Properties

Label 1850.2.a.ba.1.3
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} +3.28726 q^{7} -1.00000 q^{8} +4.15633 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} +3.28726 q^{7} -1.00000 q^{8} +4.15633 q^{9} +4.83146 q^{11} +2.67513 q^{12} +5.67513 q^{13} -3.28726 q^{14} +1.00000 q^{16} -5.63752 q^{17} -4.15633 q^{18} +4.76845 q^{19} +8.79384 q^{21} -4.83146 q^{22} +2.38787 q^{23} -2.67513 q^{24} -5.67513 q^{26} +3.09332 q^{27} +3.28726 q^{28} -7.92478 q^{29} -7.44358 q^{31} -1.00000 q^{32} +12.9248 q^{33} +5.63752 q^{34} +4.15633 q^{36} +1.00000 q^{37} -4.76845 q^{38} +15.1817 q^{39} -10.3503 q^{41} -8.79384 q^{42} -11.4314 q^{43} +4.83146 q^{44} -2.38787 q^{46} -5.92478 q^{47} +2.67513 q^{48} +3.80606 q^{49} -15.0811 q^{51} +5.67513 q^{52} +9.53690 q^{53} -3.09332 q^{54} -3.28726 q^{56} +12.7562 q^{57} +7.92478 q^{58} -11.4314 q^{59} -4.89938 q^{61} +7.44358 q^{62} +13.6629 q^{63} +1.00000 q^{64} -12.9248 q^{66} +2.10062 q^{67} -5.63752 q^{68} +6.38787 q^{69} -4.89938 q^{71} -4.15633 q^{72} +10.1939 q^{73} -1.00000 q^{74} +4.76845 q^{76} +15.8822 q^{77} -15.1817 q^{78} +7.61213 q^{79} -4.19394 q^{81} +10.3503 q^{82} +13.9199 q^{83} +8.79384 q^{84} +11.4314 q^{86} -21.1998 q^{87} -4.83146 q^{88} +4.44358 q^{89} +18.6556 q^{91} +2.38787 q^{92} -19.9126 q^{93} +5.92478 q^{94} -2.67513 q^{96} +5.35026 q^{97} -3.80606 q^{98} +20.0811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26} + 3 q^{27} + 4 q^{28} - 2 q^{29} - 6 q^{31} - 3 q^{32} + 17 q^{33} + q^{34} + 2 q^{36} + 3 q^{37} - 3 q^{38} + 20 q^{39} - 21 q^{41} + 8 q^{43} - q^{44} - 8 q^{46} + 4 q^{47} + 3 q^{48} + 11 q^{49} - 13 q^{51} + 12 q^{52} + 6 q^{53} - 3 q^{54} - 4 q^{56} + q^{57} + 2 q^{58} + 8 q^{59} - 8 q^{61} + 6 q^{62} + 10 q^{63} + 3 q^{64} - 17 q^{66} + 13 q^{67} - q^{68} + 20 q^{69} - 8 q^{71} - 2 q^{72} + 31 q^{73} - 3 q^{74} + 3 q^{76} + 2 q^{77} - 20 q^{78} + 22 q^{79} - 13 q^{81} + 21 q^{82} + 7 q^{83} - 8 q^{86} - 10 q^{87} + q^{88} - 3 q^{89} + 12 q^{91} + 8 q^{92} - 12 q^{93} - 4 q^{94} - 3 q^{96} + 6 q^{97} - 11 q^{98} + 28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.67513 1.54449 0.772244 0.635326i \(-0.219134\pi\)
0.772244 + 0.635326i \(0.219134\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.67513 −1.09212
\(7\) 3.28726 1.24247 0.621233 0.783626i \(-0.286632\pi\)
0.621233 + 0.783626i \(0.286632\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.15633 1.38544
\(10\) 0 0
\(11\) 4.83146 1.45674 0.728369 0.685185i \(-0.240279\pi\)
0.728369 + 0.685185i \(0.240279\pi\)
\(12\) 2.67513 0.772244
\(13\) 5.67513 1.57400 0.786999 0.616954i \(-0.211634\pi\)
0.786999 + 0.616954i \(0.211634\pi\)
\(14\) −3.28726 −0.878557
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.63752 −1.36730 −0.683650 0.729810i \(-0.739608\pi\)
−0.683650 + 0.729810i \(0.739608\pi\)
\(18\) −4.15633 −0.979655
\(19\) 4.76845 1.09396 0.546979 0.837146i \(-0.315778\pi\)
0.546979 + 0.837146i \(0.315778\pi\)
\(20\) 0 0
\(21\) 8.79384 1.91897
\(22\) −4.83146 −1.03007
\(23\) 2.38787 0.497906 0.248953 0.968516i \(-0.419914\pi\)
0.248953 + 0.968516i \(0.419914\pi\)
\(24\) −2.67513 −0.546059
\(25\) 0 0
\(26\) −5.67513 −1.11298
\(27\) 3.09332 0.595310
\(28\) 3.28726 0.621233
\(29\) −7.92478 −1.47159 −0.735797 0.677202i \(-0.763192\pi\)
−0.735797 + 0.677202i \(0.763192\pi\)
\(30\) 0 0
\(31\) −7.44358 −1.33691 −0.668453 0.743754i \(-0.733043\pi\)
−0.668453 + 0.743754i \(0.733043\pi\)
\(32\) −1.00000 −0.176777
\(33\) 12.9248 2.24991
\(34\) 5.63752 0.966827
\(35\) 0 0
\(36\) 4.15633 0.692721
\(37\) 1.00000 0.164399
\(38\) −4.76845 −0.773545
\(39\) 15.1817 2.43102
\(40\) 0 0
\(41\) −10.3503 −1.61644 −0.808220 0.588881i \(-0.799569\pi\)
−0.808220 + 0.588881i \(0.799569\pi\)
\(42\) −8.79384 −1.35692
\(43\) −11.4314 −1.74327 −0.871633 0.490158i \(-0.836939\pi\)
−0.871633 + 0.490158i \(0.836939\pi\)
\(44\) 4.83146 0.728369
\(45\) 0 0
\(46\) −2.38787 −0.352073
\(47\) −5.92478 −0.864218 −0.432109 0.901821i \(-0.642230\pi\)
−0.432109 + 0.901821i \(0.642230\pi\)
\(48\) 2.67513 0.386122
\(49\) 3.80606 0.543723
\(50\) 0 0
\(51\) −15.0811 −2.11178
\(52\) 5.67513 0.786999
\(53\) 9.53690 1.30999 0.654997 0.755631i \(-0.272670\pi\)
0.654997 + 0.755631i \(0.272670\pi\)
\(54\) −3.09332 −0.420948
\(55\) 0 0
\(56\) −3.28726 −0.439278
\(57\) 12.7562 1.68960
\(58\) 7.92478 1.04057
\(59\) −11.4314 −1.48824 −0.744118 0.668048i \(-0.767130\pi\)
−0.744118 + 0.668048i \(0.767130\pi\)
\(60\) 0 0
\(61\) −4.89938 −0.627302 −0.313651 0.949538i \(-0.601552\pi\)
−0.313651 + 0.949538i \(0.601552\pi\)
\(62\) 7.44358 0.945336
\(63\) 13.6629 1.72137
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −12.9248 −1.59093
\(67\) 2.10062 0.256631 0.128316 0.991733i \(-0.459043\pi\)
0.128316 + 0.991733i \(0.459043\pi\)
\(68\) −5.63752 −0.683650
\(69\) 6.38787 0.769010
\(70\) 0 0
\(71\) −4.89938 −0.581450 −0.290725 0.956807i \(-0.593896\pi\)
−0.290725 + 0.956807i \(0.593896\pi\)
\(72\) −4.15633 −0.489828
\(73\) 10.1939 1.19311 0.596555 0.802572i \(-0.296536\pi\)
0.596555 + 0.802572i \(0.296536\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 4.76845 0.546979
\(77\) 15.8822 1.80995
\(78\) −15.1817 −1.71899
\(79\) 7.61213 0.856431 0.428216 0.903677i \(-0.359142\pi\)
0.428216 + 0.903677i \(0.359142\pi\)
\(80\) 0 0
\(81\) −4.19394 −0.465993
\(82\) 10.3503 1.14300
\(83\) 13.9199 1.52790 0.763951 0.645274i \(-0.223257\pi\)
0.763951 + 0.645274i \(0.223257\pi\)
\(84\) 8.79384 0.959487
\(85\) 0 0
\(86\) 11.4314 1.23268
\(87\) −21.1998 −2.27286
\(88\) −4.83146 −0.515035
\(89\) 4.44358 0.471019 0.235509 0.971872i \(-0.424324\pi\)
0.235509 + 0.971872i \(0.424324\pi\)
\(90\) 0 0
\(91\) 18.6556 1.95564
\(92\) 2.38787 0.248953
\(93\) −19.9126 −2.06484
\(94\) 5.92478 0.611094
\(95\) 0 0
\(96\) −2.67513 −0.273029
\(97\) 5.35026 0.543237 0.271618 0.962405i \(-0.412441\pi\)
0.271618 + 0.962405i \(0.412441\pi\)
\(98\) −3.80606 −0.384470
\(99\) 20.0811 2.01823
\(100\) 0 0
\(101\) −8.12601 −0.808568 −0.404284 0.914634i \(-0.632479\pi\)
−0.404284 + 0.914634i \(0.632479\pi\)
\(102\) 15.0811 1.49325
\(103\) −1.90668 −0.187871 −0.0939353 0.995578i \(-0.529945\pi\)
−0.0939353 + 0.995578i \(0.529945\pi\)
\(104\) −5.67513 −0.556492
\(105\) 0 0
\(106\) −9.53690 −0.926306
\(107\) −2.41327 −0.233299 −0.116650 0.993173i \(-0.537215\pi\)
−0.116650 + 0.993173i \(0.537215\pi\)
\(108\) 3.09332 0.297655
\(109\) −1.73813 −0.166483 −0.0832416 0.996529i \(-0.526527\pi\)
−0.0832416 + 0.996529i \(0.526527\pi\)
\(110\) 0 0
\(111\) 2.67513 0.253912
\(112\) 3.28726 0.310617
\(113\) −5.09332 −0.479139 −0.239570 0.970879i \(-0.577006\pi\)
−0.239570 + 0.970879i \(0.577006\pi\)
\(114\) −12.7562 −1.19473
\(115\) 0 0
\(116\) −7.92478 −0.735797
\(117\) 23.5877 2.18068
\(118\) 11.4314 1.05234
\(119\) −18.5320 −1.69882
\(120\) 0 0
\(121\) 12.3430 1.12209
\(122\) 4.89938 0.443569
\(123\) −27.6883 −2.49657
\(124\) −7.44358 −0.668453
\(125\) 0 0
\(126\) −13.6629 −1.21719
\(127\) −7.59991 −0.674383 −0.337191 0.941436i \(-0.609477\pi\)
−0.337191 + 0.941436i \(0.609477\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −30.5804 −2.69245
\(130\) 0 0
\(131\) 13.3806 1.16907 0.584533 0.811370i \(-0.301278\pi\)
0.584533 + 0.811370i \(0.301278\pi\)
\(132\) 12.9248 1.12496
\(133\) 15.6751 1.35921
\(134\) −2.10062 −0.181466
\(135\) 0 0
\(136\) 5.63752 0.483413
\(137\) 11.1866 0.955739 0.477870 0.878431i \(-0.341409\pi\)
0.477870 + 0.878431i \(0.341409\pi\)
\(138\) −6.38787 −0.543772
\(139\) −17.0689 −1.44776 −0.723882 0.689924i \(-0.757644\pi\)
−0.723882 + 0.689924i \(0.757644\pi\)
\(140\) 0 0
\(141\) −15.8496 −1.33477
\(142\) 4.89938 0.411147
\(143\) 27.4191 2.29290
\(144\) 4.15633 0.346360
\(145\) 0 0
\(146\) −10.1939 −0.843656
\(147\) 10.1817 0.839774
\(148\) 1.00000 0.0821995
\(149\) −7.56959 −0.620125 −0.310063 0.950716i \(-0.600350\pi\)
−0.310063 + 0.950716i \(0.600350\pi\)
\(150\) 0 0
\(151\) 0.649738 0.0528749 0.0264375 0.999650i \(-0.491584\pi\)
0.0264375 + 0.999650i \(0.491584\pi\)
\(152\) −4.76845 −0.386773
\(153\) −23.4314 −1.89431
\(154\) −15.8822 −1.27983
\(155\) 0 0
\(156\) 15.1817 1.21551
\(157\) −3.36836 −0.268824 −0.134412 0.990926i \(-0.542915\pi\)
−0.134412 + 0.990926i \(0.542915\pi\)
\(158\) −7.61213 −0.605588
\(159\) 25.5125 2.02327
\(160\) 0 0
\(161\) 7.84955 0.618632
\(162\) 4.19394 0.329507
\(163\) −4.31994 −0.338364 −0.169182 0.985585i \(-0.554113\pi\)
−0.169182 + 0.985585i \(0.554113\pi\)
\(164\) −10.3503 −0.808220
\(165\) 0 0
\(166\) −13.9199 −1.08039
\(167\) 14.6678 1.13503 0.567516 0.823363i \(-0.307905\pi\)
0.567516 + 0.823363i \(0.307905\pi\)
\(168\) −8.79384 −0.678460
\(169\) 19.2071 1.47747
\(170\) 0 0
\(171\) 19.8192 1.51561
\(172\) −11.4314 −0.871633
\(173\) −8.05079 −0.612090 −0.306045 0.952017i \(-0.599006\pi\)
−0.306045 + 0.952017i \(0.599006\pi\)
\(174\) 21.1998 1.60715
\(175\) 0 0
\(176\) 4.83146 0.364185
\(177\) −30.5804 −2.29856
\(178\) −4.44358 −0.333061
\(179\) −5.05079 −0.377513 −0.188757 0.982024i \(-0.560446\pi\)
−0.188757 + 0.982024i \(0.560446\pi\)
\(180\) 0 0
\(181\) 3.73813 0.277853 0.138927 0.990303i \(-0.455635\pi\)
0.138927 + 0.990303i \(0.455635\pi\)
\(182\) −18.6556 −1.38285
\(183\) −13.1065 −0.968860
\(184\) −2.38787 −0.176036
\(185\) 0 0
\(186\) 19.9126 1.46006
\(187\) −27.2374 −1.99180
\(188\) −5.92478 −0.432109
\(189\) 10.1685 0.739653
\(190\) 0 0
\(191\) −12.5501 −0.908092 −0.454046 0.890978i \(-0.650020\pi\)
−0.454046 + 0.890978i \(0.650020\pi\)
\(192\) 2.67513 0.193061
\(193\) 20.8822 1.50314 0.751568 0.659655i \(-0.229298\pi\)
0.751568 + 0.659655i \(0.229298\pi\)
\(194\) −5.35026 −0.384126
\(195\) 0 0
\(196\) 3.80606 0.271862
\(197\) −1.07030 −0.0762556 −0.0381278 0.999273i \(-0.512139\pi\)
−0.0381278 + 0.999273i \(0.512139\pi\)
\(198\) −20.0811 −1.42710
\(199\) −22.1016 −1.56674 −0.783369 0.621556i \(-0.786501\pi\)
−0.783369 + 0.621556i \(0.786501\pi\)
\(200\) 0 0
\(201\) 5.61942 0.396363
\(202\) 8.12601 0.571744
\(203\) −26.0508 −1.82841
\(204\) −15.0811 −1.05589
\(205\) 0 0
\(206\) 1.90668 0.132845
\(207\) 9.92478 0.689820
\(208\) 5.67513 0.393500
\(209\) 23.0386 1.59361
\(210\) 0 0
\(211\) −16.3987 −1.12893 −0.564466 0.825456i \(-0.690918\pi\)
−0.564466 + 0.825456i \(0.690918\pi\)
\(212\) 9.53690 0.654997
\(213\) −13.1065 −0.898042
\(214\) 2.41327 0.164967
\(215\) 0 0
\(216\) −3.09332 −0.210474
\(217\) −24.4690 −1.66106
\(218\) 1.73813 0.117721
\(219\) 27.2701 1.84274
\(220\) 0 0
\(221\) −31.9937 −2.15213
\(222\) −2.67513 −0.179543
\(223\) 18.4387 1.23474 0.617372 0.786671i \(-0.288197\pi\)
0.617372 + 0.786671i \(0.288197\pi\)
\(224\) −3.28726 −0.219639
\(225\) 0 0
\(226\) 5.09332 0.338803
\(227\) −11.4558 −0.760348 −0.380174 0.924915i \(-0.624136\pi\)
−0.380174 + 0.924915i \(0.624136\pi\)
\(228\) 12.7562 0.844802
\(229\) 25.6385 1.69424 0.847119 0.531403i \(-0.178335\pi\)
0.847119 + 0.531403i \(0.178335\pi\)
\(230\) 0 0
\(231\) 42.4871 2.79544
\(232\) 7.92478 0.520287
\(233\) 9.76845 0.639953 0.319976 0.947426i \(-0.396325\pi\)
0.319976 + 0.947426i \(0.396325\pi\)
\(234\) −23.5877 −1.54198
\(235\) 0 0
\(236\) −11.4314 −0.744118
\(237\) 20.3634 1.32275
\(238\) 18.5320 1.20125
\(239\) 16.2882 1.05360 0.526798 0.849990i \(-0.323392\pi\)
0.526798 + 0.849990i \(0.323392\pi\)
\(240\) 0 0
\(241\) 13.1744 0.848639 0.424320 0.905512i \(-0.360513\pi\)
0.424320 + 0.905512i \(0.360513\pi\)
\(242\) −12.3430 −0.793436
\(243\) −20.4993 −1.31503
\(244\) −4.89938 −0.313651
\(245\) 0 0
\(246\) 27.6883 1.76534
\(247\) 27.0616 1.72189
\(248\) 7.44358 0.472668
\(249\) 37.2374 2.35983
\(250\) 0 0
\(251\) 13.0508 0.823758 0.411879 0.911238i \(-0.364873\pi\)
0.411879 + 0.911238i \(0.364873\pi\)
\(252\) 13.6629 0.860683
\(253\) 11.5369 0.725319
\(254\) 7.59991 0.476861
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.7005 1.29126 0.645632 0.763649i \(-0.276594\pi\)
0.645632 + 0.763649i \(0.276594\pi\)
\(258\) 30.5804 1.90385
\(259\) 3.28726 0.204260
\(260\) 0 0
\(261\) −32.9380 −2.03881
\(262\) −13.3806 −0.826655
\(263\) −20.4387 −1.26030 −0.630151 0.776473i \(-0.717007\pi\)
−0.630151 + 0.776473i \(0.717007\pi\)
\(264\) −12.9248 −0.795465
\(265\) 0 0
\(266\) −15.6751 −0.961104
\(267\) 11.8872 0.727483
\(268\) 2.10062 0.128316
\(269\) 6.25202 0.381192 0.190596 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(270\) 0 0
\(271\) −0.824162 −0.0500643 −0.0250321 0.999687i \(-0.507969\pi\)
−0.0250321 + 0.999687i \(0.507969\pi\)
\(272\) −5.63752 −0.341825
\(273\) 49.9062 3.02046
\(274\) −11.1866 −0.675810
\(275\) 0 0
\(276\) 6.38787 0.384505
\(277\) −18.1866 −1.09273 −0.546365 0.837547i \(-0.683989\pi\)
−0.546365 + 0.837547i \(0.683989\pi\)
\(278\) 17.0689 1.02372
\(279\) −30.9380 −1.85221
\(280\) 0 0
\(281\) 25.0435 1.49397 0.746985 0.664841i \(-0.231501\pi\)
0.746985 + 0.664841i \(0.231501\pi\)
\(282\) 15.8496 0.943827
\(283\) 13.1055 0.779043 0.389522 0.921017i \(-0.372640\pi\)
0.389522 + 0.921017i \(0.372640\pi\)
\(284\) −4.89938 −0.290725
\(285\) 0 0
\(286\) −27.4191 −1.62133
\(287\) −34.0240 −2.00837
\(288\) −4.15633 −0.244914
\(289\) 14.7816 0.869507
\(290\) 0 0
\(291\) 14.3127 0.839022
\(292\) 10.1939 0.596555
\(293\) 9.68101 0.565571 0.282785 0.959183i \(-0.408742\pi\)
0.282785 + 0.959183i \(0.408742\pi\)
\(294\) −10.1817 −0.593810
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 14.9452 0.867211
\(298\) 7.56959 0.438495
\(299\) 13.5515 0.783703
\(300\) 0 0
\(301\) −37.5778 −2.16595
\(302\) −0.649738 −0.0373882
\(303\) −21.7381 −1.24882
\(304\) 4.76845 0.273489
\(305\) 0 0
\(306\) 23.4314 1.33948
\(307\) −20.4337 −1.16621 −0.583107 0.812395i \(-0.698163\pi\)
−0.583107 + 0.812395i \(0.698163\pi\)
\(308\) 15.8822 0.904975
\(309\) −5.10062 −0.290164
\(310\) 0 0
\(311\) 10.5320 0.597214 0.298607 0.954376i \(-0.403478\pi\)
0.298607 + 0.954376i \(0.403478\pi\)
\(312\) −15.1817 −0.859496
\(313\) 9.96968 0.563520 0.281760 0.959485i \(-0.409082\pi\)
0.281760 + 0.959485i \(0.409082\pi\)
\(314\) 3.36836 0.190088
\(315\) 0 0
\(316\) 7.61213 0.428216
\(317\) −24.3961 −1.37022 −0.685111 0.728438i \(-0.740246\pi\)
−0.685111 + 0.728438i \(0.740246\pi\)
\(318\) −25.5125 −1.43067
\(319\) −38.2882 −2.14373
\(320\) 0 0
\(321\) −6.45580 −0.360328
\(322\) −7.84955 −0.437439
\(323\) −26.8822 −1.49577
\(324\) −4.19394 −0.232996
\(325\) 0 0
\(326\) 4.31994 0.239260
\(327\) −4.64974 −0.257131
\(328\) 10.3503 0.571498
\(329\) −19.4763 −1.07376
\(330\) 0 0
\(331\) 34.3561 1.88838 0.944192 0.329395i \(-0.106845\pi\)
0.944192 + 0.329395i \(0.106845\pi\)
\(332\) 13.9199 0.763951
\(333\) 4.15633 0.227765
\(334\) −14.6678 −0.802588
\(335\) 0 0
\(336\) 8.79384 0.479744
\(337\) 20.6385 1.12425 0.562125 0.827053i \(-0.309984\pi\)
0.562125 + 0.827053i \(0.309984\pi\)
\(338\) −19.2071 −1.04473
\(339\) −13.6253 −0.740025
\(340\) 0 0
\(341\) −35.9633 −1.94752
\(342\) −19.8192 −1.07170
\(343\) −10.4993 −0.566909
\(344\) 11.4314 0.616338
\(345\) 0 0
\(346\) 8.05079 0.432813
\(347\) 4.48612 0.240827 0.120414 0.992724i \(-0.461578\pi\)
0.120414 + 0.992724i \(0.461578\pi\)
\(348\) −21.1998 −1.13643
\(349\) −20.1949 −1.08101 −0.540504 0.841342i \(-0.681766\pi\)
−0.540504 + 0.841342i \(0.681766\pi\)
\(350\) 0 0
\(351\) 17.5550 0.937017
\(352\) −4.83146 −0.257517
\(353\) −25.9102 −1.37906 −0.689530 0.724257i \(-0.742183\pi\)
−0.689530 + 0.724257i \(0.742183\pi\)
\(354\) 30.5804 1.62533
\(355\) 0 0
\(356\) 4.44358 0.235509
\(357\) −49.5755 −2.62381
\(358\) 5.05079 0.266942
\(359\) 4.43866 0.234263 0.117132 0.993116i \(-0.462630\pi\)
0.117132 + 0.993116i \(0.462630\pi\)
\(360\) 0 0
\(361\) 3.73813 0.196744
\(362\) −3.73813 −0.196472
\(363\) 33.0191 1.73305
\(364\) 18.6556 0.977820
\(365\) 0 0
\(366\) 13.1065 0.685087
\(367\) 28.2858 1.47651 0.738254 0.674522i \(-0.235650\pi\)
0.738254 + 0.674522i \(0.235650\pi\)
\(368\) 2.38787 0.124476
\(369\) −43.0191 −2.23948
\(370\) 0 0
\(371\) 31.3503 1.62762
\(372\) −19.9126 −1.03242
\(373\) −2.71862 −0.140765 −0.0703825 0.997520i \(-0.522422\pi\)
−0.0703825 + 0.997520i \(0.522422\pi\)
\(374\) 27.2374 1.40841
\(375\) 0 0
\(376\) 5.92478 0.305547
\(377\) −44.9741 −2.31629
\(378\) −10.1685 −0.523013
\(379\) 18.1514 0.932375 0.466187 0.884686i \(-0.345627\pi\)
0.466187 + 0.884686i \(0.345627\pi\)
\(380\) 0 0
\(381\) −20.3307 −1.04158
\(382\) 12.5501 0.642118
\(383\) −3.90668 −0.199622 −0.0998110 0.995006i \(-0.531824\pi\)
−0.0998110 + 0.995006i \(0.531824\pi\)
\(384\) −2.67513 −0.136515
\(385\) 0 0
\(386\) −20.8822 −1.06288
\(387\) −47.5125 −2.41519
\(388\) 5.35026 0.271618
\(389\) 5.28726 0.268075 0.134037 0.990976i \(-0.457206\pi\)
0.134037 + 0.990976i \(0.457206\pi\)
\(390\) 0 0
\(391\) −13.4617 −0.680786
\(392\) −3.80606 −0.192235
\(393\) 35.7948 1.80561
\(394\) 1.07030 0.0539208
\(395\) 0 0
\(396\) 20.0811 1.00911
\(397\) 4.15396 0.208481 0.104241 0.994552i \(-0.466759\pi\)
0.104241 + 0.994552i \(0.466759\pi\)
\(398\) 22.1016 1.10785
\(399\) 41.9330 2.09928
\(400\) 0 0
\(401\) −22.1817 −1.10770 −0.553851 0.832616i \(-0.686842\pi\)
−0.553851 + 0.832616i \(0.686842\pi\)
\(402\) −5.61942 −0.280271
\(403\) −42.2433 −2.10429
\(404\) −8.12601 −0.404284
\(405\) 0 0
\(406\) 26.0508 1.29288
\(407\) 4.83146 0.239486
\(408\) 15.0811 0.746626
\(409\) 0.650693 0.0321747 0.0160874 0.999871i \(-0.494879\pi\)
0.0160874 + 0.999871i \(0.494879\pi\)
\(410\) 0 0
\(411\) 29.9257 1.47613
\(412\) −1.90668 −0.0939353
\(413\) −37.5778 −1.84908
\(414\) −9.92478 −0.487776
\(415\) 0 0
\(416\) −5.67513 −0.278246
\(417\) −45.6615 −2.23605
\(418\) −23.0386 −1.12685
\(419\) 10.8169 0.528439 0.264219 0.964463i \(-0.414886\pi\)
0.264219 + 0.964463i \(0.414886\pi\)
\(420\) 0 0
\(421\) 2.05079 0.0999492 0.0499746 0.998750i \(-0.484086\pi\)
0.0499746 + 0.998750i \(0.484086\pi\)
\(422\) 16.3987 0.798275
\(423\) −24.6253 −1.19732
\(424\) −9.53690 −0.463153
\(425\) 0 0
\(426\) 13.1065 0.635012
\(427\) −16.1055 −0.779402
\(428\) −2.41327 −0.116650
\(429\) 73.3498 3.54136
\(430\) 0 0
\(431\) 15.4861 0.745940 0.372970 0.927843i \(-0.378339\pi\)
0.372970 + 0.927843i \(0.378339\pi\)
\(432\) 3.09332 0.148827
\(433\) 35.8773 1.72415 0.862077 0.506777i \(-0.169163\pi\)
0.862077 + 0.506777i \(0.169163\pi\)
\(434\) 24.4690 1.17455
\(435\) 0 0
\(436\) −1.73813 −0.0832416
\(437\) 11.3865 0.544688
\(438\) −27.2701 −1.30302
\(439\) −26.7005 −1.27435 −0.637173 0.770721i \(-0.719896\pi\)
−0.637173 + 0.770721i \(0.719896\pi\)
\(440\) 0 0
\(441\) 15.8192 0.753297
\(442\) 31.9937 1.52178
\(443\) 5.75765 0.273554 0.136777 0.990602i \(-0.456326\pi\)
0.136777 + 0.990602i \(0.456326\pi\)
\(444\) 2.67513 0.126956
\(445\) 0 0
\(446\) −18.4387 −0.873096
\(447\) −20.2496 −0.957775
\(448\) 3.28726 0.155308
\(449\) −31.1803 −1.47149 −0.735745 0.677259i \(-0.763168\pi\)
−0.735745 + 0.677259i \(0.763168\pi\)
\(450\) 0 0
\(451\) −50.0068 −2.35473
\(452\) −5.09332 −0.239570
\(453\) 1.73813 0.0816647
\(454\) 11.4558 0.537647
\(455\) 0 0
\(456\) −12.7562 −0.597365
\(457\) 7.27011 0.340082 0.170041 0.985437i \(-0.445610\pi\)
0.170041 + 0.985437i \(0.445610\pi\)
\(458\) −25.6385 −1.19801
\(459\) −17.4387 −0.813967
\(460\) 0 0
\(461\) −18.6253 −0.867467 −0.433733 0.901041i \(-0.642804\pi\)
−0.433733 + 0.901041i \(0.642804\pi\)
\(462\) −42.4871 −1.97668
\(463\) 1.89209 0.0879329 0.0439664 0.999033i \(-0.486001\pi\)
0.0439664 + 0.999033i \(0.486001\pi\)
\(464\) −7.92478 −0.367899
\(465\) 0 0
\(466\) −9.76845 −0.452515
\(467\) 1.21440 0.0561959 0.0280980 0.999605i \(-0.491055\pi\)
0.0280980 + 0.999605i \(0.491055\pi\)
\(468\) 23.5877 1.09034
\(469\) 6.90526 0.318855
\(470\) 0 0
\(471\) −9.01080 −0.415196
\(472\) 11.4314 0.526171
\(473\) −55.2301 −2.53948
\(474\) −20.3634 −0.935324
\(475\) 0 0
\(476\) −18.5320 −0.849412
\(477\) 39.6385 1.81492
\(478\) −16.2882 −0.745006
\(479\) −40.7694 −1.86280 −0.931401 0.363995i \(-0.881412\pi\)
−0.931401 + 0.363995i \(0.881412\pi\)
\(480\) 0 0
\(481\) 5.67513 0.258764
\(482\) −13.1744 −0.600079
\(483\) 20.9986 0.955469
\(484\) 12.3430 0.561044
\(485\) 0 0
\(486\) 20.4993 0.929867
\(487\) −23.8578 −1.08110 −0.540550 0.841312i \(-0.681784\pi\)
−0.540550 + 0.841312i \(0.681784\pi\)
\(488\) 4.89938 0.221785
\(489\) −11.5564 −0.522599
\(490\) 0 0
\(491\) 36.0362 1.62629 0.813145 0.582061i \(-0.197753\pi\)
0.813145 + 0.582061i \(0.197753\pi\)
\(492\) −27.6883 −1.24829
\(493\) 44.6761 2.01211
\(494\) −27.0616 −1.21756
\(495\) 0 0
\(496\) −7.44358 −0.334227
\(497\) −16.1055 −0.722432
\(498\) −37.2374 −1.66865
\(499\) −31.4558 −1.40816 −0.704078 0.710123i \(-0.748639\pi\)
−0.704078 + 0.710123i \(0.748639\pi\)
\(500\) 0 0
\(501\) 39.2384 1.75304
\(502\) −13.0508 −0.582485
\(503\) 3.07381 0.137054 0.0685272 0.997649i \(-0.478170\pi\)
0.0685272 + 0.997649i \(0.478170\pi\)
\(504\) −13.6629 −0.608594
\(505\) 0 0
\(506\) −11.5369 −0.512878
\(507\) 51.3815 2.28193
\(508\) −7.59991 −0.337191
\(509\) −3.01951 −0.133838 −0.0669188 0.997758i \(-0.521317\pi\)
−0.0669188 + 0.997758i \(0.521317\pi\)
\(510\) 0 0
\(511\) 33.5101 1.48240
\(512\) −1.00000 −0.0441942
\(513\) 14.7504 0.651244
\(514\) −20.7005 −0.913061
\(515\) 0 0
\(516\) −30.5804 −1.34623
\(517\) −28.6253 −1.25894
\(518\) −3.28726 −0.144434
\(519\) −21.5369 −0.945365
\(520\) 0 0
\(521\) 28.8700 1.26482 0.632409 0.774634i \(-0.282066\pi\)
0.632409 + 0.774634i \(0.282066\pi\)
\(522\) 32.9380 1.44165
\(523\) 3.37328 0.147503 0.0737517 0.997277i \(-0.476503\pi\)
0.0737517 + 0.997277i \(0.476503\pi\)
\(524\) 13.3806 0.584533
\(525\) 0 0
\(526\) 20.4387 0.891168
\(527\) 41.9633 1.82795
\(528\) 12.9248 0.562479
\(529\) −17.2981 −0.752090
\(530\) 0 0
\(531\) −47.5125 −2.06187
\(532\) 15.6751 0.679603
\(533\) −58.7391 −2.54427
\(534\) −11.8872 −0.514408
\(535\) 0 0
\(536\) −2.10062 −0.0907328
\(537\) −13.5115 −0.583065
\(538\) −6.25202 −0.269544
\(539\) 18.3888 0.792063
\(540\) 0 0
\(541\) 10.3004 0.442850 0.221425 0.975177i \(-0.428929\pi\)
0.221425 + 0.975177i \(0.428929\pi\)
\(542\) 0.824162 0.0354008
\(543\) 10.0000 0.429141
\(544\) 5.63752 0.241707
\(545\) 0 0
\(546\) −49.9062 −2.13579
\(547\) 15.3576 0.656642 0.328321 0.944566i \(-0.393517\pi\)
0.328321 + 0.944566i \(0.393517\pi\)
\(548\) 11.1866 0.477870
\(549\) −20.3634 −0.869090
\(550\) 0 0
\(551\) −37.7889 −1.60986
\(552\) −6.38787 −0.271886
\(553\) 25.0230 1.06409
\(554\) 18.1866 0.772676
\(555\) 0 0
\(556\) −17.0689 −0.723882
\(557\) 14.8119 0.627602 0.313801 0.949489i \(-0.398397\pi\)
0.313801 + 0.949489i \(0.398397\pi\)
\(558\) 30.9380 1.30971
\(559\) −64.8745 −2.74390
\(560\) 0 0
\(561\) −72.8637 −3.07631
\(562\) −25.0435 −1.05640
\(563\) 3.24472 0.136749 0.0683744 0.997660i \(-0.478219\pi\)
0.0683744 + 0.997660i \(0.478219\pi\)
\(564\) −15.8496 −0.667387
\(565\) 0 0
\(566\) −13.1055 −0.550867
\(567\) −13.7866 −0.578981
\(568\) 4.89938 0.205574
\(569\) −17.1587 −0.719330 −0.359665 0.933082i \(-0.617109\pi\)
−0.359665 + 0.933082i \(0.617109\pi\)
\(570\) 0 0
\(571\) 0.0244376 0.00102268 0.000511340 1.00000i \(-0.499837\pi\)
0.000511340 1.00000i \(0.499837\pi\)
\(572\) 27.4191 1.14645
\(573\) −33.5731 −1.40254
\(574\) 34.0240 1.42013
\(575\) 0 0
\(576\) 4.15633 0.173180
\(577\) −34.5002 −1.43626 −0.718132 0.695907i \(-0.755003\pi\)
−0.718132 + 0.695907i \(0.755003\pi\)
\(578\) −14.7816 −0.614835
\(579\) 55.8627 2.32158
\(580\) 0 0
\(581\) 45.7581 1.89837
\(582\) −14.3127 −0.593278
\(583\) 46.0771 1.90832
\(584\) −10.1939 −0.421828
\(585\) 0 0
\(586\) −9.68101 −0.399919
\(587\) −4.84367 −0.199920 −0.0999599 0.994991i \(-0.531871\pi\)
−0.0999599 + 0.994991i \(0.531871\pi\)
\(588\) 10.1817 0.419887
\(589\) −35.4944 −1.46252
\(590\) 0 0
\(591\) −2.86319 −0.117776
\(592\) 1.00000 0.0410997
\(593\) 31.0263 1.27410 0.637050 0.770823i \(-0.280155\pi\)
0.637050 + 0.770823i \(0.280155\pi\)
\(594\) −14.9452 −0.613211
\(595\) 0 0
\(596\) −7.56959 −0.310063
\(597\) −59.1246 −2.41981
\(598\) −13.5515 −0.554162
\(599\) −32.0240 −1.30846 −0.654232 0.756294i \(-0.727008\pi\)
−0.654232 + 0.756294i \(0.727008\pi\)
\(600\) 0 0
\(601\) −9.14174 −0.372899 −0.186450 0.982465i \(-0.559698\pi\)
−0.186450 + 0.982465i \(0.559698\pi\)
\(602\) 37.5778 1.53156
\(603\) 8.73084 0.355547
\(604\) 0.649738 0.0264375
\(605\) 0 0
\(606\) 21.7381 0.883051
\(607\) 36.8691 1.49647 0.748235 0.663434i \(-0.230902\pi\)
0.748235 + 0.663434i \(0.230902\pi\)
\(608\) −4.76845 −0.193386
\(609\) −69.6893 −2.82395
\(610\) 0 0
\(611\) −33.6239 −1.36028
\(612\) −23.4314 −0.947157
\(613\) −27.9902 −1.13051 −0.565256 0.824916i \(-0.691223\pi\)
−0.565256 + 0.824916i \(0.691223\pi\)
\(614\) 20.4337 0.824638
\(615\) 0 0
\(616\) −15.8822 −0.639914
\(617\) −22.5804 −0.909052 −0.454526 0.890733i \(-0.650191\pi\)
−0.454526 + 0.890733i \(0.650191\pi\)
\(618\) 5.10062 0.205177
\(619\) 40.0362 1.60919 0.804595 0.593824i \(-0.202382\pi\)
0.804595 + 0.593824i \(0.202382\pi\)
\(620\) 0 0
\(621\) 7.38646 0.296408
\(622\) −10.5320 −0.422294
\(623\) 14.6072 0.585225
\(624\) 15.1817 0.607755
\(625\) 0 0
\(626\) −9.96968 −0.398469
\(627\) 61.6312 2.46131
\(628\) −3.36836 −0.134412
\(629\) −5.63752 −0.224783
\(630\) 0 0
\(631\) 26.1768 1.04208 0.521041 0.853532i \(-0.325544\pi\)
0.521041 + 0.853532i \(0.325544\pi\)
\(632\) −7.61213 −0.302794
\(633\) −43.8686 −1.74362
\(634\) 24.3961 0.968894
\(635\) 0 0
\(636\) 25.5125 1.01164
\(637\) 21.5999 0.855820
\(638\) 38.2882 1.51584
\(639\) −20.3634 −0.805565
\(640\) 0 0
\(641\) 20.9525 0.827576 0.413788 0.910373i \(-0.364206\pi\)
0.413788 + 0.910373i \(0.364206\pi\)
\(642\) 6.45580 0.254790
\(643\) 17.7586 0.700331 0.350165 0.936688i \(-0.386125\pi\)
0.350165 + 0.936688i \(0.386125\pi\)
\(644\) 7.84955 0.309316
\(645\) 0 0
\(646\) 26.8822 1.05767
\(647\) −11.5550 −0.454274 −0.227137 0.973863i \(-0.572937\pi\)
−0.227137 + 0.973863i \(0.572937\pi\)
\(648\) 4.19394 0.164753
\(649\) −55.2301 −2.16797
\(650\) 0 0
\(651\) −65.4577 −2.56549
\(652\) −4.31994 −0.169182
\(653\) 7.78655 0.304711 0.152356 0.988326i \(-0.451314\pi\)
0.152356 + 0.988326i \(0.451314\pi\)
\(654\) 4.64974 0.181819
\(655\) 0 0
\(656\) −10.3503 −0.404110
\(657\) 42.3693 1.65298
\(658\) 19.4763 0.759264
\(659\) −13.7029 −0.533789 −0.266894 0.963726i \(-0.585998\pi\)
−0.266894 + 0.963726i \(0.585998\pi\)
\(660\) 0 0
\(661\) 42.0263 1.63464 0.817318 0.576187i \(-0.195460\pi\)
0.817318 + 0.576187i \(0.195460\pi\)
\(662\) −34.3561 −1.33529
\(663\) −85.5872 −3.32393
\(664\) −13.9199 −0.540195
\(665\) 0 0
\(666\) −4.15633 −0.161054
\(667\) −18.9234 −0.732716
\(668\) 14.6678 0.567516
\(669\) 49.3258 1.90705
\(670\) 0 0
\(671\) −23.6712 −0.913815
\(672\) −8.79384 −0.339230
\(673\) −29.0943 −1.12150 −0.560751 0.827985i \(-0.689487\pi\)
−0.560751 + 0.827985i \(0.689487\pi\)
\(674\) −20.6385 −0.794964
\(675\) 0 0
\(676\) 19.2071 0.738735
\(677\) 25.3112 0.972790 0.486395 0.873739i \(-0.338312\pi\)
0.486395 + 0.873739i \(0.338312\pi\)
\(678\) 13.6253 0.523277
\(679\) 17.5877 0.674954
\(680\) 0 0
\(681\) −30.6458 −1.17435
\(682\) 35.9633 1.37711
\(683\) 4.04349 0.154720 0.0773599 0.997003i \(-0.475351\pi\)
0.0773599 + 0.997003i \(0.475351\pi\)
\(684\) 19.8192 0.757807
\(685\) 0 0
\(686\) 10.4993 0.400865
\(687\) 68.5863 2.61673
\(688\) −11.4314 −0.435817
\(689\) 54.1232 2.06193
\(690\) 0 0
\(691\) 7.92970 0.301660 0.150830 0.988560i \(-0.451805\pi\)
0.150830 + 0.988560i \(0.451805\pi\)
\(692\) −8.05079 −0.306045
\(693\) 66.0118 2.50758
\(694\) −4.48612 −0.170291
\(695\) 0 0
\(696\) 21.1998 0.803577
\(697\) 58.3498 2.21016
\(698\) 20.1949 0.764388
\(699\) 26.1319 0.988399
\(700\) 0 0
\(701\) −48.3611 −1.82657 −0.913286 0.407319i \(-0.866464\pi\)
−0.913286 + 0.407319i \(0.866464\pi\)
\(702\) −17.5550 −0.662571
\(703\) 4.76845 0.179846
\(704\) 4.83146 0.182092
\(705\) 0 0
\(706\) 25.9102 0.975143
\(707\) −26.7123 −1.00462
\(708\) −30.5804 −1.14928
\(709\) 7.45183 0.279859 0.139930 0.990161i \(-0.455312\pi\)
0.139930 + 0.990161i \(0.455312\pi\)
\(710\) 0 0
\(711\) 31.6385 1.18654
\(712\) −4.44358 −0.166530
\(713\) −17.7743 −0.665654
\(714\) 49.5755 1.85532
\(715\) 0 0
\(716\) −5.05079 −0.188757
\(717\) 43.5731 1.62727
\(718\) −4.43866 −0.165649
\(719\) −6.45088 −0.240577 −0.120289 0.992739i \(-0.538382\pi\)
−0.120289 + 0.992739i \(0.538382\pi\)
\(720\) 0 0
\(721\) −6.26774 −0.233423
\(722\) −3.73813 −0.139119
\(723\) 35.2433 1.31071
\(724\) 3.73813 0.138927
\(725\) 0 0
\(726\) −33.0191 −1.22545
\(727\) 37.1754 1.37876 0.689379 0.724401i \(-0.257883\pi\)
0.689379 + 0.724401i \(0.257883\pi\)
\(728\) −18.6556 −0.691423
\(729\) −42.2565 −1.56505
\(730\) 0 0
\(731\) 64.4445 2.38357
\(732\) −13.1065 −0.484430
\(733\) −37.1754 −1.37310 −0.686552 0.727081i \(-0.740877\pi\)
−0.686552 + 0.727081i \(0.740877\pi\)
\(734\) −28.2858 −1.04405
\(735\) 0 0
\(736\) −2.38787 −0.0880182
\(737\) 10.1490 0.373844
\(738\) 43.0191 1.58355
\(739\) 6.74940 0.248281 0.124140 0.992265i \(-0.460383\pi\)
0.124140 + 0.992265i \(0.460383\pi\)
\(740\) 0 0
\(741\) 72.3933 2.65943
\(742\) −31.3503 −1.15090
\(743\) −23.9248 −0.877715 −0.438857 0.898557i \(-0.644617\pi\)
−0.438857 + 0.898557i \(0.644617\pi\)
\(744\) 19.9126 0.730030
\(745\) 0 0
\(746\) 2.71862 0.0995358
\(747\) 57.8554 2.11682
\(748\) −27.2374 −0.995899
\(749\) −7.93303 −0.289866
\(750\) 0 0
\(751\) −5.27504 −0.192489 −0.0962445 0.995358i \(-0.530683\pi\)
−0.0962445 + 0.995358i \(0.530683\pi\)
\(752\) −5.92478 −0.216054
\(753\) 34.9126 1.27228
\(754\) 44.9741 1.63786
\(755\) 0 0
\(756\) 10.1685 0.369826
\(757\) 3.62672 0.131815 0.0659076 0.997826i \(-0.479006\pi\)
0.0659076 + 0.997826i \(0.479006\pi\)
\(758\) −18.1514 −0.659289
\(759\) 30.8627 1.12025
\(760\) 0 0
\(761\) −4.03173 −0.146150 −0.0730751 0.997326i \(-0.523281\pi\)
−0.0730751 + 0.997326i \(0.523281\pi\)
\(762\) 20.3307 0.736505
\(763\) −5.71370 −0.206850
\(764\) −12.5501 −0.454046
\(765\) 0 0
\(766\) 3.90668 0.141154
\(767\) −64.8745 −2.34248
\(768\) 2.67513 0.0965305
\(769\) −27.6072 −0.995541 −0.497771 0.867309i \(-0.665848\pi\)
−0.497771 + 0.867309i \(0.665848\pi\)
\(770\) 0 0
\(771\) 55.3766 1.99434
\(772\) 20.8822 0.751568
\(773\) 19.4798 0.700639 0.350319 0.936630i \(-0.386073\pi\)
0.350319 + 0.936630i \(0.386073\pi\)
\(774\) 47.5125 1.70780
\(775\) 0 0
\(776\) −5.35026 −0.192063
\(777\) 8.79384 0.315477
\(778\) −5.28726 −0.189557
\(779\) −49.3547 −1.76832
\(780\) 0 0
\(781\) −23.6712 −0.847021
\(782\) 13.4617 0.481389
\(783\) −24.5139 −0.876055
\(784\) 3.80606 0.135931
\(785\) 0 0
\(786\) −35.7948 −1.27676
\(787\) −14.1768 −0.505348 −0.252674 0.967551i \(-0.581310\pi\)
−0.252674 + 0.967551i \(0.581310\pi\)
\(788\) −1.07030 −0.0381278
\(789\) −54.6761 −1.94652
\(790\) 0 0
\(791\) −16.7431 −0.595315
\(792\) −20.0811 −0.713551
\(793\) −27.8046 −0.987372
\(794\) −4.15396 −0.147418
\(795\) 0 0
\(796\) −22.1016 −0.783369
\(797\) 21.8764 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(798\) −41.9330 −1.48441
\(799\) 33.4010 1.18164
\(800\) 0 0
\(801\) 18.4690 0.652569
\(802\) 22.1817 0.783264
\(803\) 49.2516 1.73805
\(804\) 5.61942 0.198182
\(805\) 0 0
\(806\) 42.2433 1.48796
\(807\) 16.7250 0.588747
\(808\) 8.12601 0.285872
\(809\) −12.2130 −0.429386 −0.214693 0.976682i \(-0.568875\pi\)
−0.214693 + 0.976682i \(0.568875\pi\)
\(810\) 0 0
\(811\) 8.64974 0.303733 0.151867 0.988401i \(-0.451472\pi\)
0.151867 + 0.988401i \(0.451472\pi\)
\(812\) −26.0508 −0.914203
\(813\) −2.20474 −0.0773236
\(814\) −4.83146 −0.169342
\(815\) 0 0
\(816\) −15.0811 −0.527944
\(817\) −54.5099 −1.90706
\(818\) −0.650693 −0.0227510
\(819\) 77.5388 2.70943
\(820\) 0 0
\(821\) −41.7972 −1.45873 −0.729366 0.684124i \(-0.760185\pi\)
−0.729366 + 0.684124i \(0.760185\pi\)
\(822\) −29.9257 −1.04378
\(823\) 41.9633 1.46275 0.731375 0.681975i \(-0.238879\pi\)
0.731375 + 0.681975i \(0.238879\pi\)
\(824\) 1.90668 0.0664223
\(825\) 0 0
\(826\) 37.5778 1.30750
\(827\) 23.5950 0.820478 0.410239 0.911978i \(-0.365445\pi\)
0.410239 + 0.911978i \(0.365445\pi\)
\(828\) 9.92478 0.344910
\(829\) 39.4396 1.36979 0.684897 0.728640i \(-0.259847\pi\)
0.684897 + 0.728640i \(0.259847\pi\)
\(830\) 0 0
\(831\) −48.6516 −1.68771
\(832\) 5.67513 0.196750
\(833\) −21.4568 −0.743433
\(834\) 45.6615 1.58113
\(835\) 0 0
\(836\) 23.0386 0.796805
\(837\) −23.0254 −0.795874
\(838\) −10.8169 −0.373662
\(839\) 33.4495 1.15480 0.577402 0.816460i \(-0.304067\pi\)
0.577402 + 0.816460i \(0.304067\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) −2.05079 −0.0706747
\(843\) 66.9946 2.30742
\(844\) −16.3987 −0.564466
\(845\) 0 0
\(846\) 24.6253 0.846635
\(847\) 40.5745 1.39416
\(848\) 9.53690 0.327499
\(849\) 35.0590 1.20322
\(850\) 0 0
\(851\) 2.38787 0.0818552
\(852\) −13.1065 −0.449021
\(853\) −29.5101 −1.01041 −0.505203 0.863000i \(-0.668582\pi\)
−0.505203 + 0.863000i \(0.668582\pi\)
\(854\) 16.1055 0.551120
\(855\) 0 0
\(856\) 2.41327 0.0824837
\(857\) 36.4133 1.24385 0.621927 0.783075i \(-0.286350\pi\)
0.621927 + 0.783075i \(0.286350\pi\)
\(858\) −73.3498 −2.50412
\(859\) −50.7685 −1.73220 −0.866099 0.499873i \(-0.833380\pi\)
−0.866099 + 0.499873i \(0.833380\pi\)
\(860\) 0 0
\(861\) −91.0186 −3.10191
\(862\) −15.4861 −0.527459
\(863\) 44.6032 1.51831 0.759156 0.650909i \(-0.225612\pi\)
0.759156 + 0.650909i \(0.225612\pi\)
\(864\) −3.09332 −0.105237
\(865\) 0 0
\(866\) −35.8773 −1.21916
\(867\) 39.5428 1.34294
\(868\) −24.4690 −0.830531
\(869\) 36.7777 1.24760
\(870\) 0 0
\(871\) 11.9213 0.403937
\(872\) 1.73813 0.0588607
\(873\) 22.2374 0.752623
\(874\) −11.3865 −0.385153
\(875\) 0 0
\(876\) 27.2701 0.921372
\(877\) −42.4079 −1.43201 −0.716006 0.698094i \(-0.754032\pi\)
−0.716006 + 0.698094i \(0.754032\pi\)
\(878\) 26.7005 0.901099
\(879\) 25.8980 0.873517
\(880\) 0 0
\(881\) −11.0435 −0.372065 −0.186032 0.982544i \(-0.559563\pi\)
−0.186032 + 0.982544i \(0.559563\pi\)
\(882\) −15.8192 −0.532661
\(883\) 3.52847 0.118742 0.0593712 0.998236i \(-0.481090\pi\)
0.0593712 + 0.998236i \(0.481090\pi\)
\(884\) −31.9937 −1.07606
\(885\) 0 0
\(886\) −5.75765 −0.193432
\(887\) −11.8035 −0.396323 −0.198162 0.980169i \(-0.563497\pi\)
−0.198162 + 0.980169i \(0.563497\pi\)
\(888\) −2.67513 −0.0897715
\(889\) −24.9829 −0.837898
\(890\) 0 0
\(891\) −20.2628 −0.678830
\(892\) 18.4387 0.617372
\(893\) −28.2520 −0.945418
\(894\) 20.2496 0.677249
\(895\) 0 0
\(896\) −3.28726 −0.109820
\(897\) 36.2520 1.21042
\(898\) 31.1803 1.04050
\(899\) 58.9887 1.96738
\(900\) 0 0
\(901\) −53.7645 −1.79115
\(902\) 50.0068 1.66505
\(903\) −100.526 −3.34528
\(904\) 5.09332 0.169401
\(905\) 0 0
\(906\) −1.73813 −0.0577457
\(907\) 34.2579 1.13751 0.568757 0.822505i \(-0.307424\pi\)
0.568757 + 0.822505i \(0.307424\pi\)
\(908\) −11.4558 −0.380174
\(909\) −33.7743 −1.12022
\(910\) 0 0
\(911\) 31.0214 1.02779 0.513893 0.857854i \(-0.328203\pi\)
0.513893 + 0.857854i \(0.328203\pi\)
\(912\) 12.7562 0.422401
\(913\) 67.2532 2.22575
\(914\) −7.27011 −0.240474
\(915\) 0 0
\(916\) 25.6385 0.847119
\(917\) 43.9854 1.45253
\(918\) 17.4387 0.575561
\(919\) 18.9986 0.626706 0.313353 0.949637i \(-0.398548\pi\)
0.313353 + 0.949637i \(0.398548\pi\)
\(920\) 0 0
\(921\) −54.6629 −1.80120
\(922\) 18.6253 0.613392
\(923\) −27.8046 −0.915201
\(924\) 42.4871 1.39772
\(925\) 0 0
\(926\) −1.89209 −0.0621779
\(927\) −7.92478 −0.260284
\(928\) 7.92478 0.260144
\(929\) 55.4530 1.81935 0.909677 0.415318i \(-0.136330\pi\)
0.909677 + 0.415318i \(0.136330\pi\)
\(930\) 0 0
\(931\) 18.1490 0.594810
\(932\) 9.76845 0.319976
\(933\) 28.1744 0.922389
\(934\) −1.21440 −0.0397365
\(935\) 0 0
\(936\) −23.5877 −0.770988
\(937\) 27.9683 0.913683 0.456842 0.889548i \(-0.348981\pi\)
0.456842 + 0.889548i \(0.348981\pi\)
\(938\) −6.90526 −0.225465
\(939\) 26.6702 0.870349
\(940\) 0 0
\(941\) 52.3390 1.70620 0.853101 0.521745i \(-0.174719\pi\)
0.853101 + 0.521745i \(0.174719\pi\)
\(942\) 9.01080 0.293588
\(943\) −24.7151 −0.804835
\(944\) −11.4314 −0.372059
\(945\) 0 0
\(946\) 55.2301 1.79569
\(947\) −10.0752 −0.327401 −0.163700 0.986510i \(-0.552343\pi\)
−0.163700 + 0.986510i \(0.552343\pi\)
\(948\) 20.3634 0.661374
\(949\) 57.8519 1.87795
\(950\) 0 0
\(951\) −65.2628 −2.11629
\(952\) 18.5320 0.600625
\(953\) 10.2809 0.333032 0.166516 0.986039i \(-0.446748\pi\)
0.166516 + 0.986039i \(0.446748\pi\)
\(954\) −39.6385 −1.28334
\(955\) 0 0
\(956\) 16.2882 0.526798
\(957\) −102.426 −3.31096
\(958\) 40.7694 1.31720
\(959\) 36.7734 1.18747
\(960\) 0 0
\(961\) 24.4069 0.787320
\(962\) −5.67513 −0.182974
\(963\) −10.0303 −0.323222
\(964\) 13.1744 0.424320
\(965\) 0 0
\(966\) −20.9986 −0.675618
\(967\) −30.7612 −0.989212 −0.494606 0.869117i \(-0.664688\pi\)
−0.494606 + 0.869117i \(0.664688\pi\)
\(968\) −12.3430 −0.396718
\(969\) −71.9135 −2.31019
\(970\) 0 0
\(971\) 52.9086 1.69792 0.848959 0.528459i \(-0.177230\pi\)
0.848959 + 0.528459i \(0.177230\pi\)
\(972\) −20.4993 −0.657515
\(973\) −56.1098 −1.79880
\(974\) 23.8578 0.764453
\(975\) 0 0
\(976\) −4.89938 −0.156825
\(977\) −16.5901 −0.530763 −0.265382 0.964143i \(-0.585498\pi\)
−0.265382 + 0.964143i \(0.585498\pi\)
\(978\) 11.5564 0.369533
\(979\) 21.4690 0.686151
\(980\) 0 0
\(981\) −7.22425 −0.230653
\(982\) −36.0362 −1.14996
\(983\) 36.3146 1.15825 0.579127 0.815237i \(-0.303393\pi\)
0.579127 + 0.815237i \(0.303393\pi\)
\(984\) 27.6883 0.882671
\(985\) 0 0
\(986\) −44.6761 −1.42278
\(987\) −52.1016 −1.65841
\(988\) 27.0616 0.860944
\(989\) −27.2966 −0.867983
\(990\) 0 0
\(991\) 3.95272 0.125562 0.0627812 0.998027i \(-0.480003\pi\)
0.0627812 + 0.998027i \(0.480003\pi\)
\(992\) 7.44358 0.236334
\(993\) 91.9072 2.91659
\(994\) 16.1055 0.510837
\(995\) 0 0
\(996\) 37.2374 1.17991
\(997\) −41.6625 −1.31946 −0.659732 0.751501i \(-0.729330\pi\)
−0.659732 + 0.751501i \(0.729330\pi\)
\(998\) 31.4558 0.995716
\(999\) 3.09332 0.0978684
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 1850.2.a.ba.1.3 3
5.2 odd 4 1850.2.b.n.149.1 6
5.3 odd 4 1850.2.b.n.149.6 6
5.4 even 2 1850.2.a.bb.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.3 3 1.1 even 1 trivial
1850.2.a.bb.1.1 yes 3 5.4 even 2
1850.2.b.n.149.1 6 5.2 odd 4
1850.2.b.n.149.6 6 5.3 odd 4