Properties

Label 8043.2.a.l.1.2
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $2$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} +2.56155 q^{5} +1.56155 q^{6} +1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} +2.56155 q^{5} +1.56155 q^{6} +1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{9} +4.00000 q^{10} -6.56155 q^{11} +0.438447 q^{12} +2.00000 q^{13} +1.56155 q^{14} +2.56155 q^{15} -4.68466 q^{16} -2.00000 q^{17} +1.56155 q^{18} -6.56155 q^{19} +1.12311 q^{20} +1.00000 q^{21} -10.2462 q^{22} +5.12311 q^{23} -2.43845 q^{24} +1.56155 q^{25} +3.12311 q^{26} +1.00000 q^{27} +0.438447 q^{28} -7.12311 q^{29} +4.00000 q^{30} -2.56155 q^{31} -2.43845 q^{32} -6.56155 q^{33} -3.12311 q^{34} +2.56155 q^{35} +0.438447 q^{36} -8.24621 q^{37} -10.2462 q^{38} +2.00000 q^{39} -6.24621 q^{40} +3.68466 q^{41} +1.56155 q^{42} -7.68466 q^{43} -2.87689 q^{44} +2.56155 q^{45} +8.00000 q^{46} +5.43845 q^{47} -4.68466 q^{48} +1.00000 q^{49} +2.43845 q^{50} -2.00000 q^{51} +0.876894 q^{52} -1.12311 q^{53} +1.56155 q^{54} -16.8078 q^{55} -2.43845 q^{56} -6.56155 q^{57} -11.1231 q^{58} +3.68466 q^{59} +1.12311 q^{60} -12.2462 q^{61} -4.00000 q^{62} +1.00000 q^{63} +5.56155 q^{64} +5.12311 q^{65} -10.2462 q^{66} -7.68466 q^{67} -0.876894 q^{68} +5.12311 q^{69} +4.00000 q^{70} +8.00000 q^{71} -2.43845 q^{72} +4.56155 q^{73} -12.8769 q^{74} +1.56155 q^{75} -2.87689 q^{76} -6.56155 q^{77} +3.12311 q^{78} +3.12311 q^{79} -12.0000 q^{80} +1.00000 q^{81} +5.75379 q^{82} +4.00000 q^{83} +0.438447 q^{84} -5.12311 q^{85} -12.0000 q^{86} -7.12311 q^{87} +16.0000 q^{88} -14.5616 q^{89} +4.00000 q^{90} +2.00000 q^{91} +2.24621 q^{92} -2.56155 q^{93} +8.49242 q^{94} -16.8078 q^{95} -2.43845 q^{96} -11.1231 q^{97} +1.56155 q^{98} -6.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9} + 8 q^{10} - 9 q^{11} + 5 q^{12} + 4 q^{13} - q^{14} + q^{15} + 3 q^{16} - 4 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} + 2 q^{21} - 4 q^{22} + 2 q^{23} - 9 q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + 5 q^{28} - 6 q^{29} + 8 q^{30} - q^{31} - 9 q^{32} - 9 q^{33} + 2 q^{34} + q^{35} + 5 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40} - 5 q^{41} - q^{42} - 3 q^{43} - 14 q^{44} + q^{45} + 16 q^{46} + 15 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 4 q^{51} + 10 q^{52} + 6 q^{53} - q^{54} - 13 q^{55} - 9 q^{56} - 9 q^{57} - 14 q^{58} - 5 q^{59} - 6 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 8 q^{70} + 16 q^{71} - 9 q^{72} + 5 q^{73} - 34 q^{74} - q^{75} - 14 q^{76} - 9 q^{77} - 2 q^{78} - 2 q^{79} - 24 q^{80} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 5 q^{84} - 2 q^{85} - 24 q^{86} - 6 q^{87} + 32 q^{88} - 25 q^{89} + 8 q^{90} + 4 q^{91} - 12 q^{92} - q^{93} - 16 q^{94} - 13 q^{95} - 9 q^{96} - 14 q^{97} - q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 1.10418 0.552092 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.438447 0.219224
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 1.56155 0.637501
\(7\) 1.00000 0.377964
\(8\) −2.43845 −0.862121
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) −6.56155 −1.97838 −0.989191 0.146631i \(-0.953157\pi\)
−0.989191 + 0.146631i \(0.953157\pi\)
\(12\) 0.438447 0.126569
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.56155 0.417343
\(15\) 2.56155 0.661390
\(16\) −4.68466 −1.17116
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.56155 0.368062
\(19\) −6.56155 −1.50532 −0.752662 0.658407i \(-0.771230\pi\)
−0.752662 + 0.658407i \(0.771230\pi\)
\(20\) 1.12311 0.251134
\(21\) 1.00000 0.218218
\(22\) −10.2462 −2.18450
\(23\) 5.12311 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(24\) −2.43845 −0.497746
\(25\) 1.56155 0.312311
\(26\) 3.12311 0.612491
\(27\) 1.00000 0.192450
\(28\) 0.438447 0.0828587
\(29\) −7.12311 −1.32273 −0.661364 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 4.00000 0.730297
\(31\) −2.56155 −0.460068 −0.230034 0.973183i \(-0.573884\pi\)
−0.230034 + 0.973183i \(0.573884\pi\)
\(32\) −2.43845 −0.431061
\(33\) −6.56155 −1.14222
\(34\) −3.12311 −0.535608
\(35\) 2.56155 0.432981
\(36\) 0.438447 0.0730745
\(37\) −8.24621 −1.35567 −0.677834 0.735215i \(-0.737081\pi\)
−0.677834 + 0.735215i \(0.737081\pi\)
\(38\) −10.2462 −1.66215
\(39\) 2.00000 0.320256
\(40\) −6.24621 −0.987613
\(41\) 3.68466 0.575447 0.287723 0.957714i \(-0.407102\pi\)
0.287723 + 0.957714i \(0.407102\pi\)
\(42\) 1.56155 0.240953
\(43\) −7.68466 −1.17190 −0.585950 0.810347i \(-0.699278\pi\)
−0.585950 + 0.810347i \(0.699278\pi\)
\(44\) −2.87689 −0.433708
\(45\) 2.56155 0.381854
\(46\) 8.00000 1.17954
\(47\) 5.43845 0.793279 0.396640 0.917974i \(-0.370176\pi\)
0.396640 + 0.917974i \(0.370176\pi\)
\(48\) −4.68466 −0.676172
\(49\) 1.00000 0.142857
\(50\) 2.43845 0.344849
\(51\) −2.00000 −0.280056
\(52\) 0.876894 0.121603
\(53\) −1.12311 −0.154270 −0.0771352 0.997021i \(-0.524577\pi\)
−0.0771352 + 0.997021i \(0.524577\pi\)
\(54\) 1.56155 0.212500
\(55\) −16.8078 −2.26636
\(56\) −2.43845 −0.325851
\(57\) −6.56155 −0.869099
\(58\) −11.1231 −1.46054
\(59\) 3.68466 0.479702 0.239851 0.970810i \(-0.422901\pi\)
0.239851 + 0.970810i \(0.422901\pi\)
\(60\) 1.12311 0.144992
\(61\) −12.2462 −1.56797 −0.783983 0.620782i \(-0.786815\pi\)
−0.783983 + 0.620782i \(0.786815\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 5.56155 0.695194
\(65\) 5.12311 0.635443
\(66\) −10.2462 −1.26122
\(67\) −7.68466 −0.938830 −0.469415 0.882978i \(-0.655535\pi\)
−0.469415 + 0.882978i \(0.655535\pi\)
\(68\) −0.876894 −0.106339
\(69\) 5.12311 0.616749
\(70\) 4.00000 0.478091
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −2.43845 −0.287374
\(73\) 4.56155 0.533889 0.266945 0.963712i \(-0.413986\pi\)
0.266945 + 0.963712i \(0.413986\pi\)
\(74\) −12.8769 −1.49691
\(75\) 1.56155 0.180313
\(76\) −2.87689 −0.330002
\(77\) −6.56155 −0.747758
\(78\) 3.12311 0.353622
\(79\) 3.12311 0.351377 0.175688 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(80\) −12.0000 −1.34164
\(81\) 1.00000 0.111111
\(82\) 5.75379 0.635400
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0.438447 0.0478385
\(85\) −5.12311 −0.555679
\(86\) −12.0000 −1.29399
\(87\) −7.12311 −0.763677
\(88\) 16.0000 1.70561
\(89\) −14.5616 −1.54352 −0.771761 0.635913i \(-0.780624\pi\)
−0.771761 + 0.635913i \(0.780624\pi\)
\(90\) 4.00000 0.421637
\(91\) 2.00000 0.209657
\(92\) 2.24621 0.234184
\(93\) −2.56155 −0.265621
\(94\) 8.49242 0.875927
\(95\) −16.8078 −1.72444
\(96\) −2.43845 −0.248873
\(97\) −11.1231 −1.12938 −0.564690 0.825303i \(-0.691004\pi\)
−0.564690 + 0.825303i \(0.691004\pi\)
\(98\) 1.56155 0.157741
\(99\) −6.56155 −0.659461
\(100\) 0.684658 0.0684658
\(101\) 4.87689 0.485269 0.242635 0.970118i \(-0.421988\pi\)
0.242635 + 0.970118i \(0.421988\pi\)
\(102\) −3.12311 −0.309234
\(103\) −5.43845 −0.535866 −0.267933 0.963438i \(-0.586341\pi\)
−0.267933 + 0.963438i \(0.586341\pi\)
\(104\) −4.87689 −0.478219
\(105\) 2.56155 0.249982
\(106\) −1.75379 −0.170343
\(107\) 14.2462 1.37723 0.688617 0.725126i \(-0.258218\pi\)
0.688617 + 0.725126i \(0.258218\pi\)
\(108\) 0.438447 0.0421896
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −26.2462 −2.50248
\(111\) −8.24621 −0.782696
\(112\) −4.68466 −0.442659
\(113\) 5.36932 0.505103 0.252551 0.967583i \(-0.418730\pi\)
0.252551 + 0.967583i \(0.418730\pi\)
\(114\) −10.2462 −0.959646
\(115\) 13.1231 1.22374
\(116\) −3.12311 −0.289973
\(117\) 2.00000 0.184900
\(118\) 5.75379 0.529679
\(119\) −2.00000 −0.183340
\(120\) −6.24621 −0.570198
\(121\) 32.0540 2.91400
\(122\) −19.1231 −1.73132
\(123\) 3.68466 0.332234
\(124\) −1.12311 −0.100858
\(125\) −8.80776 −0.787790
\(126\) 1.56155 0.139114
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 13.5616 1.19868
\(129\) −7.68466 −0.676596
\(130\) 8.00000 0.701646
\(131\) −16.8078 −1.46850 −0.734251 0.678879i \(-0.762466\pi\)
−0.734251 + 0.678879i \(0.762466\pi\)
\(132\) −2.87689 −0.250402
\(133\) −6.56155 −0.568959
\(134\) −12.0000 −1.03664
\(135\) 2.56155 0.220463
\(136\) 4.87689 0.418190
\(137\) −4.24621 −0.362778 −0.181389 0.983411i \(-0.558059\pi\)
−0.181389 + 0.983411i \(0.558059\pi\)
\(138\) 8.00000 0.681005
\(139\) 5.43845 0.461283 0.230642 0.973039i \(-0.425918\pi\)
0.230642 + 0.973039i \(0.425918\pi\)
\(140\) 1.12311 0.0949197
\(141\) 5.43845 0.458000
\(142\) 12.4924 1.04834
\(143\) −13.1231 −1.09741
\(144\) −4.68466 −0.390388
\(145\) −18.2462 −1.51527
\(146\) 7.12311 0.589512
\(147\) 1.00000 0.0824786
\(148\) −3.61553 −0.297195
\(149\) −5.36932 −0.439872 −0.219936 0.975514i \(-0.570585\pi\)
−0.219936 + 0.975514i \(0.570585\pi\)
\(150\) 2.43845 0.199098
\(151\) 16.2462 1.32210 0.661049 0.750343i \(-0.270112\pi\)
0.661049 + 0.750343i \(0.270112\pi\)
\(152\) 16.0000 1.29777
\(153\) −2.00000 −0.161690
\(154\) −10.2462 −0.825663
\(155\) −6.56155 −0.527037
\(156\) 0.876894 0.0702077
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 4.87689 0.387985
\(159\) −1.12311 −0.0890681
\(160\) −6.24621 −0.493806
\(161\) 5.12311 0.403757
\(162\) 1.56155 0.122687
\(163\) −12.8769 −1.00860 −0.504298 0.863530i \(-0.668249\pi\)
−0.504298 + 0.863530i \(0.668249\pi\)
\(164\) 1.61553 0.126152
\(165\) −16.8078 −1.30848
\(166\) 6.24621 0.484800
\(167\) −1.75379 −0.135712 −0.0678561 0.997695i \(-0.521616\pi\)
−0.0678561 + 0.997695i \(0.521616\pi\)
\(168\) −2.43845 −0.188130
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) −6.56155 −0.501774
\(172\) −3.36932 −0.256908
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −11.1231 −0.843240
\(175\) 1.56155 0.118042
\(176\) 30.7386 2.31701
\(177\) 3.68466 0.276956
\(178\) −22.7386 −1.70433
\(179\) 0.807764 0.0603751 0.0301876 0.999544i \(-0.490390\pi\)
0.0301876 + 0.999544i \(0.490390\pi\)
\(180\) 1.12311 0.0837114
\(181\) −0.876894 −0.0651790 −0.0325895 0.999469i \(-0.510375\pi\)
−0.0325895 + 0.999469i \(0.510375\pi\)
\(182\) 3.12311 0.231500
\(183\) −12.2462 −0.905266
\(184\) −12.4924 −0.920954
\(185\) −21.1231 −1.55300
\(186\) −4.00000 −0.293294
\(187\) 13.1231 0.959657
\(188\) 2.38447 0.173905
\(189\) 1.00000 0.0727393
\(190\) −26.2462 −1.90410
\(191\) −9.75379 −0.705759 −0.352880 0.935669i \(-0.614798\pi\)
−0.352880 + 0.935669i \(0.614798\pi\)
\(192\) 5.56155 0.401371
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −17.3693 −1.24704
\(195\) 5.12311 0.366873
\(196\) 0.438447 0.0313177
\(197\) 23.0540 1.64253 0.821264 0.570549i \(-0.193269\pi\)
0.821264 + 0.570549i \(0.193269\pi\)
\(198\) −10.2462 −0.728167
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) −3.80776 −0.269250
\(201\) −7.68466 −0.542034
\(202\) 7.61553 0.535827
\(203\) −7.12311 −0.499944
\(204\) −0.876894 −0.0613949
\(205\) 9.43845 0.659210
\(206\) −8.49242 −0.591695
\(207\) 5.12311 0.356080
\(208\) −9.36932 −0.649645
\(209\) 43.0540 2.97811
\(210\) 4.00000 0.276026
\(211\) 15.1231 1.04112 0.520559 0.853826i \(-0.325724\pi\)
0.520559 + 0.853826i \(0.325724\pi\)
\(212\) −0.492423 −0.0338197
\(213\) 8.00000 0.548151
\(214\) 22.2462 1.52072
\(215\) −19.6847 −1.34248
\(216\) −2.43845 −0.165915
\(217\) −2.56155 −0.173890
\(218\) −3.12311 −0.211523
\(219\) 4.56155 0.308241
\(220\) −7.36932 −0.496839
\(221\) −4.00000 −0.269069
\(222\) −12.8769 −0.864241
\(223\) 7.68466 0.514603 0.257301 0.966331i \(-0.417167\pi\)
0.257301 + 0.966331i \(0.417167\pi\)
\(224\) −2.43845 −0.162926
\(225\) 1.56155 0.104104
\(226\) 8.38447 0.557727
\(227\) −6.24621 −0.414576 −0.207288 0.978280i \(-0.566464\pi\)
−0.207288 + 0.978280i \(0.566464\pi\)
\(228\) −2.87689 −0.190527
\(229\) 12.5616 0.830091 0.415045 0.909801i \(-0.363766\pi\)
0.415045 + 0.909801i \(0.363766\pi\)
\(230\) 20.4924 1.35123
\(231\) −6.56155 −0.431718
\(232\) 17.3693 1.14035
\(233\) −5.43845 −0.356285 −0.178142 0.984005i \(-0.557009\pi\)
−0.178142 + 0.984005i \(0.557009\pi\)
\(234\) 3.12311 0.204164
\(235\) 13.9309 0.908750
\(236\) 1.61553 0.105162
\(237\) 3.12311 0.202868
\(238\) −3.12311 −0.202441
\(239\) −23.6847 −1.53203 −0.766017 0.642821i \(-0.777764\pi\)
−0.766017 + 0.642821i \(0.777764\pi\)
\(240\) −12.0000 −0.774597
\(241\) −16.2462 −1.04651 −0.523255 0.852176i \(-0.675283\pi\)
−0.523255 + 0.852176i \(0.675283\pi\)
\(242\) 50.0540 3.21759
\(243\) 1.00000 0.0641500
\(244\) −5.36932 −0.343735
\(245\) 2.56155 0.163652
\(246\) 5.75379 0.366848
\(247\) −13.1231 −0.835003
\(248\) 6.24621 0.396635
\(249\) 4.00000 0.253490
\(250\) −13.7538 −0.869866
\(251\) 1.12311 0.0708898 0.0354449 0.999372i \(-0.488715\pi\)
0.0354449 + 0.999372i \(0.488715\pi\)
\(252\) 0.438447 0.0276196
\(253\) −33.6155 −2.11339
\(254\) −9.36932 −0.587883
\(255\) −5.12311 −0.320821
\(256\) 10.0540 0.628373
\(257\) −22.5616 −1.40735 −0.703675 0.710521i \(-0.748459\pi\)
−0.703675 + 0.710521i \(0.748459\pi\)
\(258\) −12.0000 −0.747087
\(259\) −8.24621 −0.512395
\(260\) 2.24621 0.139304
\(261\) −7.12311 −0.440909
\(262\) −26.2462 −1.62150
\(263\) 31.3693 1.93431 0.967157 0.254178i \(-0.0818049\pi\)
0.967157 + 0.254178i \(0.0818049\pi\)
\(264\) 16.0000 0.984732
\(265\) −2.87689 −0.176726
\(266\) −10.2462 −0.628236
\(267\) −14.5616 −0.891153
\(268\) −3.36932 −0.205814
\(269\) 0.315342 0.0192267 0.00961336 0.999954i \(-0.496940\pi\)
0.00961336 + 0.999954i \(0.496940\pi\)
\(270\) 4.00000 0.243432
\(271\) 11.7538 0.713992 0.356996 0.934106i \(-0.383801\pi\)
0.356996 + 0.934106i \(0.383801\pi\)
\(272\) 9.36932 0.568098
\(273\) 2.00000 0.121046
\(274\) −6.63068 −0.400574
\(275\) −10.2462 −0.617870
\(276\) 2.24621 0.135206
\(277\) −0.0691303 −0.00415364 −0.00207682 0.999998i \(-0.500661\pi\)
−0.00207682 + 0.999998i \(0.500661\pi\)
\(278\) 8.49242 0.509342
\(279\) −2.56155 −0.153356
\(280\) −6.24621 −0.373283
\(281\) 15.0540 0.898045 0.449022 0.893520i \(-0.351772\pi\)
0.449022 + 0.893520i \(0.351772\pi\)
\(282\) 8.49242 0.505716
\(283\) 12.8769 0.765452 0.382726 0.923862i \(-0.374985\pi\)
0.382726 + 0.923862i \(0.374985\pi\)
\(284\) 3.50758 0.208136
\(285\) −16.8078 −0.995606
\(286\) −20.4924 −1.21174
\(287\) 3.68466 0.217499
\(288\) −2.43845 −0.143687
\(289\) −13.0000 −0.764706
\(290\) −28.4924 −1.67313
\(291\) −11.1231 −0.652048
\(292\) 2.00000 0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 1.56155 0.0910716
\(295\) 9.43845 0.549528
\(296\) 20.1080 1.16875
\(297\) −6.56155 −0.380740
\(298\) −8.38447 −0.485699
\(299\) 10.2462 0.592554
\(300\) 0.684658 0.0395288
\(301\) −7.68466 −0.442936
\(302\) 25.3693 1.45984
\(303\) 4.87689 0.280170
\(304\) 30.7386 1.76298
\(305\) −31.3693 −1.79620
\(306\) −3.12311 −0.178536
\(307\) 0.246211 0.0140520 0.00702601 0.999975i \(-0.497764\pi\)
0.00702601 + 0.999975i \(0.497764\pi\)
\(308\) −2.87689 −0.163926
\(309\) −5.43845 −0.309382
\(310\) −10.2462 −0.581946
\(311\) −33.3002 −1.88828 −0.944140 0.329544i \(-0.893105\pi\)
−0.944140 + 0.329544i \(0.893105\pi\)
\(312\) −4.87689 −0.276100
\(313\) 18.1771 1.02743 0.513715 0.857961i \(-0.328269\pi\)
0.513715 + 0.857961i \(0.328269\pi\)
\(314\) 21.8617 1.23373
\(315\) 2.56155 0.144327
\(316\) 1.36932 0.0770301
\(317\) 6.63068 0.372416 0.186208 0.982510i \(-0.440380\pi\)
0.186208 + 0.982510i \(0.440380\pi\)
\(318\) −1.75379 −0.0983476
\(319\) 46.7386 2.61686
\(320\) 14.2462 0.796387
\(321\) 14.2462 0.795146
\(322\) 8.00000 0.445823
\(323\) 13.1231 0.730189
\(324\) 0.438447 0.0243582
\(325\) 3.12311 0.173239
\(326\) −20.1080 −1.11368
\(327\) −2.00000 −0.110600
\(328\) −8.98485 −0.496105
\(329\) 5.43845 0.299831
\(330\) −26.2462 −1.44481
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) 1.75379 0.0962517
\(333\) −8.24621 −0.451890
\(334\) −2.73863 −0.149851
\(335\) −19.6847 −1.07549
\(336\) −4.68466 −0.255569
\(337\) 13.3693 0.728273 0.364137 0.931346i \(-0.381364\pi\)
0.364137 + 0.931346i \(0.381364\pi\)
\(338\) −14.0540 −0.764435
\(339\) 5.36932 0.291621
\(340\) −2.24621 −0.121818
\(341\) 16.8078 0.910191
\(342\) −10.2462 −0.554052
\(343\) 1.00000 0.0539949
\(344\) 18.7386 1.01032
\(345\) 13.1231 0.706524
\(346\) −15.6155 −0.839496
\(347\) −12.8078 −0.687557 −0.343778 0.939051i \(-0.611707\pi\)
−0.343778 + 0.939051i \(0.611707\pi\)
\(348\) −3.12311 −0.167416
\(349\) 25.8617 1.38435 0.692174 0.721731i \(-0.256653\pi\)
0.692174 + 0.721731i \(0.256653\pi\)
\(350\) 2.43845 0.130340
\(351\) 2.00000 0.106752
\(352\) 16.0000 0.852803
\(353\) −15.1231 −0.804922 −0.402461 0.915437i \(-0.631845\pi\)
−0.402461 + 0.915437i \(0.631845\pi\)
\(354\) 5.75379 0.305810
\(355\) 20.4924 1.08762
\(356\) −6.38447 −0.338376
\(357\) −2.00000 −0.105851
\(358\) 1.26137 0.0666653
\(359\) −16.3153 −0.861091 −0.430545 0.902569i \(-0.641679\pi\)
−0.430545 + 0.902569i \(0.641679\pi\)
\(360\) −6.24621 −0.329204
\(361\) 24.0540 1.26600
\(362\) −1.36932 −0.0719697
\(363\) 32.0540 1.68240
\(364\) 0.876894 0.0459618
\(365\) 11.6847 0.611603
\(366\) −19.1231 −0.999581
\(367\) 23.6155 1.23272 0.616360 0.787464i \(-0.288607\pi\)
0.616360 + 0.787464i \(0.288607\pi\)
\(368\) −24.0000 −1.25109
\(369\) 3.68466 0.191816
\(370\) −32.9848 −1.71480
\(371\) −1.12311 −0.0583087
\(372\) −1.12311 −0.0582303
\(373\) −13.6847 −0.708565 −0.354282 0.935138i \(-0.615275\pi\)
−0.354282 + 0.935138i \(0.615275\pi\)
\(374\) 20.4924 1.05964
\(375\) −8.80776 −0.454831
\(376\) −13.2614 −0.683903
\(377\) −14.2462 −0.733717
\(378\) 1.56155 0.0803176
\(379\) 10.4924 0.538960 0.269480 0.963006i \(-0.413148\pi\)
0.269480 + 0.963006i \(0.413148\pi\)
\(380\) −7.36932 −0.378038
\(381\) −6.00000 −0.307389
\(382\) −15.2311 −0.779289
\(383\) −1.00000 −0.0510976
\(384\) 13.5616 0.692060
\(385\) −16.8078 −0.856603
\(386\) 3.12311 0.158962
\(387\) −7.68466 −0.390633
\(388\) −4.87689 −0.247587
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 8.00000 0.405096
\(391\) −10.2462 −0.518173
\(392\) −2.43845 −0.123160
\(393\) −16.8078 −0.847840
\(394\) 36.0000 1.81365
\(395\) 8.00000 0.402524
\(396\) −2.87689 −0.144569
\(397\) −22.8078 −1.14469 −0.572344 0.820013i \(-0.693966\pi\)
−0.572344 + 0.820013i \(0.693966\pi\)
\(398\) 15.6155 0.782736
\(399\) −6.56155 −0.328489
\(400\) −7.31534 −0.365767
\(401\) 4.87689 0.243540 0.121770 0.992558i \(-0.461143\pi\)
0.121770 + 0.992558i \(0.461143\pi\)
\(402\) −12.0000 −0.598506
\(403\) −5.12311 −0.255200
\(404\) 2.13826 0.106382
\(405\) 2.56155 0.127285
\(406\) −11.1231 −0.552030
\(407\) 54.1080 2.68203
\(408\) 4.87689 0.241442
\(409\) −10.4924 −0.518817 −0.259408 0.965768i \(-0.583528\pi\)
−0.259408 + 0.965768i \(0.583528\pi\)
\(410\) 14.7386 0.727889
\(411\) −4.24621 −0.209450
\(412\) −2.38447 −0.117474
\(413\) 3.68466 0.181310
\(414\) 8.00000 0.393179
\(415\) 10.2462 0.502967
\(416\) −4.87689 −0.239109
\(417\) 5.43845 0.266322
\(418\) 67.2311 3.28838
\(419\) −22.2462 −1.08680 −0.543399 0.839474i \(-0.682863\pi\)
−0.543399 + 0.839474i \(0.682863\pi\)
\(420\) 1.12311 0.0548019
\(421\) −16.2462 −0.791792 −0.395896 0.918295i \(-0.629566\pi\)
−0.395896 + 0.918295i \(0.629566\pi\)
\(422\) 23.6155 1.14959
\(423\) 5.43845 0.264426
\(424\) 2.73863 0.133000
\(425\) −3.12311 −0.151493
\(426\) 12.4924 0.605260
\(427\) −12.2462 −0.592636
\(428\) 6.24621 0.301922
\(429\) −13.1231 −0.633590
\(430\) −30.7386 −1.48235
\(431\) 18.8769 0.909268 0.454634 0.890678i \(-0.349770\pi\)
0.454634 + 0.890678i \(0.349770\pi\)
\(432\) −4.68466 −0.225391
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −4.00000 −0.192006
\(435\) −18.2462 −0.874839
\(436\) −0.876894 −0.0419956
\(437\) −33.6155 −1.60805
\(438\) 7.12311 0.340355
\(439\) −11.6847 −0.557678 −0.278839 0.960338i \(-0.589950\pi\)
−0.278839 + 0.960338i \(0.589950\pi\)
\(440\) 40.9848 1.95388
\(441\) 1.00000 0.0476190
\(442\) −6.24621 −0.297102
\(443\) −27.0540 −1.28537 −0.642687 0.766129i \(-0.722180\pi\)
−0.642687 + 0.766129i \(0.722180\pi\)
\(444\) −3.61553 −0.171585
\(445\) −37.3002 −1.76820
\(446\) 12.0000 0.568216
\(447\) −5.36932 −0.253960
\(448\) 5.56155 0.262759
\(449\) 11.3693 0.536551 0.268276 0.963342i \(-0.413546\pi\)
0.268276 + 0.963342i \(0.413546\pi\)
\(450\) 2.43845 0.114950
\(451\) −24.1771 −1.13845
\(452\) 2.35416 0.110730
\(453\) 16.2462 0.763314
\(454\) −9.75379 −0.457768
\(455\) 5.12311 0.240175
\(456\) 16.0000 0.749269
\(457\) 0.246211 0.0115173 0.00575864 0.999983i \(-0.498167\pi\)
0.00575864 + 0.999983i \(0.498167\pi\)
\(458\) 19.6155 0.916573
\(459\) −2.00000 −0.0933520
\(460\) 5.75379 0.268272
\(461\) −2.87689 −0.133990 −0.0669952 0.997753i \(-0.521341\pi\)
−0.0669952 + 0.997753i \(0.521341\pi\)
\(462\) −10.2462 −0.476697
\(463\) 30.0000 1.39422 0.697109 0.716965i \(-0.254469\pi\)
0.697109 + 0.716965i \(0.254469\pi\)
\(464\) 33.3693 1.54913
\(465\) −6.56155 −0.304285
\(466\) −8.49242 −0.393404
\(467\) −29.6155 −1.37044 −0.685222 0.728335i \(-0.740295\pi\)
−0.685222 + 0.728335i \(0.740295\pi\)
\(468\) 0.876894 0.0405345
\(469\) −7.68466 −0.354845
\(470\) 21.7538 1.00343
\(471\) 14.0000 0.645086
\(472\) −8.98485 −0.413561
\(473\) 50.4233 2.31847
\(474\) 4.87689 0.224003
\(475\) −10.2462 −0.470128
\(476\) −0.876894 −0.0401924
\(477\) −1.12311 −0.0514235
\(478\) −36.9848 −1.69165
\(479\) 0.630683 0.0288166 0.0144083 0.999896i \(-0.495414\pi\)
0.0144083 + 0.999896i \(0.495414\pi\)
\(480\) −6.24621 −0.285099
\(481\) −16.4924 −0.751990
\(482\) −25.3693 −1.15554
\(483\) 5.12311 0.233109
\(484\) 14.0540 0.638817
\(485\) −28.4924 −1.29377
\(486\) 1.56155 0.0708335
\(487\) 36.2462 1.64247 0.821236 0.570588i \(-0.193285\pi\)
0.821236 + 0.570588i \(0.193285\pi\)
\(488\) 29.8617 1.35178
\(489\) −12.8769 −0.582313
\(490\) 4.00000 0.180702
\(491\) 14.8769 0.671385 0.335692 0.941972i \(-0.391030\pi\)
0.335692 + 0.941972i \(0.391030\pi\)
\(492\) 1.61553 0.0728336
\(493\) 14.2462 0.641617
\(494\) −20.4924 −0.921998
\(495\) −16.8078 −0.755453
\(496\) 12.0000 0.538816
\(497\) 8.00000 0.358849
\(498\) 6.24621 0.279899
\(499\) −22.7386 −1.01792 −0.508961 0.860790i \(-0.669970\pi\)
−0.508961 + 0.860790i \(0.669970\pi\)
\(500\) −3.86174 −0.172702
\(501\) −1.75379 −0.0783535
\(502\) 1.75379 0.0782754
\(503\) −23.6847 −1.05605 −0.528023 0.849230i \(-0.677067\pi\)
−0.528023 + 0.849230i \(0.677067\pi\)
\(504\) −2.43845 −0.108617
\(505\) 12.4924 0.555906
\(506\) −52.4924 −2.33357
\(507\) −9.00000 −0.399704
\(508\) −2.63068 −0.116718
\(509\) −21.3693 −0.947178 −0.473589 0.880746i \(-0.657042\pi\)
−0.473589 + 0.880746i \(0.657042\pi\)
\(510\) −8.00000 −0.354246
\(511\) 4.56155 0.201791
\(512\) −11.4233 −0.504843
\(513\) −6.56155 −0.289700
\(514\) −35.2311 −1.55398
\(515\) −13.9309 −0.613867
\(516\) −3.36932 −0.148326
\(517\) −35.6847 −1.56941
\(518\) −12.8769 −0.565778
\(519\) −10.0000 −0.438951
\(520\) −12.4924 −0.547829
\(521\) 4.24621 0.186030 0.0930149 0.995665i \(-0.470350\pi\)
0.0930149 + 0.995665i \(0.470350\pi\)
\(522\) −11.1231 −0.486845
\(523\) 11.6155 0.507912 0.253956 0.967216i \(-0.418268\pi\)
0.253956 + 0.967216i \(0.418268\pi\)
\(524\) −7.36932 −0.321930
\(525\) 1.56155 0.0681518
\(526\) 48.9848 2.13584
\(527\) 5.12311 0.223166
\(528\) 30.7386 1.33773
\(529\) 3.24621 0.141140
\(530\) −4.49242 −0.195138
\(531\) 3.68466 0.159901
\(532\) −2.87689 −0.124729
\(533\) 7.36932 0.319201
\(534\) −22.7386 −0.983997
\(535\) 36.4924 1.57771
\(536\) 18.7386 0.809386
\(537\) 0.807764 0.0348576
\(538\) 0.492423 0.0212298
\(539\) −6.56155 −0.282626
\(540\) 1.12311 0.0483308
\(541\) −32.7386 −1.40754 −0.703772 0.710426i \(-0.748502\pi\)
−0.703772 + 0.710426i \(0.748502\pi\)
\(542\) 18.3542 0.788379
\(543\) −0.876894 −0.0376311
\(544\) 4.87689 0.209095
\(545\) −5.12311 −0.219450
\(546\) 3.12311 0.133657
\(547\) −3.75379 −0.160500 −0.0802502 0.996775i \(-0.525572\pi\)
−0.0802502 + 0.996775i \(0.525572\pi\)
\(548\) −1.86174 −0.0795296
\(549\) −12.2462 −0.522656
\(550\) −16.0000 −0.682242
\(551\) 46.7386 1.99113
\(552\) −12.4924 −0.531713
\(553\) 3.12311 0.132808
\(554\) −0.107951 −0.00458638
\(555\) −21.1231 −0.896626
\(556\) 2.38447 0.101124
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −4.00000 −0.169334
\(559\) −15.3693 −0.650053
\(560\) −12.0000 −0.507093
\(561\) 13.1231 0.554058
\(562\) 23.5076 0.991607
\(563\) 12.4924 0.526493 0.263246 0.964729i \(-0.415207\pi\)
0.263246 + 0.964729i \(0.415207\pi\)
\(564\) 2.38447 0.100404
\(565\) 13.7538 0.578626
\(566\) 20.1080 0.845200
\(567\) 1.00000 0.0419961
\(568\) −19.5076 −0.818520
\(569\) −11.6155 −0.486948 −0.243474 0.969907i \(-0.578287\pi\)
−0.243474 + 0.969907i \(0.578287\pi\)
\(570\) −26.2462 −1.09933
\(571\) −45.3693 −1.89865 −0.949323 0.314301i \(-0.898230\pi\)
−0.949323 + 0.314301i \(0.898230\pi\)
\(572\) −5.75379 −0.240578
\(573\) −9.75379 −0.407470
\(574\) 5.75379 0.240159
\(575\) 8.00000 0.333623
\(576\) 5.56155 0.231731
\(577\) −28.8769 −1.20216 −0.601080 0.799189i \(-0.705263\pi\)
−0.601080 + 0.799189i \(0.705263\pi\)
\(578\) −20.3002 −0.844376
\(579\) 2.00000 0.0831172
\(580\) −8.00000 −0.332182
\(581\) 4.00000 0.165948
\(582\) −17.3693 −0.719981
\(583\) 7.36932 0.305206
\(584\) −11.1231 −0.460277
\(585\) 5.12311 0.211814
\(586\) −21.8617 −0.903100
\(587\) 47.2311 1.94943 0.974717 0.223442i \(-0.0717294\pi\)
0.974717 + 0.223442i \(0.0717294\pi\)
\(588\) 0.438447 0.0180813
\(589\) 16.8078 0.692552
\(590\) 14.7386 0.606780
\(591\) 23.0540 0.948314
\(592\) 38.6307 1.58771
\(593\) 11.6847 0.479831 0.239916 0.970794i \(-0.422880\pi\)
0.239916 + 0.970794i \(0.422880\pi\)
\(594\) −10.2462 −0.420407
\(595\) −5.12311 −0.210027
\(596\) −2.35416 −0.0964302
\(597\) 10.0000 0.409273
\(598\) 16.0000 0.654289
\(599\) 47.3693 1.93546 0.967729 0.251994i \(-0.0810862\pi\)
0.967729 + 0.251994i \(0.0810862\pi\)
\(600\) −3.80776 −0.155451
\(601\) 18.4924 0.754322 0.377161 0.926148i \(-0.376900\pi\)
0.377161 + 0.926148i \(0.376900\pi\)
\(602\) −12.0000 −0.489083
\(603\) −7.68466 −0.312943
\(604\) 7.12311 0.289835
\(605\) 82.1080 3.33816
\(606\) 7.61553 0.309360
\(607\) 17.4384 0.707805 0.353902 0.935282i \(-0.384855\pi\)
0.353902 + 0.935282i \(0.384855\pi\)
\(608\) 16.0000 0.648886
\(609\) −7.12311 −0.288643
\(610\) −48.9848 −1.98334
\(611\) 10.8769 0.440032
\(612\) −0.876894 −0.0354464
\(613\) 12.8769 0.520093 0.260046 0.965596i \(-0.416262\pi\)
0.260046 + 0.965596i \(0.416262\pi\)
\(614\) 0.384472 0.0155160
\(615\) 9.43845 0.380595
\(616\) 16.0000 0.644658
\(617\) −12.1771 −0.490231 −0.245115 0.969494i \(-0.578826\pi\)
−0.245115 + 0.969494i \(0.578826\pi\)
\(618\) −8.49242 −0.341615
\(619\) −10.4924 −0.421726 −0.210863 0.977516i \(-0.567627\pi\)
−0.210863 + 0.977516i \(0.567627\pi\)
\(620\) −2.87689 −0.115539
\(621\) 5.12311 0.205583
\(622\) −52.0000 −2.08501
\(623\) −14.5616 −0.583396
\(624\) −9.36932 −0.375073
\(625\) −30.3693 −1.21477
\(626\) 28.3845 1.13447
\(627\) 43.0540 1.71941
\(628\) 6.13826 0.244943
\(629\) 16.4924 0.657596
\(630\) 4.00000 0.159364
\(631\) 1.75379 0.0698172 0.0349086 0.999391i \(-0.488886\pi\)
0.0349086 + 0.999391i \(0.488886\pi\)
\(632\) −7.61553 −0.302929
\(633\) 15.1231 0.601089
\(634\) 10.3542 0.411216
\(635\) −15.3693 −0.609913
\(636\) −0.492423 −0.0195258
\(637\) 2.00000 0.0792429
\(638\) 72.9848 2.88950
\(639\) 8.00000 0.316475
\(640\) 34.7386 1.37317
\(641\) −2.49242 −0.0984448 −0.0492224 0.998788i \(-0.515674\pi\)
−0.0492224 + 0.998788i \(0.515674\pi\)
\(642\) 22.2462 0.877988
\(643\) 8.17708 0.322473 0.161236 0.986916i \(-0.448452\pi\)
0.161236 + 0.986916i \(0.448452\pi\)
\(644\) 2.24621 0.0885131
\(645\) −19.6847 −0.775083
\(646\) 20.4924 0.806264
\(647\) −9.43845 −0.371064 −0.185532 0.982638i \(-0.559401\pi\)
−0.185532 + 0.982638i \(0.559401\pi\)
\(648\) −2.43845 −0.0957913
\(649\) −24.1771 −0.949033
\(650\) 4.87689 0.191288
\(651\) −2.56155 −0.100395
\(652\) −5.64584 −0.221108
\(653\) 5.61553 0.219753 0.109876 0.993945i \(-0.464955\pi\)
0.109876 + 0.993945i \(0.464955\pi\)
\(654\) −3.12311 −0.122123
\(655\) −43.0540 −1.68226
\(656\) −17.2614 −0.673943
\(657\) 4.56155 0.177963
\(658\) 8.49242 0.331069
\(659\) −47.8617 −1.86443 −0.932214 0.361907i \(-0.882126\pi\)
−0.932214 + 0.361907i \(0.882126\pi\)
\(660\) −7.36932 −0.286850
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 16.0000 0.621858
\(663\) −4.00000 −0.155347
\(664\) −9.75379 −0.378520
\(665\) −16.8078 −0.651777
\(666\) −12.8769 −0.498970
\(667\) −36.4924 −1.41299
\(668\) −0.768944 −0.0297513
\(669\) 7.68466 0.297106
\(670\) −30.7386 −1.18754
\(671\) 80.3542 3.10204
\(672\) −2.43845 −0.0940651
\(673\) −28.7386 −1.10779 −0.553896 0.832586i \(-0.686860\pi\)
−0.553896 + 0.832586i \(0.686860\pi\)
\(674\) 20.8769 0.804148
\(675\) 1.56155 0.0601042
\(676\) −3.94602 −0.151770
\(677\) 21.8617 0.840215 0.420107 0.907474i \(-0.361992\pi\)
0.420107 + 0.907474i \(0.361992\pi\)
\(678\) 8.38447 0.322004
\(679\) −11.1231 −0.426866
\(680\) 12.4924 0.479063
\(681\) −6.24621 −0.239355
\(682\) 26.2462 1.00502
\(683\) 31.2311 1.19502 0.597512 0.801860i \(-0.296156\pi\)
0.597512 + 0.801860i \(0.296156\pi\)
\(684\) −2.87689 −0.110001
\(685\) −10.8769 −0.415585
\(686\) 1.56155 0.0596204
\(687\) 12.5616 0.479253
\(688\) 36.0000 1.37249
\(689\) −2.24621 −0.0855738
\(690\) 20.4924 0.780133
\(691\) −37.3693 −1.42160 −0.710798 0.703396i \(-0.751666\pi\)
−0.710798 + 0.703396i \(0.751666\pi\)
\(692\) −4.38447 −0.166673
\(693\) −6.56155 −0.249253
\(694\) −20.0000 −0.759190
\(695\) 13.9309 0.528428
\(696\) 17.3693 0.658382
\(697\) −7.36932 −0.279133
\(698\) 40.3845 1.52857
\(699\) −5.43845 −0.205701
\(700\) 0.684658 0.0258777
\(701\) −40.8078 −1.54129 −0.770644 0.637266i \(-0.780065\pi\)
−0.770644 + 0.637266i \(0.780065\pi\)
\(702\) 3.12311 0.117874
\(703\) 54.1080 2.04072
\(704\) −36.4924 −1.37536
\(705\) 13.9309 0.524667
\(706\) −23.6155 −0.888782
\(707\) 4.87689 0.183414
\(708\) 1.61553 0.0607153
\(709\) 27.1231 1.01863 0.509315 0.860580i \(-0.329899\pi\)
0.509315 + 0.860580i \(0.329899\pi\)
\(710\) 32.0000 1.20094
\(711\) 3.12311 0.117126
\(712\) 35.5076 1.33070
\(713\) −13.1231 −0.491464
\(714\) −3.12311 −0.116879
\(715\) −33.6155 −1.25715
\(716\) 0.354162 0.0132357
\(717\) −23.6847 −0.884520
\(718\) −25.4773 −0.950803
\(719\) −10.8769 −0.405640 −0.202820 0.979216i \(-0.565011\pi\)
−0.202820 + 0.979216i \(0.565011\pi\)
\(720\) −12.0000 −0.447214
\(721\) −5.43845 −0.202538
\(722\) 37.5616 1.39790
\(723\) −16.2462 −0.604203
\(724\) −0.384472 −0.0142888
\(725\) −11.1231 −0.413102
\(726\) 50.0540 1.85768
\(727\) 2.56155 0.0950027 0.0475014 0.998871i \(-0.484874\pi\)
0.0475014 + 0.998871i \(0.484874\pi\)
\(728\) −4.87689 −0.180750
\(729\) 1.00000 0.0370370
\(730\) 18.2462 0.675323
\(731\) 15.3693 0.568455
\(732\) −5.36932 −0.198456
\(733\) −48.7386 −1.80020 −0.900101 0.435681i \(-0.856508\pi\)
−0.900101 + 0.435681i \(0.856508\pi\)
\(734\) 36.8769 1.36115
\(735\) 2.56155 0.0944843
\(736\) −12.4924 −0.460477
\(737\) 50.4233 1.85737
\(738\) 5.75379 0.211800
\(739\) 50.9848 1.87551 0.937754 0.347300i \(-0.112902\pi\)
0.937754 + 0.347300i \(0.112902\pi\)
\(740\) −9.26137 −0.340455
\(741\) −13.1231 −0.482089
\(742\) −1.75379 −0.0643836
\(743\) −1.43845 −0.0527715 −0.0263858 0.999652i \(-0.508400\pi\)
−0.0263858 + 0.999652i \(0.508400\pi\)
\(744\) 6.24621 0.228997
\(745\) −13.7538 −0.503900
\(746\) −21.3693 −0.782386
\(747\) 4.00000 0.146352
\(748\) 5.75379 0.210379
\(749\) 14.2462 0.520545
\(750\) −13.7538 −0.502217
\(751\) −43.6847 −1.59408 −0.797038 0.603929i \(-0.793601\pi\)
−0.797038 + 0.603929i \(0.793601\pi\)
\(752\) −25.4773 −0.929060
\(753\) 1.12311 0.0409282
\(754\) −22.2462 −0.810159
\(755\) 41.6155 1.51454
\(756\) 0.438447 0.0159462
\(757\) −46.4924 −1.68980 −0.844898 0.534928i \(-0.820339\pi\)
−0.844898 + 0.534928i \(0.820339\pi\)
\(758\) 16.3845 0.595111
\(759\) −33.6155 −1.22017
\(760\) 40.9848 1.48668
\(761\) −7.75379 −0.281075 −0.140537 0.990075i \(-0.544883\pi\)
−0.140537 + 0.990075i \(0.544883\pi\)
\(762\) −9.36932 −0.339415
\(763\) −2.00000 −0.0724049
\(764\) −4.27652 −0.154719
\(765\) −5.12311 −0.185226
\(766\) −1.56155 −0.0564212
\(767\) 7.36932 0.266091
\(768\) 10.0540 0.362792
\(769\) 14.8078 0.533982 0.266991 0.963699i \(-0.413971\pi\)
0.266991 + 0.963699i \(0.413971\pi\)
\(770\) −26.2462 −0.945848
\(771\) −22.5616 −0.812534
\(772\) 0.876894 0.0315601
\(773\) −50.9848 −1.83380 −0.916899 0.399120i \(-0.869316\pi\)
−0.916899 + 0.399120i \(0.869316\pi\)
\(774\) −12.0000 −0.431331
\(775\) −4.00000 −0.143684
\(776\) 27.1231 0.973663
\(777\) −8.24621 −0.295831
\(778\) 21.8617 0.783781
\(779\) −24.1771 −0.866234
\(780\) 2.24621 0.0804273
\(781\) −52.4924 −1.87833
\(782\) −16.0000 −0.572159
\(783\) −7.12311 −0.254559
\(784\) −4.68466 −0.167309
\(785\) 35.8617 1.27996
\(786\) −26.2462 −0.936171
\(787\) 42.2462 1.50591 0.752957 0.658069i \(-0.228627\pi\)
0.752957 + 0.658069i \(0.228627\pi\)
\(788\) 10.1080 0.360081
\(789\) 31.3693 1.11678
\(790\) 12.4924 0.444460
\(791\) 5.36932 0.190911
\(792\) 16.0000 0.568535
\(793\) −24.4924 −0.869751
\(794\) −35.6155 −1.26395
\(795\) −2.87689 −0.102033
\(796\) 4.38447 0.155403
\(797\) −20.2462 −0.717158 −0.358579 0.933499i \(-0.616739\pi\)
−0.358579 + 0.933499i \(0.616739\pi\)
\(798\) −10.2462 −0.362712
\(799\) −10.8769 −0.384797
\(800\) −3.80776 −0.134625
\(801\) −14.5616 −0.514507
\(802\) 7.61553 0.268914
\(803\) −29.9309 −1.05624
\(804\) −3.36932 −0.118827
\(805\) 13.1231 0.462529
\(806\) −8.00000 −0.281788
\(807\) 0.315342 0.0111005
\(808\) −11.8920 −0.418361
\(809\) −52.1080 −1.83202 −0.916009 0.401158i \(-0.868608\pi\)
−0.916009 + 0.401158i \(0.868608\pi\)
\(810\) 4.00000 0.140546
\(811\) 18.9848 0.666648 0.333324 0.942812i \(-0.391830\pi\)
0.333324 + 0.942812i \(0.391830\pi\)
\(812\) −3.12311 −0.109600
\(813\) 11.7538 0.412223
\(814\) 84.4924 2.96146
\(815\) −32.9848 −1.15541
\(816\) 9.36932 0.327992
\(817\) 50.4233 1.76409
\(818\) −16.3845 −0.572870
\(819\) 2.00000 0.0698857
\(820\) 4.13826 0.144514
\(821\) −44.1080 −1.53938 −0.769689 0.638419i \(-0.779589\pi\)
−0.769689 + 0.638419i \(0.779589\pi\)
\(822\) −6.63068 −0.231272
\(823\) 41.3002 1.43963 0.719817 0.694164i \(-0.244226\pi\)
0.719817 + 0.694164i \(0.244226\pi\)
\(824\) 13.2614 0.461982
\(825\) −10.2462 −0.356727
\(826\) 5.75379 0.200200
\(827\) −30.2462 −1.05176 −0.525882 0.850558i \(-0.676265\pi\)
−0.525882 + 0.850558i \(0.676265\pi\)
\(828\) 2.24621 0.0780612
\(829\) −11.9309 −0.414376 −0.207188 0.978301i \(-0.566431\pi\)
−0.207188 + 0.978301i \(0.566431\pi\)
\(830\) 16.0000 0.555368
\(831\) −0.0691303 −0.00239810
\(832\) 11.1231 0.385624
\(833\) −2.00000 −0.0692959
\(834\) 8.49242 0.294069
\(835\) −4.49242 −0.155467
\(836\) 18.8769 0.652871
\(837\) −2.56155 −0.0885402
\(838\) −34.7386 −1.20003
\(839\) 2.24621 0.0775478 0.0387739 0.999248i \(-0.487655\pi\)
0.0387739 + 0.999248i \(0.487655\pi\)
\(840\) −6.24621 −0.215515
\(841\) 21.7386 0.749608
\(842\) −25.3693 −0.874284
\(843\) 15.0540 0.518486
\(844\) 6.63068 0.228238
\(845\) −23.0540 −0.793081
\(846\) 8.49242 0.291976
\(847\) 32.0540 1.10139
\(848\) 5.26137 0.180676
\(849\) 12.8769 0.441934
\(850\) −4.87689 −0.167276
\(851\) −42.2462 −1.44818
\(852\) 3.50758 0.120168
\(853\) 46.1771 1.58107 0.790537 0.612415i \(-0.209802\pi\)
0.790537 + 0.612415i \(0.209802\pi\)
\(854\) −19.1231 −0.654379
\(855\) −16.8078 −0.574813
\(856\) −34.7386 −1.18734
\(857\) 43.3693 1.48147 0.740734 0.671799i \(-0.234478\pi\)
0.740734 + 0.671799i \(0.234478\pi\)
\(858\) −20.4924 −0.699600
\(859\) −15.5076 −0.529112 −0.264556 0.964370i \(-0.585225\pi\)
−0.264556 + 0.964370i \(0.585225\pi\)
\(860\) −8.63068 −0.294304
\(861\) 3.68466 0.125573
\(862\) 29.4773 1.00400
\(863\) −12.8078 −0.435981 −0.217991 0.975951i \(-0.569950\pi\)
−0.217991 + 0.975951i \(0.569950\pi\)
\(864\) −2.43845 −0.0829577
\(865\) −25.6155 −0.870954
\(866\) −28.1080 −0.955147
\(867\) −13.0000 −0.441503
\(868\) −1.12311 −0.0381207
\(869\) −20.4924 −0.695158
\(870\) −28.4924 −0.965984
\(871\) −15.3693 −0.520769
\(872\) 4.87689 0.165152
\(873\) −11.1231 −0.376460
\(874\) −52.4924 −1.77558
\(875\) −8.80776 −0.297757
\(876\) 2.00000 0.0675737
\(877\) −35.1231 −1.18602 −0.593012 0.805194i \(-0.702061\pi\)
−0.593012 + 0.805194i \(0.702061\pi\)
\(878\) −18.2462 −0.615780
\(879\) −14.0000 −0.472208
\(880\) 78.7386 2.65428
\(881\) −24.8078 −0.835795 −0.417897 0.908494i \(-0.637233\pi\)
−0.417897 + 0.908494i \(0.637233\pi\)
\(882\) 1.56155 0.0525802
\(883\) −28.7386 −0.967132 −0.483566 0.875308i \(-0.660659\pi\)
−0.483566 + 0.875308i \(0.660659\pi\)
\(884\) −1.75379 −0.0589863
\(885\) 9.43845 0.317270
\(886\) −42.2462 −1.41929
\(887\) 4.49242 0.150841 0.0754204 0.997152i \(-0.475970\pi\)
0.0754204 + 0.997152i \(0.475970\pi\)
\(888\) 20.1080 0.674779
\(889\) −6.00000 −0.201234
\(890\) −58.2462 −1.95242
\(891\) −6.56155 −0.219820
\(892\) 3.36932 0.112813
\(893\) −35.6847 −1.19414
\(894\) −8.38447 −0.280419
\(895\) 2.06913 0.0691634
\(896\) 13.5616 0.453060
\(897\) 10.2462 0.342111
\(898\) 17.7538 0.592452
\(899\) 18.2462 0.608545
\(900\) 0.684658 0.0228219
\(901\) 2.24621 0.0748321
\(902\) −37.7538 −1.25706
\(903\) −7.68466 −0.255729
\(904\) −13.0928 −0.435460
\(905\) −2.24621 −0.0746666
\(906\) 25.3693 0.842839
\(907\) 47.4773 1.57646 0.788228 0.615383i \(-0.210999\pi\)
0.788228 + 0.615383i \(0.210999\pi\)
\(908\) −2.73863 −0.0908848
\(909\) 4.87689 0.161756
\(910\) 8.00000 0.265197
\(911\) −29.9309 −0.991654 −0.495827 0.868421i \(-0.665135\pi\)
−0.495827 + 0.868421i \(0.665135\pi\)
\(912\) 30.7386 1.01786
\(913\) −26.2462 −0.868623
\(914\) 0.384472 0.0127172
\(915\) −31.3693 −1.03704
\(916\) 5.50758 0.181975
\(917\) −16.8078 −0.555041
\(918\) −3.12311 −0.103078
\(919\) 23.0540 0.760480 0.380240 0.924888i \(-0.375841\pi\)
0.380240 + 0.924888i \(0.375841\pi\)
\(920\) −32.0000 −1.05501
\(921\) 0.246211 0.00811294
\(922\) −4.49242 −0.147950
\(923\) 16.0000 0.526646
\(924\) −2.87689 −0.0946429
\(925\) −12.8769 −0.423390
\(926\) 46.8466 1.53947
\(927\) −5.43845 −0.178622
\(928\) 17.3693 0.570176
\(929\) −49.1231 −1.61168 −0.805838 0.592136i \(-0.798285\pi\)
−0.805838 + 0.592136i \(0.798285\pi\)
\(930\) −10.2462 −0.335987
\(931\) −6.56155 −0.215046
\(932\) −2.38447 −0.0781060
\(933\) −33.3002 −1.09020
\(934\) −46.2462 −1.51322
\(935\) 33.6155 1.09935
\(936\) −4.87689 −0.159406
\(937\) 10.4924 0.342773 0.171386 0.985204i \(-0.445175\pi\)
0.171386 + 0.985204i \(0.445175\pi\)
\(938\) −12.0000 −0.391814
\(939\) 18.1771 0.593187
\(940\) 6.10795 0.199219
\(941\) 41.2311 1.34409 0.672047 0.740508i \(-0.265415\pi\)
0.672047 + 0.740508i \(0.265415\pi\)
\(942\) 21.8617 0.712294
\(943\) 18.8769 0.614716
\(944\) −17.2614 −0.561810
\(945\) 2.56155 0.0833273
\(946\) 78.7386 2.56001
\(947\) −2.06913 −0.0672377 −0.0336189 0.999435i \(-0.510703\pi\)
−0.0336189 + 0.999435i \(0.510703\pi\)
\(948\) 1.36932 0.0444733
\(949\) 9.12311 0.296149
\(950\) −16.0000 −0.519109
\(951\) 6.63068 0.215015
\(952\) 4.87689 0.158061
\(953\) 19.3693 0.627434 0.313717 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(954\) −1.75379 −0.0567810
\(955\) −24.9848 −0.808491
\(956\) −10.3845 −0.335858
\(957\) 46.7386 1.51085
\(958\) 0.984845 0.0318189
\(959\) −4.24621 −0.137117
\(960\) 14.2462 0.459794
\(961\) −24.4384 −0.788337
\(962\) −25.7538 −0.830335
\(963\) 14.2462 0.459078
\(964\) −7.12311 −0.229420
\(965\) 5.12311 0.164919
\(966\) 8.00000 0.257396
\(967\) −6.73863 −0.216700 −0.108350 0.994113i \(-0.534557\pi\)
−0.108350 + 0.994113i \(0.534557\pi\)
\(968\) −78.1619 −2.51222
\(969\) 13.1231 0.421575
\(970\) −44.4924 −1.42857
\(971\) −42.7386 −1.37155 −0.685774 0.727815i \(-0.740536\pi\)
−0.685774 + 0.727815i \(0.740536\pi\)
\(972\) 0.438447 0.0140632
\(973\) 5.43845 0.174349
\(974\) 56.6004 1.81359
\(975\) 3.12311 0.100019
\(976\) 57.3693 1.83635
\(977\) −55.6847 −1.78151 −0.890755 0.454484i \(-0.849824\pi\)
−0.890755 + 0.454484i \(0.849824\pi\)
\(978\) −20.1080 −0.642981
\(979\) 95.5464 3.05368
\(980\) 1.12311 0.0358763
\(981\) −2.00000 −0.0638551
\(982\) 23.2311 0.741333
\(983\) 52.4924 1.67425 0.837124 0.547013i \(-0.184235\pi\)
0.837124 + 0.547013i \(0.184235\pi\)
\(984\) −8.98485 −0.286426
\(985\) 59.0540 1.88162
\(986\) 22.2462 0.708464
\(987\) 5.43845 0.173108
\(988\) −5.75379 −0.183052
\(989\) −39.3693 −1.25187
\(990\) −26.2462 −0.834159
\(991\) 24.9848 0.793670 0.396835 0.917890i \(-0.370109\pi\)
0.396835 + 0.917890i \(0.370109\pi\)
\(992\) 6.24621 0.198317
\(993\) 10.2462 0.325154
\(994\) 12.4924 0.396236
\(995\) 25.6155 0.812067
\(996\) 1.75379 0.0555709
\(997\) 47.6155 1.50800 0.753999 0.656875i \(-0.228122\pi\)
0.753999 + 0.656875i \(0.228122\pi\)
\(998\) −35.5076 −1.12397
\(999\) −8.24621 −0.260899
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 8043.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.l.1.2 2 1.1 even 1 trivial