Properties

Label 8022.2.a.r.1.10
Level $8022$
Weight $2$
Character 8022.1
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $10$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 28x^{7} + 114x^{6} - 110x^{5} - 282x^{4} + 149x^{3} + 285x^{2} - 49x - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-3.07769\) of defining polynomial
Character \(\chi\) \(=\) 8022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.07769 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.07769 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.07769 q^{10} +2.56478 q^{11} -1.00000 q^{12} +1.57278 q^{13} +1.00000 q^{14} -4.07769 q^{15} +1.00000 q^{16} -5.73372 q^{17} +1.00000 q^{18} +3.75278 q^{19} +4.07769 q^{20} -1.00000 q^{21} +2.56478 q^{22} +3.96884 q^{23} -1.00000 q^{24} +11.6276 q^{25} +1.57278 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.01074 q^{29} -4.07769 q^{30} -3.45946 q^{31} +1.00000 q^{32} -2.56478 q^{33} -5.73372 q^{34} +4.07769 q^{35} +1.00000 q^{36} -2.32402 q^{37} +3.75278 q^{38} -1.57278 q^{39} +4.07769 q^{40} +9.32677 q^{41} -1.00000 q^{42} -6.29321 q^{43} +2.56478 q^{44} +4.07769 q^{45} +3.96884 q^{46} +5.11837 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.6276 q^{50} +5.73372 q^{51} +1.57278 q^{52} -1.69413 q^{53} -1.00000 q^{54} +10.4584 q^{55} +1.00000 q^{56} -3.75278 q^{57} -3.01074 q^{58} -2.33548 q^{59} -4.07769 q^{60} +0.135157 q^{61} -3.45946 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.41332 q^{65} -2.56478 q^{66} +0.892401 q^{67} -5.73372 q^{68} -3.96884 q^{69} +4.07769 q^{70} +0.594616 q^{71} +1.00000 q^{72} +7.96042 q^{73} -2.32402 q^{74} -11.6276 q^{75} +3.75278 q^{76} +2.56478 q^{77} -1.57278 q^{78} +5.28701 q^{79} +4.07769 q^{80} +1.00000 q^{81} +9.32677 q^{82} -4.59673 q^{83} -1.00000 q^{84} -23.3804 q^{85} -6.29321 q^{86} +3.01074 q^{87} +2.56478 q^{88} +0.429519 q^{89} +4.07769 q^{90} +1.57278 q^{91} +3.96884 q^{92} +3.45946 q^{93} +5.11837 q^{94} +15.3027 q^{95} -1.00000 q^{96} +1.46339 q^{97} +1.00000 q^{98} +2.56478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 8 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 8 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9} + 8 q^{10} + 4 q^{11} - 10 q^{12} + 6 q^{13} + 10 q^{14} - 8 q^{15} + 10 q^{16} + 5 q^{17} + 10 q^{18} + 15 q^{19} + 8 q^{20} - 10 q^{21} + 4 q^{22} + 12 q^{23} - 10 q^{24} - 2 q^{25} + 6 q^{26} - 10 q^{27} + 10 q^{28} + 7 q^{29} - 8 q^{30} + 28 q^{31} + 10 q^{32} - 4 q^{33} + 5 q^{34} + 8 q^{35} + 10 q^{36} - 3 q^{37} + 15 q^{38} - 6 q^{39} + 8 q^{40} + 24 q^{41} - 10 q^{42} + 4 q^{43} + 4 q^{44} + 8 q^{45} + 12 q^{46} + 16 q^{47} - 10 q^{48} + 10 q^{49} - 2 q^{50} - 5 q^{51} + 6 q^{52} + 5 q^{53} - 10 q^{54} - q^{55} + 10 q^{56} - 15 q^{57} + 7 q^{58} + 17 q^{59} - 8 q^{60} + 15 q^{61} + 28 q^{62} + 10 q^{63} + 10 q^{64} + 14 q^{65} - 4 q^{66} + 6 q^{67} + 5 q^{68} - 12 q^{69} + 8 q^{70} + 5 q^{71} + 10 q^{72} + 16 q^{73} - 3 q^{74} + 2 q^{75} + 15 q^{76} + 4 q^{77} - 6 q^{78} - 5 q^{79} + 8 q^{80} + 10 q^{81} + 24 q^{82} + 24 q^{83} - 10 q^{84} - 19 q^{85} + 4 q^{86} - 7 q^{87} + 4 q^{88} + 17 q^{89} + 8 q^{90} + 6 q^{91} + 12 q^{92} - 28 q^{93} + 16 q^{94} + 15 q^{95} - 10 q^{96} + 20 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 4.07769 1.82360 0.911800 0.410635i \(-0.134693\pi\)
0.911800 + 0.410635i \(0.134693\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.07769 1.28948
\(11\) 2.56478 0.773311 0.386656 0.922224i \(-0.373630\pi\)
0.386656 + 0.922224i \(0.373630\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.57278 0.436211 0.218105 0.975925i \(-0.430012\pi\)
0.218105 + 0.975925i \(0.430012\pi\)
\(14\) 1.00000 0.267261
\(15\) −4.07769 −1.05286
\(16\) 1.00000 0.250000
\(17\) −5.73372 −1.39063 −0.695316 0.718704i \(-0.744735\pi\)
−0.695316 + 0.718704i \(0.744735\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.75278 0.860947 0.430474 0.902603i \(-0.358347\pi\)
0.430474 + 0.902603i \(0.358347\pi\)
\(20\) 4.07769 0.911800
\(21\) −1.00000 −0.218218
\(22\) 2.56478 0.546814
\(23\) 3.96884 0.827560 0.413780 0.910377i \(-0.364208\pi\)
0.413780 + 0.910377i \(0.364208\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.6276 2.32552
\(26\) 1.57278 0.308448
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −3.01074 −0.559081 −0.279540 0.960134i \(-0.590182\pi\)
−0.279540 + 0.960134i \(0.590182\pi\)
\(30\) −4.07769 −0.744482
\(31\) −3.45946 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.56478 −0.446471
\(34\) −5.73372 −0.983325
\(35\) 4.07769 0.689256
\(36\) 1.00000 0.166667
\(37\) −2.32402 −0.382067 −0.191034 0.981584i \(-0.561184\pi\)
−0.191034 + 0.981584i \(0.561184\pi\)
\(38\) 3.75278 0.608782
\(39\) −1.57278 −0.251847
\(40\) 4.07769 0.644740
\(41\) 9.32677 1.45660 0.728298 0.685261i \(-0.240312\pi\)
0.728298 + 0.685261i \(0.240312\pi\)
\(42\) −1.00000 −0.154303
\(43\) −6.29321 −0.959706 −0.479853 0.877349i \(-0.659310\pi\)
−0.479853 + 0.877349i \(0.659310\pi\)
\(44\) 2.56478 0.386656
\(45\) 4.07769 0.607867
\(46\) 3.96884 0.585173
\(47\) 5.11837 0.746591 0.373295 0.927713i \(-0.378228\pi\)
0.373295 + 0.927713i \(0.378228\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 11.6276 1.64439
\(51\) 5.73372 0.802881
\(52\) 1.57278 0.218105
\(53\) −1.69413 −0.232706 −0.116353 0.993208i \(-0.537120\pi\)
−0.116353 + 0.993208i \(0.537120\pi\)
\(54\) −1.00000 −0.136083
\(55\) 10.4584 1.41021
\(56\) 1.00000 0.133631
\(57\) −3.75278 −0.497068
\(58\) −3.01074 −0.395330
\(59\) −2.33548 −0.304054 −0.152027 0.988376i \(-0.548580\pi\)
−0.152027 + 0.988376i \(0.548580\pi\)
\(60\) −4.07769 −0.526428
\(61\) 0.135157 0.0173050 0.00865252 0.999963i \(-0.497246\pi\)
0.00865252 + 0.999963i \(0.497246\pi\)
\(62\) −3.45946 −0.439351
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 6.41332 0.795474
\(66\) −2.56478 −0.315703
\(67\) 0.892401 0.109024 0.0545121 0.998513i \(-0.482640\pi\)
0.0545121 + 0.998513i \(0.482640\pi\)
\(68\) −5.73372 −0.695316
\(69\) −3.96884 −0.477792
\(70\) 4.07769 0.487378
\(71\) 0.594616 0.0705679 0.0352839 0.999377i \(-0.488766\pi\)
0.0352839 + 0.999377i \(0.488766\pi\)
\(72\) 1.00000 0.117851
\(73\) 7.96042 0.931697 0.465848 0.884865i \(-0.345749\pi\)
0.465848 + 0.884865i \(0.345749\pi\)
\(74\) −2.32402 −0.270162
\(75\) −11.6276 −1.34264
\(76\) 3.75278 0.430474
\(77\) 2.56478 0.292284
\(78\) −1.57278 −0.178082
\(79\) 5.28701 0.594835 0.297417 0.954748i \(-0.403875\pi\)
0.297417 + 0.954748i \(0.403875\pi\)
\(80\) 4.07769 0.455900
\(81\) 1.00000 0.111111
\(82\) 9.32677 1.02997
\(83\) −4.59673 −0.504557 −0.252278 0.967655i \(-0.581180\pi\)
−0.252278 + 0.967655i \(0.581180\pi\)
\(84\) −1.00000 −0.109109
\(85\) −23.3804 −2.53596
\(86\) −6.29321 −0.678615
\(87\) 3.01074 0.322785
\(88\) 2.56478 0.273407
\(89\) 0.429519 0.0455290 0.0227645 0.999741i \(-0.492753\pi\)
0.0227645 + 0.999741i \(0.492753\pi\)
\(90\) 4.07769 0.429827
\(91\) 1.57278 0.164872
\(92\) 3.96884 0.413780
\(93\) 3.45946 0.358729
\(94\) 5.11837 0.527919
\(95\) 15.3027 1.57002
\(96\) −1.00000 −0.102062
\(97\) 1.46339 0.148584 0.0742922 0.997237i \(-0.476330\pi\)
0.0742922 + 0.997237i \(0.476330\pi\)
\(98\) 1.00000 0.101015
\(99\) 2.56478 0.257770
\(100\) 11.6276 1.16276
\(101\) −7.75739 −0.771890 −0.385945 0.922522i \(-0.626124\pi\)
−0.385945 + 0.922522i \(0.626124\pi\)
\(102\) 5.73372 0.567723
\(103\) 7.79971 0.768528 0.384264 0.923223i \(-0.374455\pi\)
0.384264 + 0.923223i \(0.374455\pi\)
\(104\) 1.57278 0.154224
\(105\) −4.07769 −0.397942
\(106\) −1.69413 −0.164548
\(107\) 3.74273 0.361823 0.180912 0.983499i \(-0.442095\pi\)
0.180912 + 0.983499i \(0.442095\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.11979 0.203039 0.101520 0.994834i \(-0.467630\pi\)
0.101520 + 0.994834i \(0.467630\pi\)
\(110\) 10.4584 0.997169
\(111\) 2.32402 0.220587
\(112\) 1.00000 0.0944911
\(113\) 4.22957 0.397884 0.198942 0.980011i \(-0.436249\pi\)
0.198942 + 0.980011i \(0.436249\pi\)
\(114\) −3.75278 −0.351480
\(115\) 16.1837 1.50914
\(116\) −3.01074 −0.279540
\(117\) 1.57278 0.145404
\(118\) −2.33548 −0.214999
\(119\) −5.73372 −0.525609
\(120\) −4.07769 −0.372241
\(121\) −4.42189 −0.401990
\(122\) 0.135157 0.0122365
\(123\) −9.32677 −0.840966
\(124\) −3.45946 −0.310668
\(125\) 27.0252 2.41721
\(126\) 1.00000 0.0890871
\(127\) −11.6091 −1.03014 −0.515069 0.857149i \(-0.672234\pi\)
−0.515069 + 0.857149i \(0.672234\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.29321 0.554086
\(130\) 6.41332 0.562485
\(131\) 11.3478 0.991460 0.495730 0.868477i \(-0.334901\pi\)
0.495730 + 0.868477i \(0.334901\pi\)
\(132\) −2.56478 −0.223236
\(133\) 3.75278 0.325408
\(134\) 0.892401 0.0770917
\(135\) −4.07769 −0.350952
\(136\) −5.73372 −0.491662
\(137\) −7.15669 −0.611437 −0.305719 0.952122i \(-0.598897\pi\)
−0.305719 + 0.952122i \(0.598897\pi\)
\(138\) −3.96884 −0.337850
\(139\) −7.99822 −0.678400 −0.339200 0.940714i \(-0.610156\pi\)
−0.339200 + 0.940714i \(0.610156\pi\)
\(140\) 4.07769 0.344628
\(141\) −5.11837 −0.431044
\(142\) 0.594616 0.0498990
\(143\) 4.03384 0.337327
\(144\) 1.00000 0.0833333
\(145\) −12.2769 −1.01954
\(146\) 7.96042 0.658809
\(147\) −1.00000 −0.0824786
\(148\) −2.32402 −0.191034
\(149\) −12.3489 −1.01166 −0.505830 0.862633i \(-0.668814\pi\)
−0.505830 + 0.862633i \(0.668814\pi\)
\(150\) −11.6276 −0.949388
\(151\) −3.04757 −0.248008 −0.124004 0.992282i \(-0.539574\pi\)
−0.124004 + 0.992282i \(0.539574\pi\)
\(152\) 3.75278 0.304391
\(153\) −5.73372 −0.463544
\(154\) 2.56478 0.206676
\(155\) −14.1066 −1.13307
\(156\) −1.57278 −0.125923
\(157\) −16.4234 −1.31073 −0.655363 0.755314i \(-0.727484\pi\)
−0.655363 + 0.755314i \(0.727484\pi\)
\(158\) 5.28701 0.420612
\(159\) 1.69413 0.134353
\(160\) 4.07769 0.322370
\(161\) 3.96884 0.312788
\(162\) 1.00000 0.0785674
\(163\) 9.19645 0.720322 0.360161 0.932890i \(-0.382722\pi\)
0.360161 + 0.932890i \(0.382722\pi\)
\(164\) 9.32677 0.728298
\(165\) −10.4584 −0.814185
\(166\) −4.59673 −0.356776
\(167\) 23.3380 1.80595 0.902976 0.429692i \(-0.141378\pi\)
0.902976 + 0.429692i \(0.141378\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −10.5264 −0.809720
\(170\) −23.3804 −1.79319
\(171\) 3.75278 0.286982
\(172\) −6.29321 −0.479853
\(173\) −2.19648 −0.166995 −0.0834976 0.996508i \(-0.526609\pi\)
−0.0834976 + 0.996508i \(0.526609\pi\)
\(174\) 3.01074 0.228244
\(175\) 11.6276 0.878963
\(176\) 2.56478 0.193328
\(177\) 2.33548 0.175546
\(178\) 0.429519 0.0321938
\(179\) 1.59185 0.118980 0.0594902 0.998229i \(-0.481052\pi\)
0.0594902 + 0.998229i \(0.481052\pi\)
\(180\) 4.07769 0.303933
\(181\) −10.1396 −0.753668 −0.376834 0.926281i \(-0.622987\pi\)
−0.376834 + 0.926281i \(0.622987\pi\)
\(182\) 1.57278 0.116582
\(183\) −0.135157 −0.00999107
\(184\) 3.96884 0.292587
\(185\) −9.47665 −0.696737
\(186\) 3.45946 0.253660
\(187\) −14.7057 −1.07539
\(188\) 5.11837 0.373295
\(189\) −1.00000 −0.0727393
\(190\) 15.3027 1.11017
\(191\) 1.00000 0.0723575
\(192\) −1.00000 −0.0721688
\(193\) 17.6252 1.26869 0.634346 0.773049i \(-0.281269\pi\)
0.634346 + 0.773049i \(0.281269\pi\)
\(194\) 1.46339 0.105065
\(195\) −6.41332 −0.459267
\(196\) 1.00000 0.0714286
\(197\) −17.5489 −1.25031 −0.625155 0.780501i \(-0.714964\pi\)
−0.625155 + 0.780501i \(0.714964\pi\)
\(198\) 2.56478 0.182271
\(199\) −14.5178 −1.02914 −0.514569 0.857449i \(-0.672048\pi\)
−0.514569 + 0.857449i \(0.672048\pi\)
\(200\) 11.6276 0.822194
\(201\) −0.892401 −0.0629451
\(202\) −7.75739 −0.545808
\(203\) −3.01074 −0.211313
\(204\) 5.73372 0.401441
\(205\) 38.0317 2.65625
\(206\) 7.79971 0.543431
\(207\) 3.96884 0.275853
\(208\) 1.57278 0.109053
\(209\) 9.62507 0.665780
\(210\) −4.07769 −0.281388
\(211\) 13.8456 0.953171 0.476585 0.879128i \(-0.341874\pi\)
0.476585 + 0.879128i \(0.341874\pi\)
\(212\) −1.69413 −0.116353
\(213\) −0.594616 −0.0407424
\(214\) 3.74273 0.255848
\(215\) −25.6618 −1.75012
\(216\) −1.00000 −0.0680414
\(217\) −3.45946 −0.234843
\(218\) 2.11979 0.143570
\(219\) −7.96042 −0.537915
\(220\) 10.4584 0.705105
\(221\) −9.01789 −0.606609
\(222\) 2.32402 0.155978
\(223\) 5.47594 0.366696 0.183348 0.983048i \(-0.441306\pi\)
0.183348 + 0.983048i \(0.441306\pi\)
\(224\) 1.00000 0.0668153
\(225\) 11.6276 0.775172
\(226\) 4.22957 0.281347
\(227\) 13.7497 0.912601 0.456301 0.889826i \(-0.349174\pi\)
0.456301 + 0.889826i \(0.349174\pi\)
\(228\) −3.75278 −0.248534
\(229\) −2.69390 −0.178018 −0.0890089 0.996031i \(-0.528370\pi\)
−0.0890089 + 0.996031i \(0.528370\pi\)
\(230\) 16.1837 1.06712
\(231\) −2.56478 −0.168750
\(232\) −3.01074 −0.197665
\(233\) 11.4638 0.751021 0.375511 0.926818i \(-0.377467\pi\)
0.375511 + 0.926818i \(0.377467\pi\)
\(234\) 1.57278 0.102816
\(235\) 20.8711 1.36148
\(236\) −2.33548 −0.152027
\(237\) −5.28701 −0.343428
\(238\) −5.73372 −0.371662
\(239\) −20.2601 −1.31052 −0.655258 0.755406i \(-0.727440\pi\)
−0.655258 + 0.755406i \(0.727440\pi\)
\(240\) −4.07769 −0.263214
\(241\) 10.0873 0.649779 0.324890 0.945752i \(-0.394673\pi\)
0.324890 + 0.945752i \(0.394673\pi\)
\(242\) −4.42189 −0.284250
\(243\) −1.00000 −0.0641500
\(244\) 0.135157 0.00865252
\(245\) 4.07769 0.260514
\(246\) −9.32677 −0.594653
\(247\) 5.90231 0.375555
\(248\) −3.45946 −0.219676
\(249\) 4.59673 0.291306
\(250\) 27.0252 1.70923
\(251\) −17.4515 −1.10153 −0.550764 0.834661i \(-0.685664\pi\)
−0.550764 + 0.834661i \(0.685664\pi\)
\(252\) 1.00000 0.0629941
\(253\) 10.1792 0.639961
\(254\) −11.6091 −0.728418
\(255\) 23.3804 1.46413
\(256\) 1.00000 0.0625000
\(257\) −18.6271 −1.16193 −0.580965 0.813929i \(-0.697325\pi\)
−0.580965 + 0.813929i \(0.697325\pi\)
\(258\) 6.29321 0.391798
\(259\) −2.32402 −0.144408
\(260\) 6.41332 0.397737
\(261\) −3.01074 −0.186360
\(262\) 11.3478 0.701068
\(263\) 12.7520 0.786324 0.393162 0.919469i \(-0.371381\pi\)
0.393162 + 0.919469i \(0.371381\pi\)
\(264\) −2.56478 −0.157851
\(265\) −6.90814 −0.424363
\(266\) 3.75278 0.230098
\(267\) −0.429519 −0.0262862
\(268\) 0.892401 0.0545121
\(269\) −19.8773 −1.21194 −0.605972 0.795486i \(-0.707216\pi\)
−0.605972 + 0.795486i \(0.707216\pi\)
\(270\) −4.07769 −0.248161
\(271\) 0.00929113 0.000564396 0 0.000282198 1.00000i \(-0.499910\pi\)
0.000282198 1.00000i \(0.499910\pi\)
\(272\) −5.73372 −0.347658
\(273\) −1.57278 −0.0951890
\(274\) −7.15669 −0.432351
\(275\) 29.8222 1.79835
\(276\) −3.96884 −0.238896
\(277\) 20.3778 1.22438 0.612192 0.790709i \(-0.290288\pi\)
0.612192 + 0.790709i \(0.290288\pi\)
\(278\) −7.99822 −0.479701
\(279\) −3.45946 −0.207112
\(280\) 4.07769 0.243689
\(281\) −32.5466 −1.94157 −0.970784 0.239954i \(-0.922867\pi\)
−0.970784 + 0.239954i \(0.922867\pi\)
\(282\) −5.11837 −0.304794
\(283\) −12.0390 −0.715642 −0.357821 0.933790i \(-0.616480\pi\)
−0.357821 + 0.933790i \(0.616480\pi\)
\(284\) 0.594616 0.0352839
\(285\) −15.3027 −0.906454
\(286\) 4.03384 0.238526
\(287\) 9.32677 0.550542
\(288\) 1.00000 0.0589256
\(289\) 15.8755 0.933856
\(290\) −12.2769 −0.720923
\(291\) −1.46339 −0.0857852
\(292\) 7.96042 0.465848
\(293\) −7.57841 −0.442735 −0.221368 0.975190i \(-0.571052\pi\)
−0.221368 + 0.975190i \(0.571052\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −9.52338 −0.554473
\(296\) −2.32402 −0.135081
\(297\) −2.56478 −0.148824
\(298\) −12.3489 −0.715352
\(299\) 6.24211 0.360991
\(300\) −11.6276 −0.671319
\(301\) −6.29321 −0.362735
\(302\) −3.04757 −0.175368
\(303\) 7.75739 0.445651
\(304\) 3.75278 0.215237
\(305\) 0.551128 0.0315575
\(306\) −5.73372 −0.327775
\(307\) −9.76130 −0.557107 −0.278554 0.960421i \(-0.589855\pi\)
−0.278554 + 0.960421i \(0.589855\pi\)
\(308\) 2.56478 0.146142
\(309\) −7.79971 −0.443710
\(310\) −14.1066 −0.801201
\(311\) 24.7211 1.40180 0.700901 0.713258i \(-0.252781\pi\)
0.700901 + 0.713258i \(0.252781\pi\)
\(312\) −1.57278 −0.0890412
\(313\) 20.9113 1.18198 0.590989 0.806680i \(-0.298738\pi\)
0.590989 + 0.806680i \(0.298738\pi\)
\(314\) −16.4234 −0.926823
\(315\) 4.07769 0.229752
\(316\) 5.28701 0.297417
\(317\) −16.1572 −0.907478 −0.453739 0.891135i \(-0.649910\pi\)
−0.453739 + 0.891135i \(0.649910\pi\)
\(318\) 1.69413 0.0950020
\(319\) −7.72190 −0.432343
\(320\) 4.07769 0.227950
\(321\) −3.74273 −0.208899
\(322\) 3.96884 0.221175
\(323\) −21.5174 −1.19726
\(324\) 1.00000 0.0555556
\(325\) 18.2876 1.01442
\(326\) 9.19645 0.509344
\(327\) −2.11979 −0.117225
\(328\) 9.32677 0.514984
\(329\) 5.11837 0.282185
\(330\) −10.4584 −0.575716
\(331\) 5.72173 0.314495 0.157247 0.987559i \(-0.449738\pi\)
0.157247 + 0.987559i \(0.449738\pi\)
\(332\) −4.59673 −0.252278
\(333\) −2.32402 −0.127356
\(334\) 23.3380 1.27700
\(335\) 3.63894 0.198816
\(336\) −1.00000 −0.0545545
\(337\) −4.89728 −0.266772 −0.133386 0.991064i \(-0.542585\pi\)
−0.133386 + 0.991064i \(0.542585\pi\)
\(338\) −10.5264 −0.572559
\(339\) −4.22957 −0.229719
\(340\) −23.3804 −1.26798
\(341\) −8.87275 −0.480486
\(342\) 3.75278 0.202927
\(343\) 1.00000 0.0539949
\(344\) −6.29321 −0.339307
\(345\) −16.1837 −0.871301
\(346\) −2.19648 −0.118083
\(347\) −6.51461 −0.349723 −0.174861 0.984593i \(-0.555948\pi\)
−0.174861 + 0.984593i \(0.555948\pi\)
\(348\) 3.01074 0.161393
\(349\) −6.41046 −0.343144 −0.171572 0.985172i \(-0.554885\pi\)
−0.171572 + 0.985172i \(0.554885\pi\)
\(350\) 11.6276 0.621520
\(351\) −1.57278 −0.0839488
\(352\) 2.56478 0.136703
\(353\) −13.6793 −0.728075 −0.364037 0.931384i \(-0.618602\pi\)
−0.364037 + 0.931384i \(0.618602\pi\)
\(354\) 2.33548 0.124129
\(355\) 2.42466 0.128688
\(356\) 0.429519 0.0227645
\(357\) 5.73372 0.303461
\(358\) 1.59185 0.0841319
\(359\) −23.8330 −1.25786 −0.628930 0.777462i \(-0.716507\pi\)
−0.628930 + 0.777462i \(0.716507\pi\)
\(360\) 4.07769 0.214913
\(361\) −4.91662 −0.258769
\(362\) −10.1396 −0.532923
\(363\) 4.42189 0.232089
\(364\) 1.57278 0.0824361
\(365\) 32.4602 1.69904
\(366\) −0.135157 −0.00706476
\(367\) 14.5553 0.759781 0.379891 0.925031i \(-0.375962\pi\)
0.379891 + 0.925031i \(0.375962\pi\)
\(368\) 3.96884 0.206890
\(369\) 9.32677 0.485532
\(370\) −9.47665 −0.492668
\(371\) −1.69413 −0.0879548
\(372\) 3.45946 0.179364
\(373\) −28.7370 −1.48795 −0.743974 0.668209i \(-0.767061\pi\)
−0.743974 + 0.668209i \(0.767061\pi\)
\(374\) −14.7057 −0.760416
\(375\) −27.0252 −1.39558
\(376\) 5.11837 0.263960
\(377\) −4.73524 −0.243877
\(378\) −1.00000 −0.0514344
\(379\) 2.21881 0.113972 0.0569862 0.998375i \(-0.481851\pi\)
0.0569862 + 0.998375i \(0.481851\pi\)
\(380\) 15.3027 0.785012
\(381\) 11.6091 0.594751
\(382\) 1.00000 0.0511645
\(383\) 2.18019 0.111402 0.0557012 0.998447i \(-0.482261\pi\)
0.0557012 + 0.998447i \(0.482261\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 10.4584 0.533009
\(386\) 17.6252 0.897101
\(387\) −6.29321 −0.319902
\(388\) 1.46339 0.0742922
\(389\) −27.0514 −1.37156 −0.685779 0.727810i \(-0.740538\pi\)
−0.685779 + 0.727810i \(0.740538\pi\)
\(390\) −6.41332 −0.324751
\(391\) −22.7562 −1.15083
\(392\) 1.00000 0.0505076
\(393\) −11.3478 −0.572420
\(394\) −17.5489 −0.884103
\(395\) 21.5588 1.08474
\(396\) 2.56478 0.128885
\(397\) 31.7466 1.59331 0.796657 0.604432i \(-0.206600\pi\)
0.796657 + 0.604432i \(0.206600\pi\)
\(398\) −14.5178 −0.727710
\(399\) −3.75278 −0.187874
\(400\) 11.6276 0.581379
\(401\) 22.3567 1.11644 0.558221 0.829692i \(-0.311484\pi\)
0.558221 + 0.829692i \(0.311484\pi\)
\(402\) −0.892401 −0.0445089
\(403\) −5.44097 −0.271034
\(404\) −7.75739 −0.385945
\(405\) 4.07769 0.202622
\(406\) −3.01074 −0.149421
\(407\) −5.96061 −0.295457
\(408\) 5.73372 0.283861
\(409\) 24.7892 1.22575 0.612875 0.790180i \(-0.290013\pi\)
0.612875 + 0.790180i \(0.290013\pi\)
\(410\) 38.0317 1.87825
\(411\) 7.15669 0.353014
\(412\) 7.79971 0.384264
\(413\) −2.33548 −0.114922
\(414\) 3.96884 0.195058
\(415\) −18.7441 −0.920110
\(416\) 1.57278 0.0771119
\(417\) 7.99822 0.391674
\(418\) 9.62507 0.470778
\(419\) 7.62384 0.372449 0.186225 0.982507i \(-0.440375\pi\)
0.186225 + 0.982507i \(0.440375\pi\)
\(420\) −4.07769 −0.198971
\(421\) −9.35074 −0.455727 −0.227863 0.973693i \(-0.573174\pi\)
−0.227863 + 0.973693i \(0.573174\pi\)
\(422\) 13.8456 0.673993
\(423\) 5.11837 0.248864
\(424\) −1.69413 −0.0822742
\(425\) −66.6693 −3.23394
\(426\) −0.594616 −0.0288092
\(427\) 0.135157 0.00654069
\(428\) 3.74273 0.180912
\(429\) −4.03384 −0.194756
\(430\) −25.6618 −1.23752
\(431\) −4.62815 −0.222930 −0.111465 0.993768i \(-0.535554\pi\)
−0.111465 + 0.993768i \(0.535554\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −1.89413 −0.0910261 −0.0455131 0.998964i \(-0.514492\pi\)
−0.0455131 + 0.998964i \(0.514492\pi\)
\(434\) −3.45946 −0.166059
\(435\) 12.2769 0.588631
\(436\) 2.11979 0.101520
\(437\) 14.8942 0.712486
\(438\) −7.96042 −0.380364
\(439\) −13.9275 −0.664723 −0.332362 0.943152i \(-0.607845\pi\)
−0.332362 + 0.943152i \(0.607845\pi\)
\(440\) 10.4584 0.498585
\(441\) 1.00000 0.0476190
\(442\) −9.01789 −0.428937
\(443\) 8.69247 0.412992 0.206496 0.978447i \(-0.433794\pi\)
0.206496 + 0.978447i \(0.433794\pi\)
\(444\) 2.32402 0.110293
\(445\) 1.75145 0.0830266
\(446\) 5.47594 0.259293
\(447\) 12.3489 0.584082
\(448\) 1.00000 0.0472456
\(449\) 29.6829 1.40082 0.700412 0.713739i \(-0.253000\pi\)
0.700412 + 0.713739i \(0.253000\pi\)
\(450\) 11.6276 0.548129
\(451\) 23.9211 1.12640
\(452\) 4.22957 0.198942
\(453\) 3.04757 0.143187
\(454\) 13.7497 0.645307
\(455\) 6.41332 0.300661
\(456\) −3.75278 −0.175740
\(457\) 13.7098 0.641319 0.320659 0.947195i \(-0.396096\pi\)
0.320659 + 0.947195i \(0.396096\pi\)
\(458\) −2.69390 −0.125878
\(459\) 5.73372 0.267627
\(460\) 16.1837 0.754569
\(461\) −22.5901 −1.05213 −0.526063 0.850446i \(-0.676332\pi\)
−0.526063 + 0.850446i \(0.676332\pi\)
\(462\) −2.56478 −0.119324
\(463\) 20.8874 0.970721 0.485360 0.874314i \(-0.338688\pi\)
0.485360 + 0.874314i \(0.338688\pi\)
\(464\) −3.01074 −0.139770
\(465\) 14.1066 0.654178
\(466\) 11.4638 0.531052
\(467\) 17.6380 0.816188 0.408094 0.912940i \(-0.366194\pi\)
0.408094 + 0.912940i \(0.366194\pi\)
\(468\) 1.57278 0.0727018
\(469\) 0.892401 0.0412072
\(470\) 20.8711 0.962714
\(471\) 16.4234 0.756748
\(472\) −2.33548 −0.107499
\(473\) −16.1407 −0.742151
\(474\) −5.28701 −0.242840
\(475\) 43.6358 2.00215
\(476\) −5.73372 −0.262805
\(477\) −1.69413 −0.0775688
\(478\) −20.2601 −0.926674
\(479\) 28.7142 1.31199 0.655993 0.754767i \(-0.272250\pi\)
0.655993 + 0.754767i \(0.272250\pi\)
\(480\) −4.07769 −0.186120
\(481\) −3.65518 −0.166662
\(482\) 10.0873 0.459463
\(483\) −3.96884 −0.180588
\(484\) −4.42189 −0.200995
\(485\) 5.96724 0.270958
\(486\) −1.00000 −0.0453609
\(487\) −9.60371 −0.435186 −0.217593 0.976040i \(-0.569820\pi\)
−0.217593 + 0.976040i \(0.569820\pi\)
\(488\) 0.135157 0.00611826
\(489\) −9.19645 −0.415878
\(490\) 4.07769 0.184211
\(491\) 19.0198 0.858350 0.429175 0.903221i \(-0.358804\pi\)
0.429175 + 0.903221i \(0.358804\pi\)
\(492\) −9.32677 −0.420483
\(493\) 17.2627 0.777475
\(494\) 5.90231 0.265557
\(495\) 10.4584 0.470070
\(496\) −3.45946 −0.155334
\(497\) 0.594616 0.0266722
\(498\) 4.59673 0.205984
\(499\) −1.33794 −0.0598943 −0.0299472 0.999551i \(-0.509534\pi\)
−0.0299472 + 0.999551i \(0.509534\pi\)
\(500\) 27.0252 1.20861
\(501\) −23.3380 −1.04267
\(502\) −17.4515 −0.778898
\(503\) 4.43400 0.197702 0.0988511 0.995102i \(-0.468483\pi\)
0.0988511 + 0.995102i \(0.468483\pi\)
\(504\) 1.00000 0.0445435
\(505\) −31.6323 −1.40762
\(506\) 10.1792 0.452521
\(507\) 10.5264 0.467492
\(508\) −11.6091 −0.515069
\(509\) −6.59567 −0.292348 −0.146174 0.989259i \(-0.546696\pi\)
−0.146174 + 0.989259i \(0.546696\pi\)
\(510\) 23.3804 1.03530
\(511\) 7.96042 0.352148
\(512\) 1.00000 0.0441942
\(513\) −3.75278 −0.165689
\(514\) −18.6271 −0.821608
\(515\) 31.8048 1.40149
\(516\) 6.29321 0.277043
\(517\) 13.1275 0.577347
\(518\) −2.32402 −0.102112
\(519\) 2.19648 0.0964147
\(520\) 6.41332 0.281243
\(521\) 6.18280 0.270873 0.135437 0.990786i \(-0.456756\pi\)
0.135437 + 0.990786i \(0.456756\pi\)
\(522\) −3.01074 −0.131777
\(523\) 15.8645 0.693708 0.346854 0.937919i \(-0.387250\pi\)
0.346854 + 0.937919i \(0.387250\pi\)
\(524\) 11.3478 0.495730
\(525\) −11.6276 −0.507469
\(526\) 12.7520 0.556015
\(527\) 19.8355 0.864050
\(528\) −2.56478 −0.111618
\(529\) −7.24832 −0.315144
\(530\) −6.90814 −0.300070
\(531\) −2.33548 −0.101351
\(532\) 3.75278 0.162704
\(533\) 14.6690 0.635383
\(534\) −0.429519 −0.0185871
\(535\) 15.2617 0.659821
\(536\) 0.892401 0.0385458
\(537\) −1.59185 −0.0686934
\(538\) −19.8773 −0.856973
\(539\) 2.56478 0.110473
\(540\) −4.07769 −0.175476
\(541\) 4.59720 0.197649 0.0988246 0.995105i \(-0.468492\pi\)
0.0988246 + 0.995105i \(0.468492\pi\)
\(542\) 0.00929113 0.000399088 0
\(543\) 10.1396 0.435130
\(544\) −5.73372 −0.245831
\(545\) 8.64386 0.370262
\(546\) −1.57278 −0.0673088
\(547\) 6.93569 0.296549 0.148274 0.988946i \(-0.452628\pi\)
0.148274 + 0.988946i \(0.452628\pi\)
\(548\) −7.15669 −0.305719
\(549\) 0.135157 0.00576835
\(550\) 29.8222 1.27162
\(551\) −11.2987 −0.481339
\(552\) −3.96884 −0.168925
\(553\) 5.28701 0.224826
\(554\) 20.3778 0.865771
\(555\) 9.47665 0.402262
\(556\) −7.99822 −0.339200
\(557\) 43.7679 1.85450 0.927252 0.374437i \(-0.122164\pi\)
0.927252 + 0.374437i \(0.122164\pi\)
\(558\) −3.45946 −0.146450
\(559\) −9.89785 −0.418634
\(560\) 4.07769 0.172314
\(561\) 14.7057 0.620877
\(562\) −32.5466 −1.37290
\(563\) 9.14786 0.385536 0.192768 0.981244i \(-0.438253\pi\)
0.192768 + 0.981244i \(0.438253\pi\)
\(564\) −5.11837 −0.215522
\(565\) 17.2469 0.725582
\(566\) −12.0390 −0.506035
\(567\) 1.00000 0.0419961
\(568\) 0.594616 0.0249495
\(569\) 17.4815 0.732863 0.366432 0.930445i \(-0.380579\pi\)
0.366432 + 0.930445i \(0.380579\pi\)
\(570\) −15.3027 −0.640959
\(571\) 37.3185 1.56173 0.780865 0.624700i \(-0.214779\pi\)
0.780865 + 0.624700i \(0.214779\pi\)
\(572\) 4.03384 0.168663
\(573\) −1.00000 −0.0417756
\(574\) 9.32677 0.389292
\(575\) 46.1480 1.92450
\(576\) 1.00000 0.0416667
\(577\) 17.6927 0.736555 0.368277 0.929716i \(-0.379948\pi\)
0.368277 + 0.929716i \(0.379948\pi\)
\(578\) 15.8755 0.660336
\(579\) −17.6252 −0.732480
\(580\) −12.2769 −0.509770
\(581\) −4.59673 −0.190705
\(582\) −1.46339 −0.0606593
\(583\) −4.34507 −0.179954
\(584\) 7.96042 0.329405
\(585\) 6.41332 0.265158
\(586\) −7.57841 −0.313061
\(587\) −4.97625 −0.205392 −0.102696 0.994713i \(-0.532747\pi\)
−0.102696 + 0.994713i \(0.532747\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −12.9826 −0.534938
\(590\) −9.52338 −0.392071
\(591\) 17.5489 0.721867
\(592\) −2.32402 −0.0955168
\(593\) −6.09259 −0.250193 −0.125096 0.992145i \(-0.539924\pi\)
−0.125096 + 0.992145i \(0.539924\pi\)
\(594\) −2.56478 −0.105234
\(595\) −23.3804 −0.958501
\(596\) −12.3489 −0.505830
\(597\) 14.5178 0.594173
\(598\) 6.24211 0.255259
\(599\) 5.24382 0.214257 0.107128 0.994245i \(-0.465834\pi\)
0.107128 + 0.994245i \(0.465834\pi\)
\(600\) −11.6276 −0.474694
\(601\) −13.8564 −0.565213 −0.282607 0.959236i \(-0.591199\pi\)
−0.282607 + 0.959236i \(0.591199\pi\)
\(602\) −6.29321 −0.256492
\(603\) 0.892401 0.0363414
\(604\) −3.04757 −0.124004
\(605\) −18.0311 −0.733069
\(606\) 7.75739 0.315123
\(607\) −13.5037 −0.548096 −0.274048 0.961716i \(-0.588363\pi\)
−0.274048 + 0.961716i \(0.588363\pi\)
\(608\) 3.75278 0.152195
\(609\) 3.01074 0.122001
\(610\) 0.551128 0.0223145
\(611\) 8.05007 0.325671
\(612\) −5.73372 −0.231772
\(613\) −41.5318 −1.67745 −0.838727 0.544551i \(-0.816700\pi\)
−0.838727 + 0.544551i \(0.816700\pi\)
\(614\) −9.76130 −0.393934
\(615\) −38.0317 −1.53359
\(616\) 2.56478 0.103338
\(617\) 34.0369 1.37027 0.685137 0.728414i \(-0.259742\pi\)
0.685137 + 0.728414i \(0.259742\pi\)
\(618\) −7.79971 −0.313750
\(619\) −28.1786 −1.13259 −0.566297 0.824202i \(-0.691624\pi\)
−0.566297 + 0.824202i \(0.691624\pi\)
\(620\) −14.1066 −0.566535
\(621\) −3.96884 −0.159264
\(622\) 24.7211 0.991224
\(623\) 0.429519 0.0172083
\(624\) −1.57278 −0.0629616
\(625\) 52.0628 2.08251
\(626\) 20.9113 0.835784
\(627\) −9.62507 −0.384388
\(628\) −16.4234 −0.655363
\(629\) 13.3253 0.531314
\(630\) 4.07769 0.162459
\(631\) −33.2427 −1.32337 −0.661686 0.749781i \(-0.730159\pi\)
−0.661686 + 0.749781i \(0.730159\pi\)
\(632\) 5.28701 0.210306
\(633\) −13.8456 −0.550313
\(634\) −16.1572 −0.641684
\(635\) −47.3382 −1.87856
\(636\) 1.69413 0.0671766
\(637\) 1.57278 0.0623159
\(638\) −7.72190 −0.305713
\(639\) 0.594616 0.0235226
\(640\) 4.07769 0.161185
\(641\) −44.1877 −1.74531 −0.872655 0.488338i \(-0.837603\pi\)
−0.872655 + 0.488338i \(0.837603\pi\)
\(642\) −3.74273 −0.147714
\(643\) 20.4912 0.808092 0.404046 0.914739i \(-0.367604\pi\)
0.404046 + 0.914739i \(0.367604\pi\)
\(644\) 3.96884 0.156394
\(645\) 25.6618 1.01043
\(646\) −21.5174 −0.846591
\(647\) 27.5450 1.08290 0.541452 0.840732i \(-0.317875\pi\)
0.541452 + 0.840732i \(0.317875\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.99001 −0.235128
\(650\) 18.2876 0.717300
\(651\) 3.45946 0.135587
\(652\) 9.19645 0.360161
\(653\) 20.9651 0.820429 0.410214 0.911989i \(-0.365454\pi\)
0.410214 + 0.911989i \(0.365454\pi\)
\(654\) −2.11979 −0.0828904
\(655\) 46.2728 1.80803
\(656\) 9.32677 0.364149
\(657\) 7.96042 0.310566
\(658\) 5.11837 0.199535
\(659\) −6.45875 −0.251597 −0.125799 0.992056i \(-0.540149\pi\)
−0.125799 + 0.992056i \(0.540149\pi\)
\(660\) −10.4584 −0.407093
\(661\) 9.51195 0.369972 0.184986 0.982741i \(-0.440776\pi\)
0.184986 + 0.982741i \(0.440776\pi\)
\(662\) 5.72173 0.222381
\(663\) 9.01789 0.350226
\(664\) −4.59673 −0.178388
\(665\) 15.3027 0.593413
\(666\) −2.32402 −0.0900541
\(667\) −11.9491 −0.462673
\(668\) 23.3380 0.902976
\(669\) −5.47594 −0.211712
\(670\) 3.63894 0.140584
\(671\) 0.346648 0.0133822
\(672\) −1.00000 −0.0385758
\(673\) −37.5547 −1.44763 −0.723813 0.689996i \(-0.757612\pi\)
−0.723813 + 0.689996i \(0.757612\pi\)
\(674\) −4.89728 −0.188636
\(675\) −11.6276 −0.447546
\(676\) −10.5264 −0.404860
\(677\) 12.6435 0.485931 0.242965 0.970035i \(-0.421880\pi\)
0.242965 + 0.970035i \(0.421880\pi\)
\(678\) −4.22957 −0.162436
\(679\) 1.46339 0.0561596
\(680\) −23.3804 −0.896596
\(681\) −13.7497 −0.526891
\(682\) −8.87275 −0.339755
\(683\) −17.7185 −0.677979 −0.338989 0.940790i \(-0.610085\pi\)
−0.338989 + 0.940790i \(0.610085\pi\)
\(684\) 3.75278 0.143491
\(685\) −29.1828 −1.11502
\(686\) 1.00000 0.0381802
\(687\) 2.69390 0.102779
\(688\) −6.29321 −0.239926
\(689\) −2.66449 −0.101509
\(690\) −16.1837 −0.616103
\(691\) 7.09699 0.269982 0.134991 0.990847i \(-0.456899\pi\)
0.134991 + 0.990847i \(0.456899\pi\)
\(692\) −2.19648 −0.0834976
\(693\) 2.56478 0.0974280
\(694\) −6.51461 −0.247291
\(695\) −32.6143 −1.23713
\(696\) 3.01074 0.114122
\(697\) −53.4771 −2.02559
\(698\) −6.41046 −0.242640
\(699\) −11.4638 −0.433602
\(700\) 11.6276 0.439481
\(701\) −5.18003 −0.195647 −0.0978234 0.995204i \(-0.531188\pi\)
−0.0978234 + 0.995204i \(0.531188\pi\)
\(702\) −1.57278 −0.0593608
\(703\) −8.72156 −0.328940
\(704\) 2.56478 0.0966639
\(705\) −20.8711 −0.786053
\(706\) −13.6793 −0.514827
\(707\) −7.75739 −0.291747
\(708\) 2.33548 0.0877728
\(709\) −34.2080 −1.28471 −0.642355 0.766407i \(-0.722043\pi\)
−0.642355 + 0.766407i \(0.722043\pi\)
\(710\) 2.42466 0.0909959
\(711\) 5.28701 0.198278
\(712\) 0.429519 0.0160969
\(713\) −13.7300 −0.514193
\(714\) 5.73372 0.214579
\(715\) 16.4488 0.615149
\(716\) 1.59185 0.0594902
\(717\) 20.2601 0.756626
\(718\) −23.8330 −0.889441
\(719\) −3.65623 −0.136354 −0.0681771 0.997673i \(-0.521718\pi\)
−0.0681771 + 0.997673i \(0.521718\pi\)
\(720\) 4.07769 0.151967
\(721\) 7.79971 0.290476
\(722\) −4.91662 −0.182978
\(723\) −10.0873 −0.375150
\(724\) −10.1396 −0.376834
\(725\) −35.0076 −1.30015
\(726\) 4.42189 0.164112
\(727\) −36.8838 −1.36794 −0.683971 0.729509i \(-0.739749\pi\)
−0.683971 + 0.729509i \(0.739749\pi\)
\(728\) 1.57278 0.0582911
\(729\) 1.00000 0.0370370
\(730\) 32.4602 1.20140
\(731\) 36.0835 1.33460
\(732\) −0.135157 −0.00499554
\(733\) 18.0900 0.668170 0.334085 0.942543i \(-0.391573\pi\)
0.334085 + 0.942543i \(0.391573\pi\)
\(734\) 14.5553 0.537247
\(735\) −4.07769 −0.150408
\(736\) 3.96884 0.146293
\(737\) 2.28881 0.0843096
\(738\) 9.32677 0.343323
\(739\) 15.0147 0.552323 0.276162 0.961111i \(-0.410937\pi\)
0.276162 + 0.961111i \(0.410937\pi\)
\(740\) −9.47665 −0.348369
\(741\) −5.90231 −0.216827
\(742\) −1.69413 −0.0621934
\(743\) 35.6810 1.30901 0.654504 0.756059i \(-0.272878\pi\)
0.654504 + 0.756059i \(0.272878\pi\)
\(744\) 3.45946 0.126830
\(745\) −50.3550 −1.84486
\(746\) −28.7370 −1.05214
\(747\) −4.59673 −0.168186
\(748\) −14.7057 −0.537695
\(749\) 3.74273 0.136756
\(750\) −27.0252 −0.986823
\(751\) −21.6705 −0.790768 −0.395384 0.918516i \(-0.629389\pi\)
−0.395384 + 0.918516i \(0.629389\pi\)
\(752\) 5.11837 0.186648
\(753\) 17.4515 0.635968
\(754\) −4.73524 −0.172447
\(755\) −12.4271 −0.452267
\(756\) −1.00000 −0.0363696
\(757\) −34.3745 −1.24936 −0.624681 0.780880i \(-0.714771\pi\)
−0.624681 + 0.780880i \(0.714771\pi\)
\(758\) 2.21881 0.0805907
\(759\) −10.1792 −0.369482
\(760\) 15.3027 0.555087
\(761\) −38.0492 −1.37928 −0.689641 0.724152i \(-0.742232\pi\)
−0.689641 + 0.724152i \(0.742232\pi\)
\(762\) 11.6091 0.420552
\(763\) 2.11979 0.0767416
\(764\) 1.00000 0.0361787
\(765\) −23.3804 −0.845318
\(766\) 2.18019 0.0787734
\(767\) −3.67320 −0.132632
\(768\) −1.00000 −0.0360844
\(769\) 38.3242 1.38201 0.691003 0.722852i \(-0.257169\pi\)
0.691003 + 0.722852i \(0.257169\pi\)
\(770\) 10.4584 0.376894
\(771\) 18.6271 0.670840
\(772\) 17.6252 0.634346
\(773\) −51.4478 −1.85045 −0.925225 0.379419i \(-0.876124\pi\)
−0.925225 + 0.379419i \(0.876124\pi\)
\(774\) −6.29321 −0.226205
\(775\) −40.2251 −1.44493
\(776\) 1.46339 0.0525325
\(777\) 2.32402 0.0833739
\(778\) −27.0514 −0.969838
\(779\) 35.0013 1.25405
\(780\) −6.41332 −0.229634
\(781\) 1.52506 0.0545709
\(782\) −22.7562 −0.813760
\(783\) 3.01074 0.107595
\(784\) 1.00000 0.0357143
\(785\) −66.9694 −2.39024
\(786\) −11.3478 −0.404762
\(787\) 29.6509 1.05694 0.528471 0.848952i \(-0.322766\pi\)
0.528471 + 0.848952i \(0.322766\pi\)
\(788\) −17.5489 −0.625155
\(789\) −12.7520 −0.453984
\(790\) 21.5588 0.767027
\(791\) 4.22957 0.150386
\(792\) 2.56478 0.0911356
\(793\) 0.212572 0.00754865
\(794\) 31.7466 1.12664
\(795\) 6.90814 0.245006
\(796\) −14.5178 −0.514569
\(797\) 27.1118 0.960349 0.480174 0.877173i \(-0.340573\pi\)
0.480174 + 0.877173i \(0.340573\pi\)
\(798\) −3.75278 −0.132847
\(799\) −29.3473 −1.03823
\(800\) 11.6276 0.411097
\(801\) 0.429519 0.0151763
\(802\) 22.3567 0.789444
\(803\) 20.4168 0.720491
\(804\) −0.892401 −0.0314726
\(805\) 16.1837 0.570401
\(806\) −5.44097 −0.191650
\(807\) 19.8773 0.699716
\(808\) −7.75739 −0.272904
\(809\) −19.3607 −0.680685 −0.340343 0.940301i \(-0.610543\pi\)
−0.340343 + 0.940301i \(0.610543\pi\)
\(810\) 4.07769 0.143276
\(811\) −4.24813 −0.149172 −0.0745861 0.997215i \(-0.523764\pi\)
−0.0745861 + 0.997215i \(0.523764\pi\)
\(812\) −3.01074 −0.105656
\(813\) −0.00929113 −0.000325854 0
\(814\) −5.96061 −0.208919
\(815\) 37.5003 1.31358
\(816\) 5.73372 0.200720
\(817\) −23.6171 −0.826256
\(818\) 24.7892 0.866735
\(819\) 1.57278 0.0549574
\(820\) 38.0317 1.32812
\(821\) 5.13428 0.179188 0.0895939 0.995978i \(-0.471443\pi\)
0.0895939 + 0.995978i \(0.471443\pi\)
\(822\) 7.15669 0.249618
\(823\) 6.50021 0.226583 0.113291 0.993562i \(-0.463861\pi\)
0.113291 + 0.993562i \(0.463861\pi\)
\(824\) 7.79971 0.271716
\(825\) −29.8222 −1.03828
\(826\) −2.33548 −0.0812618
\(827\) −3.45720 −0.120219 −0.0601093 0.998192i \(-0.519145\pi\)
−0.0601093 + 0.998192i \(0.519145\pi\)
\(828\) 3.96884 0.137927
\(829\) 2.37343 0.0824327 0.0412164 0.999150i \(-0.486877\pi\)
0.0412164 + 0.999150i \(0.486877\pi\)
\(830\) −18.7441 −0.650616
\(831\) −20.3778 −0.706899
\(832\) 1.57278 0.0545264
\(833\) −5.73372 −0.198662
\(834\) 7.99822 0.276956
\(835\) 95.1653 3.29333
\(836\) 9.62507 0.332890
\(837\) 3.45946 0.119576
\(838\) 7.62384 0.263361
\(839\) 15.6160 0.539126 0.269563 0.962983i \(-0.413121\pi\)
0.269563 + 0.962983i \(0.413121\pi\)
\(840\) −4.07769 −0.140694
\(841\) −19.9354 −0.687429
\(842\) −9.35074 −0.322248
\(843\) 32.5466 1.12096
\(844\) 13.8456 0.476585
\(845\) −42.9233 −1.47661
\(846\) 5.11837 0.175973
\(847\) −4.42189 −0.151938
\(848\) −1.69413 −0.0581766
\(849\) 12.0390 0.413176
\(850\) −66.6693 −2.28674
\(851\) −9.22367 −0.316183
\(852\) −0.594616 −0.0203712
\(853\) 32.7032 1.11974 0.559868 0.828582i \(-0.310852\pi\)
0.559868 + 0.828582i \(0.310852\pi\)
\(854\) 0.135157 0.00462497
\(855\) 15.3027 0.523341
\(856\) 3.74273 0.127924
\(857\) 50.0858 1.71090 0.855450 0.517886i \(-0.173281\pi\)
0.855450 + 0.517886i \(0.173281\pi\)
\(858\) −4.03384 −0.137713
\(859\) 34.4917 1.17684 0.588421 0.808555i \(-0.299750\pi\)
0.588421 + 0.808555i \(0.299750\pi\)
\(860\) −25.6618 −0.875060
\(861\) −9.32677 −0.317855
\(862\) −4.62815 −0.157635
\(863\) −54.6981 −1.86194 −0.930972 0.365090i \(-0.881038\pi\)
−0.930972 + 0.365090i \(0.881038\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −8.95657 −0.304532
\(866\) −1.89413 −0.0643652
\(867\) −15.8755 −0.539162
\(868\) −3.45946 −0.117422
\(869\) 13.5600 0.459992
\(870\) 12.2769 0.416225
\(871\) 1.40355 0.0475575
\(872\) 2.11979 0.0717852
\(873\) 1.46339 0.0495281
\(874\) 14.8942 0.503803
\(875\) 27.0252 0.913620
\(876\) −7.96042 −0.268958
\(877\) −29.5356 −0.997348 −0.498674 0.866790i \(-0.666179\pi\)
−0.498674 + 0.866790i \(0.666179\pi\)
\(878\) −13.9275 −0.470030
\(879\) 7.57841 0.255613
\(880\) 10.4584 0.352553
\(881\) −50.7049 −1.70829 −0.854145 0.520035i \(-0.825919\pi\)
−0.854145 + 0.520035i \(0.825919\pi\)
\(882\) 1.00000 0.0336718
\(883\) 4.30851 0.144993 0.0724965 0.997369i \(-0.476903\pi\)
0.0724965 + 0.997369i \(0.476903\pi\)
\(884\) −9.01789 −0.303304
\(885\) 9.52338 0.320125
\(886\) 8.69247 0.292029
\(887\) 15.3016 0.513778 0.256889 0.966441i \(-0.417302\pi\)
0.256889 + 0.966441i \(0.417302\pi\)
\(888\) 2.32402 0.0779891
\(889\) −11.6091 −0.389356
\(890\) 1.75145 0.0587087
\(891\) 2.56478 0.0859235
\(892\) 5.47594 0.183348
\(893\) 19.2081 0.642776
\(894\) 12.3489 0.413009
\(895\) 6.49108 0.216973
\(896\) 1.00000 0.0334077
\(897\) −6.24211 −0.208418
\(898\) 29.6829 0.990532
\(899\) 10.4155 0.347377
\(900\) 11.6276 0.387586
\(901\) 9.71366 0.323609
\(902\) 23.9211 0.796486
\(903\) 6.29321 0.209425
\(904\) 4.22957 0.140673
\(905\) −41.3460 −1.37439
\(906\) 3.04757 0.101249
\(907\) 4.70366 0.156182 0.0780912 0.996946i \(-0.475117\pi\)
0.0780912 + 0.996946i \(0.475117\pi\)
\(908\) 13.7497 0.456301
\(909\) −7.75739 −0.257297
\(910\) 6.41332 0.212599
\(911\) −6.11806 −0.202700 −0.101350 0.994851i \(-0.532316\pi\)
−0.101350 + 0.994851i \(0.532316\pi\)
\(912\) −3.75278 −0.124267
\(913\) −11.7896 −0.390179
\(914\) 13.7098 0.453481
\(915\) −0.551128 −0.0182197
\(916\) −2.69390 −0.0890089
\(917\) 11.3478 0.374737
\(918\) 5.73372 0.189241
\(919\) −28.7886 −0.949647 −0.474823 0.880081i \(-0.657488\pi\)
−0.474823 + 0.880081i \(0.657488\pi\)
\(920\) 16.1837 0.533561
\(921\) 9.76130 0.321646
\(922\) −22.5901 −0.743965
\(923\) 0.935200 0.0307825
\(924\) −2.56478 −0.0843752
\(925\) −27.0228 −0.888503
\(926\) 20.8874 0.686403
\(927\) 7.79971 0.256176
\(928\) −3.01074 −0.0988324
\(929\) −8.07840 −0.265044 −0.132522 0.991180i \(-0.542307\pi\)
−0.132522 + 0.991180i \(0.542307\pi\)
\(930\) 14.1066 0.462574
\(931\) 3.75278 0.122992
\(932\) 11.4638 0.375511
\(933\) −24.7211 −0.809331
\(934\) 17.6380 0.577132
\(935\) −59.9655 −1.96108
\(936\) 1.57278 0.0514080
\(937\) −51.6971 −1.68887 −0.844436 0.535657i \(-0.820064\pi\)
−0.844436 + 0.535657i \(0.820064\pi\)
\(938\) 0.892401 0.0291379
\(939\) −20.9113 −0.682415
\(940\) 20.8711 0.680742
\(941\) 9.47125 0.308754 0.154377 0.988012i \(-0.450663\pi\)
0.154377 + 0.988012i \(0.450663\pi\)
\(942\) 16.4234 0.535102
\(943\) 37.0164 1.20542
\(944\) −2.33548 −0.0760135
\(945\) −4.07769 −0.132647
\(946\) −16.1407 −0.524780
\(947\) −8.38667 −0.272530 −0.136265 0.990672i \(-0.543510\pi\)
−0.136265 + 0.990672i \(0.543510\pi\)
\(948\) −5.28701 −0.171714
\(949\) 12.5200 0.406416
\(950\) 43.6358 1.41573
\(951\) 16.1572 0.523932
\(952\) −5.73372 −0.185831
\(953\) −5.93373 −0.192212 −0.0961061 0.995371i \(-0.530639\pi\)
−0.0961061 + 0.995371i \(0.530639\pi\)
\(954\) −1.69413 −0.0548494
\(955\) 4.07769 0.131951
\(956\) −20.2601 −0.655258
\(957\) 7.72190 0.249613
\(958\) 28.7142 0.927714
\(959\) −7.15669 −0.231102
\(960\) −4.07769 −0.131607
\(961\) −19.0322 −0.613941
\(962\) −3.65518 −0.117848
\(963\) 3.74273 0.120608
\(964\) 10.0873 0.324890
\(965\) 71.8703 2.31359
\(966\) −3.96884 −0.127695
\(967\) −54.0140 −1.73697 −0.868487 0.495713i \(-0.834907\pi\)
−0.868487 + 0.495713i \(0.834907\pi\)
\(968\) −4.42189 −0.142125
\(969\) 21.5174 0.691239
\(970\) 5.96724 0.191597
\(971\) 42.2203 1.35491 0.677457 0.735562i \(-0.263082\pi\)
0.677457 + 0.735562i \(0.263082\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −7.99822 −0.256411
\(974\) −9.60371 −0.307723
\(975\) −18.2876 −0.585673
\(976\) 0.135157 0.00432626
\(977\) 11.4062 0.364915 0.182458 0.983214i \(-0.441595\pi\)
0.182458 + 0.983214i \(0.441595\pi\)
\(978\) −9.19645 −0.294070
\(979\) 1.10162 0.0352081
\(980\) 4.07769 0.130257
\(981\) 2.11979 0.0676798
\(982\) 19.0198 0.606945
\(983\) −40.3992 −1.28853 −0.644267 0.764801i \(-0.722837\pi\)
−0.644267 + 0.764801i \(0.722837\pi\)
\(984\) −9.32677 −0.297326
\(985\) −71.5592 −2.28007
\(986\) 17.2627 0.549758
\(987\) −5.11837 −0.162919
\(988\) 5.90231 0.187777
\(989\) −24.9767 −0.794214
\(990\) 10.4584 0.332390
\(991\) −8.41459 −0.267298 −0.133649 0.991029i \(-0.542670\pi\)
−0.133649 + 0.991029i \(0.542670\pi\)
\(992\) −3.45946 −0.109838
\(993\) −5.72173 −0.181574
\(994\) 0.594616 0.0188601
\(995\) −59.1990 −1.87674
\(996\) 4.59673 0.145653
\(997\) −46.9059 −1.48552 −0.742762 0.669556i \(-0.766485\pi\)
−0.742762 + 0.669556i \(0.766485\pi\)
\(998\) −1.33794 −0.0423517
\(999\) 2.32402 0.0735288
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 8022.2.a.r.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8022.2.a.r.1.10 10 1.1 even 1 trivial