Properties

Label 8001.2.a.x.1.8
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.691771 q^{2} -1.52145 q^{4} -0.236679 q^{5} +1.00000 q^{7} +2.43604 q^{8} +O(q^{10})\) \(q-0.691771 q^{2} -1.52145 q^{4} -0.236679 q^{5} +1.00000 q^{7} +2.43604 q^{8} +0.163728 q^{10} -4.73037 q^{11} -1.51889 q^{13} -0.691771 q^{14} +1.35773 q^{16} +4.88453 q^{17} +2.13777 q^{19} +0.360096 q^{20} +3.27233 q^{22} +1.62008 q^{23} -4.94398 q^{25} +1.05072 q^{26} -1.52145 q^{28} -0.481543 q^{29} -9.10709 q^{31} -5.81131 q^{32} -3.37897 q^{34} -0.236679 q^{35} +10.9544 q^{37} -1.47885 q^{38} -0.576559 q^{40} -4.78089 q^{41} +1.26680 q^{43} +7.19704 q^{44} -1.12073 q^{46} +6.43162 q^{47} +1.00000 q^{49} +3.42010 q^{50} +2.31092 q^{52} +11.2045 q^{53} +1.11958 q^{55} +2.43604 q^{56} +0.333117 q^{58} -7.68081 q^{59} +5.26239 q^{61} +6.30002 q^{62} +1.30465 q^{64} +0.359489 q^{65} -15.1697 q^{67} -7.43158 q^{68} +0.163728 q^{70} +2.66621 q^{71} +10.5894 q^{73} -7.57791 q^{74} -3.25252 q^{76} -4.73037 q^{77} -13.1527 q^{79} -0.321345 q^{80} +3.30728 q^{82} -11.1108 q^{83} -1.15606 q^{85} -0.876336 q^{86} -11.5234 q^{88} +11.0913 q^{89} -1.51889 q^{91} -2.46488 q^{92} -4.44921 q^{94} -0.505966 q^{95} -9.66578 q^{97} -0.691771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.691771 −0.489156 −0.244578 0.969630i \(-0.578649\pi\)
−0.244578 + 0.969630i \(0.578649\pi\)
\(3\) 0 0
\(4\) −1.52145 −0.760727
\(5\) −0.236679 −0.105846 −0.0529230 0.998599i \(-0.516854\pi\)
−0.0529230 + 0.998599i \(0.516854\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.43604 0.861270
\(9\) 0 0
\(10\) 0.163728 0.0517752
\(11\) −4.73037 −1.42626 −0.713130 0.701032i \(-0.752723\pi\)
−0.713130 + 0.701032i \(0.752723\pi\)
\(12\) 0 0
\(13\) −1.51889 −0.421264 −0.210632 0.977565i \(-0.567552\pi\)
−0.210632 + 0.977565i \(0.567552\pi\)
\(14\) −0.691771 −0.184884
\(15\) 0 0
\(16\) 1.35773 0.339431
\(17\) 4.88453 1.18467 0.592336 0.805691i \(-0.298206\pi\)
0.592336 + 0.805691i \(0.298206\pi\)
\(18\) 0 0
\(19\) 2.13777 0.490439 0.245219 0.969468i \(-0.421140\pi\)
0.245219 + 0.969468i \(0.421140\pi\)
\(20\) 0.360096 0.0805199
\(21\) 0 0
\(22\) 3.27233 0.697664
\(23\) 1.62008 0.337811 0.168905 0.985632i \(-0.445977\pi\)
0.168905 + 0.985632i \(0.445977\pi\)
\(24\) 0 0
\(25\) −4.94398 −0.988797
\(26\) 1.05072 0.206064
\(27\) 0 0
\(28\) −1.52145 −0.287528
\(29\) −0.481543 −0.0894203 −0.0447101 0.999000i \(-0.514236\pi\)
−0.0447101 + 0.999000i \(0.514236\pi\)
\(30\) 0 0
\(31\) −9.10709 −1.63568 −0.817841 0.575445i \(-0.804829\pi\)
−0.817841 + 0.575445i \(0.804829\pi\)
\(32\) −5.81131 −1.02730
\(33\) 0 0
\(34\) −3.37897 −0.579489
\(35\) −0.236679 −0.0400060
\(36\) 0 0
\(37\) 10.9544 1.80089 0.900443 0.434974i \(-0.143242\pi\)
0.900443 + 0.434974i \(0.143242\pi\)
\(38\) −1.47885 −0.239901
\(39\) 0 0
\(40\) −0.576559 −0.0911620
\(41\) −4.78089 −0.746650 −0.373325 0.927701i \(-0.621782\pi\)
−0.373325 + 0.927701i \(0.621782\pi\)
\(42\) 0 0
\(43\) 1.26680 0.193185 0.0965927 0.995324i \(-0.469206\pi\)
0.0965927 + 0.995324i \(0.469206\pi\)
\(44\) 7.19704 1.08499
\(45\) 0 0
\(46\) −1.12073 −0.165242
\(47\) 6.43162 0.938148 0.469074 0.883159i \(-0.344588\pi\)
0.469074 + 0.883159i \(0.344588\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.42010 0.483676
\(51\) 0 0
\(52\) 2.31092 0.320467
\(53\) 11.2045 1.53906 0.769531 0.638610i \(-0.220490\pi\)
0.769531 + 0.638610i \(0.220490\pi\)
\(54\) 0 0
\(55\) 1.11958 0.150964
\(56\) 2.43604 0.325529
\(57\) 0 0
\(58\) 0.333117 0.0437405
\(59\) −7.68081 −0.999956 −0.499978 0.866038i \(-0.666659\pi\)
−0.499978 + 0.866038i \(0.666659\pi\)
\(60\) 0 0
\(61\) 5.26239 0.673780 0.336890 0.941544i \(-0.390625\pi\)
0.336890 + 0.941544i \(0.390625\pi\)
\(62\) 6.30002 0.800103
\(63\) 0 0
\(64\) 1.30465 0.163081
\(65\) 0.359489 0.0445891
\(66\) 0 0
\(67\) −15.1697 −1.85327 −0.926635 0.375963i \(-0.877312\pi\)
−0.926635 + 0.375963i \(0.877312\pi\)
\(68\) −7.43158 −0.901211
\(69\) 0 0
\(70\) 0.163728 0.0195692
\(71\) 2.66621 0.316421 0.158210 0.987405i \(-0.449428\pi\)
0.158210 + 0.987405i \(0.449428\pi\)
\(72\) 0 0
\(73\) 10.5894 1.23939 0.619695 0.784843i \(-0.287256\pi\)
0.619695 + 0.784843i \(0.287256\pi\)
\(74\) −7.57791 −0.880914
\(75\) 0 0
\(76\) −3.25252 −0.373090
\(77\) −4.73037 −0.539076
\(78\) 0 0
\(79\) −13.1527 −1.47979 −0.739895 0.672722i \(-0.765125\pi\)
−0.739895 + 0.672722i \(0.765125\pi\)
\(80\) −0.321345 −0.0359275
\(81\) 0 0
\(82\) 3.30728 0.365228
\(83\) −11.1108 −1.21957 −0.609783 0.792568i \(-0.708743\pi\)
−0.609783 + 0.792568i \(0.708743\pi\)
\(84\) 0 0
\(85\) −1.15606 −0.125393
\(86\) −0.876336 −0.0944978
\(87\) 0 0
\(88\) −11.5234 −1.22839
\(89\) 11.0913 1.17568 0.587838 0.808979i \(-0.299979\pi\)
0.587838 + 0.808979i \(0.299979\pi\)
\(90\) 0 0
\(91\) −1.51889 −0.159223
\(92\) −2.46488 −0.256982
\(93\) 0 0
\(94\) −4.44921 −0.458901
\(95\) −0.505966 −0.0519110
\(96\) 0 0
\(97\) −9.66578 −0.981411 −0.490706 0.871325i \(-0.663261\pi\)
−0.490706 + 0.871325i \(0.663261\pi\)
\(98\) −0.691771 −0.0698794
\(99\) 0 0
\(100\) 7.52204 0.752204
\(101\) 13.6540 1.35863 0.679314 0.733848i \(-0.262277\pi\)
0.679314 + 0.733848i \(0.262277\pi\)
\(102\) 0 0
\(103\) 6.04299 0.595434 0.297717 0.954654i \(-0.403775\pi\)
0.297717 + 0.954654i \(0.403775\pi\)
\(104\) −3.70007 −0.362822
\(105\) 0 0
\(106\) −7.75097 −0.752841
\(107\) −6.94638 −0.671532 −0.335766 0.941945i \(-0.608995\pi\)
−0.335766 + 0.941945i \(0.608995\pi\)
\(108\) 0 0
\(109\) −7.77730 −0.744930 −0.372465 0.928046i \(-0.621487\pi\)
−0.372465 + 0.928046i \(0.621487\pi\)
\(110\) −0.774492 −0.0738449
\(111\) 0 0
\(112\) 1.35773 0.128293
\(113\) 5.69588 0.535823 0.267911 0.963444i \(-0.413667\pi\)
0.267911 + 0.963444i \(0.413667\pi\)
\(114\) 0 0
\(115\) −0.383440 −0.0357559
\(116\) 0.732645 0.0680244
\(117\) 0 0
\(118\) 5.31336 0.489134
\(119\) 4.88453 0.447764
\(120\) 0 0
\(121\) 11.3764 1.03422
\(122\) −3.64037 −0.329583
\(123\) 0 0
\(124\) 13.8560 1.24431
\(125\) 2.35353 0.210506
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 10.7201 0.947533
\(129\) 0 0
\(130\) −0.248684 −0.0218110
\(131\) −2.77660 −0.242593 −0.121296 0.992616i \(-0.538705\pi\)
−0.121296 + 0.992616i \(0.538705\pi\)
\(132\) 0 0
\(133\) 2.13777 0.185368
\(134\) 10.4939 0.906538
\(135\) 0 0
\(136\) 11.8989 1.02032
\(137\) −5.85349 −0.500097 −0.250048 0.968233i \(-0.580447\pi\)
−0.250048 + 0.968233i \(0.580447\pi\)
\(138\) 0 0
\(139\) 12.8728 1.09186 0.545929 0.837832i \(-0.316177\pi\)
0.545929 + 0.837832i \(0.316177\pi\)
\(140\) 0.360096 0.0304337
\(141\) 0 0
\(142\) −1.84440 −0.154779
\(143\) 7.18490 0.600832
\(144\) 0 0
\(145\) 0.113971 0.00946478
\(146\) −7.32541 −0.606255
\(147\) 0 0
\(148\) −16.6665 −1.36998
\(149\) 22.2004 1.81873 0.909364 0.416001i \(-0.136569\pi\)
0.909364 + 0.416001i \(0.136569\pi\)
\(150\) 0 0
\(151\) 13.5226 1.10045 0.550225 0.835016i \(-0.314542\pi\)
0.550225 + 0.835016i \(0.314542\pi\)
\(152\) 5.20770 0.422400
\(153\) 0 0
\(154\) 3.27233 0.263692
\(155\) 2.15546 0.173130
\(156\) 0 0
\(157\) 1.03369 0.0824973 0.0412487 0.999149i \(-0.486866\pi\)
0.0412487 + 0.999149i \(0.486866\pi\)
\(158\) 9.09863 0.723848
\(159\) 0 0
\(160\) 1.37542 0.108736
\(161\) 1.62008 0.127680
\(162\) 0 0
\(163\) −21.0280 −1.64704 −0.823519 0.567289i \(-0.807992\pi\)
−0.823519 + 0.567289i \(0.807992\pi\)
\(164\) 7.27391 0.567997
\(165\) 0 0
\(166\) 7.68611 0.596558
\(167\) −18.2548 −1.41260 −0.706300 0.707913i \(-0.749637\pi\)
−0.706300 + 0.707913i \(0.749637\pi\)
\(168\) 0 0
\(169\) −10.6930 −0.822537
\(170\) 0.799732 0.0613366
\(171\) 0 0
\(172\) −1.92738 −0.146961
\(173\) −10.4129 −0.791679 −0.395840 0.918320i \(-0.629546\pi\)
−0.395840 + 0.918320i \(0.629546\pi\)
\(174\) 0 0
\(175\) −4.94398 −0.373730
\(176\) −6.42254 −0.484117
\(177\) 0 0
\(178\) −7.67264 −0.575089
\(179\) −5.46552 −0.408513 −0.204256 0.978917i \(-0.565478\pi\)
−0.204256 + 0.978917i \(0.565478\pi\)
\(180\) 0 0
\(181\) 2.91504 0.216673 0.108337 0.994114i \(-0.465448\pi\)
0.108337 + 0.994114i \(0.465448\pi\)
\(182\) 1.05072 0.0778848
\(183\) 0 0
\(184\) 3.94659 0.290946
\(185\) −2.59267 −0.190617
\(186\) 0 0
\(187\) −23.1056 −1.68965
\(188\) −9.78540 −0.713674
\(189\) 0 0
\(190\) 0.350012 0.0253926
\(191\) 4.76963 0.345118 0.172559 0.984999i \(-0.444796\pi\)
0.172559 + 0.984999i \(0.444796\pi\)
\(192\) 0 0
\(193\) 0.130634 0.00940327 0.00470163 0.999989i \(-0.498503\pi\)
0.00470163 + 0.999989i \(0.498503\pi\)
\(194\) 6.68651 0.480063
\(195\) 0 0
\(196\) −1.52145 −0.108675
\(197\) 19.2367 1.37056 0.685280 0.728279i \(-0.259680\pi\)
0.685280 + 0.728279i \(0.259680\pi\)
\(198\) 0 0
\(199\) −11.8797 −0.842128 −0.421064 0.907031i \(-0.638343\pi\)
−0.421064 + 0.907031i \(0.638343\pi\)
\(200\) −12.0437 −0.851621
\(201\) 0 0
\(202\) −9.44547 −0.664581
\(203\) −0.481543 −0.0337977
\(204\) 0 0
\(205\) 1.13154 0.0790300
\(206\) −4.18037 −0.291260
\(207\) 0 0
\(208\) −2.06223 −0.142990
\(209\) −10.1125 −0.699493
\(210\) 0 0
\(211\) 14.3794 0.989915 0.494958 0.868917i \(-0.335184\pi\)
0.494958 + 0.868917i \(0.335184\pi\)
\(212\) −17.0472 −1.17080
\(213\) 0 0
\(214\) 4.80531 0.328484
\(215\) −0.299825 −0.0204479
\(216\) 0 0
\(217\) −9.10709 −0.618229
\(218\) 5.38011 0.364387
\(219\) 0 0
\(220\) −1.70339 −0.114842
\(221\) −7.41905 −0.499059
\(222\) 0 0
\(223\) −8.11238 −0.543245 −0.271623 0.962404i \(-0.587560\pi\)
−0.271623 + 0.962404i \(0.587560\pi\)
\(224\) −5.81131 −0.388285
\(225\) 0 0
\(226\) −3.94024 −0.262101
\(227\) 24.9231 1.65420 0.827101 0.562054i \(-0.189989\pi\)
0.827101 + 0.562054i \(0.189989\pi\)
\(228\) 0 0
\(229\) −9.85102 −0.650974 −0.325487 0.945547i \(-0.605528\pi\)
−0.325487 + 0.945547i \(0.605528\pi\)
\(230\) 0.265252 0.0174902
\(231\) 0 0
\(232\) −1.17306 −0.0770150
\(233\) −27.7502 −1.81797 −0.908987 0.416823i \(-0.863143\pi\)
−0.908987 + 0.416823i \(0.863143\pi\)
\(234\) 0 0
\(235\) −1.52223 −0.0992992
\(236\) 11.6860 0.760693
\(237\) 0 0
\(238\) −3.37897 −0.219026
\(239\) 0.812448 0.0525529 0.0262765 0.999655i \(-0.491635\pi\)
0.0262765 + 0.999655i \(0.491635\pi\)
\(240\) 0 0
\(241\) −5.39668 −0.347631 −0.173815 0.984778i \(-0.555610\pi\)
−0.173815 + 0.984778i \(0.555610\pi\)
\(242\) −7.86986 −0.505894
\(243\) 0 0
\(244\) −8.00648 −0.512562
\(245\) −0.236679 −0.0151209
\(246\) 0 0
\(247\) −3.24704 −0.206604
\(248\) −22.1852 −1.40876
\(249\) 0 0
\(250\) −1.62810 −0.102970
\(251\) −13.4081 −0.846314 −0.423157 0.906056i \(-0.639078\pi\)
−0.423157 + 0.906056i \(0.639078\pi\)
\(252\) 0 0
\(253\) −7.66359 −0.481806
\(254\) −0.691771 −0.0434056
\(255\) 0 0
\(256\) −10.0252 −0.626572
\(257\) −23.1837 −1.44616 −0.723079 0.690765i \(-0.757274\pi\)
−0.723079 + 0.690765i \(0.757274\pi\)
\(258\) 0 0
\(259\) 10.9544 0.680671
\(260\) −0.546946 −0.0339201
\(261\) 0 0
\(262\) 1.92077 0.118666
\(263\) 29.6493 1.82825 0.914126 0.405430i \(-0.132878\pi\)
0.914126 + 0.405430i \(0.132878\pi\)
\(264\) 0 0
\(265\) −2.65188 −0.162904
\(266\) −1.47885 −0.0906741
\(267\) 0 0
\(268\) 23.0799 1.40983
\(269\) 12.4588 0.759624 0.379812 0.925064i \(-0.375989\pi\)
0.379812 + 0.925064i \(0.375989\pi\)
\(270\) 0 0
\(271\) −7.16036 −0.434961 −0.217481 0.976065i \(-0.569784\pi\)
−0.217481 + 0.976065i \(0.569784\pi\)
\(272\) 6.63185 0.402115
\(273\) 0 0
\(274\) 4.04927 0.244625
\(275\) 23.3869 1.41028
\(276\) 0 0
\(277\) −29.1691 −1.75260 −0.876299 0.481767i \(-0.839995\pi\)
−0.876299 + 0.481767i \(0.839995\pi\)
\(278\) −8.90504 −0.534089
\(279\) 0 0
\(280\) −0.576559 −0.0344560
\(281\) 2.42921 0.144914 0.0724572 0.997372i \(-0.476916\pi\)
0.0724572 + 0.997372i \(0.476916\pi\)
\(282\) 0 0
\(283\) −13.0816 −0.777621 −0.388810 0.921318i \(-0.627114\pi\)
−0.388810 + 0.921318i \(0.627114\pi\)
\(284\) −4.05651 −0.240710
\(285\) 0 0
\(286\) −4.97031 −0.293900
\(287\) −4.78089 −0.282207
\(288\) 0 0
\(289\) 6.85859 0.403447
\(290\) −0.0788419 −0.00462975
\(291\) 0 0
\(292\) −16.1112 −0.942837
\(293\) −11.9508 −0.698174 −0.349087 0.937090i \(-0.613508\pi\)
−0.349087 + 0.937090i \(0.613508\pi\)
\(294\) 0 0
\(295\) 1.81788 0.105841
\(296\) 26.6852 1.55105
\(297\) 0 0
\(298\) −15.3576 −0.889642
\(299\) −2.46073 −0.142307
\(300\) 0 0
\(301\) 1.26680 0.0730172
\(302\) −9.35451 −0.538292
\(303\) 0 0
\(304\) 2.90251 0.166470
\(305\) −1.24550 −0.0713169
\(306\) 0 0
\(307\) −33.3233 −1.90186 −0.950931 0.309402i \(-0.899871\pi\)
−0.950931 + 0.309402i \(0.899871\pi\)
\(308\) 7.19704 0.410089
\(309\) 0 0
\(310\) −1.49108 −0.0846877
\(311\) 17.5249 0.993747 0.496874 0.867823i \(-0.334481\pi\)
0.496874 + 0.867823i \(0.334481\pi\)
\(312\) 0 0
\(313\) −10.4399 −0.590099 −0.295050 0.955482i \(-0.595336\pi\)
−0.295050 + 0.955482i \(0.595336\pi\)
\(314\) −0.715076 −0.0403540
\(315\) 0 0
\(316\) 20.0112 1.12572
\(317\) 17.5619 0.986377 0.493188 0.869923i \(-0.335831\pi\)
0.493188 + 0.869923i \(0.335831\pi\)
\(318\) 0 0
\(319\) 2.27788 0.127537
\(320\) −0.308782 −0.0172614
\(321\) 0 0
\(322\) −1.12073 −0.0624556
\(323\) 10.4420 0.581009
\(324\) 0 0
\(325\) 7.50936 0.416544
\(326\) 14.5465 0.805658
\(327\) 0 0
\(328\) −11.6464 −0.643067
\(329\) 6.43162 0.354587
\(330\) 0 0
\(331\) −23.3554 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(332\) 16.9045 0.927756
\(333\) 0 0
\(334\) 12.6281 0.690981
\(335\) 3.59034 0.196161
\(336\) 0 0
\(337\) −4.90547 −0.267218 −0.133609 0.991034i \(-0.542657\pi\)
−0.133609 + 0.991034i \(0.542657\pi\)
\(338\) 7.39709 0.402349
\(339\) 0 0
\(340\) 1.75890 0.0953896
\(341\) 43.0799 2.33291
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 3.08598 0.166385
\(345\) 0 0
\(346\) 7.20335 0.387255
\(347\) 10.4466 0.560805 0.280403 0.959882i \(-0.409532\pi\)
0.280403 + 0.959882i \(0.409532\pi\)
\(348\) 0 0
\(349\) −19.3334 −1.03489 −0.517446 0.855716i \(-0.673117\pi\)
−0.517446 + 0.855716i \(0.673117\pi\)
\(350\) 3.42010 0.182812
\(351\) 0 0
\(352\) 27.4897 1.46520
\(353\) −4.03886 −0.214967 −0.107483 0.994207i \(-0.534279\pi\)
−0.107483 + 0.994207i \(0.534279\pi\)
\(354\) 0 0
\(355\) −0.631035 −0.0334919
\(356\) −16.8749 −0.894368
\(357\) 0 0
\(358\) 3.78089 0.199826
\(359\) −0.855429 −0.0451478 −0.0225739 0.999745i \(-0.507186\pi\)
−0.0225739 + 0.999745i \(0.507186\pi\)
\(360\) 0 0
\(361\) −14.4299 −0.759470
\(362\) −2.01654 −0.105987
\(363\) 0 0
\(364\) 2.31092 0.121125
\(365\) −2.50628 −0.131185
\(366\) 0 0
\(367\) 24.7046 1.28957 0.644784 0.764365i \(-0.276947\pi\)
0.644784 + 0.764365i \(0.276947\pi\)
\(368\) 2.19963 0.114664
\(369\) 0 0
\(370\) 1.79353 0.0932412
\(371\) 11.2045 0.581711
\(372\) 0 0
\(373\) 0.282770 0.0146413 0.00732065 0.999973i \(-0.497670\pi\)
0.00732065 + 0.999973i \(0.497670\pi\)
\(374\) 15.9838 0.826502
\(375\) 0 0
\(376\) 15.6677 0.807998
\(377\) 0.731410 0.0376695
\(378\) 0 0
\(379\) −12.3555 −0.634657 −0.317329 0.948316i \(-0.602786\pi\)
−0.317329 + 0.948316i \(0.602786\pi\)
\(380\) 0.769803 0.0394901
\(381\) 0 0
\(382\) −3.29949 −0.168817
\(383\) −22.1873 −1.13372 −0.566859 0.823815i \(-0.691841\pi\)
−0.566859 + 0.823815i \(0.691841\pi\)
\(384\) 0 0
\(385\) 1.11958 0.0570590
\(386\) −0.0903691 −0.00459966
\(387\) 0 0
\(388\) 14.7060 0.746586
\(389\) 27.5859 1.39866 0.699331 0.714798i \(-0.253481\pi\)
0.699331 + 0.714798i \(0.253481\pi\)
\(390\) 0 0
\(391\) 7.91334 0.400195
\(392\) 2.43604 0.123039
\(393\) 0 0
\(394\) −13.3074 −0.670418
\(395\) 3.11296 0.156630
\(396\) 0 0
\(397\) −34.4752 −1.73026 −0.865130 0.501547i \(-0.832764\pi\)
−0.865130 + 0.501547i \(0.832764\pi\)
\(398\) 8.21801 0.411932
\(399\) 0 0
\(400\) −6.71257 −0.335629
\(401\) 18.3464 0.916174 0.458087 0.888907i \(-0.348535\pi\)
0.458087 + 0.888907i \(0.348535\pi\)
\(402\) 0 0
\(403\) 13.8327 0.689053
\(404\) −20.7740 −1.03354
\(405\) 0 0
\(406\) 0.333117 0.0165323
\(407\) −51.8182 −2.56853
\(408\) 0 0
\(409\) −11.2920 −0.558355 −0.279178 0.960239i \(-0.590062\pi\)
−0.279178 + 0.960239i \(0.590062\pi\)
\(410\) −0.782764 −0.0386580
\(411\) 0 0
\(412\) −9.19413 −0.452962
\(413\) −7.68081 −0.377948
\(414\) 0 0
\(415\) 2.62969 0.129086
\(416\) 8.82674 0.432766
\(417\) 0 0
\(418\) 6.99550 0.342161
\(419\) −4.64156 −0.226755 −0.113377 0.993552i \(-0.536167\pi\)
−0.113377 + 0.993552i \(0.536167\pi\)
\(420\) 0 0
\(421\) −18.1742 −0.885756 −0.442878 0.896582i \(-0.646042\pi\)
−0.442878 + 0.896582i \(0.646042\pi\)
\(422\) −9.94722 −0.484223
\(423\) 0 0
\(424\) 27.2947 1.32555
\(425\) −24.1490 −1.17140
\(426\) 0 0
\(427\) 5.26239 0.254665
\(428\) 10.5686 0.510852
\(429\) 0 0
\(430\) 0.207410 0.0100022
\(431\) −13.9712 −0.672969 −0.336485 0.941689i \(-0.609238\pi\)
−0.336485 + 0.941689i \(0.609238\pi\)
\(432\) 0 0
\(433\) −1.48040 −0.0711437 −0.0355719 0.999367i \(-0.511325\pi\)
−0.0355719 + 0.999367i \(0.511325\pi\)
\(434\) 6.30002 0.302411
\(435\) 0 0
\(436\) 11.8328 0.566688
\(437\) 3.46337 0.165675
\(438\) 0 0
\(439\) −40.9084 −1.95245 −0.976225 0.216758i \(-0.930452\pi\)
−0.976225 + 0.216758i \(0.930452\pi\)
\(440\) 2.72734 0.130021
\(441\) 0 0
\(442\) 5.13228 0.244118
\(443\) −2.15526 −0.102400 −0.0511998 0.998688i \(-0.516305\pi\)
−0.0511998 + 0.998688i \(0.516305\pi\)
\(444\) 0 0
\(445\) −2.62508 −0.124441
\(446\) 5.61191 0.265732
\(447\) 0 0
\(448\) 1.30465 0.0616387
\(449\) −1.25094 −0.0590356 −0.0295178 0.999564i \(-0.509397\pi\)
−0.0295178 + 0.999564i \(0.509397\pi\)
\(450\) 0 0
\(451\) 22.6154 1.06492
\(452\) −8.66601 −0.407615
\(453\) 0 0
\(454\) −17.2410 −0.809162
\(455\) 0.359489 0.0168531
\(456\) 0 0
\(457\) −9.33789 −0.436808 −0.218404 0.975858i \(-0.570085\pi\)
−0.218404 + 0.975858i \(0.570085\pi\)
\(458\) 6.81465 0.318428
\(459\) 0 0
\(460\) 0.583385 0.0272005
\(461\) 9.40350 0.437965 0.218982 0.975729i \(-0.429726\pi\)
0.218982 + 0.975729i \(0.429726\pi\)
\(462\) 0 0
\(463\) 35.8889 1.66790 0.833950 0.551840i \(-0.186074\pi\)
0.833950 + 0.551840i \(0.186074\pi\)
\(464\) −0.653803 −0.0303520
\(465\) 0 0
\(466\) 19.1968 0.889273
\(467\) 9.27065 0.428994 0.214497 0.976725i \(-0.431189\pi\)
0.214497 + 0.976725i \(0.431189\pi\)
\(468\) 0 0
\(469\) −15.1697 −0.700470
\(470\) 1.05303 0.0485728
\(471\) 0 0
\(472\) −18.7107 −0.861232
\(473\) −5.99244 −0.275533
\(474\) 0 0
\(475\) −10.5691 −0.484944
\(476\) −7.43158 −0.340626
\(477\) 0 0
\(478\) −0.562028 −0.0257066
\(479\) −23.0284 −1.05220 −0.526098 0.850424i \(-0.676345\pi\)
−0.526098 + 0.850424i \(0.676345\pi\)
\(480\) 0 0
\(481\) −16.6385 −0.758648
\(482\) 3.73327 0.170046
\(483\) 0 0
\(484\) −17.3087 −0.786757
\(485\) 2.28769 0.103879
\(486\) 0 0
\(487\) −11.9762 −0.542694 −0.271347 0.962482i \(-0.587469\pi\)
−0.271347 + 0.962482i \(0.587469\pi\)
\(488\) 12.8194 0.580306
\(489\) 0 0
\(490\) 0.163728 0.00739646
\(491\) −42.2730 −1.90775 −0.953877 0.300199i \(-0.902947\pi\)
−0.953877 + 0.300199i \(0.902947\pi\)
\(492\) 0 0
\(493\) −2.35211 −0.105934
\(494\) 2.24621 0.101062
\(495\) 0 0
\(496\) −12.3649 −0.555201
\(497\) 2.66621 0.119596
\(498\) 0 0
\(499\) 3.88448 0.173893 0.0869465 0.996213i \(-0.472289\pi\)
0.0869465 + 0.996213i \(0.472289\pi\)
\(500\) −3.58079 −0.160138
\(501\) 0 0
\(502\) 9.27536 0.413980
\(503\) 7.79134 0.347399 0.173699 0.984799i \(-0.444428\pi\)
0.173699 + 0.984799i \(0.444428\pi\)
\(504\) 0 0
\(505\) −3.23162 −0.143805
\(506\) 5.30145 0.235678
\(507\) 0 0
\(508\) −1.52145 −0.0675036
\(509\) 25.8939 1.14773 0.573864 0.818951i \(-0.305444\pi\)
0.573864 + 0.818951i \(0.305444\pi\)
\(510\) 0 0
\(511\) 10.5894 0.468445
\(512\) −14.5051 −0.641041
\(513\) 0 0
\(514\) 16.0378 0.707397
\(515\) −1.43025 −0.0630243
\(516\) 0 0
\(517\) −30.4239 −1.33804
\(518\) −7.57791 −0.332954
\(519\) 0 0
\(520\) 0.875729 0.0384033
\(521\) −7.34696 −0.321876 −0.160938 0.986964i \(-0.551452\pi\)
−0.160938 + 0.986964i \(0.551452\pi\)
\(522\) 0 0
\(523\) 2.81126 0.122928 0.0614639 0.998109i \(-0.480423\pi\)
0.0614639 + 0.998109i \(0.480423\pi\)
\(524\) 4.22447 0.184547
\(525\) 0 0
\(526\) −20.5105 −0.894300
\(527\) −44.4838 −1.93774
\(528\) 0 0
\(529\) −20.3753 −0.885884
\(530\) 1.83449 0.0796852
\(531\) 0 0
\(532\) −3.25252 −0.141015
\(533\) 7.26165 0.314537
\(534\) 0 0
\(535\) 1.64406 0.0710790
\(536\) −36.9539 −1.59616
\(537\) 0 0
\(538\) −8.61860 −0.371574
\(539\) −4.73037 −0.203751
\(540\) 0 0
\(541\) 4.76871 0.205023 0.102511 0.994732i \(-0.467312\pi\)
0.102511 + 0.994732i \(0.467312\pi\)
\(542\) 4.95333 0.212764
\(543\) 0 0
\(544\) −28.3855 −1.21702
\(545\) 1.84072 0.0788479
\(546\) 0 0
\(547\) 4.47547 0.191357 0.0956786 0.995412i \(-0.469498\pi\)
0.0956786 + 0.995412i \(0.469498\pi\)
\(548\) 8.90580 0.380437
\(549\) 0 0
\(550\) −16.1784 −0.689847
\(551\) −1.02943 −0.0438552
\(552\) 0 0
\(553\) −13.1527 −0.559308
\(554\) 20.1783 0.857294
\(555\) 0 0
\(556\) −19.5854 −0.830605
\(557\) −30.3570 −1.28627 −0.643135 0.765753i \(-0.722366\pi\)
−0.643135 + 0.765753i \(0.722366\pi\)
\(558\) 0 0
\(559\) −1.92413 −0.0813820
\(560\) −0.321345 −0.0135793
\(561\) 0 0
\(562\) −1.68046 −0.0708858
\(563\) −40.3405 −1.70015 −0.850076 0.526661i \(-0.823444\pi\)
−0.850076 + 0.526661i \(0.823444\pi\)
\(564\) 0 0
\(565\) −1.34809 −0.0567147
\(566\) 9.04947 0.380378
\(567\) 0 0
\(568\) 6.49499 0.272524
\(569\) 1.96447 0.0823550 0.0411775 0.999152i \(-0.486889\pi\)
0.0411775 + 0.999152i \(0.486889\pi\)
\(570\) 0 0
\(571\) 17.7620 0.743315 0.371658 0.928370i \(-0.378790\pi\)
0.371658 + 0.928370i \(0.378790\pi\)
\(572\) −10.9315 −0.457069
\(573\) 0 0
\(574\) 3.30728 0.138043
\(575\) −8.00966 −0.334026
\(576\) 0 0
\(577\) 8.06360 0.335692 0.167846 0.985813i \(-0.446319\pi\)
0.167846 + 0.985813i \(0.446319\pi\)
\(578\) −4.74457 −0.197348
\(579\) 0 0
\(580\) −0.173402 −0.00720011
\(581\) −11.1108 −0.460953
\(582\) 0 0
\(583\) −53.0016 −2.19510
\(584\) 25.7961 1.06745
\(585\) 0 0
\(586\) 8.26723 0.341516
\(587\) 2.82469 0.116588 0.0582938 0.998299i \(-0.481434\pi\)
0.0582938 + 0.998299i \(0.481434\pi\)
\(588\) 0 0
\(589\) −19.4689 −0.802202
\(590\) −1.25756 −0.0517729
\(591\) 0 0
\(592\) 14.8730 0.611277
\(593\) 25.7116 1.05585 0.527925 0.849291i \(-0.322970\pi\)
0.527925 + 0.849291i \(0.322970\pi\)
\(594\) 0 0
\(595\) −1.15606 −0.0473940
\(596\) −33.7769 −1.38355
\(597\) 0 0
\(598\) 1.70226 0.0696105
\(599\) 34.6917 1.41746 0.708732 0.705478i \(-0.249268\pi\)
0.708732 + 0.705478i \(0.249268\pi\)
\(600\) 0 0
\(601\) 12.8762 0.525230 0.262615 0.964901i \(-0.415415\pi\)
0.262615 + 0.964901i \(0.415415\pi\)
\(602\) −0.876336 −0.0357168
\(603\) 0 0
\(604\) −20.5739 −0.837142
\(605\) −2.69255 −0.109468
\(606\) 0 0
\(607\) 25.8987 1.05120 0.525599 0.850733i \(-0.323841\pi\)
0.525599 + 0.850733i \(0.323841\pi\)
\(608\) −12.4233 −0.503830
\(609\) 0 0
\(610\) 0.861598 0.0348851
\(611\) −9.76891 −0.395208
\(612\) 0 0
\(613\) −36.6547 −1.48047 −0.740234 0.672349i \(-0.765285\pi\)
−0.740234 + 0.672349i \(0.765285\pi\)
\(614\) 23.0521 0.930307
\(615\) 0 0
\(616\) −11.5234 −0.464290
\(617\) −49.5324 −1.99410 −0.997050 0.0767545i \(-0.975544\pi\)
−0.997050 + 0.0767545i \(0.975544\pi\)
\(618\) 0 0
\(619\) −43.3703 −1.74320 −0.871599 0.490219i \(-0.836917\pi\)
−0.871599 + 0.490219i \(0.836917\pi\)
\(620\) −3.27942 −0.131705
\(621\) 0 0
\(622\) −12.1232 −0.486097
\(623\) 11.0913 0.444364
\(624\) 0 0
\(625\) 24.1629 0.966515
\(626\) 7.22203 0.288651
\(627\) 0 0
\(628\) −1.57271 −0.0627579
\(629\) 53.5069 2.13346
\(630\) 0 0
\(631\) −31.7213 −1.26280 −0.631402 0.775455i \(-0.717520\pi\)
−0.631402 + 0.775455i \(0.717520\pi\)
\(632\) −32.0404 −1.27450
\(633\) 0 0
\(634\) −12.1488 −0.482492
\(635\) −0.236679 −0.00939232
\(636\) 0 0
\(637\) −1.51889 −0.0601806
\(638\) −1.57577 −0.0623853
\(639\) 0 0
\(640\) −2.53722 −0.100293
\(641\) 6.12263 0.241829 0.120915 0.992663i \(-0.461417\pi\)
0.120915 + 0.992663i \(0.461417\pi\)
\(642\) 0 0
\(643\) −1.36807 −0.0539513 −0.0269757 0.999636i \(-0.508588\pi\)
−0.0269757 + 0.999636i \(0.508588\pi\)
\(644\) −2.46488 −0.0971299
\(645\) 0 0
\(646\) −7.22348 −0.284204
\(647\) −23.2875 −0.915525 −0.457762 0.889075i \(-0.651349\pi\)
−0.457762 + 0.889075i \(0.651349\pi\)
\(648\) 0 0
\(649\) 36.3330 1.42620
\(650\) −5.19476 −0.203755
\(651\) 0 0
\(652\) 31.9931 1.25295
\(653\) −26.1315 −1.02260 −0.511302 0.859401i \(-0.670836\pi\)
−0.511302 + 0.859401i \(0.670836\pi\)
\(654\) 0 0
\(655\) 0.657163 0.0256775
\(656\) −6.49114 −0.253436
\(657\) 0 0
\(658\) −4.44921 −0.173448
\(659\) −10.1531 −0.395510 −0.197755 0.980252i \(-0.563365\pi\)
−0.197755 + 0.980252i \(0.563365\pi\)
\(660\) 0 0
\(661\) 2.42067 0.0941533 0.0470766 0.998891i \(-0.485009\pi\)
0.0470766 + 0.998891i \(0.485009\pi\)
\(662\) 16.1566 0.627944
\(663\) 0 0
\(664\) −27.0663 −1.05038
\(665\) −0.505966 −0.0196205
\(666\) 0 0
\(667\) −0.780140 −0.0302071
\(668\) 27.7738 1.07460
\(669\) 0 0
\(670\) −2.48369 −0.0959534
\(671\) −24.8930 −0.960985
\(672\) 0 0
\(673\) −26.0790 −1.00527 −0.502636 0.864498i \(-0.667636\pi\)
−0.502636 + 0.864498i \(0.667636\pi\)
\(674\) 3.39346 0.130711
\(675\) 0 0
\(676\) 16.2689 0.625725
\(677\) −45.3731 −1.74383 −0.871915 0.489657i \(-0.837122\pi\)
−0.871915 + 0.489657i \(0.837122\pi\)
\(678\) 0 0
\(679\) −9.66578 −0.370939
\(680\) −2.81622 −0.107997
\(681\) 0 0
\(682\) −29.8014 −1.14116
\(683\) 11.9507 0.457282 0.228641 0.973511i \(-0.426572\pi\)
0.228641 + 0.973511i \(0.426572\pi\)
\(684\) 0 0
\(685\) 1.38540 0.0529333
\(686\) −0.691771 −0.0264119
\(687\) 0 0
\(688\) 1.71997 0.0655732
\(689\) −17.0184 −0.648351
\(690\) 0 0
\(691\) −11.9466 −0.454469 −0.227235 0.973840i \(-0.572968\pi\)
−0.227235 + 0.973840i \(0.572968\pi\)
\(692\) 15.8428 0.602251
\(693\) 0 0
\(694\) −7.22669 −0.274321
\(695\) −3.04672 −0.115569
\(696\) 0 0
\(697\) −23.3524 −0.884535
\(698\) 13.3743 0.506223
\(699\) 0 0
\(700\) 7.52204 0.284306
\(701\) −11.7682 −0.444478 −0.222239 0.974992i \(-0.571337\pi\)
−0.222239 + 0.974992i \(0.571337\pi\)
\(702\) 0 0
\(703\) 23.4179 0.883224
\(704\) −6.17146 −0.232596
\(705\) 0 0
\(706\) 2.79396 0.105152
\(707\) 13.6540 0.513513
\(708\) 0 0
\(709\) −21.7838 −0.818108 −0.409054 0.912510i \(-0.634141\pi\)
−0.409054 + 0.912510i \(0.634141\pi\)
\(710\) 0.436532 0.0163827
\(711\) 0 0
\(712\) 27.0189 1.01257
\(713\) −14.7542 −0.552551
\(714\) 0 0
\(715\) −1.70052 −0.0635957
\(716\) 8.31554 0.310766
\(717\) 0 0
\(718\) 0.591761 0.0220843
\(719\) −49.2724 −1.83755 −0.918774 0.394783i \(-0.870820\pi\)
−0.918774 + 0.394783i \(0.870820\pi\)
\(720\) 0 0
\(721\) 6.04299 0.225053
\(722\) 9.98220 0.371499
\(723\) 0 0
\(724\) −4.43510 −0.164829
\(725\) 2.38074 0.0884185
\(726\) 0 0
\(727\) 12.7353 0.472327 0.236164 0.971713i \(-0.424110\pi\)
0.236164 + 0.971713i \(0.424110\pi\)
\(728\) −3.70007 −0.137134
\(729\) 0 0
\(730\) 1.73377 0.0641697
\(731\) 6.18772 0.228861
\(732\) 0 0
\(733\) 22.6644 0.837130 0.418565 0.908187i \(-0.362533\pi\)
0.418565 + 0.908187i \(0.362533\pi\)
\(734\) −17.0899 −0.630799
\(735\) 0 0
\(736\) −9.41481 −0.347035
\(737\) 71.7581 2.64324
\(738\) 0 0
\(739\) −7.56804 −0.278395 −0.139197 0.990265i \(-0.544452\pi\)
−0.139197 + 0.990265i \(0.544452\pi\)
\(740\) 3.94462 0.145007
\(741\) 0 0
\(742\) −7.75097 −0.284547
\(743\) 2.44813 0.0898132 0.0449066 0.998991i \(-0.485701\pi\)
0.0449066 + 0.998991i \(0.485701\pi\)
\(744\) 0 0
\(745\) −5.25437 −0.192505
\(746\) −0.195612 −0.00716188
\(747\) 0 0
\(748\) 35.1541 1.28536
\(749\) −6.94638 −0.253815
\(750\) 0 0
\(751\) 36.3013 1.32465 0.662327 0.749215i \(-0.269569\pi\)
0.662327 + 0.749215i \(0.269569\pi\)
\(752\) 8.73237 0.318437
\(753\) 0 0
\(754\) −0.505968 −0.0184263
\(755\) −3.20050 −0.116478
\(756\) 0 0
\(757\) 19.0015 0.690620 0.345310 0.938489i \(-0.387774\pi\)
0.345310 + 0.938489i \(0.387774\pi\)
\(758\) 8.54714 0.310446
\(759\) 0 0
\(760\) −1.23255 −0.0447094
\(761\) −11.8346 −0.429005 −0.214503 0.976723i \(-0.568813\pi\)
−0.214503 + 0.976723i \(0.568813\pi\)
\(762\) 0 0
\(763\) −7.77730 −0.281557
\(764\) −7.25677 −0.262541
\(765\) 0 0
\(766\) 15.3485 0.554564
\(767\) 11.6663 0.421245
\(768\) 0 0
\(769\) 12.9124 0.465634 0.232817 0.972521i \(-0.425206\pi\)
0.232817 + 0.972521i \(0.425206\pi\)
\(770\) −0.774492 −0.0279108
\(771\) 0 0
\(772\) −0.198754 −0.00715332
\(773\) −44.8628 −1.61360 −0.806801 0.590823i \(-0.798803\pi\)
−0.806801 + 0.590823i \(0.798803\pi\)
\(774\) 0 0
\(775\) 45.0253 1.61736
\(776\) −23.5462 −0.845260
\(777\) 0 0
\(778\) −19.0832 −0.684164
\(779\) −10.2205 −0.366186
\(780\) 0 0
\(781\) −12.6121 −0.451298
\(782\) −5.47422 −0.195758
\(783\) 0 0
\(784\) 1.35773 0.0484902
\(785\) −0.244652 −0.00873201
\(786\) 0 0
\(787\) 25.5191 0.909658 0.454829 0.890579i \(-0.349700\pi\)
0.454829 + 0.890579i \(0.349700\pi\)
\(788\) −29.2678 −1.04262
\(789\) 0 0
\(790\) −2.15345 −0.0766165
\(791\) 5.69588 0.202522
\(792\) 0 0
\(793\) −7.99298 −0.283839
\(794\) 23.8489 0.846367
\(795\) 0 0
\(796\) 18.0744 0.640629
\(797\) 35.4729 1.25651 0.628257 0.778006i \(-0.283769\pi\)
0.628257 + 0.778006i \(0.283769\pi\)
\(798\) 0 0
\(799\) 31.4154 1.11140
\(800\) 28.7310 1.01580
\(801\) 0 0
\(802\) −12.6915 −0.448152
\(803\) −50.0916 −1.76769
\(804\) 0 0
\(805\) −0.383440 −0.0135145
\(806\) −9.56903 −0.337055
\(807\) 0 0
\(808\) 33.2618 1.17015
\(809\) −39.9982 −1.40626 −0.703131 0.711061i \(-0.748215\pi\)
−0.703131 + 0.711061i \(0.748215\pi\)
\(810\) 0 0
\(811\) −21.3383 −0.749287 −0.374644 0.927169i \(-0.622235\pi\)
−0.374644 + 0.927169i \(0.622235\pi\)
\(812\) 0.732645 0.0257108
\(813\) 0 0
\(814\) 35.8463 1.25641
\(815\) 4.97688 0.174332
\(816\) 0 0
\(817\) 2.70813 0.0947456
\(818\) 7.81150 0.273123
\(819\) 0 0
\(820\) −1.72158 −0.0601202
\(821\) −0.612120 −0.0213631 −0.0106816 0.999943i \(-0.503400\pi\)
−0.0106816 + 0.999943i \(0.503400\pi\)
\(822\) 0 0
\(823\) −37.7565 −1.31611 −0.658055 0.752970i \(-0.728621\pi\)
−0.658055 + 0.752970i \(0.728621\pi\)
\(824\) 14.7210 0.512829
\(825\) 0 0
\(826\) 5.31336 0.184875
\(827\) −38.2190 −1.32900 −0.664502 0.747287i \(-0.731356\pi\)
−0.664502 + 0.747287i \(0.731356\pi\)
\(828\) 0 0
\(829\) −23.0302 −0.799871 −0.399935 0.916543i \(-0.630967\pi\)
−0.399935 + 0.916543i \(0.630967\pi\)
\(830\) −1.81914 −0.0631433
\(831\) 0 0
\(832\) −1.98161 −0.0687000
\(833\) 4.88453 0.169239
\(834\) 0 0
\(835\) 4.32053 0.149518
\(836\) 15.3856 0.532123
\(837\) 0 0
\(838\) 3.21089 0.110919
\(839\) 34.5464 1.19268 0.596338 0.802734i \(-0.296622\pi\)
0.596338 + 0.802734i \(0.296622\pi\)
\(840\) 0 0
\(841\) −28.7681 −0.992004
\(842\) 12.5724 0.433273
\(843\) 0 0
\(844\) −21.8775 −0.753055
\(845\) 2.53080 0.0870623
\(846\) 0 0
\(847\) 11.3764 0.390898
\(848\) 15.2127 0.522406
\(849\) 0 0
\(850\) 16.7056 0.572997
\(851\) 17.7470 0.608359
\(852\) 0 0
\(853\) 34.0512 1.16589 0.582946 0.812511i \(-0.301900\pi\)
0.582946 + 0.812511i \(0.301900\pi\)
\(854\) −3.64037 −0.124571
\(855\) 0 0
\(856\) −16.9217 −0.578370
\(857\) −33.3224 −1.13827 −0.569136 0.822243i \(-0.692722\pi\)
−0.569136 + 0.822243i \(0.692722\pi\)
\(858\) 0 0
\(859\) 34.3511 1.17204 0.586022 0.810295i \(-0.300693\pi\)
0.586022 + 0.810295i \(0.300693\pi\)
\(860\) 0.456170 0.0155553
\(861\) 0 0
\(862\) 9.66487 0.329187
\(863\) −0.252328 −0.00858935 −0.00429467 0.999991i \(-0.501367\pi\)
−0.00429467 + 0.999991i \(0.501367\pi\)
\(864\) 0 0
\(865\) 2.46452 0.0837961
\(866\) 1.02410 0.0348004
\(867\) 0 0
\(868\) 13.8560 0.470303
\(869\) 62.2170 2.11057
\(870\) 0 0
\(871\) 23.0410 0.780715
\(872\) −18.9458 −0.641586
\(873\) 0 0
\(874\) −2.39586 −0.0810411
\(875\) 2.35353 0.0795639
\(876\) 0 0
\(877\) 38.6171 1.30401 0.652004 0.758216i \(-0.273929\pi\)
0.652004 + 0.758216i \(0.273929\pi\)
\(878\) 28.2992 0.955053
\(879\) 0 0
\(880\) 1.52008 0.0512419
\(881\) 0.798968 0.0269179 0.0134590 0.999909i \(-0.495716\pi\)
0.0134590 + 0.999909i \(0.495716\pi\)
\(882\) 0 0
\(883\) −8.06536 −0.271421 −0.135711 0.990749i \(-0.543332\pi\)
−0.135711 + 0.990749i \(0.543332\pi\)
\(884\) 11.2877 0.379648
\(885\) 0 0
\(886\) 1.49095 0.0500894
\(887\) −16.3375 −0.548559 −0.274280 0.961650i \(-0.588439\pi\)
−0.274280 + 0.961650i \(0.588439\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 1.81595 0.0608709
\(891\) 0 0
\(892\) 12.3426 0.413261
\(893\) 13.7493 0.460104
\(894\) 0 0
\(895\) 1.29357 0.0432394
\(896\) 10.7201 0.358134
\(897\) 0 0
\(898\) 0.865365 0.0288776
\(899\) 4.38545 0.146263
\(900\) 0 0
\(901\) 54.7289 1.82328
\(902\) −15.6447 −0.520911
\(903\) 0 0
\(904\) 13.8754 0.461488
\(905\) −0.689928 −0.0229340
\(906\) 0 0
\(907\) −9.06870 −0.301121 −0.150561 0.988601i \(-0.548108\pi\)
−0.150561 + 0.988601i \(0.548108\pi\)
\(908\) −37.9193 −1.25839
\(909\) 0 0
\(910\) −0.248684 −0.00824379
\(911\) −10.9544 −0.362936 −0.181468 0.983397i \(-0.558085\pi\)
−0.181468 + 0.983397i \(0.558085\pi\)
\(912\) 0 0
\(913\) 52.5581 1.73942
\(914\) 6.45968 0.213667
\(915\) 0 0
\(916\) 14.9879 0.495213
\(917\) −2.77660 −0.0916914
\(918\) 0 0
\(919\) −3.67443 −0.121208 −0.0606041 0.998162i \(-0.519303\pi\)
−0.0606041 + 0.998162i \(0.519303\pi\)
\(920\) −0.934074 −0.0307955
\(921\) 0 0
\(922\) −6.50507 −0.214233
\(923\) −4.04967 −0.133297
\(924\) 0 0
\(925\) −54.1582 −1.78071
\(926\) −24.8269 −0.815863
\(927\) 0 0
\(928\) 2.79840 0.0918619
\(929\) −44.8141 −1.47030 −0.735151 0.677903i \(-0.762889\pi\)
−0.735151 + 0.677903i \(0.762889\pi\)
\(930\) 0 0
\(931\) 2.13777 0.0700627
\(932\) 42.2206 1.38298
\(933\) 0 0
\(934\) −6.41316 −0.209845
\(935\) 5.46861 0.178843
\(936\) 0 0
\(937\) 7.40715 0.241981 0.120990 0.992654i \(-0.461393\pi\)
0.120990 + 0.992654i \(0.461393\pi\)
\(938\) 10.4939 0.342639
\(939\) 0 0
\(940\) 2.31600 0.0755396
\(941\) −50.7499 −1.65440 −0.827199 0.561909i \(-0.810067\pi\)
−0.827199 + 0.561909i \(0.810067\pi\)
\(942\) 0 0
\(943\) −7.74545 −0.252226
\(944\) −10.4284 −0.339416
\(945\) 0 0
\(946\) 4.14539 0.134778
\(947\) −40.3285 −1.31050 −0.655250 0.755412i \(-0.727437\pi\)
−0.655250 + 0.755412i \(0.727437\pi\)
\(948\) 0 0
\(949\) −16.0840 −0.522110
\(950\) 7.31141 0.237213
\(951\) 0 0
\(952\) 11.8989 0.385645
\(953\) 3.51760 0.113946 0.0569731 0.998376i \(-0.481855\pi\)
0.0569731 + 0.998376i \(0.481855\pi\)
\(954\) 0 0
\(955\) −1.12887 −0.0365294
\(956\) −1.23610 −0.0399784
\(957\) 0 0
\(958\) 15.9304 0.514688
\(959\) −5.85349 −0.189019
\(960\) 0 0
\(961\) 51.9390 1.67545
\(962\) 11.5100 0.371097
\(963\) 0 0
\(964\) 8.21080 0.264452
\(965\) −0.0309184 −0.000995299 0
\(966\) 0 0
\(967\) −45.1545 −1.45207 −0.726035 0.687658i \(-0.758639\pi\)
−0.726035 + 0.687658i \(0.758639\pi\)
\(968\) 27.7133 0.890741
\(969\) 0 0
\(970\) −1.58256 −0.0508128
\(971\) 38.2249 1.22669 0.613347 0.789813i \(-0.289823\pi\)
0.613347 + 0.789813i \(0.289823\pi\)
\(972\) 0 0
\(973\) 12.8728 0.412683
\(974\) 8.28480 0.265462
\(975\) 0 0
\(976\) 7.14488 0.228702
\(977\) 32.0622 1.02576 0.512880 0.858460i \(-0.328579\pi\)
0.512880 + 0.858460i \(0.328579\pi\)
\(978\) 0 0
\(979\) −52.4660 −1.67682
\(980\) 0.360096 0.0115028
\(981\) 0 0
\(982\) 29.2432 0.933189
\(983\) −31.8590 −1.01614 −0.508072 0.861315i \(-0.669642\pi\)
−0.508072 + 0.861315i \(0.669642\pi\)
\(984\) 0 0
\(985\) −4.55293 −0.145068
\(986\) 1.62712 0.0518181
\(987\) 0 0
\(988\) 4.94022 0.157169
\(989\) 2.05232 0.0652601
\(990\) 0 0
\(991\) 24.9795 0.793500 0.396750 0.917927i \(-0.370138\pi\)
0.396750 + 0.917927i \(0.370138\pi\)
\(992\) 52.9241 1.68034
\(993\) 0 0
\(994\) −1.84440 −0.0585010
\(995\) 2.81167 0.0891359
\(996\) 0 0
\(997\) 34.0534 1.07848 0.539241 0.842151i \(-0.318711\pi\)
0.539241 + 0.842151i \(0.318711\pi\)
\(998\) −2.68717 −0.0850608
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.x.1.8 22
3.2 odd 2 inner 8001.2.a.x.1.15 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.x.1.8 22 1.1 even 1 trivial
8001.2.a.x.1.15 yes 22 3.2 odd 2 inner