 0.992878i \(-0.538013\pi\)
−0.119138 + 0.992878i \(0.538013\pi\)
\(312\) −2.44949 −0.138675
\(313\) −22.1010 −1.24922 −0.624612 0.780935i \(-0.714743\pi\)
−0.624612 + 0.780935i \(0.714743\pi\)
\(314\) 4.89898 0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 9.24745 0.519388 0.259694 0.965691i \(-0.416378\pi\)
0.259694 + 0.965691i \(0.416378\pi\)
\(318\) 2.55051 0.143026
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 2.65153 0.147994
\(322\) −0.449490 −0.0250491
\(323\) 0.449490 0.0250103
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0.202041 0.0111900
\(327\) −6.89898 −0.381514
\(328\) −8.89898 −0.491364
\(329\) −5.30306 −0.292367
\(330\) 0 0
\(331\) −7.65153 −0.420566 −0.210283 0.977641i \(-0.567439\pi\)
−0.210283 + 0.977641i \(0.567439\pi\)
\(332\) 6.34847 0.348418
\(333\) −11.7980 −0.646524
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −0.449490 −0.0245217
\(337\) 15.1010 0.822605 0.411303 0.911499i \(-0.365074\pi\)
0.411303 + 0.911499i \(0.365074\pi\)
\(338\) 7.00000 0.380750
\(339\) 2.79796 0.151964
\(340\) 0 0
\(341\) 4.34847 0.235483
\(342\) −1.00000 −0.0540738
\(343\) −6.20204 −0.334879
\(344\) 2.44949 0.132068
\(345\) 0 0
\(346\) −16.3485 −0.878899
\(347\) −15.5959 −0.837233 −0.418616 0.908163i \(-0.637485\pi\)
−0.418616 + 0.908163i \(0.637485\pi\)
\(348\) −10.3485 −0.554736
\(349\) 27.9444 1.49583 0.747914 0.663795i \(-0.231055\pi\)
0.747914 + 0.663795i \(0.231055\pi\)
\(350\) 0 0
\(351\) 2.44949 0.130744
\(352\) 1.44949 0.0772581
\(353\) −6.24745 −0.332518 −0.166259 0.986082i \(-0.553169\pi\)
−0.166259 + 0.986082i \(0.553169\pi\)
\(354\) 1.55051 0.0824087
\(355\) 0 0
\(356\) 11.8990 0.630645
\(357\) −0.202041 −0.0106931
\(358\) −6.20204 −0.327788
\(359\) −13.1010 −0.691445 −0.345723 0.938337i \(-0.612366\pi\)
−0.345723 + 0.938337i \(0.612366\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 9.55051 0.501964
\(363\) 8.89898 0.467075
\(364\) −1.10102 −0.0577092
\(365\) 0 0
\(366\) −4.55051 −0.237859
\(367\) 10.4495 0.545459 0.272729 0.962091i \(-0.412074\pi\)
0.272729 + 0.962091i \(0.412074\pi\)
\(368\) 1.00000 0.0521286
\(369\) 8.89898 0.463262
\(370\) 0 0
\(371\) 1.14643 0.0595196
\(372\) 3.00000 0.155543
\(373\) 17.5505 0.908731 0.454365 0.890815i \(-0.349866\pi\)
0.454365 + 0.890815i \(0.349866\pi\)
\(374\) 0.651531 0.0336899
\(375\) 0 0
\(376\) 11.7980 0.608433
\(377\) −25.3485 −1.30551
\(378\) 0.449490 0.0231193
\(379\) −30.6969 −1.57680 −0.788398 0.615166i \(-0.789089\pi\)
−0.788398 + 0.615166i \(0.789089\pi\)
\(380\) 0 0
\(381\) 9.89898 0.507140
\(382\) 9.89898 0.506476
\(383\) 8.24745 0.421425 0.210712 0.977548i \(-0.432422\pi\)
0.210712 + 0.977548i \(0.432422\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −0.651531 −0.0331620
\(387\) −2.44949 −0.124515
\(388\) −6.44949 −0.327423
\(389\) −17.5959 −0.892148 −0.446074 0.894996i \(-0.647178\pi\)
−0.446074 + 0.894996i \(0.647178\pi\)
\(390\) 0 0
\(391\) 0.449490 0.0227317
\(392\) 6.79796 0.343349
\(393\) 15.2474 0.769132
\(394\) 12.6515 0.637375
\(395\) 0 0
\(396\) −1.44949 −0.0728396
\(397\) 29.9444 1.50287 0.751433 0.659810i \(-0.229363\pi\)
0.751433 + 0.659810i \(0.229363\pi\)
\(398\) 10.0000 0.501255
\(399\) −0.449490 −0.0225026
\(400\) 0 0
\(401\) 30.7980 1.53798 0.768988 0.639263i \(-0.220760\pi\)
0.768988 + 0.639263i \(0.220760\pi\)
\(402\) 9.24745 0.461221
\(403\) 7.34847 0.366053
\(404\) −6.44949 −0.320874
\(405\) 0 0
\(406\) −4.65153 −0.230852
\(407\) 17.1010 0.847666
\(408\) 0.449490 0.0222531
\(409\) −13.1010 −0.647804 −0.323902 0.946091i \(-0.604995\pi\)
−0.323902 + 0.946091i \(0.604995\pi\)
\(410\) 0 0
\(411\) 4.89898 0.241649
\(412\) 7.89898 0.389155
\(413\) 0.696938 0.0342941
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.44949 0.120096
\(417\) −2.24745 −0.110058
\(418\) 1.44949 0.0708969
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 18.3485 0.893190
\(423\) −11.7980 −0.573636
\(424\) −2.55051 −0.123864
\(425\) 0 0
\(426\) −6.44949 −0.312479
\(427\) −2.04541 −0.0989842
\(428\) −2.65153 −0.128167
\(429\) −3.55051 −0.171420
\(430\) 0 0
\(431\) −14.8990 −0.717659 −0.358829 0.933403i \(-0.616824\pi\)
−0.358829 + 0.933403i \(0.616824\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.3485 −0.929828 −0.464914 0.885356i \(-0.653915\pi\)
−0.464914 + 0.885356i \(0.653915\pi\)
\(434\) 1.34847 0.0647286
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) 1.00000 0.0478365
\(438\) 1.00000 0.0477818
\(439\) −21.8990 −1.04518 −0.522591 0.852584i \(-0.675034\pi\)
−0.522591 + 0.852584i \(0.675034\pi\)
\(440\) 0 0
\(441\) −6.79796 −0.323712
\(442\) 1.10102 0.0523702
\(443\) 3.24745 0.154291 0.0771455 0.997020i \(-0.475419\pi\)
0.0771455 + 0.997020i \(0.475419\pi\)
\(444\) 11.7980 0.559906
\(445\) 0 0
\(446\) 25.8990 1.22635
\(447\) 13.7980 0.652621
\(448\) 0.449490 0.0212364
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −12.8990 −0.607389
\(452\) −2.79796 −0.131605
\(453\) 4.20204 0.197429
\(454\) −9.59592 −0.450359
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 15.1010 0.706396 0.353198 0.935549i \(-0.385094\pi\)
0.353198 + 0.935549i \(0.385094\pi\)
\(458\) 17.2474 0.805920
\(459\) −0.449490 −0.0209804
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) −0.651531 −0.0303120
\(463\) −14.6969 −0.683025 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(464\) 10.3485 0.480416
\(465\) 0 0
\(466\) −18.2474 −0.845297
\(467\) −8.34847 −0.386321 −0.193161 0.981167i \(-0.561874\pi\)
−0.193161 + 0.981167i \(0.561874\pi\)
\(468\) −2.44949 −0.113228
\(469\) 4.15663 0.191935
\(470\) 0 0
\(471\) 4.89898 0.225733
\(472\) −1.55051 −0.0713680
\(473\) 3.55051 0.163253
\(474\) −5.00000 −0.229658
\(475\) 0 0
\(476\) 0.202041 0.00926054
\(477\) 2.55051 0.116780
\(478\) −20.6969 −0.946656
\(479\) −22.5959 −1.03243 −0.516217 0.856458i \(-0.672660\pi\)
−0.516217 + 0.856458i \(0.672660\pi\)
\(480\) 0 0
\(481\) 28.8990 1.31768
\(482\) −0.449490 −0.0204737
\(483\) −0.449490 −0.0204525
\(484\) −8.89898 −0.404499
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 4.55051 0.205992
\(489\) 0.202041 0.00913661
\(490\) 0 0
\(491\) 4.40408 0.198753 0.0993767 0.995050i \(-0.468315\pi\)
0.0993767 + 0.995050i \(0.468315\pi\)
\(492\) −8.89898 −0.401197
\(493\) 4.65153 0.209494
\(494\) 2.44949 0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −2.89898 −0.130037
\(498\) 6.34847 0.284482
\(499\) 39.8434 1.78363 0.891817 0.452396i \(-0.149431\pi\)
0.891817 + 0.452396i \(0.149431\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 18.0000 0.803379
\(503\) −10.2020 −0.454887 −0.227443 0.973791i \(-0.573037\pi\)
−0.227443 + 0.973791i \(0.573037\pi\)
\(504\) −0.449490 −0.0200219
\(505\) 0 0
\(506\) 1.44949 0.0644377
\(507\) 7.00000 0.310881
\(508\) −9.89898 −0.439196
\(509\) 4.14643 0.183787 0.0918936 0.995769i \(-0.470708\pi\)
0.0918936 + 0.995769i \(0.470708\pi\)
\(510\) 0 0
\(511\) 0.449490 0.0198843
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 27.4949 1.21275
\(515\) 0 0
\(516\) 2.44949 0.107833
\(517\) 17.1010 0.752102
\(518\) 5.30306 0.233003
\(519\) −16.3485 −0.717618
\(520\) 0 0
\(521\) −13.6969 −0.600074 −0.300037 0.953928i \(-0.596999\pi\)
−0.300037 + 0.953928i \(0.596999\pi\)
\(522\) −10.3485 −0.452940
\(523\) −25.3939 −1.11040 −0.555198 0.831718i \(-0.687358\pi\)
−0.555198 + 0.831718i \(0.687358\pi\)
\(524\) −15.2474 −0.666088
\(525\) 0 0
\(526\) 12.7980 0.558018
\(527\) −1.34847 −0.0587402
\(528\) 1.44949 0.0630809
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 1.55051 0.0672864
\(532\) 0.449490 0.0194879
\(533\) −21.7980 −0.944174
\(534\) 11.8990 0.514919
\(535\) 0 0
\(536\) −9.24745 −0.399429
\(537\) −6.20204 −0.267638
\(538\) −6.89898 −0.297436
\(539\) 9.85357 0.424423
\(540\) 0 0
\(541\) −12.1464 −0.522216 −0.261108 0.965310i \(-0.584088\pi\)
−0.261108 + 0.965310i \(0.584088\pi\)
\(542\) 30.9444 1.32918
\(543\) 9.55051 0.409852
\(544\) −0.449490 −0.0192717
\(545\) 0 0
\(546\) −1.10102 −0.0471193
\(547\) −36.6413 −1.56667 −0.783335 0.621600i \(-0.786483\pi\)
−0.783335 + 0.621600i \(0.786483\pi\)
\(548\) −4.89898 −0.209274
\(549\) −4.55051 −0.194211
\(550\) 0 0
\(551\) 10.3485 0.440860
\(552\) 1.00000 0.0425628
\(553\) −2.24745 −0.0955712
\(554\) −23.0454 −0.979106
\(555\) 0 0
\(556\) 2.24745 0.0953131
\(557\) −12.6515 −0.536063 −0.268031 0.963410i \(-0.586373\pi\)
−0.268031 + 0.963410i \(0.586373\pi\)
\(558\) 3.00000 0.127000
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0.651531 0.0275077
\(562\) −7.00000 −0.295277
\(563\) 18.2474 0.769038 0.384519 0.923117i \(-0.374367\pi\)
0.384519 + 0.923117i \(0.374367\pi\)
\(564\) 11.7980 0.496784
\(565\) 0 0
\(566\) 10.2020 0.428824
\(567\) 0.449490 0.0188768
\(568\) 6.44949 0.270615
\(569\) 45.1918 1.89454 0.947270 0.320436i \(-0.103829\pi\)
0.947270 + 0.320436i \(0.103829\pi\)
\(570\) 0 0
\(571\) 14.2474 0.596237 0.298119 0.954529i \(-0.403641\pi\)
0.298119 + 0.954529i \(0.403641\pi\)
\(572\) 3.55051 0.148454
\(573\) 9.89898 0.413536
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −9.89898 −0.412100 −0.206050 0.978541i \(-0.566061\pi\)
−0.206050 + 0.978541i \(0.566061\pi\)
\(578\) 16.7980 0.698703
\(579\) −0.651531 −0.0270767
\(580\) 0 0
\(581\) 2.85357 0.118386
\(582\) −6.44949 −0.267340
\(583\) −3.69694 −0.153112
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) 1.24745 0.0515316
\(587\) 28.5505 1.17841 0.589203 0.807985i \(-0.299442\pi\)
0.589203 + 0.807985i \(0.299442\pi\)
\(588\) 6.79796 0.280343
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 12.6515 0.520414
\(592\) −11.7980 −0.484893
\(593\) 40.4949 1.66293 0.831463 0.555580i \(-0.187504\pi\)
0.831463 + 0.555580i \(0.187504\pi\)
\(594\) −1.44949 −0.0594733
\(595\) 0 0
\(596\) −13.7980 −0.565186
\(597\) 10.0000 0.409273
\(598\) 2.44949 0.100167
\(599\) 11.5505 0.471941 0.235971 0.971760i \(-0.424173\pi\)
0.235971 + 0.971760i \(0.424173\pi\)
\(600\) 0 0
\(601\) −17.1464 −0.699417 −0.349709 0.936858i \(-0.613719\pi\)
−0.349709 + 0.936858i \(0.613719\pi\)
\(602\) 1.10102 0.0448742
\(603\) 9.24745 0.376585
\(604\) −4.20204 −0.170979
\(605\) 0 0
\(606\) −6.44949 −0.261993
\(607\) 32.1918 1.30663 0.653313 0.757088i \(-0.273379\pi\)
0.653313 + 0.757088i \(0.273379\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −4.65153 −0.188490
\(610\) 0 0
\(611\) 28.8990 1.16913
\(612\) 0.449490 0.0181695
\(613\) 41.1918 1.66372 0.831861 0.554984i \(-0.187275\pi\)
0.831861 + 0.554984i \(0.187275\pi\)
\(614\) −1.65153 −0.0666504
\(615\) 0 0
\(616\) 0.651531 0.0262509
\(617\) 40.2929 1.62213 0.811065 0.584957i \(-0.198888\pi\)
0.811065 + 0.584957i \(0.198888\pi\)
\(618\) 7.89898 0.317744
\(619\) 10.8536 0.436242 0.218121 0.975922i \(-0.430007\pi\)
0.218121 + 0.975922i \(0.430007\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 4.20204 0.168486
\(623\) 5.34847 0.214282
\(624\) 2.44949 0.0980581
\(625\) 0 0
\(626\) 22.1010 0.883334
\(627\) 1.44949 0.0578870
\(628\) −4.89898 −0.195491
\(629\) −5.30306 −0.211447
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 5.00000 0.198889
\(633\) 18.3485 0.729286
\(634\) −9.24745 −0.367263
\(635\) 0 0
\(636\) −2.55051 −0.101134
\(637\) 16.6515 0.659758
\(638\) 15.0000 0.593856
\(639\) −6.44949 −0.255138
\(640\) 0 0
\(641\) 18.2020 0.718937 0.359469 0.933157i \(-0.382958\pi\)
0.359469 + 0.933157i \(0.382958\pi\)
\(642\) −2.65153 −0.104648
\(643\) 19.7980 0.780755 0.390378 0.920655i \(-0.372344\pi\)
0.390378 + 0.920655i \(0.372344\pi\)
\(644\) 0.449490 0.0177124
\(645\) 0 0
\(646\) −0.449490 −0.0176849
\(647\) −16.1010 −0.632996 −0.316498 0.948593i \(-0.602507\pi\)
−0.316498 + 0.948593i \(0.602507\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.24745 −0.0882201
\(650\) 0 0
\(651\) 1.34847 0.0528507
\(652\) −0.202041 −0.00791254
\(653\) −14.6969 −0.575136 −0.287568 0.957760i \(-0.592847\pi\)
−0.287568 + 0.957760i \(0.592847\pi\)
\(654\) 6.89898 0.269771
\(655\) 0 0
\(656\) 8.89898 0.347447
\(657\) 1.00000 0.0390137
\(658\) 5.30306 0.206735
\(659\) 26.0454 1.01459 0.507293 0.861774i \(-0.330646\pi\)
0.507293 + 0.861774i \(0.330646\pi\)
\(660\) 0 0
\(661\) −15.5959 −0.606611 −0.303305 0.952893i \(-0.598090\pi\)
−0.303305 + 0.952893i \(0.598090\pi\)
\(662\) 7.65153 0.297385
\(663\) 1.10102 0.0427601
\(664\) −6.34847 −0.246368
\(665\) 0 0
\(666\) 11.7980 0.457162
\(667\) 10.3485 0.400694
\(668\) −18.0000 −0.696441
\(669\) 25.8990 1.00131
\(670\) 0 0
\(671\) 6.59592 0.254633
\(672\) 0.449490 0.0173394
\(673\) 30.6515 1.18153 0.590765 0.806844i \(-0.298826\pi\)
0.590765 + 0.806844i \(0.298826\pi\)
\(674\) −15.1010 −0.581670
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) −7.65153 −0.294072 −0.147036 0.989131i \(-0.546973\pi\)
−0.147036 + 0.989131i \(0.546973\pi\)
\(678\) −2.79796 −0.107455
\(679\) −2.89898 −0.111253
\(680\) 0 0
\(681\) −9.59592 −0.367716
\(682\) −4.34847 −0.166511
\(683\) −47.6413 −1.82294 −0.911472 0.411361i \(-0.865053\pi\)
−0.911472 + 0.411361i \(0.865053\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 6.20204 0.236795
\(687\) 17.2474 0.658031
\(688\) −2.44949 −0.0933859
\(689\) −6.24745 −0.238009
\(690\) 0 0
\(691\) 34.2474 1.30283 0.651417 0.758720i \(-0.274175\pi\)
0.651417 + 0.758720i \(0.274175\pi\)
\(692\) 16.3485 0.621476
\(693\) −0.651531 −0.0247496
\(694\) 15.5959 0.592013
\(695\) 0 0
\(696\) 10.3485 0.392258
\(697\) 4.00000 0.151511
\(698\) −27.9444 −1.05771
\(699\) −18.2474 −0.690182
\(700\) 0 0
\(701\) 16.4949 0.623004 0.311502 0.950246i \(-0.399168\pi\)
0.311502 + 0.950246i \(0.399168\pi\)
\(702\) −2.44949 −0.0924500
\(703\) −11.7980 −0.444968
\(704\) −1.44949 −0.0546297
\(705\) 0 0
\(706\) 6.24745 0.235126
\(707\) −2.89898 −0.109027
\(708\) −1.55051 −0.0582717
\(709\) −37.2474 −1.39886 −0.699429 0.714702i \(-0.746562\pi\)
−0.699429 + 0.714702i \(0.746562\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) −11.8990 −0.445933
\(713\) −3.00000 −0.112351
\(714\) 0.202041 0.00756120
\(715\) 0 0
\(716\) 6.20204 0.231781
\(717\) −20.6969 −0.772941
\(718\) 13.1010 0.488926
\(719\) −38.7980 −1.44692 −0.723460 0.690366i \(-0.757449\pi\)
−0.723460 + 0.690366i \(0.757449\pi\)
\(720\) 0 0
\(721\) 3.55051 0.132228
\(722\) −1.00000 −0.0372161
\(723\) −0.449490 −0.0167167
\(724\) −9.55051 −0.354942
\(725\) 0 0
\(726\) −8.89898 −0.330272
\(727\) 26.4949 0.982641 0.491321 0.870979i \(-0.336514\pi\)
0.491321 + 0.870979i \(0.336514\pi\)
\(728\) 1.10102 0.0408065
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.10102 −0.0407227
\(732\) 4.55051 0.168192
\(733\) 30.1464 1.11348 0.556742 0.830686i \(-0.312051\pi\)
0.556742 + 0.830686i \(0.312051\pi\)
\(734\) −10.4495 −0.385698
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −13.4041 −0.493746
\(738\) −8.89898 −0.327576
\(739\) 23.1010 0.849785 0.424892 0.905244i \(-0.360312\pi\)
0.424892 + 0.905244i \(0.360312\pi\)
\(740\) 0 0
\(741\) 2.44949 0.0899843
\(742\) −1.14643 −0.0420867
\(743\) −5.39388 −0.197882 −0.0989411 0.995093i \(-0.531546\pi\)
−0.0989411 + 0.995093i \(0.531546\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −17.5505 −0.642570
\(747\) 6.34847 0.232278
\(748\) −0.651531 −0.0238223
\(749\) −1.19184 −0.0435487
\(750\) 0 0
\(751\) −4.20204 −0.153335 −0.0766673 0.997057i \(-0.524428\pi\)
−0.0766673 + 0.997057i \(0.524428\pi\)
\(752\) −11.7980 −0.430227
\(753\) 18.0000 0.655956
\(754\) 25.3485 0.923137
\(755\) 0 0
\(756\) −0.449490 −0.0163478
\(757\) −13.8536 −0.503517 −0.251758 0.967790i \(-0.581009\pi\)
−0.251758 + 0.967790i \(0.581009\pi\)
\(758\) 30.6969 1.11496
\(759\) 1.44949 0.0526131
\(760\) 0 0
\(761\) 36.6515 1.32862 0.664308 0.747459i \(-0.268726\pi\)
0.664308 + 0.747459i \(0.268726\pi\)
\(762\) −9.89898 −0.358602
\(763\) 3.10102 0.112264
\(764\) −9.89898 −0.358133
\(765\) 0 0
\(766\) −8.24745 −0.297992
\(767\) −3.79796 −0.137136
\(768\) −1.00000 −0.0360844
\(769\) 1.89898 0.0684790 0.0342395 0.999414i \(-0.489099\pi\)
0.0342395 + 0.999414i \(0.489099\pi\)
\(770\) 0 0
\(771\) 27.4949 0.990205
\(772\) 0.651531 0.0234491
\(773\) 30.4949 1.09683 0.548413 0.836208i \(-0.315232\pi\)
0.548413 + 0.836208i \(0.315232\pi\)
\(774\) 2.44949 0.0880451
\(775\) 0 0
\(776\) 6.44949 0.231523
\(777\) 5.30306 0.190246
\(778\) 17.5959 0.630844
\(779\) 8.89898 0.318839
\(780\) 0 0
\(781\) 9.34847 0.334515
\(782\) −0.449490 −0.0160737
\(783\) −10.3485 −0.369824
\(784\) −6.79796 −0.242784
\(785\) 0 0
\(786\) −15.2474 −0.543858
\(787\) 28.5505 1.01772 0.508858 0.860851i \(-0.330068\pi\)
0.508858 + 0.860851i \(0.330068\pi\)
\(788\) −12.6515 −0.450692
\(789\) 12.7980 0.455619
\(790\) 0 0
\(791\) −1.25765 −0.0447170
\(792\) 1.44949 0.0515054
\(793\) 11.1464 0.395821
\(794\) −29.9444 −1.06269
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −52.4949 −1.85946 −0.929732 0.368236i \(-0.879962\pi\)
−0.929732 + 0.368236i \(0.879962\pi\)
\(798\) 0.449490 0.0159118
\(799\) −5.30306 −0.187609
\(800\) 0 0
\(801\) 11.8990 0.420430
\(802\) −30.7980 −1.08751
\(803\) −1.44949 −0.0511514
\(804\) −9.24745 −0.326132
\(805\) 0 0
\(806\) −7.34847 −0.258839
\(807\) −6.89898 −0.242856
\(808\) 6.44949 0.226892
\(809\) −8.44949 −0.297068 −0.148534 0.988907i \(-0.547455\pi\)
−0.148534 + 0.988907i \(0.547455\pi\)
\(810\) 0 0
\(811\) −14.5505 −0.510938 −0.255469 0.966817i \(-0.582230\pi\)
−0.255469 + 0.966817i \(0.582230\pi\)
\(812\) 4.65153 0.163237
\(813\) 30.9444 1.08527
\(814\) −17.1010 −0.599390
\(815\) 0 0
\(816\) −0.449490 −0.0157353
\(817\) −2.44949 −0.0856968
\(818\) 13.1010 0.458066
\(819\) −1.10102 −0.0384728
\(820\) 0 0
\(821\) 11.1464 0.389013 0.194507 0.980901i \(-0.437689\pi\)
0.194507 + 0.980901i \(0.437689\pi\)
\(822\) −4.89898 −0.170872
\(823\) 12.2020 0.425336 0.212668 0.977124i \(-0.431785\pi\)
0.212668 + 0.977124i \(0.431785\pi\)
\(824\) −7.89898 −0.275174
\(825\) 0 0
\(826\) −0.696938 −0.0242496
\(827\) 24.2474 0.843166 0.421583 0.906790i \(-0.361475\pi\)
0.421583 + 0.906790i \(0.361475\pi\)
\(828\) 1.00000 0.0347524
\(829\) 43.6413 1.51573 0.757863 0.652414i \(-0.226244\pi\)
0.757863 + 0.652414i \(0.226244\pi\)
\(830\) 0 0
\(831\) −23.0454 −0.799436
\(832\) −2.44949 −0.0849208
\(833\) −3.05561 −0.105871
\(834\) 2.24745 0.0778228
\(835\) 0 0
\(836\) −1.44949 −0.0501317
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) 25.3485 0.875126 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(840\) 0 0
\(841\) 78.0908 2.69279
\(842\) −22.0000 −0.758170
\(843\) −7.00000 −0.241093
\(844\) −18.3485 −0.631580
\(845\) 0 0
\(846\) 11.7980 0.405622
\(847\) −4.00000 −0.137442
\(848\) 2.55051 0.0875849
\(849\) 10.2020 0.350133
\(850\) 0 0
\(851\) −11.7980 −0.404429
\(852\) 6.44949 0.220956
\(853\) −10.8990 −0.373174 −0.186587 0.982438i \(-0.559743\pi\)
−0.186587 + 0.982438i \(0.559743\pi\)
\(854\) 2.04541 0.0699924
\(855\) 0 0
\(856\) 2.65153 0.0906275
\(857\) −12.4949 −0.426818 −0.213409 0.976963i \(-0.568457\pi\)
−0.213409 + 0.976963i \(0.568457\pi\)
\(858\) 3.55051 0.121212
\(859\) −23.7980 −0.811976 −0.405988 0.913878i \(-0.633073\pi\)
−0.405988 + 0.913878i \(0.633073\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 14.8990 0.507461
\(863\) 46.6969 1.58958 0.794791 0.606883i \(-0.207580\pi\)
0.794791 + 0.606883i \(0.207580\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 19.3485 0.657488
\(867\) 16.7980 0.570489
\(868\) −1.34847 −0.0457700
\(869\) 7.24745 0.245853
\(870\) 0 0
\(871\) −22.6515 −0.767518
\(872\) −6.89898 −0.233629
\(873\) −6.44949 −0.218282
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −1.00000 −0.0337869
\(877\) 18.8990 0.638173 0.319087 0.947726i \(-0.396624\pi\)
0.319087 + 0.947726i \(0.396624\pi\)
\(878\) 21.8990 0.739055
\(879\) 1.24745 0.0420754
\(880\) 0 0
\(881\) 21.3031 0.717718 0.358859 0.933392i \(-0.383166\pi\)
0.358859 + 0.933392i \(0.383166\pi\)
\(882\) 6.79796 0.228899
\(883\) −49.3485 −1.66071 −0.830354 0.557236i \(-0.811862\pi\)
−0.830354 + 0.557236i \(0.811862\pi\)
\(884\) −1.10102 −0.0370313
\(885\) 0 0
\(886\) −3.24745 −0.109100
\(887\) −4.89898 −0.164492 −0.0822458 0.996612i \(-0.526209\pi\)
−0.0822458 + 0.996612i \(0.526209\pi\)
\(888\) −11.7980 −0.395914
\(889\) −4.44949 −0.149231
\(890\) 0 0
\(891\) −1.44949 −0.0485597
\(892\) −25.8990 −0.867162
\(893\) −11.7980 −0.394804
\(894\) −13.7980 −0.461473
\(895\) 0 0
\(896\) −0.449490 −0.0150164
\(897\) 2.44949 0.0817861
\(898\) 15.0000 0.500556
\(899\) −31.0454 −1.03542
\(900\) 0 0
\(901\) 1.14643 0.0381931
\(902\) 12.8990 0.429489
\(903\) 1.10102 0.0366397
\(904\) 2.79796 0.0930587
\(905\) 0 0
\(906\) −4.20204 −0.139603
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 9.59592 0.318452
\(909\) −6.44949 −0.213916
\(910\) 0 0
\(911\) −42.4949 −1.40792 −0.703959 0.710240i \(-0.748586\pi\)
−0.703959 + 0.710240i \(0.748586\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −9.20204 −0.304543
\(914\) −15.1010 −0.499497
\(915\) 0 0
\(916\) −17.2474 −0.569872
\(917\) −6.85357 −0.226325
\(918\) 0.449490 0.0148354
\(919\) −39.8434 −1.31431 −0.657156 0.753755i \(-0.728241\pi\)
−0.657156 + 0.753755i \(0.728241\pi\)
\(920\) 0 0
\(921\) −1.65153 −0.0544198
\(922\) −12.0000 −0.395199
\(923\) 15.7980 0.519996
\(924\) 0.651531 0.0214338
\(925\) 0 0
\(926\) 14.6969 0.482971
\(927\) 7.89898 0.259437
\(928\) −10.3485 −0.339705
\(929\) 7.59592 0.249214 0.124607 0.992206i \(-0.460233\pi\)
0.124607 + 0.992206i \(0.460233\pi\)
\(930\) 0 0
\(931\) −6.79796 −0.222794
\(932\) 18.2474 0.597715
\(933\) 4.20204 0.137569
\(934\) 8.34847 0.273170
\(935\) 0 0
\(936\) 2.44949 0.0800641
\(937\) −39.3939 −1.28694 −0.643471 0.765471i \(-0.722506\pi\)
−0.643471 + 0.765471i \(0.722506\pi\)
\(938\) −4.15663 −0.135719
\(939\) 22.1010 0.721240
\(940\) 0 0
\(941\) −46.6413 −1.52046 −0.760232 0.649652i \(-0.774915\pi\)
−0.760232 + 0.649652i \(0.774915\pi\)
\(942\) −4.89898 −0.159617
\(943\) 8.89898 0.289791
\(944\) 1.55051 0.0504648
\(945\) 0 0
\(946\) −3.55051 −0.115437
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 5.00000 0.162392
\(949\) −2.44949 −0.0795138
\(950\) 0 0
\(951\) −9.24745 −0.299869
\(952\) −0.202041 −0.00654819
\(953\) −33.4949 −1.08501 −0.542503 0.840054i \(-0.682523\pi\)
−0.542503 + 0.840054i \(0.682523\pi\)
\(954\) −2.55051 −0.0825758
\(955\) 0 0
\(956\) 20.6969 0.669387
\(957\) 15.0000 0.484881
\(958\) 22.5959 0.730041
\(959\) −2.20204 −0.0711076
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −28.8990 −0.931740
\(963\) −2.65153 −0.0854444
\(964\) 0.449490 0.0144771
\(965\) 0 0
\(966\) 0.449490 0.0144621
\(967\) 25.1010 0.807194 0.403597 0.914937i \(-0.367760\pi\)
0.403597 + 0.914937i \(0.367760\pi\)
\(968\) 8.89898 0.286024
\(969\) −0.449490 −0.0144397
\(970\) 0 0
\(971\) −41.6413 −1.33633 −0.668167 0.744011i \(-0.732921\pi\)
−0.668167 + 0.744011i \(0.732921\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 1.01021 0.0323857
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −4.55051 −0.145658
\(977\) 19.5959 0.626929 0.313464 0.949600i \(-0.398510\pi\)
0.313464 + 0.949600i \(0.398510\pi\)
\(978\) −0.202041 −0.00646056
\(979\) −17.2474 −0.551231
\(980\) 0 0
\(981\) 6.89898 0.220267
\(982\) −4.40408 −0.140540
\(983\) −45.5505 −1.45284 −0.726418 0.687253i \(-0.758816\pi\)
−0.726418 + 0.687253i \(0.758816\pi\)
\(984\) 8.89898 0.283689
\(985\) 0 0
\(986\) −4.65153 −0.148135
\(987\) 5.30306 0.168798
\(988\) −2.44949 −0.0779287
\(989\) −2.44949 −0.0778892
\(990\) 0 0
\(991\) 53.8990 1.71216 0.856079 0.516845i \(-0.172894\pi\)
0.856079 + 0.516845i \(0.172894\pi\)
\(992\) 3.00000 0.0952501
\(993\) 7.65153 0.242814
\(994\) 2.89898 0.0919500
\(995\) 0 0
\(996\) −6.34847 −0.201159
\(997\) −34.5505 −1.09423 −0.547113 0.837059i \(-0.684273\pi\)
−0.547113 + 0.837059i \(0.684273\pi\)
\(998\) −39.8434 −1.26122
\(999\) 11.7980 0.373271
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 2850.2.a.bc.1.2 2
3.2 odd 2 8550.2.a.bv.1.2 2
5.2 odd 4 2850.2.d.w.799.2 4
5.3 odd 4 2850.2.d.w.799.3 4
5.4 even 2 2850.2.a.bj.1.1 yes 2
15.14 odd 2 8550.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.2 2 1.1 even 1 trivial
2850.2.a.bj.1.1 yes 2 5.4 even 2
2850.2.d.w.799.2 4 5.2 odd 4
2850.2.d.w.799.3 4 5.3 odd 4
8550.2.a.bu.1.1 2 15.14 odd 2
8550.2.a.bv.1.2 2 3.2 odd 2