Properties

Label 8001.2.a.s.1.15
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} - 298 x^{8} - 5422 x^{7} + 2075 x^{6} + 5385 x^{5} - 3163 x^{4} - 1882 x^{3} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-2.35145\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.35145 q^{2} +3.52933 q^{4} +2.01529 q^{5} -1.00000 q^{7} +3.59614 q^{8} +O(q^{10})\) \(q+2.35145 q^{2} +3.52933 q^{4} +2.01529 q^{5} -1.00000 q^{7} +3.59614 q^{8} +4.73886 q^{10} -0.0911513 q^{11} -2.36528 q^{13} -2.35145 q^{14} +1.39749 q^{16} +4.89667 q^{17} -0.257348 q^{19} +7.11262 q^{20} -0.214338 q^{22} -1.91105 q^{23} -0.938597 q^{25} -5.56184 q^{26} -3.52933 q^{28} +5.21144 q^{29} +2.91024 q^{31} -3.90614 q^{32} +11.5143 q^{34} -2.01529 q^{35} +7.61218 q^{37} -0.605141 q^{38} +7.24727 q^{40} +4.73944 q^{41} +9.38103 q^{43} -0.321703 q^{44} -4.49374 q^{46} +1.49159 q^{47} +1.00000 q^{49} -2.20707 q^{50} -8.34785 q^{52} +11.8399 q^{53} -0.183697 q^{55} -3.59614 q^{56} +12.2544 q^{58} +5.10974 q^{59} +7.51881 q^{61} +6.84329 q^{62} -11.9801 q^{64} -4.76673 q^{65} +6.09291 q^{67} +17.2819 q^{68} -4.73886 q^{70} +5.28553 q^{71} +2.70061 q^{73} +17.8997 q^{74} -0.908264 q^{76} +0.0911513 q^{77} +15.8809 q^{79} +2.81635 q^{80} +11.1446 q^{82} -12.3986 q^{83} +9.86822 q^{85} +22.0590 q^{86} -0.327793 q^{88} -2.65998 q^{89} +2.36528 q^{91} -6.74472 q^{92} +3.50740 q^{94} -0.518631 q^{95} -14.3630 q^{97} +2.35145 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35145 1.66273 0.831364 0.555729i \(-0.187561\pi\)
0.831364 + 0.555729i \(0.187561\pi\)
\(3\) 0 0
\(4\) 3.52933 1.76466
\(5\) 2.01529 0.901266 0.450633 0.892709i \(-0.351198\pi\)
0.450633 + 0.892709i \(0.351198\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.59614 1.27143
\(9\) 0 0
\(10\) 4.73886 1.49856
\(11\) −0.0911513 −0.0274832 −0.0137416 0.999906i \(-0.504374\pi\)
−0.0137416 + 0.999906i \(0.504374\pi\)
\(12\) 0 0
\(13\) −2.36528 −0.656011 −0.328005 0.944676i \(-0.606376\pi\)
−0.328005 + 0.944676i \(0.606376\pi\)
\(14\) −2.35145 −0.628452
\(15\) 0 0
\(16\) 1.39749 0.349373
\(17\) 4.89667 1.18762 0.593808 0.804607i \(-0.297624\pi\)
0.593808 + 0.804607i \(0.297624\pi\)
\(18\) 0 0
\(19\) −0.257348 −0.0590396 −0.0295198 0.999564i \(-0.509398\pi\)
−0.0295198 + 0.999564i \(0.509398\pi\)
\(20\) 7.11262 1.59043
\(21\) 0 0
\(22\) −0.214338 −0.0456970
\(23\) −1.91105 −0.398482 −0.199241 0.979951i \(-0.563848\pi\)
−0.199241 + 0.979951i \(0.563848\pi\)
\(24\) 0 0
\(25\) −0.938597 −0.187719
\(26\) −5.56184 −1.09077
\(27\) 0 0
\(28\) −3.52933 −0.666980
\(29\) 5.21144 0.967740 0.483870 0.875140i \(-0.339231\pi\)
0.483870 + 0.875140i \(0.339231\pi\)
\(30\) 0 0
\(31\) 2.91024 0.522694 0.261347 0.965245i \(-0.415833\pi\)
0.261347 + 0.965245i \(0.415833\pi\)
\(32\) −3.90614 −0.690514
\(33\) 0 0
\(34\) 11.5143 1.97468
\(35\) −2.01529 −0.340647
\(36\) 0 0
\(37\) 7.61218 1.25143 0.625717 0.780050i \(-0.284806\pi\)
0.625717 + 0.780050i \(0.284806\pi\)
\(38\) −0.605141 −0.0981668
\(39\) 0 0
\(40\) 7.24727 1.14589
\(41\) 4.73944 0.740177 0.370088 0.928997i \(-0.379327\pi\)
0.370088 + 0.928997i \(0.379327\pi\)
\(42\) 0 0
\(43\) 9.38103 1.43059 0.715297 0.698821i \(-0.246291\pi\)
0.715297 + 0.698821i \(0.246291\pi\)
\(44\) −0.321703 −0.0484985
\(45\) 0 0
\(46\) −4.49374 −0.662566
\(47\) 1.49159 0.217571 0.108785 0.994065i \(-0.465304\pi\)
0.108785 + 0.994065i \(0.465304\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.20707 −0.312126
\(51\) 0 0
\(52\) −8.34785 −1.15764
\(53\) 11.8399 1.62634 0.813171 0.582025i \(-0.197740\pi\)
0.813171 + 0.582025i \(0.197740\pi\)
\(54\) 0 0
\(55\) −0.183697 −0.0247696
\(56\) −3.59614 −0.480554
\(57\) 0 0
\(58\) 12.2544 1.60909
\(59\) 5.10974 0.665231 0.332616 0.943063i \(-0.392069\pi\)
0.332616 + 0.943063i \(0.392069\pi\)
\(60\) 0 0
\(61\) 7.51881 0.962686 0.481343 0.876532i \(-0.340149\pi\)
0.481343 + 0.876532i \(0.340149\pi\)
\(62\) 6.84329 0.869098
\(63\) 0 0
\(64\) −11.9801 −1.49751
\(65\) −4.76673 −0.591240
\(66\) 0 0
\(67\) 6.09291 0.744368 0.372184 0.928159i \(-0.378609\pi\)
0.372184 + 0.928159i \(0.378609\pi\)
\(68\) 17.2819 2.09574
\(69\) 0 0
\(70\) −4.73886 −0.566402
\(71\) 5.28553 0.627277 0.313639 0.949542i \(-0.398452\pi\)
0.313639 + 0.949542i \(0.398452\pi\)
\(72\) 0 0
\(73\) 2.70061 0.316082 0.158041 0.987433i \(-0.449482\pi\)
0.158041 + 0.987433i \(0.449482\pi\)
\(74\) 17.8997 2.08079
\(75\) 0 0
\(76\) −0.908264 −0.104185
\(77\) 0.0911513 0.0103877
\(78\) 0 0
\(79\) 15.8809 1.78674 0.893370 0.449322i \(-0.148335\pi\)
0.893370 + 0.449322i \(0.148335\pi\)
\(80\) 2.81635 0.314878
\(81\) 0 0
\(82\) 11.1446 1.23071
\(83\) −12.3986 −1.36092 −0.680462 0.732783i \(-0.738221\pi\)
−0.680462 + 0.732783i \(0.738221\pi\)
\(84\) 0 0
\(85\) 9.86822 1.07036
\(86\) 22.0590 2.37869
\(87\) 0 0
\(88\) −0.327793 −0.0349428
\(89\) −2.65998 −0.281958 −0.140979 0.990013i \(-0.545025\pi\)
−0.140979 + 0.990013i \(0.545025\pi\)
\(90\) 0 0
\(91\) 2.36528 0.247949
\(92\) −6.74472 −0.703186
\(93\) 0 0
\(94\) 3.50740 0.361761
\(95\) −0.518631 −0.0532104
\(96\) 0 0
\(97\) −14.3630 −1.45834 −0.729170 0.684332i \(-0.760094\pi\)
−0.729170 + 0.684332i \(0.760094\pi\)
\(98\) 2.35145 0.237533
\(99\) 0 0
\(100\) −3.31262 −0.331262
\(101\) 3.72951 0.371100 0.185550 0.982635i \(-0.440593\pi\)
0.185550 + 0.982635i \(0.440593\pi\)
\(102\) 0 0
\(103\) 10.8352 1.06762 0.533812 0.845603i \(-0.320759\pi\)
0.533812 + 0.845603i \(0.320759\pi\)
\(104\) −8.50587 −0.834070
\(105\) 0 0
\(106\) 27.8411 2.70416
\(107\) −17.2741 −1.66995 −0.834976 0.550287i \(-0.814518\pi\)
−0.834976 + 0.550287i \(0.814518\pi\)
\(108\) 0 0
\(109\) −10.5617 −1.01163 −0.505814 0.862643i \(-0.668808\pi\)
−0.505814 + 0.862643i \(0.668808\pi\)
\(110\) −0.431954 −0.0411852
\(111\) 0 0
\(112\) −1.39749 −0.132051
\(113\) −12.0474 −1.13333 −0.566663 0.823949i \(-0.691766\pi\)
−0.566663 + 0.823949i \(0.691766\pi\)
\(114\) 0 0
\(115\) −3.85132 −0.359138
\(116\) 18.3929 1.70773
\(117\) 0 0
\(118\) 12.0153 1.10610
\(119\) −4.89667 −0.448877
\(120\) 0 0
\(121\) −10.9917 −0.999245
\(122\) 17.6801 1.60068
\(123\) 0 0
\(124\) 10.2712 0.922379
\(125\) −11.9680 −1.07045
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −20.3583 −1.79944
\(129\) 0 0
\(130\) −11.2087 −0.983072
\(131\) 20.8416 1.82094 0.910468 0.413579i \(-0.135721\pi\)
0.910468 + 0.413579i \(0.135721\pi\)
\(132\) 0 0
\(133\) 0.257348 0.0223149
\(134\) 14.3272 1.23768
\(135\) 0 0
\(136\) 17.6091 1.50997
\(137\) −4.51432 −0.385685 −0.192842 0.981230i \(-0.561771\pi\)
−0.192842 + 0.981230i \(0.561771\pi\)
\(138\) 0 0
\(139\) 5.83048 0.494535 0.247267 0.968947i \(-0.420467\pi\)
0.247267 + 0.968947i \(0.420467\pi\)
\(140\) −7.11262 −0.601126
\(141\) 0 0
\(142\) 12.4287 1.04299
\(143\) 0.215598 0.0180292
\(144\) 0 0
\(145\) 10.5026 0.872191
\(146\) 6.35035 0.525559
\(147\) 0 0
\(148\) 26.8659 2.20836
\(149\) −8.09500 −0.663168 −0.331584 0.943426i \(-0.607583\pi\)
−0.331584 + 0.943426i \(0.607583\pi\)
\(150\) 0 0
\(151\) 8.34672 0.679247 0.339623 0.940562i \(-0.389700\pi\)
0.339623 + 0.940562i \(0.389700\pi\)
\(152\) −0.925458 −0.0750645
\(153\) 0 0
\(154\) 0.214338 0.0172718
\(155\) 5.86498 0.471087
\(156\) 0 0
\(157\) −13.0614 −1.04242 −0.521208 0.853430i \(-0.674518\pi\)
−0.521208 + 0.853430i \(0.674518\pi\)
\(158\) 37.3431 2.97086
\(159\) 0 0
\(160\) −7.87201 −0.622337
\(161\) 1.91105 0.150612
\(162\) 0 0
\(163\) −6.88655 −0.539396 −0.269698 0.962945i \(-0.586924\pi\)
−0.269698 + 0.962945i \(0.586924\pi\)
\(164\) 16.7270 1.30616
\(165\) 0 0
\(166\) −29.1547 −2.26285
\(167\) −12.0030 −0.928817 −0.464408 0.885621i \(-0.653733\pi\)
−0.464408 + 0.885621i \(0.653733\pi\)
\(168\) 0 0
\(169\) −7.40545 −0.569650
\(170\) 23.2046 1.77971
\(171\) 0 0
\(172\) 33.1087 2.52452
\(173\) −19.3994 −1.47491 −0.737456 0.675396i \(-0.763973\pi\)
−0.737456 + 0.675396i \(0.763973\pi\)
\(174\) 0 0
\(175\) 0.938597 0.0709513
\(176\) −0.127383 −0.00960187
\(177\) 0 0
\(178\) −6.25482 −0.468819
\(179\) 15.3418 1.14670 0.573349 0.819311i \(-0.305644\pi\)
0.573349 + 0.819311i \(0.305644\pi\)
\(180\) 0 0
\(181\) −9.81836 −0.729793 −0.364896 0.931048i \(-0.618896\pi\)
−0.364896 + 0.931048i \(0.618896\pi\)
\(182\) 5.56184 0.412271
\(183\) 0 0
\(184\) −6.87240 −0.506640
\(185\) 15.3408 1.12788
\(186\) 0 0
\(187\) −0.446338 −0.0326394
\(188\) 5.26431 0.383939
\(189\) 0 0
\(190\) −1.21954 −0.0884744
\(191\) −21.9444 −1.58784 −0.793921 0.608020i \(-0.791964\pi\)
−0.793921 + 0.608020i \(0.791964\pi\)
\(192\) 0 0
\(193\) −7.55014 −0.543471 −0.271736 0.962372i \(-0.587598\pi\)
−0.271736 + 0.962372i \(0.587598\pi\)
\(194\) −33.7739 −2.42482
\(195\) 0 0
\(196\) 3.52933 0.252095
\(197\) −19.1551 −1.36475 −0.682374 0.731004i \(-0.739052\pi\)
−0.682374 + 0.731004i \(0.739052\pi\)
\(198\) 0 0
\(199\) 20.4063 1.44656 0.723281 0.690554i \(-0.242633\pi\)
0.723281 + 0.690554i \(0.242633\pi\)
\(200\) −3.37532 −0.238671
\(201\) 0 0
\(202\) 8.76976 0.617038
\(203\) −5.21144 −0.365771
\(204\) 0 0
\(205\) 9.55136 0.667096
\(206\) 25.4785 1.77517
\(207\) 0 0
\(208\) −3.30546 −0.229192
\(209\) 0.0234576 0.00162260
\(210\) 0 0
\(211\) 4.34925 0.299415 0.149707 0.988730i \(-0.452167\pi\)
0.149707 + 0.988730i \(0.452167\pi\)
\(212\) 41.7870 2.86994
\(213\) 0 0
\(214\) −40.6192 −2.77667
\(215\) 18.9055 1.28935
\(216\) 0 0
\(217\) −2.91024 −0.197560
\(218\) −24.8353 −1.68206
\(219\) 0 0
\(220\) −0.648325 −0.0437101
\(221\) −11.5820 −0.779089
\(222\) 0 0
\(223\) 2.11821 0.141846 0.0709229 0.997482i \(-0.477406\pi\)
0.0709229 + 0.997482i \(0.477406\pi\)
\(224\) 3.90614 0.260990
\(225\) 0 0
\(226\) −28.3290 −1.88441
\(227\) −4.20493 −0.279091 −0.139546 0.990216i \(-0.544564\pi\)
−0.139546 + 0.990216i \(0.544564\pi\)
\(228\) 0 0
\(229\) −3.69447 −0.244137 −0.122069 0.992522i \(-0.538953\pi\)
−0.122069 + 0.992522i \(0.538953\pi\)
\(230\) −9.05620 −0.597148
\(231\) 0 0
\(232\) 18.7410 1.23041
\(233\) −13.1218 −0.859635 −0.429818 0.902916i \(-0.641422\pi\)
−0.429818 + 0.902916i \(0.641422\pi\)
\(234\) 0 0
\(235\) 3.00599 0.196089
\(236\) 18.0339 1.17391
\(237\) 0 0
\(238\) −11.5143 −0.746360
\(239\) −17.2224 −1.11402 −0.557012 0.830504i \(-0.688052\pi\)
−0.557012 + 0.830504i \(0.688052\pi\)
\(240\) 0 0
\(241\) 15.0726 0.970912 0.485456 0.874261i \(-0.338654\pi\)
0.485456 + 0.874261i \(0.338654\pi\)
\(242\) −25.8464 −1.66147
\(243\) 0 0
\(244\) 26.5364 1.69882
\(245\) 2.01529 0.128752
\(246\) 0 0
\(247\) 0.608700 0.0387306
\(248\) 10.4656 0.664567
\(249\) 0 0
\(250\) −28.1422 −1.77987
\(251\) 17.3625 1.09591 0.547956 0.836507i \(-0.315406\pi\)
0.547956 + 0.836507i \(0.315406\pi\)
\(252\) 0 0
\(253\) 0.174195 0.0109515
\(254\) −2.35145 −0.147543
\(255\) 0 0
\(256\) −23.9114 −1.49446
\(257\) 15.7891 0.984895 0.492448 0.870342i \(-0.336102\pi\)
0.492448 + 0.870342i \(0.336102\pi\)
\(258\) 0 0
\(259\) −7.61218 −0.472998
\(260\) −16.8234 −1.04334
\(261\) 0 0
\(262\) 49.0080 3.02772
\(263\) −11.0415 −0.680850 −0.340425 0.940272i \(-0.610571\pi\)
−0.340425 + 0.940272i \(0.610571\pi\)
\(264\) 0 0
\(265\) 23.8609 1.46577
\(266\) 0.605141 0.0371036
\(267\) 0 0
\(268\) 21.5039 1.31356
\(269\) 0.794820 0.0484610 0.0242305 0.999706i \(-0.492286\pi\)
0.0242305 + 0.999706i \(0.492286\pi\)
\(270\) 0 0
\(271\) 3.86100 0.234539 0.117269 0.993100i \(-0.462586\pi\)
0.117269 + 0.993100i \(0.462586\pi\)
\(272\) 6.84305 0.414921
\(273\) 0 0
\(274\) −10.6152 −0.641289
\(275\) 0.0855544 0.00515912
\(276\) 0 0
\(277\) −10.9785 −0.659636 −0.329818 0.944045i \(-0.606987\pi\)
−0.329818 + 0.944045i \(0.606987\pi\)
\(278\) 13.7101 0.822277
\(279\) 0 0
\(280\) −7.24727 −0.433107
\(281\) −0.451302 −0.0269224 −0.0134612 0.999909i \(-0.504285\pi\)
−0.0134612 + 0.999909i \(0.504285\pi\)
\(282\) 0 0
\(283\) 8.68388 0.516203 0.258102 0.966118i \(-0.416903\pi\)
0.258102 + 0.966118i \(0.416903\pi\)
\(284\) 18.6544 1.10693
\(285\) 0 0
\(286\) 0.506969 0.0299777
\(287\) −4.73944 −0.279761
\(288\) 0 0
\(289\) 6.97735 0.410432
\(290\) 24.6963 1.45022
\(291\) 0 0
\(292\) 9.53132 0.557779
\(293\) −10.6634 −0.622963 −0.311482 0.950252i \(-0.600825\pi\)
−0.311482 + 0.950252i \(0.600825\pi\)
\(294\) 0 0
\(295\) 10.2976 0.599550
\(296\) 27.3744 1.59111
\(297\) 0 0
\(298\) −19.0350 −1.10267
\(299\) 4.52017 0.261408
\(300\) 0 0
\(301\) −9.38103 −0.540714
\(302\) 19.6269 1.12940
\(303\) 0 0
\(304\) −0.359641 −0.0206269
\(305\) 15.1526 0.867636
\(306\) 0 0
\(307\) 18.4364 1.05222 0.526110 0.850417i \(-0.323650\pi\)
0.526110 + 0.850417i \(0.323650\pi\)
\(308\) 0.321703 0.0183307
\(309\) 0 0
\(310\) 13.7912 0.783289
\(311\) −4.89702 −0.277684 −0.138842 0.990315i \(-0.544338\pi\)
−0.138842 + 0.990315i \(0.544338\pi\)
\(312\) 0 0
\(313\) −22.9637 −1.29798 −0.648991 0.760796i \(-0.724809\pi\)
−0.648991 + 0.760796i \(0.724809\pi\)
\(314\) −30.7133 −1.73325
\(315\) 0 0
\(316\) 56.0488 3.15299
\(317\) −27.7698 −1.55971 −0.779853 0.625963i \(-0.784706\pi\)
−0.779853 + 0.625963i \(0.784706\pi\)
\(318\) 0 0
\(319\) −0.475029 −0.0265965
\(320\) −24.1434 −1.34966
\(321\) 0 0
\(322\) 4.49374 0.250426
\(323\) −1.26015 −0.0701164
\(324\) 0 0
\(325\) 2.22005 0.123146
\(326\) −16.1934 −0.896869
\(327\) 0 0
\(328\) 17.0437 0.941080
\(329\) −1.49159 −0.0822340
\(330\) 0 0
\(331\) 18.7608 1.03118 0.515592 0.856834i \(-0.327572\pi\)
0.515592 + 0.856834i \(0.327572\pi\)
\(332\) −43.7587 −2.40157
\(333\) 0 0
\(334\) −28.2244 −1.54437
\(335\) 12.2790 0.670874
\(336\) 0 0
\(337\) 27.0118 1.47143 0.735714 0.677292i \(-0.236847\pi\)
0.735714 + 0.677292i \(0.236847\pi\)
\(338\) −17.4136 −0.947173
\(339\) 0 0
\(340\) 34.8282 1.88882
\(341\) −0.265272 −0.0143653
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 33.7355 1.81890
\(345\) 0 0
\(346\) −45.6168 −2.45238
\(347\) 10.4238 0.559579 0.279790 0.960061i \(-0.409735\pi\)
0.279790 + 0.960061i \(0.409735\pi\)
\(348\) 0 0
\(349\) 7.23034 0.387031 0.193516 0.981097i \(-0.438011\pi\)
0.193516 + 0.981097i \(0.438011\pi\)
\(350\) 2.20707 0.117973
\(351\) 0 0
\(352\) 0.356050 0.0189775
\(353\) −28.2757 −1.50497 −0.752483 0.658612i \(-0.771144\pi\)
−0.752483 + 0.658612i \(0.771144\pi\)
\(354\) 0 0
\(355\) 10.6519 0.565344
\(356\) −9.38795 −0.497560
\(357\) 0 0
\(358\) 36.0755 1.90665
\(359\) −19.7759 −1.04373 −0.521867 0.853027i \(-0.674764\pi\)
−0.521867 + 0.853027i \(0.674764\pi\)
\(360\) 0 0
\(361\) −18.9338 −0.996514
\(362\) −23.0874 −1.21345
\(363\) 0 0
\(364\) 8.34785 0.437546
\(365\) 5.44251 0.284874
\(366\) 0 0
\(367\) 2.12967 0.111168 0.0555839 0.998454i \(-0.482298\pi\)
0.0555839 + 0.998454i \(0.482298\pi\)
\(368\) −2.67068 −0.139219
\(369\) 0 0
\(370\) 36.0731 1.87535
\(371\) −11.8399 −0.614699
\(372\) 0 0
\(373\) 14.5519 0.753470 0.376735 0.926321i \(-0.377047\pi\)
0.376735 + 0.926321i \(0.377047\pi\)
\(374\) −1.04954 −0.0542705
\(375\) 0 0
\(376\) 5.36396 0.276625
\(377\) −12.3265 −0.634848
\(378\) 0 0
\(379\) 24.5680 1.26198 0.630988 0.775793i \(-0.282650\pi\)
0.630988 + 0.775793i \(0.282650\pi\)
\(380\) −1.83042 −0.0938985
\(381\) 0 0
\(382\) −51.6013 −2.64015
\(383\) 3.12168 0.159510 0.0797552 0.996814i \(-0.474586\pi\)
0.0797552 + 0.996814i \(0.474586\pi\)
\(384\) 0 0
\(385\) 0.183697 0.00936204
\(386\) −17.7538 −0.903645
\(387\) 0 0
\(388\) −50.6917 −2.57348
\(389\) −28.2302 −1.43133 −0.715665 0.698444i \(-0.753876\pi\)
−0.715665 + 0.698444i \(0.753876\pi\)
\(390\) 0 0
\(391\) −9.35778 −0.473243
\(392\) 3.59614 0.181632
\(393\) 0 0
\(394\) −45.0424 −2.26920
\(395\) 32.0046 1.61033
\(396\) 0 0
\(397\) 28.6300 1.43690 0.718450 0.695579i \(-0.244852\pi\)
0.718450 + 0.695579i \(0.244852\pi\)
\(398\) 47.9844 2.40524
\(399\) 0 0
\(400\) −1.31168 −0.0655841
\(401\) −19.6061 −0.979081 −0.489540 0.871981i \(-0.662835\pi\)
−0.489540 + 0.871981i \(0.662835\pi\)
\(402\) 0 0
\(403\) −6.88353 −0.342893
\(404\) 13.1627 0.654866
\(405\) 0 0
\(406\) −12.2544 −0.608178
\(407\) −0.693860 −0.0343934
\(408\) 0 0
\(409\) 1.62163 0.0801844 0.0400922 0.999196i \(-0.487235\pi\)
0.0400922 + 0.999196i \(0.487235\pi\)
\(410\) 22.4596 1.10920
\(411\) 0 0
\(412\) 38.2410 1.88400
\(413\) −5.10974 −0.251434
\(414\) 0 0
\(415\) −24.9868 −1.22655
\(416\) 9.23912 0.452985
\(417\) 0 0
\(418\) 0.0551594 0.00269793
\(419\) 7.91033 0.386445 0.193223 0.981155i \(-0.438106\pi\)
0.193223 + 0.981155i \(0.438106\pi\)
\(420\) 0 0
\(421\) −21.5793 −1.05171 −0.525855 0.850574i \(-0.676255\pi\)
−0.525855 + 0.850574i \(0.676255\pi\)
\(422\) 10.2271 0.497845
\(423\) 0 0
\(424\) 42.5781 2.06777
\(425\) −4.59600 −0.222939
\(426\) 0 0
\(427\) −7.51881 −0.363861
\(428\) −60.9660 −2.94690
\(429\) 0 0
\(430\) 44.4554 2.14383
\(431\) −16.6583 −0.802402 −0.401201 0.915990i \(-0.631407\pi\)
−0.401201 + 0.915990i \(0.631407\pi\)
\(432\) 0 0
\(433\) 6.31119 0.303296 0.151648 0.988435i \(-0.451542\pi\)
0.151648 + 0.988435i \(0.451542\pi\)
\(434\) −6.84329 −0.328488
\(435\) 0 0
\(436\) −37.2757 −1.78518
\(437\) 0.491804 0.0235262
\(438\) 0 0
\(439\) −18.8300 −0.898706 −0.449353 0.893354i \(-0.648346\pi\)
−0.449353 + 0.893354i \(0.648346\pi\)
\(440\) −0.660598 −0.0314928
\(441\) 0 0
\(442\) −27.2345 −1.29541
\(443\) −8.24573 −0.391766 −0.195883 0.980627i \(-0.562757\pi\)
−0.195883 + 0.980627i \(0.562757\pi\)
\(444\) 0 0
\(445\) −5.36064 −0.254119
\(446\) 4.98087 0.235851
\(447\) 0 0
\(448\) 11.9801 0.566006
\(449\) −0.272917 −0.0128798 −0.00643988 0.999979i \(-0.502050\pi\)
−0.00643988 + 0.999979i \(0.502050\pi\)
\(450\) 0 0
\(451\) −0.432007 −0.0203424
\(452\) −42.5193 −1.99994
\(453\) 0 0
\(454\) −9.88770 −0.464053
\(455\) 4.76673 0.223468
\(456\) 0 0
\(457\) 34.2477 1.60204 0.801019 0.598639i \(-0.204291\pi\)
0.801019 + 0.598639i \(0.204291\pi\)
\(458\) −8.68736 −0.405934
\(459\) 0 0
\(460\) −13.5926 −0.633757
\(461\) 39.4342 1.83664 0.918318 0.395844i \(-0.129548\pi\)
0.918318 + 0.395844i \(0.129548\pi\)
\(462\) 0 0
\(463\) −21.1924 −0.984895 −0.492447 0.870342i \(-0.663898\pi\)
−0.492447 + 0.870342i \(0.663898\pi\)
\(464\) 7.28294 0.338102
\(465\) 0 0
\(466\) −30.8552 −1.42934
\(467\) 12.1148 0.560605 0.280303 0.959912i \(-0.409565\pi\)
0.280303 + 0.959912i \(0.409565\pi\)
\(468\) 0 0
\(469\) −6.09291 −0.281345
\(470\) 7.06844 0.326043
\(471\) 0 0
\(472\) 18.3753 0.845792
\(473\) −0.855094 −0.0393172
\(474\) 0 0
\(475\) 0.241546 0.0110829
\(476\) −17.2819 −0.792116
\(477\) 0 0
\(478\) −40.4976 −1.85232
\(479\) −29.5864 −1.35184 −0.675918 0.736976i \(-0.736253\pi\)
−0.675918 + 0.736976i \(0.736253\pi\)
\(480\) 0 0
\(481\) −18.0049 −0.820954
\(482\) 35.4425 1.61436
\(483\) 0 0
\(484\) −38.7933 −1.76333
\(485\) −28.9456 −1.31435
\(486\) 0 0
\(487\) −10.4625 −0.474100 −0.237050 0.971497i \(-0.576180\pi\)
−0.237050 + 0.971497i \(0.576180\pi\)
\(488\) 27.0387 1.22398
\(489\) 0 0
\(490\) 4.73886 0.214080
\(491\) −26.9269 −1.21519 −0.607597 0.794245i \(-0.707867\pi\)
−0.607597 + 0.794245i \(0.707867\pi\)
\(492\) 0 0
\(493\) 25.5187 1.14930
\(494\) 1.43133 0.0643985
\(495\) 0 0
\(496\) 4.06703 0.182615
\(497\) −5.28553 −0.237089
\(498\) 0 0
\(499\) 23.5790 1.05554 0.527771 0.849387i \(-0.323028\pi\)
0.527771 + 0.849387i \(0.323028\pi\)
\(500\) −42.2390 −1.88899
\(501\) 0 0
\(502\) 40.8271 1.82220
\(503\) 20.5970 0.918374 0.459187 0.888340i \(-0.348141\pi\)
0.459187 + 0.888340i \(0.348141\pi\)
\(504\) 0 0
\(505\) 7.51605 0.334460
\(506\) 0.409611 0.0182094
\(507\) 0 0
\(508\) −3.52933 −0.156589
\(509\) 24.5972 1.09025 0.545126 0.838354i \(-0.316482\pi\)
0.545126 + 0.838354i \(0.316482\pi\)
\(510\) 0 0
\(511\) −2.70061 −0.119468
\(512\) −15.5099 −0.685449
\(513\) 0 0
\(514\) 37.1272 1.63761
\(515\) 21.8361 0.962213
\(516\) 0 0
\(517\) −0.135960 −0.00597953
\(518\) −17.8997 −0.786466
\(519\) 0 0
\(520\) −17.1418 −0.751719
\(521\) 33.6158 1.47274 0.736368 0.676582i \(-0.236539\pi\)
0.736368 + 0.676582i \(0.236539\pi\)
\(522\) 0 0
\(523\) −27.8429 −1.21748 −0.608742 0.793368i \(-0.708326\pi\)
−0.608742 + 0.793368i \(0.708326\pi\)
\(524\) 73.5567 3.21334
\(525\) 0 0
\(526\) −25.9636 −1.13207
\(527\) 14.2505 0.620760
\(528\) 0 0
\(529\) −19.3479 −0.841212
\(530\) 56.1079 2.43717
\(531\) 0 0
\(532\) 0.908264 0.0393782
\(533\) −11.2101 −0.485564
\(534\) 0 0
\(535\) −34.8124 −1.50507
\(536\) 21.9110 0.946409
\(537\) 0 0
\(538\) 1.86898 0.0805775
\(539\) −0.0911513 −0.00392617
\(540\) 0 0
\(541\) −15.4932 −0.666104 −0.333052 0.942908i \(-0.608078\pi\)
−0.333052 + 0.942908i \(0.608078\pi\)
\(542\) 9.07895 0.389974
\(543\) 0 0
\(544\) −19.1271 −0.820066
\(545\) −21.2849 −0.911746
\(546\) 0 0
\(547\) 31.9739 1.36710 0.683552 0.729902i \(-0.260434\pi\)
0.683552 + 0.729902i \(0.260434\pi\)
\(548\) −15.9325 −0.680604
\(549\) 0 0
\(550\) 0.201177 0.00857822
\(551\) −1.34115 −0.0571350
\(552\) 0 0
\(553\) −15.8809 −0.675324
\(554\) −25.8155 −1.09680
\(555\) 0 0
\(556\) 20.5777 0.872688
\(557\) 15.6867 0.664666 0.332333 0.943162i \(-0.392164\pi\)
0.332333 + 0.943162i \(0.392164\pi\)
\(558\) 0 0
\(559\) −22.1888 −0.938485
\(560\) −2.81635 −0.119013
\(561\) 0 0
\(562\) −1.06121 −0.0447646
\(563\) 4.58303 0.193152 0.0965758 0.995326i \(-0.469211\pi\)
0.0965758 + 0.995326i \(0.469211\pi\)
\(564\) 0 0
\(565\) −24.2791 −1.02143
\(566\) 20.4197 0.858305
\(567\) 0 0
\(568\) 19.0075 0.797537
\(569\) 12.8151 0.537239 0.268619 0.963246i \(-0.413433\pi\)
0.268619 + 0.963246i \(0.413433\pi\)
\(570\) 0 0
\(571\) −31.1522 −1.30368 −0.651839 0.758357i \(-0.726002\pi\)
−0.651839 + 0.758357i \(0.726002\pi\)
\(572\) 0.760917 0.0318156
\(573\) 0 0
\(574\) −11.1446 −0.465166
\(575\) 1.79371 0.0748027
\(576\) 0 0
\(577\) −2.42009 −0.100750 −0.0503748 0.998730i \(-0.516042\pi\)
−0.0503748 + 0.998730i \(0.516042\pi\)
\(578\) 16.4069 0.682437
\(579\) 0 0
\(580\) 37.0670 1.53912
\(581\) 12.3986 0.514381
\(582\) 0 0
\(583\) −1.07923 −0.0446970
\(584\) 9.71175 0.401875
\(585\) 0 0
\(586\) −25.0745 −1.03582
\(587\) 18.6165 0.768386 0.384193 0.923253i \(-0.374480\pi\)
0.384193 + 0.923253i \(0.374480\pi\)
\(588\) 0 0
\(589\) −0.748943 −0.0308597
\(590\) 24.2143 0.996889
\(591\) 0 0
\(592\) 10.6380 0.437217
\(593\) 12.2093 0.501375 0.250688 0.968068i \(-0.419343\pi\)
0.250688 + 0.968068i \(0.419343\pi\)
\(594\) 0 0
\(595\) −9.86822 −0.404557
\(596\) −28.5699 −1.17027
\(597\) 0 0
\(598\) 10.6290 0.434651
\(599\) −22.9849 −0.939137 −0.469569 0.882896i \(-0.655591\pi\)
−0.469569 + 0.882896i \(0.655591\pi\)
\(600\) 0 0
\(601\) 0.719373 0.0293438 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(602\) −22.0590 −0.899060
\(603\) 0 0
\(604\) 29.4583 1.19864
\(605\) −22.1515 −0.900585
\(606\) 0 0
\(607\) 3.71769 0.150897 0.0754483 0.997150i \(-0.475961\pi\)
0.0754483 + 0.997150i \(0.475961\pi\)
\(608\) 1.00524 0.0407677
\(609\) 0 0
\(610\) 35.6306 1.44264
\(611\) −3.52803 −0.142729
\(612\) 0 0
\(613\) −9.97409 −0.402850 −0.201425 0.979504i \(-0.564557\pi\)
−0.201425 + 0.979504i \(0.564557\pi\)
\(614\) 43.3522 1.74955
\(615\) 0 0
\(616\) 0.327793 0.0132071
\(617\) 25.4593 1.02495 0.512476 0.858702i \(-0.328728\pi\)
0.512476 + 0.858702i \(0.328728\pi\)
\(618\) 0 0
\(619\) 36.0978 1.45089 0.725446 0.688280i \(-0.241634\pi\)
0.725446 + 0.688280i \(0.241634\pi\)
\(620\) 20.6994 0.831309
\(621\) 0 0
\(622\) −11.5151 −0.461713
\(623\) 2.65998 0.106570
\(624\) 0 0
\(625\) −19.4261 −0.777042
\(626\) −53.9979 −2.15819
\(627\) 0 0
\(628\) −46.0981 −1.83951
\(629\) 37.2743 1.48622
\(630\) 0 0
\(631\) 3.17238 0.126291 0.0631453 0.998004i \(-0.479887\pi\)
0.0631453 + 0.998004i \(0.479887\pi\)
\(632\) 57.1099 2.27171
\(633\) 0 0
\(634\) −65.2993 −2.59337
\(635\) −2.01529 −0.0799744
\(636\) 0 0
\(637\) −2.36528 −0.0937158
\(638\) −1.11701 −0.0442228
\(639\) 0 0
\(640\) −41.0280 −1.62177
\(641\) 32.5163 1.28432 0.642159 0.766571i \(-0.278039\pi\)
0.642159 + 0.766571i \(0.278039\pi\)
\(642\) 0 0
\(643\) −7.62164 −0.300568 −0.150284 0.988643i \(-0.548019\pi\)
−0.150284 + 0.988643i \(0.548019\pi\)
\(644\) 6.74472 0.265779
\(645\) 0 0
\(646\) −2.96317 −0.116585
\(647\) 16.9965 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(648\) 0 0
\(649\) −0.465759 −0.0182827
\(650\) 5.22033 0.204758
\(651\) 0 0
\(652\) −24.3049 −0.951853
\(653\) 32.8417 1.28520 0.642598 0.766203i \(-0.277856\pi\)
0.642598 + 0.766203i \(0.277856\pi\)
\(654\) 0 0
\(655\) 42.0019 1.64115
\(656\) 6.62333 0.258598
\(657\) 0 0
\(658\) −3.50740 −0.136733
\(659\) −12.1276 −0.472422 −0.236211 0.971702i \(-0.575906\pi\)
−0.236211 + 0.971702i \(0.575906\pi\)
\(660\) 0 0
\(661\) 3.30251 0.128453 0.0642263 0.997935i \(-0.479542\pi\)
0.0642263 + 0.997935i \(0.479542\pi\)
\(662\) 44.1150 1.71458
\(663\) 0 0
\(664\) −44.5871 −1.73032
\(665\) 0.518631 0.0201116
\(666\) 0 0
\(667\) −9.95932 −0.385626
\(668\) −42.3624 −1.63905
\(669\) 0 0
\(670\) 28.8735 1.11548
\(671\) −0.685350 −0.0264576
\(672\) 0 0
\(673\) −24.8503 −0.957907 −0.478954 0.877840i \(-0.658984\pi\)
−0.478954 + 0.877840i \(0.658984\pi\)
\(674\) 63.5170 2.44658
\(675\) 0 0
\(676\) −26.1362 −1.00524
\(677\) −15.2521 −0.586185 −0.293092 0.956084i \(-0.594684\pi\)
−0.293092 + 0.956084i \(0.594684\pi\)
\(678\) 0 0
\(679\) 14.3630 0.551201
\(680\) 35.4875 1.36088
\(681\) 0 0
\(682\) −0.623775 −0.0238856
\(683\) −7.38509 −0.282583 −0.141291 0.989968i \(-0.545125\pi\)
−0.141291 + 0.989968i \(0.545125\pi\)
\(684\) 0 0
\(685\) −9.09768 −0.347605
\(686\) −2.35145 −0.0897789
\(687\) 0 0
\(688\) 13.1099 0.499811
\(689\) −28.0048 −1.06690
\(690\) 0 0
\(691\) 17.7340 0.674634 0.337317 0.941391i \(-0.390481\pi\)
0.337317 + 0.941391i \(0.390481\pi\)
\(692\) −68.4669 −2.60272
\(693\) 0 0
\(694\) 24.5111 0.930428
\(695\) 11.7501 0.445708
\(696\) 0 0
\(697\) 23.2075 0.879046
\(698\) 17.0018 0.643528
\(699\) 0 0
\(700\) 3.31262 0.125205
\(701\) 42.2300 1.59500 0.797502 0.603316i \(-0.206154\pi\)
0.797502 + 0.603316i \(0.206154\pi\)
\(702\) 0 0
\(703\) −1.95898 −0.0738842
\(704\) 1.09200 0.0411563
\(705\) 0 0
\(706\) −66.4890 −2.50235
\(707\) −3.72951 −0.140263
\(708\) 0 0
\(709\) 8.71968 0.327474 0.163737 0.986504i \(-0.447645\pi\)
0.163737 + 0.986504i \(0.447645\pi\)
\(710\) 25.0474 0.940013
\(711\) 0 0
\(712\) −9.56567 −0.358489
\(713\) −5.56161 −0.208284
\(714\) 0 0
\(715\) 0.434494 0.0162492
\(716\) 54.1462 2.02354
\(717\) 0 0
\(718\) −46.5021 −1.73544
\(719\) −41.1755 −1.53559 −0.767793 0.640698i \(-0.778645\pi\)
−0.767793 + 0.640698i \(0.778645\pi\)
\(720\) 0 0
\(721\) −10.8352 −0.403524
\(722\) −44.5219 −1.65693
\(723\) 0 0
\(724\) −34.6522 −1.28784
\(725\) −4.89144 −0.181664
\(726\) 0 0
\(727\) 29.4228 1.09123 0.545615 0.838036i \(-0.316296\pi\)
0.545615 + 0.838036i \(0.316296\pi\)
\(728\) 8.50587 0.315249
\(729\) 0 0
\(730\) 12.7978 0.473668
\(731\) 45.9358 1.69900
\(732\) 0 0
\(733\) 24.0969 0.890040 0.445020 0.895521i \(-0.353197\pi\)
0.445020 + 0.895521i \(0.353197\pi\)
\(734\) 5.00782 0.184842
\(735\) 0 0
\(736\) 7.46483 0.275157
\(737\) −0.555377 −0.0204576
\(738\) 0 0
\(739\) 13.4406 0.494420 0.247210 0.968962i \(-0.420486\pi\)
0.247210 + 0.968962i \(0.420486\pi\)
\(740\) 54.1426 1.99032
\(741\) 0 0
\(742\) −27.8411 −1.02208
\(743\) 39.8981 1.46372 0.731860 0.681455i \(-0.238652\pi\)
0.731860 + 0.681455i \(0.238652\pi\)
\(744\) 0 0
\(745\) −16.3138 −0.597691
\(746\) 34.2182 1.25282
\(747\) 0 0
\(748\) −1.57527 −0.0575976
\(749\) 17.2741 0.631182
\(750\) 0 0
\(751\) −22.5801 −0.823959 −0.411979 0.911193i \(-0.635162\pi\)
−0.411979 + 0.911193i \(0.635162\pi\)
\(752\) 2.08449 0.0760134
\(753\) 0 0
\(754\) −28.9852 −1.05558
\(755\) 16.8211 0.612182
\(756\) 0 0
\(757\) −15.3023 −0.556171 −0.278086 0.960556i \(-0.589700\pi\)
−0.278086 + 0.960556i \(0.589700\pi\)
\(758\) 57.7706 2.09832
\(759\) 0 0
\(760\) −1.86507 −0.0676531
\(761\) −1.97788 −0.0716980 −0.0358490 0.999357i \(-0.511414\pi\)
−0.0358490 + 0.999357i \(0.511414\pi\)
\(762\) 0 0
\(763\) 10.5617 0.382359
\(764\) −77.4490 −2.80201
\(765\) 0 0
\(766\) 7.34048 0.265222
\(767\) −12.0860 −0.436399
\(768\) 0 0
\(769\) −37.2480 −1.34320 −0.671598 0.740916i \(-0.734392\pi\)
−0.671598 + 0.740916i \(0.734392\pi\)
\(770\) 0.431954 0.0155665
\(771\) 0 0
\(772\) −26.6469 −0.959044
\(773\) −32.5873 −1.17208 −0.586042 0.810281i \(-0.699315\pi\)
−0.586042 + 0.810281i \(0.699315\pi\)
\(774\) 0 0
\(775\) −2.73154 −0.0981199
\(776\) −51.6513 −1.85417
\(777\) 0 0
\(778\) −66.3821 −2.37991
\(779\) −1.21969 −0.0436998
\(780\) 0 0
\(781\) −0.481783 −0.0172396
\(782\) −22.0044 −0.786874
\(783\) 0 0
\(784\) 1.39749 0.0499104
\(785\) −26.3226 −0.939494
\(786\) 0 0
\(787\) 7.02321 0.250350 0.125175 0.992135i \(-0.460051\pi\)
0.125175 + 0.992135i \(0.460051\pi\)
\(788\) −67.6047 −2.40832
\(789\) 0 0
\(790\) 75.2573 2.67754
\(791\) 12.0474 0.428357
\(792\) 0 0
\(793\) −17.7841 −0.631532
\(794\) 67.3221 2.38917
\(795\) 0 0
\(796\) 72.0204 2.55269
\(797\) −43.7885 −1.55107 −0.775535 0.631305i \(-0.782520\pi\)
−0.775535 + 0.631305i \(0.782520\pi\)
\(798\) 0 0
\(799\) 7.30382 0.258391
\(800\) 3.66629 0.129623
\(801\) 0 0
\(802\) −46.1027 −1.62794
\(803\) −0.246164 −0.00868694
\(804\) 0 0
\(805\) 3.85132 0.135741
\(806\) −16.1863 −0.570138
\(807\) 0 0
\(808\) 13.4118 0.471826
\(809\) −11.3663 −0.399618 −0.199809 0.979835i \(-0.564032\pi\)
−0.199809 + 0.979835i \(0.564032\pi\)
\(810\) 0 0
\(811\) −39.3608 −1.38214 −0.691072 0.722786i \(-0.742861\pi\)
−0.691072 + 0.722786i \(0.742861\pi\)
\(812\) −18.3929 −0.645463
\(813\) 0 0
\(814\) −1.63158 −0.0571868
\(815\) −13.8784 −0.486140
\(816\) 0 0
\(817\) −2.41419 −0.0844617
\(818\) 3.81318 0.133325
\(819\) 0 0
\(820\) 33.7099 1.17720
\(821\) −11.6097 −0.405183 −0.202591 0.979263i \(-0.564936\pi\)
−0.202591 + 0.979263i \(0.564936\pi\)
\(822\) 0 0
\(823\) −14.3330 −0.499619 −0.249809 0.968295i \(-0.580368\pi\)
−0.249809 + 0.968295i \(0.580368\pi\)
\(824\) 38.9649 1.35741
\(825\) 0 0
\(826\) −12.0153 −0.418066
\(827\) −37.2619 −1.29572 −0.647862 0.761758i \(-0.724337\pi\)
−0.647862 + 0.761758i \(0.724337\pi\)
\(828\) 0 0
\(829\) 39.9915 1.38896 0.694481 0.719511i \(-0.255634\pi\)
0.694481 + 0.719511i \(0.255634\pi\)
\(830\) −58.7553 −2.03943
\(831\) 0 0
\(832\) 28.3363 0.982383
\(833\) 4.89667 0.169659
\(834\) 0 0
\(835\) −24.1895 −0.837111
\(836\) 0.0827895 0.00286333
\(837\) 0 0
\(838\) 18.6008 0.642553
\(839\) −10.8582 −0.374868 −0.187434 0.982277i \(-0.560017\pi\)
−0.187434 + 0.982277i \(0.560017\pi\)
\(840\) 0 0
\(841\) −1.84092 −0.0634801
\(842\) −50.7427 −1.74871
\(843\) 0 0
\(844\) 15.3499 0.528366
\(845\) −14.9241 −0.513406
\(846\) 0 0
\(847\) 10.9917 0.377679
\(848\) 16.5462 0.568200
\(849\) 0 0
\(850\) −10.8073 −0.370686
\(851\) −14.5473 −0.498673
\(852\) 0 0
\(853\) 3.22075 0.110276 0.0551382 0.998479i \(-0.482440\pi\)
0.0551382 + 0.998479i \(0.482440\pi\)
\(854\) −17.6801 −0.605002
\(855\) 0 0
\(856\) −62.1201 −2.12322
\(857\) −27.1752 −0.928288 −0.464144 0.885760i \(-0.653638\pi\)
−0.464144 + 0.885760i \(0.653638\pi\)
\(858\) 0 0
\(859\) −7.73152 −0.263796 −0.131898 0.991263i \(-0.542107\pi\)
−0.131898 + 0.991263i \(0.542107\pi\)
\(860\) 66.7238 2.27526
\(861\) 0 0
\(862\) −39.1712 −1.33418
\(863\) 40.5701 1.38102 0.690511 0.723322i \(-0.257386\pi\)
0.690511 + 0.723322i \(0.257386\pi\)
\(864\) 0 0
\(865\) −39.0955 −1.32929
\(866\) 14.8405 0.504299
\(867\) 0 0
\(868\) −10.2712 −0.348627
\(869\) −1.44756 −0.0491052
\(870\) 0 0
\(871\) −14.4115 −0.488313
\(872\) −37.9813 −1.28621
\(873\) 0 0
\(874\) 1.15645 0.0391177
\(875\) 11.9680 0.404593
\(876\) 0 0
\(877\) −25.5801 −0.863780 −0.431890 0.901926i \(-0.642153\pi\)
−0.431890 + 0.901926i \(0.642153\pi\)
\(878\) −44.2778 −1.49430
\(879\) 0 0
\(880\) −0.256715 −0.00865384
\(881\) −2.14538 −0.0722797 −0.0361398 0.999347i \(-0.511506\pi\)
−0.0361398 + 0.999347i \(0.511506\pi\)
\(882\) 0 0
\(883\) −30.7069 −1.03337 −0.516685 0.856176i \(-0.672834\pi\)
−0.516685 + 0.856176i \(0.672834\pi\)
\(884\) −40.8766 −1.37483
\(885\) 0 0
\(886\) −19.3894 −0.651401
\(887\) −3.09857 −0.104040 −0.0520199 0.998646i \(-0.516566\pi\)
−0.0520199 + 0.998646i \(0.516566\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −12.6053 −0.422531
\(891\) 0 0
\(892\) 7.47585 0.250310
\(893\) −0.383857 −0.0128453
\(894\) 0 0
\(895\) 30.9182 1.03348
\(896\) 20.3583 0.680124
\(897\) 0 0
\(898\) −0.641752 −0.0214155
\(899\) 15.1665 0.505832
\(900\) 0 0
\(901\) 57.9763 1.93147
\(902\) −1.01584 −0.0338239
\(903\) 0 0
\(904\) −43.3242 −1.44094
\(905\) −19.7869 −0.657737
\(906\) 0 0
\(907\) 14.1556 0.470029 0.235015 0.971992i \(-0.424486\pi\)
0.235015 + 0.971992i \(0.424486\pi\)
\(908\) −14.8406 −0.492502
\(909\) 0 0
\(910\) 11.2087 0.371566
\(911\) 29.6771 0.983247 0.491624 0.870808i \(-0.336404\pi\)
0.491624 + 0.870808i \(0.336404\pi\)
\(912\) 0 0
\(913\) 1.13015 0.0374025
\(914\) 80.5318 2.66375
\(915\) 0 0
\(916\) −13.0390 −0.430820
\(917\) −20.8416 −0.688249
\(918\) 0 0
\(919\) −53.9253 −1.77883 −0.889415 0.457100i \(-0.848888\pi\)
−0.889415 + 0.457100i \(0.848888\pi\)
\(920\) −13.8499 −0.456617
\(921\) 0 0
\(922\) 92.7277 3.05382
\(923\) −12.5018 −0.411501
\(924\) 0 0
\(925\) −7.14477 −0.234918
\(926\) −49.8329 −1.63761
\(927\) 0 0
\(928\) −20.3566 −0.668238
\(929\) −36.9355 −1.21181 −0.605906 0.795536i \(-0.707189\pi\)
−0.605906 + 0.795536i \(0.707189\pi\)
\(930\) 0 0
\(931\) −0.257348 −0.00843423
\(932\) −46.3110 −1.51697
\(933\) 0 0
\(934\) 28.4873 0.932134
\(935\) −0.899501 −0.0294168
\(936\) 0 0
\(937\) 3.40870 0.111357 0.0556787 0.998449i \(-0.482268\pi\)
0.0556787 + 0.998449i \(0.482268\pi\)
\(938\) −14.3272 −0.467799
\(939\) 0 0
\(940\) 10.6091 0.346031
\(941\) 47.8418 1.55960 0.779799 0.626029i \(-0.215321\pi\)
0.779799 + 0.626029i \(0.215321\pi\)
\(942\) 0 0
\(943\) −9.05731 −0.294947
\(944\) 7.14082 0.232414
\(945\) 0 0
\(946\) −2.01071 −0.0653739
\(947\) 5.90288 0.191818 0.0959088 0.995390i \(-0.469424\pi\)
0.0959088 + 0.995390i \(0.469424\pi\)
\(948\) 0 0
\(949\) −6.38769 −0.207353
\(950\) 0.567983 0.0184278
\(951\) 0 0
\(952\) −17.6091 −0.570714
\(953\) 35.7438 1.15785 0.578927 0.815379i \(-0.303472\pi\)
0.578927 + 0.815379i \(0.303472\pi\)
\(954\) 0 0
\(955\) −44.2244 −1.43107
\(956\) −60.7835 −1.96588
\(957\) 0 0
\(958\) −69.5710 −2.24774
\(959\) 4.51432 0.145775
\(960\) 0 0
\(961\) −22.5305 −0.726791
\(962\) −42.3377 −1.36502
\(963\) 0 0
\(964\) 53.1962 1.71333
\(965\) −15.2157 −0.489812
\(966\) 0 0
\(967\) 30.5305 0.981796 0.490898 0.871217i \(-0.336669\pi\)
0.490898 + 0.871217i \(0.336669\pi\)
\(968\) −39.5276 −1.27047
\(969\) 0 0
\(970\) −68.0643 −2.18541
\(971\) 24.6280 0.790351 0.395176 0.918606i \(-0.370684\pi\)
0.395176 + 0.918606i \(0.370684\pi\)
\(972\) 0 0
\(973\) −5.83048 −0.186917
\(974\) −24.6020 −0.788299
\(975\) 0 0
\(976\) 10.5075 0.336336
\(977\) 42.8486 1.37085 0.685423 0.728145i \(-0.259617\pi\)
0.685423 + 0.728145i \(0.259617\pi\)
\(978\) 0 0
\(979\) 0.242461 0.00774909
\(980\) 7.11262 0.227204
\(981\) 0 0
\(982\) −63.3173 −2.02054
\(983\) 48.7528 1.55497 0.777486 0.628900i \(-0.216495\pi\)
0.777486 + 0.628900i \(0.216495\pi\)
\(984\) 0 0
\(985\) −38.6032 −1.23000
\(986\) 60.0059 1.91098
\(987\) 0 0
\(988\) 2.14830 0.0683465
\(989\) −17.9276 −0.570065
\(990\) 0 0
\(991\) 27.0445 0.859096 0.429548 0.903044i \(-0.358673\pi\)
0.429548 + 0.903044i \(0.358673\pi\)
\(992\) −11.3678 −0.360928
\(993\) 0 0
\(994\) −12.4287 −0.394214
\(995\) 41.1246 1.30374
\(996\) 0 0
\(997\) −32.1373 −1.01780 −0.508900 0.860826i \(-0.669948\pi\)
−0.508900 + 0.860826i \(0.669948\pi\)
\(998\) 55.4449 1.75508
\(999\) 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.s.1.15 16
3.2 odd 2 2667.2.a.n.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.2 16 3.2 odd 2
8001.2.a.s.1.15 16 1.1 even 1 trivial