Properties

Label 8007.2.a.h.1.20
Level $8007$
Weight $2$
Character 8007.1
Self dual yes
Analytic conductor $63.936$
Analytic rank $0$
Dimension $56$
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] = [8007,2,Mod(1,8007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, 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("8007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8007.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: \(63.9362168984\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 8007.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.892239 q^{2} +1.00000 q^{3} -1.20391 q^{4} -0.989381 q^{5} -0.892239 q^{6} +3.60802 q^{7} +2.85865 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.892239 q^{2} +1.00000 q^{3} -1.20391 q^{4} -0.989381 q^{5} -0.892239 q^{6} +3.60802 q^{7} +2.85865 q^{8} +1.00000 q^{9} +0.882764 q^{10} -0.219603 q^{11} -1.20391 q^{12} -2.77618 q^{13} -3.21922 q^{14} -0.989381 q^{15} -0.142779 q^{16} -1.00000 q^{17} -0.892239 q^{18} +4.91780 q^{19} +1.19113 q^{20} +3.60802 q^{21} +0.195938 q^{22} -3.55820 q^{23} +2.85865 q^{24} -4.02112 q^{25} +2.47701 q^{26} +1.00000 q^{27} -4.34373 q^{28} +6.73135 q^{29} +0.882764 q^{30} +0.993688 q^{31} -5.58991 q^{32} -0.219603 q^{33} +0.892239 q^{34} -3.56971 q^{35} -1.20391 q^{36} +1.79684 q^{37} -4.38785 q^{38} -2.77618 q^{39} -2.82830 q^{40} +0.0934982 q^{41} -3.21922 q^{42} -5.95800 q^{43} +0.264382 q^{44} -0.989381 q^{45} +3.17476 q^{46} +1.73754 q^{47} -0.142779 q^{48} +6.01782 q^{49} +3.58780 q^{50} -1.00000 q^{51} +3.34227 q^{52} +6.41191 q^{53} -0.892239 q^{54} +0.217271 q^{55} +10.3141 q^{56} +4.91780 q^{57} -6.00597 q^{58} +11.9429 q^{59} +1.19113 q^{60} -4.93840 q^{61} -0.886607 q^{62} +3.60802 q^{63} +5.27309 q^{64} +2.74670 q^{65} +0.195938 q^{66} -5.94225 q^{67} +1.20391 q^{68} -3.55820 q^{69} +3.18503 q^{70} +11.2788 q^{71} +2.85865 q^{72} +7.86008 q^{73} -1.60321 q^{74} -4.02112 q^{75} -5.92059 q^{76} -0.792331 q^{77} +2.47701 q^{78} +5.00470 q^{79} +0.141263 q^{80} +1.00000 q^{81} -0.0834227 q^{82} +1.97285 q^{83} -4.34373 q^{84} +0.989381 q^{85} +5.31596 q^{86} +6.73135 q^{87} -0.627768 q^{88} -6.03345 q^{89} +0.882764 q^{90} -10.0165 q^{91} +4.28375 q^{92} +0.993688 q^{93} -1.55030 q^{94} -4.86558 q^{95} -5.58991 q^{96} -6.26191 q^{97} -5.36933 q^{98} -0.219603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.892239 −0.630908 −0.315454 0.948941i \(-0.602157\pi\)
−0.315454 + 0.948941i \(0.602157\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.20391 −0.601955
\(5\) −0.989381 −0.442465 −0.221232 0.975221i \(-0.571008\pi\)
−0.221232 + 0.975221i \(0.571008\pi\)
\(6\) −0.892239 −0.364255
\(7\) 3.60802 1.36370 0.681852 0.731490i \(-0.261175\pi\)
0.681852 + 0.731490i \(0.261175\pi\)
\(8\) 2.85865 1.01069
\(9\) 1.00000 0.333333
\(10\) 0.882764 0.279155
\(11\) −0.219603 −0.0662127 −0.0331063 0.999452i \(-0.510540\pi\)
−0.0331063 + 0.999452i \(0.510540\pi\)
\(12\) −1.20391 −0.347539
\(13\) −2.77618 −0.769974 −0.384987 0.922922i \(-0.625794\pi\)
−0.384987 + 0.922922i \(0.625794\pi\)
\(14\) −3.21922 −0.860372
\(15\) −0.989381 −0.255457
\(16\) −0.142779 −0.0356948
\(17\) −1.00000 −0.242536
\(18\) −0.892239 −0.210303
\(19\) 4.91780 1.12822 0.564110 0.825700i \(-0.309219\pi\)
0.564110 + 0.825700i \(0.309219\pi\)
\(20\) 1.19113 0.266344
\(21\) 3.60802 0.787335
\(22\) 0.195938 0.0417741
\(23\) −3.55820 −0.741936 −0.370968 0.928646i \(-0.620974\pi\)
−0.370968 + 0.928646i \(0.620974\pi\)
\(24\) 2.85865 0.583520
\(25\) −4.02112 −0.804225
\(26\) 2.47701 0.485782
\(27\) 1.00000 0.192450
\(28\) −4.34373 −0.820889
\(29\) 6.73135 1.24998 0.624990 0.780633i \(-0.285103\pi\)
0.624990 + 0.780633i \(0.285103\pi\)
\(30\) 0.882764 0.161170
\(31\) 0.993688 0.178472 0.0892358 0.996011i \(-0.471558\pi\)
0.0892358 + 0.996011i \(0.471558\pi\)
\(32\) −5.58991 −0.988166
\(33\) −0.219603 −0.0382279
\(34\) 0.892239 0.153018
\(35\) −3.56971 −0.603391
\(36\) −1.20391 −0.200652
\(37\) 1.79684 0.295399 0.147700 0.989032i \(-0.452813\pi\)
0.147700 + 0.989032i \(0.452813\pi\)
\(38\) −4.38785 −0.711803
\(39\) −2.77618 −0.444544
\(40\) −2.82830 −0.447193
\(41\) 0.0934982 0.0146020 0.00730098 0.999973i \(-0.497676\pi\)
0.00730098 + 0.999973i \(0.497676\pi\)
\(42\) −3.21922 −0.496736
\(43\) −5.95800 −0.908586 −0.454293 0.890852i \(-0.650108\pi\)
−0.454293 + 0.890852i \(0.650108\pi\)
\(44\) 0.264382 0.0398571
\(45\) −0.989381 −0.147488
\(46\) 3.17476 0.468093
\(47\) 1.73754 0.253446 0.126723 0.991938i \(-0.459554\pi\)
0.126723 + 0.991938i \(0.459554\pi\)
\(48\) −0.142779 −0.0206084
\(49\) 6.01782 0.859688
\(50\) 3.58780 0.507392
\(51\) −1.00000 −0.140028
\(52\) 3.34227 0.463490
\(53\) 6.41191 0.880743 0.440372 0.897816i \(-0.354847\pi\)
0.440372 + 0.897816i \(0.354847\pi\)
\(54\) −0.892239 −0.121418
\(55\) 0.217271 0.0292968
\(56\) 10.3141 1.37828
\(57\) 4.91780 0.651378
\(58\) −6.00597 −0.788622
\(59\) 11.9429 1.55483 0.777413 0.628990i \(-0.216531\pi\)
0.777413 + 0.628990i \(0.216531\pi\)
\(60\) 1.19113 0.153774
\(61\) −4.93840 −0.632298 −0.316149 0.948710i \(-0.602390\pi\)
−0.316149 + 0.948710i \(0.602390\pi\)
\(62\) −0.886607 −0.112599
\(63\) 3.60802 0.454568
\(64\) 5.27309 0.659137
\(65\) 2.74670 0.340686
\(66\) 0.195938 0.0241183
\(67\) −5.94225 −0.725961 −0.362981 0.931797i \(-0.618241\pi\)
−0.362981 + 0.931797i \(0.618241\pi\)
\(68\) 1.20391 0.145996
\(69\) −3.55820 −0.428357
\(70\) 3.18503 0.380684
\(71\) 11.2788 1.33855 0.669273 0.743016i \(-0.266606\pi\)
0.669273 + 0.743016i \(0.266606\pi\)
\(72\) 2.85865 0.336895
\(73\) 7.86008 0.919953 0.459976 0.887931i \(-0.347858\pi\)
0.459976 + 0.887931i \(0.347858\pi\)
\(74\) −1.60321 −0.186370
\(75\) −4.02112 −0.464319
\(76\) −5.92059 −0.679138
\(77\) −0.792331 −0.0902945
\(78\) 2.47701 0.280467
\(79\) 5.00470 0.563072 0.281536 0.959551i \(-0.409156\pi\)
0.281536 + 0.959551i \(0.409156\pi\)
\(80\) 0.141263 0.0157937
\(81\) 1.00000 0.111111
\(82\) −0.0834227 −0.00921249
\(83\) 1.97285 0.216548 0.108274 0.994121i \(-0.465468\pi\)
0.108274 + 0.994121i \(0.465468\pi\)
\(84\) −4.34373 −0.473940
\(85\) 0.989381 0.107313
\(86\) 5.31596 0.573234
\(87\) 6.73135 0.721676
\(88\) −0.627768 −0.0669203
\(89\) −6.03345 −0.639544 −0.319772 0.947495i \(-0.603606\pi\)
−0.319772 + 0.947495i \(0.603606\pi\)
\(90\) 0.882764 0.0930515
\(91\) −10.0165 −1.05002
\(92\) 4.28375 0.446612
\(93\) 0.993688 0.103041
\(94\) −1.55030 −0.159901
\(95\) −4.86558 −0.499198
\(96\) −5.58991 −0.570518
\(97\) −6.26191 −0.635800 −0.317900 0.948124i \(-0.602978\pi\)
−0.317900 + 0.948124i \(0.602978\pi\)
\(98\) −5.36933 −0.542384
\(99\) −0.219603 −0.0220709
\(100\) 4.84107 0.484107
\(101\) 13.5411 1.34739 0.673697 0.739008i \(-0.264705\pi\)
0.673697 + 0.739008i \(0.264705\pi\)
\(102\) 0.892239 0.0883448
\(103\) 13.0169 1.28259 0.641296 0.767293i \(-0.278397\pi\)
0.641296 + 0.767293i \(0.278397\pi\)
\(104\) −7.93613 −0.778202
\(105\) −3.56971 −0.348368
\(106\) −5.72095 −0.555668
\(107\) 0.435274 0.0420795 0.0210397 0.999779i \(-0.493302\pi\)
0.0210397 + 0.999779i \(0.493302\pi\)
\(108\) −1.20391 −0.115846
\(109\) −13.8854 −1.32998 −0.664991 0.746851i \(-0.731565\pi\)
−0.664991 + 0.746851i \(0.731565\pi\)
\(110\) −0.193857 −0.0184836
\(111\) 1.79684 0.170549
\(112\) −0.515150 −0.0486771
\(113\) −10.8080 −1.01673 −0.508367 0.861141i \(-0.669750\pi\)
−0.508367 + 0.861141i \(0.669750\pi\)
\(114\) −4.38785 −0.410960
\(115\) 3.52042 0.328280
\(116\) −8.10394 −0.752432
\(117\) −2.77618 −0.256658
\(118\) −10.6559 −0.980953
\(119\) −3.60802 −0.330747
\(120\) −2.82830 −0.258187
\(121\) −10.9518 −0.995616
\(122\) 4.40623 0.398922
\(123\) 0.0934982 0.00843045
\(124\) −1.19631 −0.107432
\(125\) 8.92533 0.798306
\(126\) −3.21922 −0.286791
\(127\) −9.20742 −0.817027 −0.408513 0.912752i \(-0.633953\pi\)
−0.408513 + 0.912752i \(0.633953\pi\)
\(128\) 6.47497 0.572312
\(129\) −5.95800 −0.524573
\(130\) −2.45071 −0.214942
\(131\) −9.10545 −0.795547 −0.397773 0.917484i \(-0.630217\pi\)
−0.397773 + 0.917484i \(0.630217\pi\)
\(132\) 0.264382 0.0230115
\(133\) 17.7435 1.53856
\(134\) 5.30190 0.458015
\(135\) −0.989381 −0.0851524
\(136\) −2.85865 −0.245127
\(137\) −2.08470 −0.178108 −0.0890538 0.996027i \(-0.528384\pi\)
−0.0890538 + 0.996027i \(0.528384\pi\)
\(138\) 3.17476 0.270254
\(139\) 19.3869 1.64437 0.822186 0.569218i \(-0.192754\pi\)
0.822186 + 0.569218i \(0.192754\pi\)
\(140\) 4.29761 0.363214
\(141\) 1.73754 0.146327
\(142\) −10.0634 −0.844500
\(143\) 0.609656 0.0509820
\(144\) −0.142779 −0.0118983
\(145\) −6.65987 −0.553072
\(146\) −7.01307 −0.580406
\(147\) 6.01782 0.496341
\(148\) −2.16324 −0.177817
\(149\) 16.5666 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(150\) 3.58780 0.292943
\(151\) 0.909183 0.0739882 0.0369941 0.999315i \(-0.488222\pi\)
0.0369941 + 0.999315i \(0.488222\pi\)
\(152\) 14.0583 1.14028
\(153\) −1.00000 −0.0808452
\(154\) 0.706948 0.0569675
\(155\) −0.983137 −0.0789674
\(156\) 3.34227 0.267596
\(157\) −1.00000 −0.0798087
\(158\) −4.46538 −0.355247
\(159\) 6.41191 0.508497
\(160\) 5.53056 0.437229
\(161\) −12.8381 −1.01178
\(162\) −0.892239 −0.0701009
\(163\) 8.99214 0.704319 0.352160 0.935940i \(-0.385447\pi\)
0.352160 + 0.935940i \(0.385447\pi\)
\(164\) −0.112563 −0.00878973
\(165\) 0.217271 0.0169145
\(166\) −1.76025 −0.136622
\(167\) 17.4735 1.35214 0.676069 0.736838i \(-0.263682\pi\)
0.676069 + 0.736838i \(0.263682\pi\)
\(168\) 10.3141 0.795748
\(169\) −5.29283 −0.407141
\(170\) −0.882764 −0.0677049
\(171\) 4.91780 0.376073
\(172\) 7.17290 0.546928
\(173\) 9.75269 0.741483 0.370742 0.928736i \(-0.379104\pi\)
0.370742 + 0.928736i \(0.379104\pi\)
\(174\) −6.00597 −0.455311
\(175\) −14.5083 −1.09672
\(176\) 0.0313547 0.00236345
\(177\) 11.9429 0.897680
\(178\) 5.38327 0.403493
\(179\) −12.3126 −0.920289 −0.460144 0.887844i \(-0.652202\pi\)
−0.460144 + 0.887844i \(0.652202\pi\)
\(180\) 1.19113 0.0887813
\(181\) 17.2539 1.28247 0.641235 0.767345i \(-0.278423\pi\)
0.641235 + 0.767345i \(0.278423\pi\)
\(182\) 8.93712 0.662463
\(183\) −4.93840 −0.365057
\(184\) −10.1717 −0.749864
\(185\) −1.77776 −0.130704
\(186\) −0.886607 −0.0650092
\(187\) 0.219603 0.0160589
\(188\) −2.09184 −0.152563
\(189\) 3.60802 0.262445
\(190\) 4.34125 0.314948
\(191\) −18.5318 −1.34092 −0.670458 0.741948i \(-0.733902\pi\)
−0.670458 + 0.741948i \(0.733902\pi\)
\(192\) 5.27309 0.380553
\(193\) 13.6803 0.984729 0.492365 0.870389i \(-0.336133\pi\)
0.492365 + 0.870389i \(0.336133\pi\)
\(194\) 5.58712 0.401132
\(195\) 2.74670 0.196695
\(196\) −7.24491 −0.517494
\(197\) 2.30949 0.164544 0.0822721 0.996610i \(-0.473782\pi\)
0.0822721 + 0.996610i \(0.473782\pi\)
\(198\) 0.195938 0.0139247
\(199\) −12.9433 −0.917524 −0.458762 0.888559i \(-0.651707\pi\)
−0.458762 + 0.888559i \(0.651707\pi\)
\(200\) −11.4950 −0.812819
\(201\) −5.94225 −0.419134
\(202\) −12.0819 −0.850081
\(203\) 24.2869 1.70460
\(204\) 1.20391 0.0842906
\(205\) −0.0925054 −0.00646086
\(206\) −11.6142 −0.809198
\(207\) −3.55820 −0.247312
\(208\) 0.396380 0.0274840
\(209\) −1.07996 −0.0747025
\(210\) 3.18503 0.219788
\(211\) −25.7974 −1.77597 −0.887985 0.459873i \(-0.847895\pi\)
−0.887985 + 0.459873i \(0.847895\pi\)
\(212\) −7.71936 −0.530168
\(213\) 11.2788 0.772810
\(214\) −0.388368 −0.0265483
\(215\) 5.89473 0.402018
\(216\) 2.85865 0.194507
\(217\) 3.58525 0.243382
\(218\) 12.3891 0.839096
\(219\) 7.86008 0.531135
\(220\) −0.261575 −0.0176354
\(221\) 2.77618 0.186746
\(222\) −1.60321 −0.107601
\(223\) 18.8319 1.26108 0.630539 0.776158i \(-0.282834\pi\)
0.630539 + 0.776158i \(0.282834\pi\)
\(224\) −20.1685 −1.34757
\(225\) −4.02112 −0.268075
\(226\) 9.64333 0.641465
\(227\) −7.11795 −0.472435 −0.236218 0.971700i \(-0.575908\pi\)
−0.236218 + 0.971700i \(0.575908\pi\)
\(228\) −5.92059 −0.392100
\(229\) −5.55949 −0.367382 −0.183691 0.982984i \(-0.558805\pi\)
−0.183691 + 0.982984i \(0.558805\pi\)
\(230\) −3.14105 −0.207115
\(231\) −0.792331 −0.0521316
\(232\) 19.2426 1.26334
\(233\) 5.61253 0.367689 0.183845 0.982955i \(-0.441146\pi\)
0.183845 + 0.982955i \(0.441146\pi\)
\(234\) 2.47701 0.161927
\(235\) −1.71909 −0.112141
\(236\) −14.3781 −0.935936
\(237\) 5.00470 0.325090
\(238\) 3.21922 0.208671
\(239\) −11.4166 −0.738478 −0.369239 0.929335i \(-0.620382\pi\)
−0.369239 + 0.929335i \(0.620382\pi\)
\(240\) 0.141263 0.00911848
\(241\) 12.2964 0.792081 0.396040 0.918233i \(-0.370384\pi\)
0.396040 + 0.918233i \(0.370384\pi\)
\(242\) 9.77160 0.628142
\(243\) 1.00000 0.0641500
\(244\) 5.94539 0.380615
\(245\) −5.95392 −0.380382
\(246\) −0.0834227 −0.00531884
\(247\) −13.6527 −0.868699
\(248\) 2.84061 0.180379
\(249\) 1.97285 0.125024
\(250\) −7.96353 −0.503658
\(251\) 28.2372 1.78232 0.891159 0.453691i \(-0.149893\pi\)
0.891159 + 0.453691i \(0.149893\pi\)
\(252\) −4.34373 −0.273630
\(253\) 0.781390 0.0491256
\(254\) 8.21522 0.515469
\(255\) 0.989381 0.0619575
\(256\) −16.3234 −1.02021
\(257\) 13.1349 0.819330 0.409665 0.912236i \(-0.365646\pi\)
0.409665 + 0.912236i \(0.365646\pi\)
\(258\) 5.31596 0.330957
\(259\) 6.48304 0.402837
\(260\) −3.30678 −0.205078
\(261\) 6.73135 0.416660
\(262\) 8.12423 0.501917
\(263\) 4.67604 0.288337 0.144169 0.989553i \(-0.453949\pi\)
0.144169 + 0.989553i \(0.453949\pi\)
\(264\) −0.627768 −0.0386364
\(265\) −6.34382 −0.389698
\(266\) −15.8314 −0.970688
\(267\) −6.03345 −0.369241
\(268\) 7.15393 0.436996
\(269\) −16.2954 −0.993548 −0.496774 0.867880i \(-0.665482\pi\)
−0.496774 + 0.867880i \(0.665482\pi\)
\(270\) 0.882764 0.0537233
\(271\) 3.98510 0.242078 0.121039 0.992648i \(-0.461377\pi\)
0.121039 + 0.992648i \(0.461377\pi\)
\(272\) 0.142779 0.00865725
\(273\) −10.0165 −0.606227
\(274\) 1.86005 0.112369
\(275\) 0.883050 0.0532499
\(276\) 4.28375 0.257852
\(277\) −21.3862 −1.28497 −0.642485 0.766298i \(-0.722096\pi\)
−0.642485 + 0.766298i \(0.722096\pi\)
\(278\) −17.2977 −1.03745
\(279\) 0.993688 0.0594905
\(280\) −10.2046 −0.609839
\(281\) 14.8841 0.887914 0.443957 0.896048i \(-0.353574\pi\)
0.443957 + 0.896048i \(0.353574\pi\)
\(282\) −1.55030 −0.0923188
\(283\) 4.58113 0.272320 0.136160 0.990687i \(-0.456524\pi\)
0.136160 + 0.990687i \(0.456524\pi\)
\(284\) −13.5787 −0.805745
\(285\) −4.86558 −0.288212
\(286\) −0.543959 −0.0321650
\(287\) 0.337343 0.0199128
\(288\) −5.58991 −0.329389
\(289\) 1.00000 0.0588235
\(290\) 5.94219 0.348938
\(291\) −6.26191 −0.367080
\(292\) −9.46283 −0.553770
\(293\) −16.3215 −0.953510 −0.476755 0.879036i \(-0.658187\pi\)
−0.476755 + 0.879036i \(0.658187\pi\)
\(294\) −5.36933 −0.313146
\(295\) −11.8160 −0.687956
\(296\) 5.13655 0.298556
\(297\) −0.219603 −0.0127426
\(298\) −14.7814 −0.856261
\(299\) 9.87819 0.571271
\(300\) 4.84107 0.279499
\(301\) −21.4966 −1.23904
\(302\) −0.811208 −0.0466798
\(303\) 13.5411 0.777918
\(304\) −0.702158 −0.0402715
\(305\) 4.88596 0.279770
\(306\) 0.892239 0.0510059
\(307\) 15.1210 0.863003 0.431502 0.902112i \(-0.357984\pi\)
0.431502 + 0.902112i \(0.357984\pi\)
\(308\) 0.953896 0.0543532
\(309\) 13.0169 0.740505
\(310\) 0.877192 0.0498212
\(311\) −14.0721 −0.797953 −0.398977 0.916961i \(-0.630635\pi\)
−0.398977 + 0.916961i \(0.630635\pi\)
\(312\) −7.93613 −0.449295
\(313\) −18.0182 −1.01845 −0.509226 0.860633i \(-0.670068\pi\)
−0.509226 + 0.860633i \(0.670068\pi\)
\(314\) 0.892239 0.0503519
\(315\) −3.56971 −0.201130
\(316\) −6.02521 −0.338944
\(317\) 30.4718 1.71147 0.855734 0.517417i \(-0.173106\pi\)
0.855734 + 0.517417i \(0.173106\pi\)
\(318\) −5.72095 −0.320815
\(319\) −1.47822 −0.0827646
\(320\) −5.21710 −0.291645
\(321\) 0.435274 0.0242946
\(322\) 11.4546 0.638340
\(323\) −4.91780 −0.273633
\(324\) −1.20391 −0.0668839
\(325\) 11.1634 0.619232
\(326\) −8.02314 −0.444360
\(327\) −13.8854 −0.767866
\(328\) 0.267279 0.0147580
\(329\) 6.26907 0.345625
\(330\) −0.193857 −0.0106715
\(331\) 3.00303 0.165062 0.0825308 0.996589i \(-0.473700\pi\)
0.0825308 + 0.996589i \(0.473700\pi\)
\(332\) −2.37513 −0.130352
\(333\) 1.79684 0.0984664
\(334\) −15.5905 −0.853074
\(335\) 5.87915 0.321212
\(336\) −0.515150 −0.0281037
\(337\) −29.6425 −1.61473 −0.807366 0.590051i \(-0.799108\pi\)
−0.807366 + 0.590051i \(0.799108\pi\)
\(338\) 4.72247 0.256868
\(339\) −10.8080 −0.587011
\(340\) −1.19113 −0.0645979
\(341\) −0.218217 −0.0118171
\(342\) −4.38785 −0.237268
\(343\) −3.54374 −0.191344
\(344\) −17.0319 −0.918296
\(345\) 3.52042 0.189533
\(346\) −8.70172 −0.467808
\(347\) 12.8339 0.688957 0.344479 0.938794i \(-0.388056\pi\)
0.344479 + 0.938794i \(0.388056\pi\)
\(348\) −8.10394 −0.434417
\(349\) −13.3069 −0.712302 −0.356151 0.934428i \(-0.615911\pi\)
−0.356151 + 0.934428i \(0.615911\pi\)
\(350\) 12.9449 0.691932
\(351\) −2.77618 −0.148181
\(352\) 1.22756 0.0654291
\(353\) 27.5704 1.46743 0.733713 0.679459i \(-0.237786\pi\)
0.733713 + 0.679459i \(0.237786\pi\)
\(354\) −10.6559 −0.566353
\(355\) −11.1590 −0.592260
\(356\) 7.26373 0.384977
\(357\) −3.60802 −0.190957
\(358\) 10.9858 0.580617
\(359\) −3.57962 −0.188925 −0.0944625 0.995528i \(-0.530113\pi\)
−0.0944625 + 0.995528i \(0.530113\pi\)
\(360\) −2.82830 −0.149064
\(361\) 5.18472 0.272880
\(362\) −15.3946 −0.809120
\(363\) −10.9518 −0.574819
\(364\) 12.0590 0.632063
\(365\) −7.77662 −0.407047
\(366\) 4.40623 0.230318
\(367\) 37.8608 1.97632 0.988160 0.153429i \(-0.0490317\pi\)
0.988160 + 0.153429i \(0.0490317\pi\)
\(368\) 0.508036 0.0264832
\(369\) 0.0934982 0.00486732
\(370\) 1.58619 0.0824620
\(371\) 23.1343 1.20107
\(372\) −1.19631 −0.0620259
\(373\) 23.2424 1.20345 0.601724 0.798704i \(-0.294481\pi\)
0.601724 + 0.798704i \(0.294481\pi\)
\(374\) −0.195938 −0.0101317
\(375\) 8.92533 0.460902
\(376\) 4.96701 0.256154
\(377\) −18.6874 −0.962452
\(378\) −3.21922 −0.165579
\(379\) −6.52900 −0.335372 −0.167686 0.985840i \(-0.553630\pi\)
−0.167686 + 0.985840i \(0.553630\pi\)
\(380\) 5.85772 0.300495
\(381\) −9.20742 −0.471711
\(382\) 16.5348 0.845994
\(383\) 25.9296 1.32494 0.662470 0.749089i \(-0.269508\pi\)
0.662470 + 0.749089i \(0.269508\pi\)
\(384\) 6.47497 0.330424
\(385\) 0.783918 0.0399521
\(386\) −12.2061 −0.621273
\(387\) −5.95800 −0.302862
\(388\) 7.53878 0.382723
\(389\) 15.4376 0.782717 0.391359 0.920238i \(-0.372005\pi\)
0.391359 + 0.920238i \(0.372005\pi\)
\(390\) −2.45071 −0.124097
\(391\) 3.55820 0.179946
\(392\) 17.2028 0.868875
\(393\) −9.10545 −0.459309
\(394\) −2.06062 −0.103812
\(395\) −4.95155 −0.249140
\(396\) 0.264382 0.0132857
\(397\) 4.36306 0.218976 0.109488 0.993988i \(-0.465079\pi\)
0.109488 + 0.993988i \(0.465079\pi\)
\(398\) 11.5485 0.578873
\(399\) 17.7435 0.888287
\(400\) 0.574132 0.0287066
\(401\) 7.15335 0.357221 0.178611 0.983920i \(-0.442840\pi\)
0.178611 + 0.983920i \(0.442840\pi\)
\(402\) 5.30190 0.264435
\(403\) −2.75866 −0.137418
\(404\) −16.3023 −0.811071
\(405\) −0.989381 −0.0491628
\(406\) −21.6697 −1.07545
\(407\) −0.394591 −0.0195592
\(408\) −2.85865 −0.141524
\(409\) −19.4140 −0.959960 −0.479980 0.877279i \(-0.659356\pi\)
−0.479980 + 0.877279i \(0.659356\pi\)
\(410\) 0.0825369 0.00407621
\(411\) −2.08470 −0.102830
\(412\) −15.6712 −0.772063
\(413\) 43.0901 2.12032
\(414\) 3.17476 0.156031
\(415\) −1.95190 −0.0958150
\(416\) 15.5186 0.760862
\(417\) 19.3869 0.949379
\(418\) 0.963583 0.0471304
\(419\) −22.5842 −1.10331 −0.551655 0.834072i \(-0.686004\pi\)
−0.551655 + 0.834072i \(0.686004\pi\)
\(420\) 4.29761 0.209702
\(421\) 16.4010 0.799334 0.399667 0.916660i \(-0.369126\pi\)
0.399667 + 0.916660i \(0.369126\pi\)
\(422\) 23.0175 1.12047
\(423\) 1.73754 0.0844819
\(424\) 18.3294 0.890155
\(425\) 4.02112 0.195053
\(426\) −10.0634 −0.487572
\(427\) −17.8179 −0.862267
\(428\) −0.524030 −0.0253300
\(429\) 0.609656 0.0294345
\(430\) −5.25951 −0.253636
\(431\) −6.10254 −0.293949 −0.146974 0.989140i \(-0.546953\pi\)
−0.146974 + 0.989140i \(0.546953\pi\)
\(432\) −0.142779 −0.00686946
\(433\) 2.88811 0.138794 0.0693968 0.997589i \(-0.477893\pi\)
0.0693968 + 0.997589i \(0.477893\pi\)
\(434\) −3.19890 −0.153552
\(435\) −6.65987 −0.319316
\(436\) 16.7168 0.800590
\(437\) −17.4985 −0.837066
\(438\) −7.01307 −0.335097
\(439\) 33.2795 1.58834 0.794171 0.607694i \(-0.207905\pi\)
0.794171 + 0.607694i \(0.207905\pi\)
\(440\) 0.621102 0.0296099
\(441\) 6.01782 0.286563
\(442\) −2.47701 −0.117820
\(443\) 35.6344 1.69304 0.846520 0.532356i \(-0.178693\pi\)
0.846520 + 0.532356i \(0.178693\pi\)
\(444\) −2.16324 −0.102663
\(445\) 5.96938 0.282976
\(446\) −16.8026 −0.795624
\(447\) 16.5666 0.783573
\(448\) 19.0254 0.898867
\(449\) 39.3240 1.85582 0.927908 0.372810i \(-0.121606\pi\)
0.927908 + 0.372810i \(0.121606\pi\)
\(450\) 3.58780 0.169131
\(451\) −0.0205325 −0.000966835 0
\(452\) 13.0119 0.612028
\(453\) 0.909183 0.0427171
\(454\) 6.35091 0.298063
\(455\) 9.91015 0.464595
\(456\) 14.0583 0.658339
\(457\) 14.6188 0.683840 0.341920 0.939729i \(-0.388923\pi\)
0.341920 + 0.939729i \(0.388923\pi\)
\(458\) 4.96039 0.231784
\(459\) −1.00000 −0.0466760
\(460\) −4.23826 −0.197610
\(461\) 12.7576 0.594183 0.297091 0.954849i \(-0.403983\pi\)
0.297091 + 0.954849i \(0.403983\pi\)
\(462\) 0.706948 0.0328902
\(463\) 12.7574 0.592888 0.296444 0.955050i \(-0.404199\pi\)
0.296444 + 0.955050i \(0.404199\pi\)
\(464\) −0.961095 −0.0446177
\(465\) −0.983137 −0.0455919
\(466\) −5.00772 −0.231978
\(467\) −0.394515 −0.0182560 −0.00912798 0.999958i \(-0.502906\pi\)
−0.00912798 + 0.999958i \(0.502906\pi\)
\(468\) 3.34227 0.154497
\(469\) −21.4398 −0.989996
\(470\) 1.53383 0.0707505
\(471\) −1.00000 −0.0460776
\(472\) 34.1405 1.57144
\(473\) 1.30839 0.0601600
\(474\) −4.46538 −0.205102
\(475\) −19.7751 −0.907342
\(476\) 4.34373 0.199095
\(477\) 6.41191 0.293581
\(478\) 10.1863 0.465911
\(479\) 38.1001 1.74084 0.870420 0.492311i \(-0.163848\pi\)
0.870420 + 0.492311i \(0.163848\pi\)
\(480\) 5.53056 0.252434
\(481\) −4.98836 −0.227449
\(482\) −10.9713 −0.499730
\(483\) −12.8381 −0.584152
\(484\) 13.1850 0.599316
\(485\) 6.19542 0.281319
\(486\) −0.892239 −0.0404728
\(487\) 8.44055 0.382478 0.191239 0.981544i \(-0.438749\pi\)
0.191239 + 0.981544i \(0.438749\pi\)
\(488\) −14.1172 −0.639055
\(489\) 8.99214 0.406639
\(490\) 5.31231 0.239986
\(491\) 7.86877 0.355113 0.177556 0.984111i \(-0.443181\pi\)
0.177556 + 0.984111i \(0.443181\pi\)
\(492\) −0.112563 −0.00507475
\(493\) −6.73135 −0.303165
\(494\) 12.1814 0.548069
\(495\) 0.217271 0.00976560
\(496\) −0.141878 −0.00637050
\(497\) 40.6941 1.82538
\(498\) −1.76025 −0.0788787
\(499\) 6.91656 0.309628 0.154814 0.987944i \(-0.450522\pi\)
0.154814 + 0.987944i \(0.450522\pi\)
\(500\) −10.7453 −0.480544
\(501\) 17.4735 0.780657
\(502\) −25.1943 −1.12448
\(503\) −6.55482 −0.292265 −0.146133 0.989265i \(-0.546683\pi\)
−0.146133 + 0.989265i \(0.546683\pi\)
\(504\) 10.3141 0.459426
\(505\) −13.3974 −0.596174
\(506\) −0.697186 −0.0309937
\(507\) −5.29283 −0.235063
\(508\) 11.0849 0.491813
\(509\) −17.3609 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(510\) −0.882764 −0.0390895
\(511\) 28.3593 1.25454
\(512\) 1.61443 0.0713485
\(513\) 4.91780 0.217126
\(514\) −11.7194 −0.516921
\(515\) −12.8787 −0.567502
\(516\) 7.17290 0.315769
\(517\) −0.381568 −0.0167813
\(518\) −5.78442 −0.254153
\(519\) 9.75269 0.428096
\(520\) 7.85186 0.344327
\(521\) −37.5898 −1.64684 −0.823419 0.567434i \(-0.807936\pi\)
−0.823419 + 0.567434i \(0.807936\pi\)
\(522\) −6.00597 −0.262874
\(523\) 3.38403 0.147973 0.0739867 0.997259i \(-0.476428\pi\)
0.0739867 + 0.997259i \(0.476428\pi\)
\(524\) 10.9621 0.478883
\(525\) −14.5083 −0.633194
\(526\) −4.17215 −0.181914
\(527\) −0.993688 −0.0432857
\(528\) 0.0313547 0.00136454
\(529\) −10.3392 −0.449532
\(530\) 5.66020 0.245863
\(531\) 11.9429 0.518276
\(532\) −21.3616 −0.926143
\(533\) −0.259568 −0.0112431
\(534\) 5.38327 0.232957
\(535\) −0.430652 −0.0186187
\(536\) −16.9868 −0.733719
\(537\) −12.3126 −0.531329
\(538\) 14.5394 0.626837
\(539\) −1.32153 −0.0569223
\(540\) 1.19113 0.0512579
\(541\) −10.9698 −0.471630 −0.235815 0.971798i \(-0.575776\pi\)
−0.235815 + 0.971798i \(0.575776\pi\)
\(542\) −3.55566 −0.152729
\(543\) 17.2539 0.740434
\(544\) 5.58991 0.239665
\(545\) 13.7380 0.588470
\(546\) 8.93712 0.382473
\(547\) 16.0799 0.687528 0.343764 0.939056i \(-0.388298\pi\)
0.343764 + 0.939056i \(0.388298\pi\)
\(548\) 2.50979 0.107213
\(549\) −4.93840 −0.210766
\(550\) −0.787891 −0.0335958
\(551\) 33.1034 1.41025
\(552\) −10.1717 −0.432934
\(553\) 18.0570 0.767864
\(554\) 19.0816 0.810697
\(555\) −1.77776 −0.0754618
\(556\) −23.3401 −0.989839
\(557\) −10.2436 −0.434037 −0.217019 0.976167i \(-0.569633\pi\)
−0.217019 + 0.976167i \(0.569633\pi\)
\(558\) −0.886607 −0.0375331
\(559\) 16.5405 0.699588
\(560\) 0.509680 0.0215379
\(561\) 0.219603 0.00927163
\(562\) −13.2802 −0.560192
\(563\) 5.66551 0.238773 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(564\) −2.09184 −0.0880823
\(565\) 10.6933 0.449869
\(566\) −4.08746 −0.171809
\(567\) 3.60802 0.151523
\(568\) 32.2421 1.35285
\(569\) 8.01376 0.335954 0.167977 0.985791i \(-0.446277\pi\)
0.167977 + 0.985791i \(0.446277\pi\)
\(570\) 4.34125 0.181835
\(571\) 25.7496 1.07759 0.538794 0.842438i \(-0.318880\pi\)
0.538794 + 0.842438i \(0.318880\pi\)
\(572\) −0.733972 −0.0306889
\(573\) −18.5318 −0.774178
\(574\) −0.300991 −0.0125631
\(575\) 14.3080 0.596683
\(576\) 5.27309 0.219712
\(577\) −14.9926 −0.624152 −0.312076 0.950057i \(-0.601024\pi\)
−0.312076 + 0.950057i \(0.601024\pi\)
\(578\) −0.892239 −0.0371122
\(579\) 13.6803 0.568534
\(580\) 8.01789 0.332925
\(581\) 7.11808 0.295308
\(582\) 5.58712 0.231593
\(583\) −1.40807 −0.0583164
\(584\) 22.4692 0.929784
\(585\) 2.74670 0.113562
\(586\) 14.5626 0.601577
\(587\) 29.7529 1.22803 0.614016 0.789293i \(-0.289553\pi\)
0.614016 + 0.789293i \(0.289553\pi\)
\(588\) −7.24491 −0.298775
\(589\) 4.88675 0.201355
\(590\) 10.5427 0.434037
\(591\) 2.30949 0.0949997
\(592\) −0.256551 −0.0105442
\(593\) −33.1252 −1.36029 −0.680145 0.733078i \(-0.738083\pi\)
−0.680145 + 0.733078i \(0.738083\pi\)
\(594\) 0.195938 0.00803943
\(595\) 3.56971 0.146344
\(596\) −19.9447 −0.816967
\(597\) −12.9433 −0.529733
\(598\) −8.81371 −0.360419
\(599\) −22.1396 −0.904600 −0.452300 0.891866i \(-0.649396\pi\)
−0.452300 + 0.891866i \(0.649396\pi\)
\(600\) −11.4950 −0.469281
\(601\) 12.6996 0.518028 0.259014 0.965874i \(-0.416602\pi\)
0.259014 + 0.965874i \(0.416602\pi\)
\(602\) 19.1801 0.781722
\(603\) −5.94225 −0.241987
\(604\) −1.09457 −0.0445376
\(605\) 10.8355 0.440525
\(606\) −12.0819 −0.490795
\(607\) 6.50596 0.264069 0.132034 0.991245i \(-0.457849\pi\)
0.132034 + 0.991245i \(0.457849\pi\)
\(608\) −27.4900 −1.11487
\(609\) 24.2869 0.984153
\(610\) −4.35945 −0.176509
\(611\) −4.82371 −0.195146
\(612\) 1.20391 0.0486652
\(613\) −13.4815 −0.544512 −0.272256 0.962225i \(-0.587770\pi\)
−0.272256 + 0.962225i \(0.587770\pi\)
\(614\) −13.4916 −0.544476
\(615\) −0.0925054 −0.00373018
\(616\) −2.26500 −0.0912594
\(617\) 3.49246 0.140601 0.0703006 0.997526i \(-0.477604\pi\)
0.0703006 + 0.997526i \(0.477604\pi\)
\(618\) −11.6142 −0.467191
\(619\) 42.2390 1.69773 0.848863 0.528612i \(-0.177287\pi\)
0.848863 + 0.528612i \(0.177287\pi\)
\(620\) 1.18361 0.0475349
\(621\) −3.55820 −0.142786
\(622\) 12.5556 0.503435
\(623\) −21.7688 −0.872149
\(624\) 0.396380 0.0158679
\(625\) 11.2751 0.451002
\(626\) 16.0766 0.642549
\(627\) −1.07996 −0.0431295
\(628\) 1.20391 0.0480413
\(629\) −1.79684 −0.0716448
\(630\) 3.18503 0.126895
\(631\) 20.0697 0.798963 0.399482 0.916741i \(-0.369190\pi\)
0.399482 + 0.916741i \(0.369190\pi\)
\(632\) 14.3067 0.569089
\(633\) −25.7974 −1.02536
\(634\) −27.1881 −1.07978
\(635\) 9.10965 0.361506
\(636\) −7.71936 −0.306093
\(637\) −16.7065 −0.661937
\(638\) 1.31893 0.0522168
\(639\) 11.2788 0.446182
\(640\) −6.40621 −0.253228
\(641\) −12.2217 −0.482729 −0.241364 0.970435i \(-0.577595\pi\)
−0.241364 + 0.970435i \(0.577595\pi\)
\(642\) −0.388368 −0.0153277
\(643\) −6.11753 −0.241252 −0.120626 0.992698i \(-0.538490\pi\)
−0.120626 + 0.992698i \(0.538490\pi\)
\(644\) 15.4559 0.609046
\(645\) 5.89473 0.232105
\(646\) 4.38785 0.172638
\(647\) 48.1369 1.89246 0.946228 0.323501i \(-0.104860\pi\)
0.946228 + 0.323501i \(0.104860\pi\)
\(648\) 2.85865 0.112298
\(649\) −2.62268 −0.102949
\(650\) −9.96038 −0.390678
\(651\) 3.58525 0.140517
\(652\) −10.8257 −0.423969
\(653\) −18.4826 −0.723282 −0.361641 0.932318i \(-0.617783\pi\)
−0.361641 + 0.932318i \(0.617783\pi\)
\(654\) 12.3891 0.484452
\(655\) 9.00876 0.352001
\(656\) −0.0133496 −0.000521214 0
\(657\) 7.86008 0.306651
\(658\) −5.59350 −0.218057
\(659\) −18.9864 −0.739604 −0.369802 0.929111i \(-0.620574\pi\)
−0.369802 + 0.929111i \(0.620574\pi\)
\(660\) −0.261575 −0.0101818
\(661\) −4.50535 −0.175238 −0.0876189 0.996154i \(-0.527926\pi\)
−0.0876189 + 0.996154i \(0.527926\pi\)
\(662\) −2.67942 −0.104139
\(663\) 2.77618 0.107818
\(664\) 5.63969 0.218862
\(665\) −17.5551 −0.680758
\(666\) −1.60321 −0.0621232
\(667\) −23.9515 −0.927405
\(668\) −21.0365 −0.813926
\(669\) 18.8319 0.728084
\(670\) −5.24560 −0.202655
\(671\) 1.08449 0.0418661
\(672\) −20.1685 −0.778018
\(673\) −8.05013 −0.310310 −0.155155 0.987890i \(-0.549588\pi\)
−0.155155 + 0.987890i \(0.549588\pi\)
\(674\) 26.4482 1.01875
\(675\) −4.02112 −0.154773
\(676\) 6.37209 0.245080
\(677\) 37.9685 1.45925 0.729625 0.683848i \(-0.239695\pi\)
0.729625 + 0.683848i \(0.239695\pi\)
\(678\) 9.64333 0.370350
\(679\) −22.5931 −0.867044
\(680\) 2.82830 0.108460
\(681\) −7.11795 −0.272761
\(682\) 0.194701 0.00745549
\(683\) 27.6502 1.05801 0.529003 0.848620i \(-0.322566\pi\)
0.529003 + 0.848620i \(0.322566\pi\)
\(684\) −5.92059 −0.226379
\(685\) 2.06256 0.0788063
\(686\) 3.16186 0.120720
\(687\) −5.55949 −0.212108
\(688\) 0.850677 0.0324318
\(689\) −17.8006 −0.678149
\(690\) −3.14105 −0.119578
\(691\) 6.55651 0.249421 0.124711 0.992193i \(-0.460200\pi\)
0.124711 + 0.992193i \(0.460200\pi\)
\(692\) −11.7414 −0.446340
\(693\) −0.792331 −0.0300982
\(694\) −11.4509 −0.434669
\(695\) −19.1810 −0.727577
\(696\) 19.2426 0.729388
\(697\) −0.0934982 −0.00354150
\(698\) 11.8729 0.449397
\(699\) 5.61253 0.212285
\(700\) 17.4667 0.660179
\(701\) 1.60435 0.0605955 0.0302977 0.999541i \(-0.490354\pi\)
0.0302977 + 0.999541i \(0.490354\pi\)
\(702\) 2.47701 0.0934889
\(703\) 8.83650 0.333275
\(704\) −1.15799 −0.0436432
\(705\) −1.71909 −0.0647445
\(706\) −24.5994 −0.925811
\(707\) 48.8567 1.83745
\(708\) −14.3781 −0.540363
\(709\) −40.2894 −1.51310 −0.756550 0.653935i \(-0.773117\pi\)
−0.756550 + 0.653935i \(0.773117\pi\)
\(710\) 9.95652 0.373661
\(711\) 5.00470 0.187691
\(712\) −17.2475 −0.646378
\(713\) −3.53574 −0.132414
\(714\) 3.21922 0.120476
\(715\) −0.603183 −0.0225578
\(716\) 14.8233 0.553973
\(717\) −11.4166 −0.426360
\(718\) 3.19387 0.119194
\(719\) 28.4225 1.05998 0.529990 0.848004i \(-0.322196\pi\)
0.529990 + 0.848004i \(0.322196\pi\)
\(720\) 0.141263 0.00526456
\(721\) 46.9652 1.74908
\(722\) −4.62600 −0.172162
\(723\) 12.2964 0.457308
\(724\) −20.7721 −0.771989
\(725\) −27.0676 −1.00527
\(726\) 9.77160 0.362658
\(727\) 41.6640 1.54523 0.772617 0.634873i \(-0.218947\pi\)
0.772617 + 0.634873i \(0.218947\pi\)
\(728\) −28.6337 −1.06124
\(729\) 1.00000 0.0370370
\(730\) 6.93860 0.256809
\(731\) 5.95800 0.220365
\(732\) 5.94539 0.219748
\(733\) −40.5778 −1.49878 −0.749389 0.662131i \(-0.769652\pi\)
−0.749389 + 0.662131i \(0.769652\pi\)
\(734\) −33.7809 −1.24688
\(735\) −5.95392 −0.219613
\(736\) 19.8900 0.733156
\(737\) 1.30493 0.0480678
\(738\) −0.0834227 −0.00307083
\(739\) −14.1032 −0.518793 −0.259396 0.965771i \(-0.583524\pi\)
−0.259396 + 0.965771i \(0.583524\pi\)
\(740\) 2.14027 0.0786778
\(741\) −13.6527 −0.501544
\(742\) −20.6413 −0.757766
\(743\) 18.0932 0.663775 0.331887 0.943319i \(-0.392315\pi\)
0.331887 + 0.943319i \(0.392315\pi\)
\(744\) 2.84061 0.104142
\(745\) −16.3907 −0.600508
\(746\) −20.7378 −0.759265
\(747\) 1.97285 0.0721827
\(748\) −0.264382 −0.00966676
\(749\) 1.57048 0.0573840
\(750\) −7.96353 −0.290787
\(751\) 50.4062 1.83935 0.919674 0.392683i \(-0.128453\pi\)
0.919674 + 0.392683i \(0.128453\pi\)
\(752\) −0.248084 −0.00904668
\(753\) 28.2372 1.02902
\(754\) 16.6736 0.607218
\(755\) −0.899529 −0.0327372
\(756\) −4.34373 −0.157980
\(757\) −46.7539 −1.69930 −0.849650 0.527348i \(-0.823187\pi\)
−0.849650 + 0.527348i \(0.823187\pi\)
\(758\) 5.82543 0.211589
\(759\) 0.781390 0.0283627
\(760\) −13.9090 −0.504532
\(761\) 37.0638 1.34356 0.671781 0.740750i \(-0.265530\pi\)
0.671781 + 0.740750i \(0.265530\pi\)
\(762\) 8.21522 0.297606
\(763\) −50.0989 −1.81370
\(764\) 22.3107 0.807171
\(765\) 0.989381 0.0357712
\(766\) −23.1354 −0.835915
\(767\) −33.1555 −1.19718
\(768\) −16.3234 −0.589020
\(769\) −20.6870 −0.745991 −0.372995 0.927833i \(-0.621669\pi\)
−0.372995 + 0.927833i \(0.621669\pi\)
\(770\) −0.699442 −0.0252061
\(771\) 13.1349 0.473040
\(772\) −16.4698 −0.592763
\(773\) 10.9866 0.395161 0.197581 0.980287i \(-0.436692\pi\)
0.197581 + 0.980287i \(0.436692\pi\)
\(774\) 5.31596 0.191078
\(775\) −3.99574 −0.143531
\(776\) −17.9006 −0.642595
\(777\) 6.48304 0.232578
\(778\) −13.7740 −0.493823
\(779\) 0.459805 0.0164742
\(780\) −3.30678 −0.118402
\(781\) −2.47685 −0.0886288
\(782\) −3.17476 −0.113529
\(783\) 6.73135 0.240559
\(784\) −0.859218 −0.0306864
\(785\) 0.989381 0.0353125
\(786\) 8.12423 0.289782
\(787\) 41.4498 1.47752 0.738762 0.673966i \(-0.235411\pi\)
0.738762 + 0.673966i \(0.235411\pi\)
\(788\) −2.78042 −0.0990483
\(789\) 4.67604 0.166472
\(790\) 4.41797 0.157184
\(791\) −38.9956 −1.38652
\(792\) −0.627768 −0.0223068
\(793\) 13.7099 0.486853
\(794\) −3.89289 −0.138154
\(795\) −6.34382 −0.224992
\(796\) 15.5825 0.552308
\(797\) 41.1032 1.45595 0.727974 0.685604i \(-0.240462\pi\)
0.727974 + 0.685604i \(0.240462\pi\)
\(798\) −15.8314 −0.560427
\(799\) −1.73754 −0.0614696
\(800\) 22.4777 0.794708
\(801\) −6.03345 −0.213181
\(802\) −6.38249 −0.225374
\(803\) −1.72609 −0.0609126
\(804\) 7.15393 0.252300
\(805\) 12.7017 0.447677
\(806\) 2.46138 0.0866984
\(807\) −16.2954 −0.573625
\(808\) 38.7094 1.36179
\(809\) 13.4962 0.474500 0.237250 0.971449i \(-0.423754\pi\)
0.237250 + 0.971449i \(0.423754\pi\)
\(810\) 0.882764 0.0310172
\(811\) −8.99366 −0.315810 −0.157905 0.987454i \(-0.550474\pi\)
−0.157905 + 0.987454i \(0.550474\pi\)
\(812\) −29.2392 −1.02609
\(813\) 3.98510 0.139764
\(814\) 0.352070 0.0123400
\(815\) −8.89666 −0.311636
\(816\) 0.142779 0.00499827
\(817\) −29.3002 −1.02509
\(818\) 17.3219 0.605647
\(819\) −10.0165 −0.350005
\(820\) 0.111368 0.00388915
\(821\) −18.1826 −0.634576 −0.317288 0.948329i \(-0.602772\pi\)
−0.317288 + 0.948329i \(0.602772\pi\)
\(822\) 1.86005 0.0648765
\(823\) −15.8361 −0.552013 −0.276007 0.961156i \(-0.589011\pi\)
−0.276007 + 0.961156i \(0.589011\pi\)
\(824\) 37.2108 1.29630
\(825\) 0.883050 0.0307438
\(826\) −38.4466 −1.33773
\(827\) 18.8956 0.657064 0.328532 0.944493i \(-0.393446\pi\)
0.328532 + 0.944493i \(0.393446\pi\)
\(828\) 4.28375 0.148871
\(829\) −14.9754 −0.520118 −0.260059 0.965593i \(-0.583742\pi\)
−0.260059 + 0.965593i \(0.583742\pi\)
\(830\) 1.74156 0.0604504
\(831\) −21.3862 −0.741878
\(832\) −14.6391 −0.507518
\(833\) −6.01782 −0.208505
\(834\) −17.2977 −0.598971
\(835\) −17.2879 −0.598273
\(836\) 1.30018 0.0449675
\(837\) 0.993688 0.0343469
\(838\) 20.1505 0.696087
\(839\) 52.1989 1.80210 0.901052 0.433710i \(-0.142796\pi\)
0.901052 + 0.433710i \(0.142796\pi\)
\(840\) −10.2046 −0.352091
\(841\) 16.3111 0.562451
\(842\) −14.6336 −0.504306
\(843\) 14.8841 0.512637
\(844\) 31.0578 1.06905
\(845\) 5.23663 0.180145
\(846\) −1.55030 −0.0533003
\(847\) −39.5142 −1.35773
\(848\) −0.915486 −0.0314379
\(849\) 4.58113 0.157224
\(850\) −3.58780 −0.123061
\(851\) −6.39352 −0.219167
\(852\) −13.5787 −0.465197
\(853\) 42.1342 1.44265 0.721324 0.692598i \(-0.243534\pi\)
0.721324 + 0.692598i \(0.243534\pi\)
\(854\) 15.8978 0.544011
\(855\) −4.86558 −0.166399
\(856\) 1.24430 0.0425292
\(857\) 15.9513 0.544887 0.272443 0.962172i \(-0.412168\pi\)
0.272443 + 0.962172i \(0.412168\pi\)
\(858\) −0.543959 −0.0185705
\(859\) 3.59920 0.122803 0.0614016 0.998113i \(-0.480443\pi\)
0.0614016 + 0.998113i \(0.480443\pi\)
\(860\) −7.09673 −0.241997
\(861\) 0.337343 0.0114966
\(862\) 5.44492 0.185455
\(863\) −0.0465595 −0.00158490 −0.000792452 1.00000i \(-0.500252\pi\)
−0.000792452 1.00000i \(0.500252\pi\)
\(864\) −5.58991 −0.190173
\(865\) −9.64913 −0.328080
\(866\) −2.57688 −0.0875660
\(867\) 1.00000 0.0339618
\(868\) −4.31632 −0.146505
\(869\) −1.09904 −0.0372825
\(870\) 5.94219 0.201459
\(871\) 16.4967 0.558971
\(872\) −39.6936 −1.34419
\(873\) −6.26191 −0.211933
\(874\) 15.6128 0.528112
\(875\) 32.2028 1.08865
\(876\) −9.46283 −0.319720
\(877\) −46.6436 −1.57504 −0.787522 0.616286i \(-0.788636\pi\)
−0.787522 + 0.616286i \(0.788636\pi\)
\(878\) −29.6932 −1.00210
\(879\) −16.3215 −0.550509
\(880\) −0.0310217 −0.00104574
\(881\) 17.9980 0.606369 0.303184 0.952932i \(-0.401950\pi\)
0.303184 + 0.952932i \(0.401950\pi\)
\(882\) −5.36933 −0.180795
\(883\) −44.2503 −1.48914 −0.744570 0.667545i \(-0.767345\pi\)
−0.744570 + 0.667545i \(0.767345\pi\)
\(884\) −3.34227 −0.112413
\(885\) −11.8160 −0.397192
\(886\) −31.7944 −1.06815
\(887\) −33.7261 −1.13241 −0.566206 0.824264i \(-0.691589\pi\)
−0.566206 + 0.824264i \(0.691589\pi\)
\(888\) 5.13655 0.172371
\(889\) −33.2206 −1.11418
\(890\) −5.32611 −0.178532
\(891\) −0.219603 −0.00735697
\(892\) −22.6719 −0.759112
\(893\) 8.54485 0.285942
\(894\) −14.7814 −0.494363
\(895\) 12.1819 0.407195
\(896\) 23.3618 0.780464
\(897\) 9.87819 0.329823
\(898\) −35.0864 −1.17085
\(899\) 6.68886 0.223086
\(900\) 4.84107 0.161369
\(901\) −6.41191 −0.213612
\(902\) 0.0183198 0.000609984 0
\(903\) −21.4966 −0.715362
\(904\) −30.8964 −1.02760
\(905\) −17.0706 −0.567448
\(906\) −0.811208 −0.0269506
\(907\) −43.4126 −1.44149 −0.720745 0.693200i \(-0.756200\pi\)
−0.720745 + 0.693200i \(0.756200\pi\)
\(908\) 8.56938 0.284385
\(909\) 13.5411 0.449131
\(910\) −8.84222 −0.293117
\(911\) −26.5779 −0.880566 −0.440283 0.897859i \(-0.645122\pi\)
−0.440283 + 0.897859i \(0.645122\pi\)
\(912\) −0.702158 −0.0232508
\(913\) −0.433243 −0.0143382
\(914\) −13.0435 −0.431440
\(915\) 4.88596 0.161525
\(916\) 6.69313 0.221147
\(917\) −32.8526 −1.08489
\(918\) 0.892239 0.0294483
\(919\) −33.5636 −1.10716 −0.553581 0.832796i \(-0.686739\pi\)
−0.553581 + 0.832796i \(0.686739\pi\)
\(920\) 10.0636 0.331788
\(921\) 15.1210 0.498255
\(922\) −11.3829 −0.374875
\(923\) −31.3120 −1.03065
\(924\) 0.953896 0.0313809
\(925\) −7.22533 −0.237567
\(926\) −11.3827 −0.374058
\(927\) 13.0169 0.427531
\(928\) −37.6276 −1.23519
\(929\) −29.9598 −0.982951 −0.491475 0.870892i \(-0.663542\pi\)
−0.491475 + 0.870892i \(0.663542\pi\)
\(930\) 0.877192 0.0287643
\(931\) 29.5944 0.969917
\(932\) −6.75698 −0.221332
\(933\) −14.0721 −0.460699
\(934\) 0.352001 0.0115178
\(935\) −0.217271 −0.00710552
\(936\) −7.93613 −0.259401
\(937\) 11.3522 0.370859 0.185429 0.982658i \(-0.440632\pi\)
0.185429 + 0.982658i \(0.440632\pi\)
\(938\) 19.1294 0.624596
\(939\) −18.0182 −0.588003
\(940\) 2.06963 0.0675037
\(941\) −29.7945 −0.971274 −0.485637 0.874160i \(-0.661412\pi\)
−0.485637 + 0.874160i \(0.661412\pi\)
\(942\) 0.892239 0.0290707
\(943\) −0.332685 −0.0108337
\(944\) −1.70519 −0.0554992
\(945\) −3.56971 −0.116123
\(946\) −1.16740 −0.0379554
\(947\) −19.3568 −0.629011 −0.314506 0.949256i \(-0.601839\pi\)
−0.314506 + 0.949256i \(0.601839\pi\)
\(948\) −6.02521 −0.195690
\(949\) −21.8210 −0.708339
\(950\) 17.6441 0.572449
\(951\) 30.4718 0.988116
\(952\) −10.3141 −0.334281
\(953\) 14.0591 0.455420 0.227710 0.973729i \(-0.426876\pi\)
0.227710 + 0.973729i \(0.426876\pi\)
\(954\) −5.72095 −0.185223
\(955\) 18.3350 0.593308
\(956\) 13.7445 0.444530
\(957\) −1.47822 −0.0477841
\(958\) −33.9944 −1.09831
\(959\) −7.52163 −0.242886
\(960\) −5.21710 −0.168381
\(961\) −30.0126 −0.968148
\(962\) 4.45080 0.143500
\(963\) 0.435274 0.0140265
\(964\) −14.8038 −0.476797
\(965\) −13.5350 −0.435708
\(966\) 11.4546 0.368546
\(967\) 19.9898 0.642828 0.321414 0.946939i \(-0.395842\pi\)
0.321414 + 0.946939i \(0.395842\pi\)
\(968\) −31.3073 −1.00626
\(969\) −4.91780 −0.157982
\(970\) −5.52779 −0.177487
\(971\) 32.0285 1.02784 0.513921 0.857837i \(-0.328192\pi\)
0.513921 + 0.857837i \(0.328192\pi\)
\(972\) −1.20391 −0.0386154
\(973\) 69.9482 2.24244
\(974\) −7.53099 −0.241308
\(975\) 11.1634 0.357514
\(976\) 0.705100 0.0225697
\(977\) 51.1067 1.63505 0.817523 0.575895i \(-0.195346\pi\)
0.817523 + 0.575895i \(0.195346\pi\)
\(978\) −8.02314 −0.256552
\(979\) 1.32496 0.0423459
\(980\) 7.16798 0.228973
\(981\) −13.8854 −0.443327
\(982\) −7.02082 −0.224043
\(983\) −49.2635 −1.57126 −0.785631 0.618695i \(-0.787662\pi\)
−0.785631 + 0.618695i \(0.787662\pi\)
\(984\) 0.267279 0.00852054
\(985\) −2.28497 −0.0728051
\(986\) 6.00597 0.191269
\(987\) 6.26907 0.199547
\(988\) 16.4366 0.522918
\(989\) 21.1997 0.674113
\(990\) −0.193857 −0.00616119
\(991\) −29.3376 −0.931941 −0.465970 0.884800i \(-0.654295\pi\)
−0.465970 + 0.884800i \(0.654295\pi\)
\(992\) −5.55463 −0.176360
\(993\) 3.00303 0.0952983
\(994\) −36.3089 −1.15165
\(995\) 12.8058 0.405972
\(996\) −2.37513 −0.0752589
\(997\) −50.3103 −1.59334 −0.796671 0.604413i \(-0.793408\pi\)
−0.796671 + 0.604413i \(0.793408\pi\)
\(998\) −6.17122 −0.195347
\(999\) 1.79684 0.0568496
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 8007.2.a.h.1.20 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.h.1.20 56 1.1 even 1 trivial