Properties

Label 1850.2.a.x.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.46410 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} +4.46410 q^{18} -4.19615 q^{19} +3.46410 q^{21} +1.46410 q^{22} +8.00000 q^{23} +2.73205 q^{24} -1.46410 q^{26} +4.00000 q^{27} +1.26795 q^{28} -8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.46410 q^{34} +4.46410 q^{36} -1.00000 q^{37} -4.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} +1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} +2.73205 q^{48} -5.39230 q^{49} +4.00000 q^{51} -1.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} -11.4641 q^{57} -8.92820 q^{58} +0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +5.66025 q^{63} +1.00000 q^{64} +4.00000 q^{66} -13.6603 q^{67} +1.46410 q^{68} +21.8564 q^{69} -10.9282 q^{71} +4.46410 q^{72} -12.9282 q^{73} -1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} -4.00000 q^{78} +5.26795 q^{79} -2.46410 q^{81} -2.00000 q^{82} +5.26795 q^{83} +3.46410 q^{84} +6.92820 q^{86} -24.3923 q^{87} +1.46410 q^{88} -2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} -7.46410 q^{93} +1.26795 q^{94} +2.73205 q^{96} +2.00000 q^{97} -5.39230 q^{98} +6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26} + 8 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{33} - 4 q^{34} + 2 q^{36} - 2 q^{37} + 2 q^{38} - 8 q^{39} - 4 q^{41} - 4 q^{44} + 16 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 8 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 6 q^{56} - 16 q^{57} - 4 q^{58} - 10 q^{59} + 4 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 8 q^{66} - 10 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 2 q^{74} + 2 q^{76} - 24 q^{77} - 8 q^{78} + 14 q^{79} + 2 q^{81} - 4 q^{82} + 14 q^{83} - 28 q^{87} - 4 q^{88} - 4 q^{89} + 24 q^{91} + 16 q^{92} - 8 q^{93} + 6 q^{94} + 2 q^{96} + 4 q^{97} + 10 q^{98} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.73205 1.11536
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) 2.73205 0.788675
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 1.26795 0.338874
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.46410 0.355097 0.177548 0.984112i \(-0.443183\pi\)
0.177548 + 0.984112i \(0.443183\pi\)
\(18\) 4.46410 1.05220
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 1.46410 0.312148
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 2.73205 0.557678
\(25\) 0 0
\(26\) −1.46410 −0.287134
\(27\) 4.00000 0.769800
\(28\) 1.26795 0.239620
\(29\) −8.92820 −1.65793 −0.828963 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 0 0
\(31\) −2.73205 −0.490691 −0.245345 0.969436i \(-0.578901\pi\)
−0.245345 + 0.969436i \(0.578901\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 1.46410 0.251091
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) −1.00000 −0.164399
\(38\) −4.19615 −0.680706
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 3.46410 0.534522
\(43\) 6.92820 1.05654 0.528271 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 1.46410 0.220722
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 1.26795 0.184949 0.0924747 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) 2.73205 0.394338
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) −1.46410 −0.203034
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.26795 0.169437
\(57\) −11.4641 −1.51846
\(58\) −8.92820 −1.17233
\(59\) 0.196152 0.0255369 0.0127684 0.999918i \(-0.495936\pi\)
0.0127684 + 0.999918i \(0.495936\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) −2.73205 −0.346971
\(63\) 5.66025 0.713125
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −13.6603 −1.66887 −0.834433 0.551110i \(-0.814205\pi\)
−0.834433 + 0.551110i \(0.814205\pi\)
\(68\) 1.46410 0.177548
\(69\) 21.8564 2.63120
\(70\) 0 0
\(71\) −10.9282 −1.29694 −0.648470 0.761241i \(-0.724591\pi\)
−0.648470 + 0.761241i \(0.724591\pi\)
\(72\) 4.46410 0.526099
\(73\) −12.9282 −1.51313 −0.756566 0.653917i \(-0.773124\pi\)
−0.756566 + 0.653917i \(0.773124\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −4.19615 −0.481332
\(77\) 1.85641 0.211557
\(78\) −4.00000 −0.452911
\(79\) 5.26795 0.592691 0.296345 0.955081i \(-0.404232\pi\)
0.296345 + 0.955081i \(0.404232\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) −2.00000 −0.220863
\(83\) 5.26795 0.578233 0.289116 0.957294i \(-0.406639\pi\)
0.289116 + 0.957294i \(0.406639\pi\)
\(84\) 3.46410 0.377964
\(85\) 0 0
\(86\) 6.92820 0.747087
\(87\) −24.3923 −2.61513
\(88\) 1.46410 0.156074
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.85641 −0.194604
\(92\) 8.00000 0.834058
\(93\) −7.46410 −0.773991
\(94\) 1.26795 0.130779
\(95\) 0 0
\(96\) 2.73205 0.278839
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −5.39230 −0.544705
\(99\) 6.53590 0.656883
\(100\) 0 0
\(101\) −2.53590 −0.252331 −0.126166 0.992009i \(-0.540267\pi\)
−0.126166 + 0.992009i \(0.540267\pi\)
\(102\) 4.00000 0.396059
\(103\) 13.4641 1.32666 0.663329 0.748328i \(-0.269143\pi\)
0.663329 + 0.748328i \(0.269143\pi\)
\(104\) −1.46410 −0.143567
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −6.73205 −0.650812 −0.325406 0.945574i \(-0.605501\pi\)
−0.325406 + 0.945574i \(0.605501\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −2.73205 −0.259315
\(112\) 1.26795 0.119810
\(113\) 17.4641 1.64288 0.821442 0.570292i \(-0.193170\pi\)
0.821442 + 0.570292i \(0.193170\pi\)
\(114\) −11.4641 −1.07371
\(115\) 0 0
\(116\) −8.92820 −0.828963
\(117\) −6.53590 −0.604244
\(118\) 0.196152 0.0180573
\(119\) 1.85641 0.170177
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 8.92820 0.808322
\(123\) −5.46410 −0.492681
\(124\) −2.73205 −0.245345
\(125\) 0 0
\(126\) 5.66025 0.504256
\(127\) 13.6603 1.21215 0.606076 0.795407i \(-0.292743\pi\)
0.606076 + 0.795407i \(0.292743\pi\)
\(128\) 1.00000 0.0883883
\(129\) 18.9282 1.66654
\(130\) 0 0
\(131\) 12.5885 1.09986 0.549929 0.835211i \(-0.314655\pi\)
0.549929 + 0.835211i \(0.314655\pi\)
\(132\) 4.00000 0.348155
\(133\) −5.32051 −0.461347
\(134\) −13.6603 −1.18007
\(135\) 0 0
\(136\) 1.46410 0.125546
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 21.8564 1.86054
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) −10.9282 −0.917074
\(143\) −2.14359 −0.179256
\(144\) 4.46410 0.372008
\(145\) 0 0
\(146\) −12.9282 −1.06995
\(147\) −14.7321 −1.21508
\(148\) −1.00000 −0.0821995
\(149\) −16.3923 −1.34291 −0.671455 0.741045i \(-0.734330\pi\)
−0.671455 + 0.741045i \(0.734330\pi\)
\(150\) 0 0
\(151\) 8.39230 0.682956 0.341478 0.939890i \(-0.389073\pi\)
0.341478 + 0.939890i \(0.389073\pi\)
\(152\) −4.19615 −0.340353
\(153\) 6.53590 0.528396
\(154\) 1.85641 0.149593
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 5.26795 0.419096
\(159\) 16.3923 1.29999
\(160\) 0 0
\(161\) 10.1436 0.799427
\(162\) −2.46410 −0.193598
\(163\) −23.3205 −1.82660 −0.913302 0.407284i \(-0.866476\pi\)
−0.913302 + 0.407284i \(0.866476\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 5.26795 0.408872
\(167\) −5.46410 −0.422825 −0.211412 0.977397i \(-0.567806\pi\)
−0.211412 + 0.977397i \(0.567806\pi\)
\(168\) 3.46410 0.267261
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) −18.7321 −1.43248
\(172\) 6.92820 0.528271
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −24.3923 −1.84918
\(175\) 0 0
\(176\) 1.46410 0.110361
\(177\) 0.535898 0.0402806
\(178\) −2.00000 −0.149906
\(179\) −17.6603 −1.31999 −0.659995 0.751270i \(-0.729441\pi\)
−0.659995 + 0.751270i \(0.729441\pi\)
\(180\) 0 0
\(181\) −1.46410 −0.108826 −0.0544129 0.998519i \(-0.517329\pi\)
−0.0544129 + 0.998519i \(0.517329\pi\)
\(182\) −1.85641 −0.137606
\(183\) 24.3923 1.80313
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) −7.46410 −0.547294
\(187\) 2.14359 0.156755
\(188\) 1.26795 0.0924747
\(189\) 5.07180 0.368919
\(190\) 0 0
\(191\) −5.26795 −0.381175 −0.190588 0.981670i \(-0.561039\pi\)
−0.190588 + 0.981670i \(0.561039\pi\)
\(192\) 2.73205 0.197169
\(193\) 11.8564 0.853443 0.426721 0.904383i \(-0.359668\pi\)
0.426721 + 0.904383i \(0.359668\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −5.39230 −0.385165
\(197\) −18.7846 −1.33835 −0.669174 0.743106i \(-0.733352\pi\)
−0.669174 + 0.743106i \(0.733352\pi\)
\(198\) 6.53590 0.464486
\(199\) 26.0526 1.84682 0.923408 0.383819i \(-0.125391\pi\)
0.923408 + 0.383819i \(0.125391\pi\)
\(200\) 0 0
\(201\) −37.3205 −2.63239
\(202\) −2.53590 −0.178425
\(203\) −11.3205 −0.794544
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 13.4641 0.938088
\(207\) 35.7128 2.48221
\(208\) −1.46410 −0.101517
\(209\) −6.14359 −0.424961
\(210\) 0 0
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) 6.00000 0.412082
\(213\) −29.8564 −2.04573
\(214\) −6.73205 −0.460194
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −3.46410 −0.235159
\(218\) 2.00000 0.135457
\(219\) −35.3205 −2.38674
\(220\) 0 0
\(221\) −2.14359 −0.144194
\(222\) −2.73205 −0.183363
\(223\) 22.0526 1.47675 0.738374 0.674391i \(-0.235594\pi\)
0.738374 + 0.674391i \(0.235594\pi\)
\(224\) 1.26795 0.0847184
\(225\) 0 0
\(226\) 17.4641 1.16169
\(227\) −3.60770 −0.239451 −0.119726 0.992807i \(-0.538201\pi\)
−0.119726 + 0.992807i \(0.538201\pi\)
\(228\) −11.4641 −0.759229
\(229\) 15.8564 1.04782 0.523910 0.851773i \(-0.324473\pi\)
0.523910 + 0.851773i \(0.324473\pi\)
\(230\) 0 0
\(231\) 5.07180 0.333700
\(232\) −8.92820 −0.586165
\(233\) −15.0718 −0.987386 −0.493693 0.869636i \(-0.664353\pi\)
−0.493693 + 0.869636i \(0.664353\pi\)
\(234\) −6.53590 −0.427265
\(235\) 0 0
\(236\) 0.196152 0.0127684
\(237\) 14.3923 0.934881
\(238\) 1.85641 0.120333
\(239\) −17.2679 −1.11697 −0.558485 0.829514i \(-0.688617\pi\)
−0.558485 + 0.829514i \(0.688617\pi\)
\(240\) 0 0
\(241\) −8.92820 −0.575116 −0.287558 0.957763i \(-0.592843\pi\)
−0.287558 + 0.957763i \(0.592843\pi\)
\(242\) −8.85641 −0.569311
\(243\) −18.7321 −1.20166
\(244\) 8.92820 0.571570
\(245\) 0 0
\(246\) −5.46410 −0.348378
\(247\) 6.14359 0.390907
\(248\) −2.73205 −0.173485
\(249\) 14.3923 0.912075
\(250\) 0 0
\(251\) 22.7321 1.43483 0.717417 0.696644i \(-0.245324\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(252\) 5.66025 0.356562
\(253\) 11.7128 0.736378
\(254\) 13.6603 0.857121
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.4641 1.58841 0.794204 0.607652i \(-0.207888\pi\)
0.794204 + 0.607652i \(0.207888\pi\)
\(258\) 18.9282 1.17842
\(259\) −1.26795 −0.0787865
\(260\) 0 0
\(261\) −39.8564 −2.46705
\(262\) 12.5885 0.777717
\(263\) −30.0526 −1.85312 −0.926560 0.376147i \(-0.877249\pi\)
−0.926560 + 0.376147i \(0.877249\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −5.32051 −0.326221
\(267\) −5.46410 −0.334398
\(268\) −13.6603 −0.834433
\(269\) −0.392305 −0.0239192 −0.0119596 0.999928i \(-0.503807\pi\)
−0.0119596 + 0.999928i \(0.503807\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) 1.46410 0.0887742
\(273\) −5.07180 −0.306959
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 21.8564 1.31560
\(277\) 26.2487 1.57713 0.788566 0.614950i \(-0.210824\pi\)
0.788566 + 0.614950i \(0.210824\pi\)
\(278\) −6.92820 −0.415526
\(279\) −12.1962 −0.730165
\(280\) 0 0
\(281\) 4.92820 0.293992 0.146996 0.989137i \(-0.453040\pi\)
0.146996 + 0.989137i \(0.453040\pi\)
\(282\) 3.46410 0.206284
\(283\) 4.39230 0.261095 0.130548 0.991442i \(-0.458326\pi\)
0.130548 + 0.991442i \(0.458326\pi\)
\(284\) −10.9282 −0.648470
\(285\) 0 0
\(286\) −2.14359 −0.126753
\(287\) −2.53590 −0.149689
\(288\) 4.46410 0.263050
\(289\) −14.8564 −0.873906
\(290\) 0 0
\(291\) 5.46410 0.320311
\(292\) −12.9282 −0.756566
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −14.7321 −0.859191
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 5.85641 0.339823
\(298\) −16.3923 −0.949581
\(299\) −11.7128 −0.677369
\(300\) 0 0
\(301\) 8.78461 0.506336
\(302\) 8.39230 0.482923
\(303\) −6.92820 −0.398015
\(304\) −4.19615 −0.240666
\(305\) 0 0
\(306\) 6.53590 0.373632
\(307\) 12.5885 0.718461 0.359231 0.933249i \(-0.383039\pi\)
0.359231 + 0.933249i \(0.383039\pi\)
\(308\) 1.85641 0.105779
\(309\) 36.7846 2.09260
\(310\) 0 0
\(311\) 27.1244 1.53808 0.769041 0.639200i \(-0.220734\pi\)
0.769041 + 0.639200i \(0.220734\pi\)
\(312\) −4.00000 −0.226455
\(313\) −3.85641 −0.217977 −0.108988 0.994043i \(-0.534761\pi\)
−0.108988 + 0.994043i \(0.534761\pi\)
\(314\) −16.9282 −0.955314
\(315\) 0 0
\(316\) 5.26795 0.296345
\(317\) 31.8564 1.78923 0.894617 0.446834i \(-0.147448\pi\)
0.894617 + 0.446834i \(0.147448\pi\)
\(318\) 16.3923 0.919235
\(319\) −13.0718 −0.731880
\(320\) 0 0
\(321\) −18.3923 −1.02656
\(322\) 10.1436 0.565280
\(323\) −6.14359 −0.341839
\(324\) −2.46410 −0.136895
\(325\) 0 0
\(326\) −23.3205 −1.29160
\(327\) 5.46410 0.302166
\(328\) −2.00000 −0.110432
\(329\) 1.60770 0.0886351
\(330\) 0 0
\(331\) −8.87564 −0.487850 −0.243925 0.969794i \(-0.578435\pi\)
−0.243925 + 0.969794i \(0.578435\pi\)
\(332\) 5.26795 0.289116
\(333\) −4.46410 −0.244631
\(334\) −5.46410 −0.298982
\(335\) 0 0
\(336\) 3.46410 0.188982
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −10.8564 −0.590511
\(339\) 47.7128 2.59140
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) −18.7321 −1.01291
\(343\) −15.7128 −0.848412
\(344\) 6.92820 0.373544
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −17.0718 −0.916462 −0.458231 0.888833i \(-0.651517\pi\)
−0.458231 + 0.888833i \(0.651517\pi\)
\(348\) −24.3923 −1.30756
\(349\) 19.3205 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) 1.46410 0.0780369
\(353\) −15.8564 −0.843951 −0.421976 0.906607i \(-0.638663\pi\)
−0.421976 + 0.906607i \(0.638663\pi\)
\(354\) 0.535898 0.0284827
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 5.07180 0.268428
\(358\) −17.6603 −0.933373
\(359\) 8.39230 0.442929 0.221464 0.975168i \(-0.428916\pi\)
0.221464 + 0.975168i \(0.428916\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) −1.46410 −0.0769515
\(363\) −24.1962 −1.26997
\(364\) −1.85641 −0.0973021
\(365\) 0 0
\(366\) 24.3923 1.27501
\(367\) 21.6603 1.13066 0.565328 0.824866i \(-0.308750\pi\)
0.565328 + 0.824866i \(0.308750\pi\)
\(368\) 8.00000 0.417029
\(369\) −8.92820 −0.464784
\(370\) 0 0
\(371\) 7.60770 0.394972
\(372\) −7.46410 −0.386996
\(373\) −30.7846 −1.59397 −0.796983 0.604001i \(-0.793572\pi\)
−0.796983 + 0.604001i \(0.793572\pi\)
\(374\) 2.14359 0.110843
\(375\) 0 0
\(376\) 1.26795 0.0653895
\(377\) 13.0718 0.673232
\(378\) 5.07180 0.260865
\(379\) 11.6077 0.596247 0.298124 0.954527i \(-0.403639\pi\)
0.298124 + 0.954527i \(0.403639\pi\)
\(380\) 0 0
\(381\) 37.3205 1.91199
\(382\) −5.26795 −0.269532
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 2.73205 0.139419
\(385\) 0 0
\(386\) 11.8564 0.603475
\(387\) 30.9282 1.57217
\(388\) 2.00000 0.101535
\(389\) 15.8564 0.803952 0.401976 0.915650i \(-0.368324\pi\)
0.401976 + 0.915650i \(0.368324\pi\)
\(390\) 0 0
\(391\) 11.7128 0.592342
\(392\) −5.39230 −0.272353
\(393\) 34.3923 1.73486
\(394\) −18.7846 −0.946355
\(395\) 0 0
\(396\) 6.53590 0.328441
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 26.0526 1.30590
\(399\) −14.5359 −0.727705
\(400\) 0 0
\(401\) −19.0718 −0.952400 −0.476200 0.879337i \(-0.657986\pi\)
−0.476200 + 0.879337i \(0.657986\pi\)
\(402\) −37.3205 −1.86138
\(403\) 4.00000 0.199254
\(404\) −2.53590 −0.126166
\(405\) 0 0
\(406\) −11.3205 −0.561827
\(407\) −1.46410 −0.0725728
\(408\) 4.00000 0.198030
\(409\) 16.9282 0.837046 0.418523 0.908206i \(-0.362548\pi\)
0.418523 + 0.908206i \(0.362548\pi\)
\(410\) 0 0
\(411\) −5.46410 −0.269524
\(412\) 13.4641 0.663329
\(413\) 0.248711 0.0122383
\(414\) 35.7128 1.75519
\(415\) 0 0
\(416\) −1.46410 −0.0717835
\(417\) −18.9282 −0.926918
\(418\) −6.14359 −0.300493
\(419\) −34.2487 −1.67316 −0.836580 0.547846i \(-0.815448\pi\)
−0.836580 + 0.547846i \(0.815448\pi\)
\(420\) 0 0
\(421\) −27.8564 −1.35764 −0.678819 0.734306i \(-0.737508\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(422\) 9.85641 0.479802
\(423\) 5.66025 0.275211
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −29.8564 −1.44655
\(427\) 11.3205 0.547838
\(428\) −6.73205 −0.325406
\(429\) −5.85641 −0.282750
\(430\) 0 0
\(431\) 8.19615 0.394795 0.197397 0.980324i \(-0.436751\pi\)
0.197397 + 0.980324i \(0.436751\pi\)
\(432\) 4.00000 0.192450
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −3.46410 −0.166282
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −33.5692 −1.60583
\(438\) −35.3205 −1.68768
\(439\) −31.5167 −1.50421 −0.752104 0.659044i \(-0.770961\pi\)
−0.752104 + 0.659044i \(0.770961\pi\)
\(440\) 0 0
\(441\) −24.0718 −1.14628
\(442\) −2.14359 −0.101960
\(443\) 39.1244 1.85885 0.929427 0.369006i \(-0.120302\pi\)
0.929427 + 0.369006i \(0.120302\pi\)
\(444\) −2.73205 −0.129657
\(445\) 0 0
\(446\) 22.0526 1.04422
\(447\) −44.7846 −2.11824
\(448\) 1.26795 0.0599050
\(449\) 33.7128 1.59101 0.795503 0.605950i \(-0.207207\pi\)
0.795503 + 0.605950i \(0.207207\pi\)
\(450\) 0 0
\(451\) −2.92820 −0.137884
\(452\) 17.4641 0.821442
\(453\) 22.9282 1.07726
\(454\) −3.60770 −0.169318
\(455\) 0 0
\(456\) −11.4641 −0.536856
\(457\) 4.14359 0.193829 0.0969146 0.995293i \(-0.469103\pi\)
0.0969146 + 0.995293i \(0.469103\pi\)
\(458\) 15.8564 0.740921
\(459\) 5.85641 0.273354
\(460\) 0 0
\(461\) −26.7846 −1.24748 −0.623742 0.781630i \(-0.714388\pi\)
−0.623742 + 0.781630i \(0.714388\pi\)
\(462\) 5.07180 0.235961
\(463\) −5.07180 −0.235706 −0.117853 0.993031i \(-0.537601\pi\)
−0.117853 + 0.993031i \(0.537601\pi\)
\(464\) −8.92820 −0.414481
\(465\) 0 0
\(466\) −15.0718 −0.698188
\(467\) −37.1769 −1.72034 −0.860171 0.510005i \(-0.829643\pi\)
−0.860171 + 0.510005i \(0.829643\pi\)
\(468\) −6.53590 −0.302122
\(469\) −17.3205 −0.799787
\(470\) 0 0
\(471\) −46.2487 −2.13103
\(472\) 0.196152 0.00902865
\(473\) 10.1436 0.466403
\(474\) 14.3923 0.661060
\(475\) 0 0
\(476\) 1.85641 0.0850883
\(477\) 26.7846 1.22638
\(478\) −17.2679 −0.789818
\(479\) −34.0526 −1.55590 −0.777951 0.628325i \(-0.783741\pi\)
−0.777951 + 0.628325i \(0.783741\pi\)
\(480\) 0 0
\(481\) 1.46410 0.0667573
\(482\) −8.92820 −0.406669
\(483\) 27.7128 1.26098
\(484\) −8.85641 −0.402564
\(485\) 0 0
\(486\) −18.7321 −0.849703
\(487\) 16.3923 0.742806 0.371403 0.928472i \(-0.378877\pi\)
0.371403 + 0.928472i \(0.378877\pi\)
\(488\) 8.92820 0.404161
\(489\) −63.7128 −2.88119
\(490\) 0 0
\(491\) −1.07180 −0.0483695 −0.0241848 0.999708i \(-0.507699\pi\)
−0.0241848 + 0.999708i \(0.507699\pi\)
\(492\) −5.46410 −0.246341
\(493\) −13.0718 −0.588724
\(494\) 6.14359 0.276413
\(495\) 0 0
\(496\) −2.73205 −0.122673
\(497\) −13.8564 −0.621545
\(498\) 14.3923 0.644935
\(499\) 40.5885 1.81699 0.908494 0.417897i \(-0.137233\pi\)
0.908494 + 0.417897i \(0.137233\pi\)
\(500\) 0 0
\(501\) −14.9282 −0.666943
\(502\) 22.7321 1.01458
\(503\) 5.07180 0.226140 0.113070 0.993587i \(-0.463932\pi\)
0.113070 + 0.993587i \(0.463932\pi\)
\(504\) 5.66025 0.252128
\(505\) 0 0
\(506\) 11.7128 0.520698
\(507\) −29.6603 −1.31726
\(508\) 13.6603 0.606076
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −16.3923 −0.725153
\(512\) 1.00000 0.0441942
\(513\) −16.7846 −0.741059
\(514\) 25.4641 1.12317
\(515\) 0 0
\(516\) 18.9282 0.833268
\(517\) 1.85641 0.0816447
\(518\) −1.26795 −0.0557105
\(519\) −27.3205 −1.19924
\(520\) 0 0
\(521\) 33.4641 1.46609 0.733044 0.680181i \(-0.238099\pi\)
0.733044 + 0.680181i \(0.238099\pi\)
\(522\) −39.8564 −1.74447
\(523\) 4.78461 0.209216 0.104608 0.994514i \(-0.466641\pi\)
0.104608 + 0.994514i \(0.466641\pi\)
\(524\) 12.5885 0.549929
\(525\) 0 0
\(526\) −30.0526 −1.31035
\(527\) −4.00000 −0.174243
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0.875644 0.0379997
\(532\) −5.32051 −0.230673
\(533\) 2.92820 0.126835
\(534\) −5.46410 −0.236455
\(535\) 0 0
\(536\) −13.6603 −0.590033
\(537\) −48.2487 −2.08209
\(538\) −0.392305 −0.0169135
\(539\) −7.89488 −0.340057
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 16.7846 0.720961
\(543\) −4.00000 −0.171656
\(544\) 1.46410 0.0627728
\(545\) 0 0
\(546\) −5.07180 −0.217053
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 39.8564 1.70103
\(550\) 0 0
\(551\) 37.4641 1.59602
\(552\) 21.8564 0.930270
\(553\) 6.67949 0.284041
\(554\) 26.2487 1.11520
\(555\) 0 0
\(556\) −6.92820 −0.293821
\(557\) 32.1051 1.36034 0.680169 0.733056i \(-0.261906\pi\)
0.680169 + 0.733056i \(0.261906\pi\)
\(558\) −12.1962 −0.516304
\(559\) −10.1436 −0.429028
\(560\) 0 0
\(561\) 5.85641 0.247258
\(562\) 4.92820 0.207884
\(563\) −17.0718 −0.719490 −0.359745 0.933051i \(-0.617136\pi\)
−0.359745 + 0.933051i \(0.617136\pi\)
\(564\) 3.46410 0.145865
\(565\) 0 0
\(566\) 4.39230 0.184622
\(567\) −3.12436 −0.131211
\(568\) −10.9282 −0.458537
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −2.14359 −0.0896281
\(573\) −14.3923 −0.601247
\(574\) −2.53590 −0.105846
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) 3.60770 0.150190 0.0750952 0.997176i \(-0.476074\pi\)
0.0750952 + 0.997176i \(0.476074\pi\)
\(578\) −14.8564 −0.617945
\(579\) 32.3923 1.34618
\(580\) 0 0
\(581\) 6.67949 0.277112
\(582\) 5.46410 0.226494
\(583\) 8.78461 0.363821
\(584\) −12.9282 −0.534973
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −3.21539 −0.132713 −0.0663567 0.997796i \(-0.521138\pi\)
−0.0663567 + 0.997796i \(0.521138\pi\)
\(588\) −14.7321 −0.607540
\(589\) 11.4641 0.472370
\(590\) 0 0
\(591\) −51.3205 −2.11104
\(592\) −1.00000 −0.0410997
\(593\) −6.78461 −0.278611 −0.139305 0.990249i \(-0.544487\pi\)
−0.139305 + 0.990249i \(0.544487\pi\)
\(594\) 5.85641 0.240291
\(595\) 0 0
\(596\) −16.3923 −0.671455
\(597\) 71.1769 2.91308
\(598\) −11.7128 −0.478973
\(599\) 2.53590 0.103614 0.0518070 0.998657i \(-0.483502\pi\)
0.0518070 + 0.998657i \(0.483502\pi\)
\(600\) 0 0
\(601\) 48.3923 1.97396 0.986982 0.160833i \(-0.0514180\pi\)
0.986982 + 0.160833i \(0.0514180\pi\)
\(602\) 8.78461 0.358034
\(603\) −60.9808 −2.48333
\(604\) 8.39230 0.341478
\(605\) 0 0
\(606\) −6.92820 −0.281439
\(607\) 40.7846 1.65540 0.827698 0.561174i \(-0.189650\pi\)
0.827698 + 0.561174i \(0.189650\pi\)
\(608\) −4.19615 −0.170176
\(609\) −30.9282 −1.25327
\(610\) 0 0
\(611\) −1.85641 −0.0751022
\(612\) 6.53590 0.264198
\(613\) −3.07180 −0.124069 −0.0620344 0.998074i \(-0.519759\pi\)
−0.0620344 + 0.998074i \(0.519759\pi\)
\(614\) 12.5885 0.508029
\(615\) 0 0
\(616\) 1.85641 0.0747967
\(617\) −12.9282 −0.520470 −0.260235 0.965545i \(-0.583800\pi\)
−0.260235 + 0.965545i \(0.583800\pi\)
\(618\) 36.7846 1.47969
\(619\) 9.85641 0.396162 0.198081 0.980186i \(-0.436529\pi\)
0.198081 + 0.980186i \(0.436529\pi\)
\(620\) 0 0
\(621\) 32.0000 1.28412
\(622\) 27.1244 1.08759
\(623\) −2.53590 −0.101599
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −3.85641 −0.154133
\(627\) −16.7846 −0.670313
\(628\) −16.9282 −0.675509
\(629\) −1.46410 −0.0583776
\(630\) 0 0
\(631\) −20.9808 −0.835231 −0.417615 0.908624i \(-0.637134\pi\)
−0.417615 + 0.908624i \(0.637134\pi\)
\(632\) 5.26795 0.209548
\(633\) 26.9282 1.07030
\(634\) 31.8564 1.26518
\(635\) 0 0
\(636\) 16.3923 0.649997
\(637\) 7.89488 0.312807
\(638\) −13.0718 −0.517517
\(639\) −48.7846 −1.92989
\(640\) 0 0
\(641\) −13.4641 −0.531800 −0.265900 0.964001i \(-0.585669\pi\)
−0.265900 + 0.964001i \(0.585669\pi\)
\(642\) −18.3923 −0.725886
\(643\) −41.4641 −1.63518 −0.817592 0.575798i \(-0.804692\pi\)
−0.817592 + 0.575798i \(0.804692\pi\)
\(644\) 10.1436 0.399714
\(645\) 0 0
\(646\) −6.14359 −0.241716
\(647\) −19.7128 −0.774991 −0.387495 0.921872i \(-0.626660\pi\)
−0.387495 + 0.921872i \(0.626660\pi\)
\(648\) −2.46410 −0.0967991
\(649\) 0.287187 0.0112731
\(650\) 0 0
\(651\) −9.46410 −0.370927
\(652\) −23.3205 −0.913302
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 5.46410 0.213663
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −57.7128 −2.25159
\(658\) 1.60770 0.0626745
\(659\) 6.92820 0.269884 0.134942 0.990853i \(-0.456915\pi\)
0.134942 + 0.990853i \(0.456915\pi\)
\(660\) 0 0
\(661\) −44.9282 −1.74750 −0.873752 0.486371i \(-0.838320\pi\)
−0.873752 + 0.486371i \(0.838320\pi\)
\(662\) −8.87564 −0.344962
\(663\) −5.85641 −0.227444
\(664\) 5.26795 0.204436
\(665\) 0 0
\(666\) −4.46410 −0.172980
\(667\) −71.4256 −2.76561
\(668\) −5.46410 −0.211412
\(669\) 60.2487 2.32935
\(670\) 0 0
\(671\) 13.0718 0.504631
\(672\) 3.46410 0.133631
\(673\) 19.0718 0.735164 0.367582 0.929991i \(-0.380186\pi\)
0.367582 + 0.929991i \(0.380186\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 31.8564 1.22434 0.612171 0.790726i \(-0.290297\pi\)
0.612171 + 0.790726i \(0.290297\pi\)
\(678\) 47.7128 1.83240
\(679\) 2.53590 0.0973188
\(680\) 0 0
\(681\) −9.85641 −0.377698
\(682\) −4.00000 −0.153168
\(683\) −36.7846 −1.40752 −0.703762 0.710436i \(-0.748498\pi\)
−0.703762 + 0.710436i \(0.748498\pi\)
\(684\) −18.7321 −0.716238
\(685\) 0 0
\(686\) −15.7128 −0.599918
\(687\) 43.3205 1.65278
\(688\) 6.92820 0.264135
\(689\) −8.78461 −0.334667
\(690\) 0 0
\(691\) 20.3923 0.775760 0.387880 0.921710i \(-0.373208\pi\)
0.387880 + 0.921710i \(0.373208\pi\)
\(692\) −10.0000 −0.380143
\(693\) 8.28719 0.314804
\(694\) −17.0718 −0.648037
\(695\) 0 0
\(696\) −24.3923 −0.924588
\(697\) −2.92820 −0.110914
\(698\) 19.3205 0.731292
\(699\) −41.1769 −1.55745
\(700\) 0 0
\(701\) 15.8564 0.598888 0.299444 0.954114i \(-0.403199\pi\)
0.299444 + 0.954114i \(0.403199\pi\)
\(702\) −5.85641 −0.221036
\(703\) 4.19615 0.158261
\(704\) 1.46410 0.0551804
\(705\) 0 0
\(706\) −15.8564 −0.596764
\(707\) −3.21539 −0.120927
\(708\) 0.535898 0.0201403
\(709\) 16.1436 0.606285 0.303143 0.952945i \(-0.401964\pi\)
0.303143 + 0.952945i \(0.401964\pi\)
\(710\) 0 0
\(711\) 23.5167 0.881944
\(712\) −2.00000 −0.0749532
\(713\) −21.8564 −0.818529
\(714\) 5.07180 0.189807
\(715\) 0 0
\(716\) −17.6603 −0.659995
\(717\) −47.1769 −1.76185
\(718\) 8.39230 0.313198
\(719\) 8.39230 0.312980 0.156490 0.987680i \(-0.449982\pi\)
0.156490 + 0.987680i \(0.449982\pi\)
\(720\) 0 0
\(721\) 17.0718 0.635787
\(722\) −1.39230 −0.0518162
\(723\) −24.3923 −0.907160
\(724\) −1.46410 −0.0544129
\(725\) 0 0
\(726\) −24.1962 −0.898003
\(727\) 32.7846 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(728\) −1.85641 −0.0688030
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) 10.1436 0.375174
\(732\) 24.3923 0.901566
\(733\) −0.143594 −0.00530375 −0.00265187 0.999996i \(-0.500844\pi\)
−0.00265187 + 0.999996i \(0.500844\pi\)
\(734\) 21.6603 0.799495
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −20.0000 −0.736709
\(738\) −8.92820 −0.328652
\(739\) −6.92820 −0.254858 −0.127429 0.991848i \(-0.540673\pi\)
−0.127429 + 0.991848i \(0.540673\pi\)
\(740\) 0 0
\(741\) 16.7846 0.616598
\(742\) 7.60770 0.279287
\(743\) 7.12436 0.261367 0.130684 0.991424i \(-0.458283\pi\)
0.130684 + 0.991424i \(0.458283\pi\)
\(744\) −7.46410 −0.273647
\(745\) 0 0
\(746\) −30.7846 −1.12710
\(747\) 23.5167 0.860430
\(748\) 2.14359 0.0783775
\(749\) −8.53590 −0.311895
\(750\) 0 0
\(751\) 0.392305 0.0143154 0.00715770 0.999974i \(-0.497722\pi\)
0.00715770 + 0.999974i \(0.497722\pi\)
\(752\) 1.26795 0.0462373
\(753\) 62.1051 2.26324
\(754\) 13.0718 0.476047
\(755\) 0 0
\(756\) 5.07180 0.184459
\(757\) 53.7128 1.95223 0.976113 0.217265i \(-0.0697135\pi\)
0.976113 + 0.217265i \(0.0697135\pi\)
\(758\) 11.6077 0.421610
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 37.3205 1.35198
\(763\) 2.53590 0.0918057
\(764\) −5.26795 −0.190588
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −0.287187 −0.0103697
\(768\) 2.73205 0.0985844
\(769\) 20.9282 0.754690 0.377345 0.926073i \(-0.376837\pi\)
0.377345 + 0.926073i \(0.376837\pi\)
\(770\) 0 0
\(771\) 69.5692 2.50547
\(772\) 11.8564 0.426721
\(773\) 38.7846 1.39499 0.697493 0.716592i \(-0.254299\pi\)
0.697493 + 0.716592i \(0.254299\pi\)
\(774\) 30.9282 1.11169
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) −3.46410 −0.124274
\(778\) 15.8564 0.568480
\(779\) 8.39230 0.300686
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 11.7128 0.418849
\(783\) −35.7128 −1.27627
\(784\) −5.39230 −0.192582
\(785\) 0 0
\(786\) 34.3923 1.22673
\(787\) −11.8038 −0.420762 −0.210381 0.977620i \(-0.567470\pi\)
−0.210381 + 0.977620i \(0.567470\pi\)
\(788\) −18.7846 −0.669174
\(789\) −82.1051 −2.92302
\(790\) 0 0
\(791\) 22.1436 0.787336
\(792\) 6.53590 0.232243
\(793\) −13.0718 −0.464193
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 26.0526 0.923408
\(797\) −17.4641 −0.618610 −0.309305 0.950963i \(-0.600096\pi\)
−0.309305 + 0.950963i \(0.600096\pi\)
\(798\) −14.5359 −0.514565
\(799\) 1.85641 0.0656749
\(800\) 0 0
\(801\) −8.92820 −0.315463
\(802\) −19.0718 −0.673449
\(803\) −18.9282 −0.667962
\(804\) −37.3205 −1.31619
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −1.07180 −0.0377290
\(808\) −2.53590 −0.0892126
\(809\) −39.5692 −1.39118 −0.695590 0.718439i \(-0.744857\pi\)
−0.695590 + 0.718439i \(0.744857\pi\)
\(810\) 0 0
\(811\) −14.1436 −0.496649 −0.248324 0.968677i \(-0.579880\pi\)
−0.248324 + 0.968677i \(0.579880\pi\)
\(812\) −11.3205 −0.397272
\(813\) 45.8564 1.60825
\(814\) −1.46410 −0.0513167
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) −29.0718 −1.01709
\(818\) 16.9282 0.591881
\(819\) −8.28719 −0.289578
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −5.46410 −0.190582
\(823\) 10.4449 0.364085 0.182043 0.983291i \(-0.441729\pi\)
0.182043 + 0.983291i \(0.441729\pi\)
\(824\) 13.4641 0.469044
\(825\) 0 0
\(826\) 0.248711 0.00865377
\(827\) −4.39230 −0.152735 −0.0763677 0.997080i \(-0.524332\pi\)
−0.0763677 + 0.997080i \(0.524332\pi\)
\(828\) 35.7128 1.24111
\(829\) 31.5692 1.09644 0.548222 0.836333i \(-0.315305\pi\)
0.548222 + 0.836333i \(0.315305\pi\)
\(830\) 0 0
\(831\) 71.7128 2.48769
\(832\) −1.46410 −0.0507586
\(833\) −7.89488 −0.273541
\(834\) −18.9282 −0.655430
\(835\) 0 0
\(836\) −6.14359 −0.212481
\(837\) −10.9282 −0.377734
\(838\) −34.2487 −1.18310
\(839\) 32.7846 1.13185 0.565925 0.824457i \(-0.308519\pi\)
0.565925 + 0.824457i \(0.308519\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) −27.8564 −0.959995
\(843\) 13.4641 0.463728
\(844\) 9.85641 0.339272
\(845\) 0 0
\(846\) 5.66025 0.194604
\(847\) −11.2295 −0.385849
\(848\) 6.00000 0.206041
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) −29.8564 −1.02286
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 11.3205 0.387380
\(855\) 0 0
\(856\) −6.73205 −0.230097
\(857\) 4.14359 0.141542 0.0707712 0.997493i \(-0.477454\pi\)
0.0707712 + 0.997493i \(0.477454\pi\)
\(858\) −5.85641 −0.199934
\(859\) −14.4449 −0.492852 −0.246426 0.969162i \(-0.579256\pi\)
−0.246426 + 0.969162i \(0.579256\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) 8.19615 0.279162
\(863\) −38.4449 −1.30868 −0.654339 0.756201i \(-0.727053\pi\)
−0.654339 + 0.756201i \(0.727053\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) −40.5885 −1.37846
\(868\) −3.46410 −0.117579
\(869\) 7.71281 0.261639
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 2.00000 0.0677285
\(873\) 8.92820 0.302174
\(874\) −33.5692 −1.13550
\(875\) 0 0
\(876\) −35.3205 −1.19337
\(877\) 50.7846 1.71487 0.857437 0.514589i \(-0.172055\pi\)
0.857437 + 0.514589i \(0.172055\pi\)
\(878\) −31.5167 −1.06364
\(879\) −16.3923 −0.552899
\(880\) 0 0
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) −24.0718 −0.810540
\(883\) 34.2487 1.15256 0.576280 0.817252i \(-0.304504\pi\)
0.576280 + 0.817252i \(0.304504\pi\)
\(884\) −2.14359 −0.0720969
\(885\) 0 0
\(886\) 39.1244 1.31441
\(887\) −20.1962 −0.678120 −0.339060 0.940765i \(-0.610109\pi\)
−0.339060 + 0.940765i \(0.610109\pi\)
\(888\) −2.73205 −0.0916816
\(889\) 17.3205 0.580911
\(890\) 0 0
\(891\) −3.60770 −0.120862
\(892\) 22.0526 0.738374
\(893\) −5.32051 −0.178044
\(894\) −44.7846 −1.49782
\(895\) 0 0
\(896\) 1.26795 0.0423592
\(897\) −32.0000 −1.06845
\(898\) 33.7128 1.12501
\(899\) 24.3923 0.813529
\(900\) 0 0
\(901\) 8.78461 0.292658
\(902\) −2.92820 −0.0974985
\(903\) 24.0000 0.798670
\(904\) 17.4641 0.580847
\(905\) 0 0
\(906\) 22.9282 0.761739
\(907\) −5.75129 −0.190968 −0.0954842 0.995431i \(-0.530440\pi\)
−0.0954842 + 0.995431i \(0.530440\pi\)
\(908\) −3.60770 −0.119726
\(909\) −11.3205 −0.375478
\(910\) 0 0
\(911\) −27.9090 −0.924665 −0.462333 0.886707i \(-0.652987\pi\)
−0.462333 + 0.886707i \(0.652987\pi\)
\(912\) −11.4641 −0.379614
\(913\) 7.71281 0.255257
\(914\) 4.14359 0.137058
\(915\) 0 0
\(916\) 15.8564 0.523910
\(917\) 15.9615 0.527096
\(918\) 5.85641 0.193290
\(919\) −13.2679 −0.437669 −0.218835 0.975762i \(-0.570226\pi\)
−0.218835 + 0.975762i \(0.570226\pi\)
\(920\) 0 0
\(921\) 34.3923 1.13326
\(922\) −26.7846 −0.882104
\(923\) 16.0000 0.526646
\(924\) 5.07180 0.166850
\(925\) 0 0
\(926\) −5.07180 −0.166670
\(927\) 60.1051 1.97411
\(928\) −8.92820 −0.293083
\(929\) −15.8564 −0.520232 −0.260116 0.965577i \(-0.583761\pi\)
−0.260116 + 0.965577i \(0.583761\pi\)
\(930\) 0 0
\(931\) 22.6269 0.741568
\(932\) −15.0718 −0.493693
\(933\) 74.1051 2.42609
\(934\) −37.1769 −1.21647
\(935\) 0 0
\(936\) −6.53590 −0.213633
\(937\) −3.85641 −0.125983 −0.0629917 0.998014i \(-0.520064\pi\)
−0.0629917 + 0.998014i \(0.520064\pi\)
\(938\) −17.3205 −0.565535
\(939\) −10.5359 −0.343826
\(940\) 0 0
\(941\) 28.3923 0.925563 0.462781 0.886472i \(-0.346852\pi\)
0.462781 + 0.886472i \(0.346852\pi\)
\(942\) −46.2487 −1.50686
\(943\) −16.0000 −0.521032
\(944\) 0.196152 0.00638422
\(945\) 0 0
\(946\) 10.1436 0.329797
\(947\) 45.1769 1.46805 0.734026 0.679121i \(-0.237639\pi\)
0.734026 + 0.679121i \(0.237639\pi\)
\(948\) 14.3923 0.467440
\(949\) 18.9282 0.614435
\(950\) 0 0
\(951\) 87.0333 2.82225
\(952\) 1.85641 0.0601665
\(953\) 11.8564 0.384067 0.192033 0.981388i \(-0.438492\pi\)
0.192033 + 0.981388i \(0.438492\pi\)
\(954\) 26.7846 0.867184
\(955\) 0 0
\(956\) −17.2679 −0.558485
\(957\) −35.7128 −1.15443
\(958\) −34.0526 −1.10019
\(959\) −2.53590 −0.0818884
\(960\) 0 0
\(961\) −23.5359 −0.759223
\(962\) 1.46410 0.0472045
\(963\) −30.0526 −0.968430
\(964\) −8.92820 −0.287558
\(965\) 0 0
\(966\) 27.7128 0.891645
\(967\) 23.6077 0.759172 0.379586 0.925156i \(-0.376066\pi\)
0.379586 + 0.925156i \(0.376066\pi\)
\(968\) −8.85641 −0.284656
\(969\) −16.7846 −0.539199
\(970\) 0 0
\(971\) 14.9282 0.479069 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(972\) −18.7321 −0.600831
\(973\) −8.78461 −0.281622
\(974\) 16.3923 0.525243
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −63.7128 −2.03731
\(979\) −2.92820 −0.0935858
\(980\) 0 0
\(981\) 8.92820 0.285056
\(982\) −1.07180 −0.0342024
\(983\) 38.4449 1.22620 0.613100 0.790005i \(-0.289922\pi\)
0.613100 + 0.790005i \(0.289922\pi\)
\(984\) −5.46410 −0.174189
\(985\) 0 0
\(986\) −13.0718 −0.416291
\(987\) 4.39230 0.139809
\(988\) 6.14359 0.195454
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) 5.94744 0.188927 0.0944633 0.995528i \(-0.469886\pi\)
0.0944633 + 0.995528i \(0.469886\pi\)
\(992\) −2.73205 −0.0867427
\(993\) −24.2487 −0.769510
\(994\) −13.8564 −0.439499
\(995\) 0 0
\(996\) 14.3923 0.456038
\(997\) 4.14359 0.131229 0.0656145 0.997845i \(-0.479099\pi\)
0.0656145 + 0.997845i \(0.479099\pi\)
\(998\) 40.5885 1.28481
\(999\) −4.00000 −0.126554
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.x.1.2 2
5.2 odd 4 1850.2.b.l.149.3 4
5.3 odd 4 1850.2.b.l.149.2 4
5.4 even 2 370.2.a.e.1.1 2
15.14 odd 2 3330.2.a.bd.1.2 2
20.19 odd 2 2960.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 5.4 even 2
1850.2.a.x.1.2 2 1.1 even 1 trivial
1850.2.b.l.149.2 4 5.3 odd 4
1850.2.b.l.149.3 4 5.2 odd 4
2960.2.a.q.1.2 2 20.19 odd 2
3330.2.a.bd.1.2 2 15.14 odd 2