Properties

Label 8002.2.a.e.1.72
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $77$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(77\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.72
Character \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.92694 q^{3} +1.00000 q^{4} +3.51878 q^{5} -2.92694 q^{6} -2.15404 q^{7} -1.00000 q^{8} +5.56701 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.92694 q^{3} +1.00000 q^{4} +3.51878 q^{5} -2.92694 q^{6} -2.15404 q^{7} -1.00000 q^{8} +5.56701 q^{9} -3.51878 q^{10} -1.41443 q^{11} +2.92694 q^{12} +1.01538 q^{13} +2.15404 q^{14} +10.2993 q^{15} +1.00000 q^{16} +2.94396 q^{17} -5.56701 q^{18} +6.03206 q^{19} +3.51878 q^{20} -6.30477 q^{21} +1.41443 q^{22} -2.14527 q^{23} -2.92694 q^{24} +7.38184 q^{25} -1.01538 q^{26} +7.51348 q^{27} -2.15404 q^{28} -0.271618 q^{29} -10.2993 q^{30} +5.07483 q^{31} -1.00000 q^{32} -4.13995 q^{33} -2.94396 q^{34} -7.57961 q^{35} +5.56701 q^{36} -8.31027 q^{37} -6.03206 q^{38} +2.97197 q^{39} -3.51878 q^{40} +9.15654 q^{41} +6.30477 q^{42} +5.82028 q^{43} -1.41443 q^{44} +19.5891 q^{45} +2.14527 q^{46} -9.09485 q^{47} +2.92694 q^{48} -2.36010 q^{49} -7.38184 q^{50} +8.61680 q^{51} +1.01538 q^{52} +7.65817 q^{53} -7.51348 q^{54} -4.97706 q^{55} +2.15404 q^{56} +17.6555 q^{57} +0.271618 q^{58} +6.11808 q^{59} +10.2993 q^{60} -1.98794 q^{61} -5.07483 q^{62} -11.9916 q^{63} +1.00000 q^{64} +3.57291 q^{65} +4.13995 q^{66} +4.12217 q^{67} +2.94396 q^{68} -6.27908 q^{69} +7.57961 q^{70} +15.1780 q^{71} -5.56701 q^{72} -2.71306 q^{73} +8.31027 q^{74} +21.6062 q^{75} +6.03206 q^{76} +3.04674 q^{77} -2.97197 q^{78} -4.28520 q^{79} +3.51878 q^{80} +5.29054 q^{81} -9.15654 q^{82} -11.1344 q^{83} -6.30477 q^{84} +10.3591 q^{85} -5.82028 q^{86} -0.795009 q^{87} +1.41443 q^{88} -15.1343 q^{89} -19.5891 q^{90} -2.18718 q^{91} -2.14527 q^{92} +14.8537 q^{93} +9.09485 q^{94} +21.2255 q^{95} -2.92694 q^{96} -15.5772 q^{97} +2.36010 q^{98} -7.87412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 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
\(3\) 2.92694 1.68987 0.844936 0.534867i \(-0.179638\pi\)
0.844936 + 0.534867i \(0.179638\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.51878 1.57365 0.786824 0.617177i \(-0.211724\pi\)
0.786824 + 0.617177i \(0.211724\pi\)
\(6\) −2.92694 −1.19492
\(7\) −2.15404 −0.814152 −0.407076 0.913394i \(-0.633452\pi\)
−0.407076 + 0.913394i \(0.633452\pi\)
\(8\) −1.00000 −0.353553
\(9\) 5.56701 1.85567
\(10\) −3.51878 −1.11274
\(11\) −1.41443 −0.426466 −0.213233 0.977001i \(-0.568399\pi\)
−0.213233 + 0.977001i \(0.568399\pi\)
\(12\) 2.92694 0.844936
\(13\) 1.01538 0.281616 0.140808 0.990037i \(-0.455030\pi\)
0.140808 + 0.990037i \(0.455030\pi\)
\(14\) 2.15404 0.575692
\(15\) 10.2993 2.65926
\(16\) 1.00000 0.250000
\(17\) 2.94396 0.714015 0.357007 0.934102i \(-0.383797\pi\)
0.357007 + 0.934102i \(0.383797\pi\)
\(18\) −5.56701 −1.31216
\(19\) 6.03206 1.38385 0.691925 0.721970i \(-0.256763\pi\)
0.691925 + 0.721970i \(0.256763\pi\)
\(20\) 3.51878 0.786824
\(21\) −6.30477 −1.37581
\(22\) 1.41443 0.301557
\(23\) −2.14527 −0.447319 −0.223660 0.974667i \(-0.571800\pi\)
−0.223660 + 0.974667i \(0.571800\pi\)
\(24\) −2.92694 −0.597460
\(25\) 7.38184 1.47637
\(26\) −1.01538 −0.199133
\(27\) 7.51348 1.44597
\(28\) −2.15404 −0.407076
\(29\) −0.271618 −0.0504381 −0.0252191 0.999682i \(-0.508028\pi\)
−0.0252191 + 0.999682i \(0.508028\pi\)
\(30\) −10.2993 −1.88038
\(31\) 5.07483 0.911466 0.455733 0.890117i \(-0.349377\pi\)
0.455733 + 0.890117i \(0.349377\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.13995 −0.720673
\(34\) −2.94396 −0.504885
\(35\) −7.57961 −1.28119
\(36\) 5.56701 0.927834
\(37\) −8.31027 −1.36620 −0.683100 0.730325i \(-0.739369\pi\)
−0.683100 + 0.730325i \(0.739369\pi\)
\(38\) −6.03206 −0.978529
\(39\) 2.97197 0.475896
\(40\) −3.51878 −0.556369
\(41\) 9.15654 1.43001 0.715006 0.699118i \(-0.246424\pi\)
0.715006 + 0.699118i \(0.246424\pi\)
\(42\) 6.30477 0.972846
\(43\) 5.82028 0.887584 0.443792 0.896130i \(-0.353633\pi\)
0.443792 + 0.896130i \(0.353633\pi\)
\(44\) −1.41443 −0.213233
\(45\) 19.5891 2.92017
\(46\) 2.14527 0.316302
\(47\) −9.09485 −1.32662 −0.663310 0.748345i \(-0.730849\pi\)
−0.663310 + 0.748345i \(0.730849\pi\)
\(48\) 2.92694 0.422468
\(49\) −2.36010 −0.337157
\(50\) −7.38184 −1.04395
\(51\) 8.61680 1.20659
\(52\) 1.01538 0.140808
\(53\) 7.65817 1.05193 0.525965 0.850506i \(-0.323704\pi\)
0.525965 + 0.850506i \(0.323704\pi\)
\(54\) −7.51348 −1.02246
\(55\) −4.97706 −0.671107
\(56\) 2.15404 0.287846
\(57\) 17.6555 2.33853
\(58\) 0.271618 0.0356651
\(59\) 6.11808 0.796506 0.398253 0.917276i \(-0.369617\pi\)
0.398253 + 0.917276i \(0.369617\pi\)
\(60\) 10.2993 1.32963
\(61\) −1.98794 −0.254530 −0.127265 0.991869i \(-0.540620\pi\)
−0.127265 + 0.991869i \(0.540620\pi\)
\(62\) −5.07483 −0.644504
\(63\) −11.9916 −1.51080
\(64\) 1.00000 0.125000
\(65\) 3.57291 0.443165
\(66\) 4.13995 0.509593
\(67\) 4.12217 0.503604 0.251802 0.967779i \(-0.418977\pi\)
0.251802 + 0.967779i \(0.418977\pi\)
\(68\) 2.94396 0.357007
\(69\) −6.27908 −0.755912
\(70\) 7.57961 0.905937
\(71\) 15.1780 1.80129 0.900647 0.434552i \(-0.143093\pi\)
0.900647 + 0.434552i \(0.143093\pi\)
\(72\) −5.56701 −0.656078
\(73\) −2.71306 −0.317539 −0.158770 0.987316i \(-0.550753\pi\)
−0.158770 + 0.987316i \(0.550753\pi\)
\(74\) 8.31027 0.966049
\(75\) 21.6062 2.49487
\(76\) 6.03206 0.691925
\(77\) 3.04674 0.347208
\(78\) −2.97197 −0.336509
\(79\) −4.28520 −0.482122 −0.241061 0.970510i \(-0.577495\pi\)
−0.241061 + 0.970510i \(0.577495\pi\)
\(80\) 3.51878 0.393412
\(81\) 5.29054 0.587838
\(82\) −9.15654 −1.01117
\(83\) −11.1344 −1.22215 −0.611077 0.791571i \(-0.709264\pi\)
−0.611077 + 0.791571i \(0.709264\pi\)
\(84\) −6.30477 −0.687906
\(85\) 10.3591 1.12361
\(86\) −5.82028 −0.627617
\(87\) −0.795009 −0.0852340
\(88\) 1.41443 0.150778
\(89\) −15.1343 −1.60424 −0.802119 0.597165i \(-0.796294\pi\)
−0.802119 + 0.597165i \(0.796294\pi\)
\(90\) −19.5891 −2.06487
\(91\) −2.18718 −0.229279
\(92\) −2.14527 −0.223660
\(93\) 14.8537 1.54026
\(94\) 9.09485 0.938062
\(95\) 21.2255 2.17769
\(96\) −2.92694 −0.298730
\(97\) −15.5772 −1.58163 −0.790813 0.612057i \(-0.790342\pi\)
−0.790813 + 0.612057i \(0.790342\pi\)
\(98\) 2.36010 0.238406
\(99\) −7.87412 −0.791379
\(100\) 7.38184 0.738184
\(101\) −0.540634 −0.0537951 −0.0268975 0.999638i \(-0.508563\pi\)
−0.0268975 + 0.999638i \(0.508563\pi\)
\(102\) −8.61680 −0.853190
\(103\) 6.79230 0.669265 0.334632 0.942349i \(-0.391388\pi\)
0.334632 + 0.942349i \(0.391388\pi\)
\(104\) −1.01538 −0.0995665
\(105\) −22.1851 −2.16504
\(106\) −7.65817 −0.743827
\(107\) 13.4644 1.30165 0.650826 0.759227i \(-0.274423\pi\)
0.650826 + 0.759227i \(0.274423\pi\)
\(108\) 7.51348 0.722985
\(109\) 6.72962 0.644580 0.322290 0.946641i \(-0.395547\pi\)
0.322290 + 0.946641i \(0.395547\pi\)
\(110\) 4.97706 0.474544
\(111\) −24.3237 −2.30870
\(112\) −2.15404 −0.203538
\(113\) −10.2145 −0.960899 −0.480450 0.877022i \(-0.659526\pi\)
−0.480450 + 0.877022i \(0.659526\pi\)
\(114\) −17.6555 −1.65359
\(115\) −7.54873 −0.703923
\(116\) −0.271618 −0.0252191
\(117\) 5.65264 0.522587
\(118\) −6.11808 −0.563215
\(119\) −6.34141 −0.581316
\(120\) −10.2993 −0.940192
\(121\) −8.99940 −0.818127
\(122\) 1.98794 0.179980
\(123\) 26.8007 2.41654
\(124\) 5.07483 0.455733
\(125\) 8.38118 0.749635
\(126\) 11.9916 1.06829
\(127\) 8.20947 0.728473 0.364236 0.931306i \(-0.381330\pi\)
0.364236 + 0.931306i \(0.381330\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 17.0356 1.49990
\(130\) −3.57291 −0.313365
\(131\) −12.4641 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(132\) −4.13995 −0.360336
\(133\) −12.9933 −1.12666
\(134\) −4.12217 −0.356102
\(135\) 26.4383 2.27545
\(136\) −2.94396 −0.252442
\(137\) 0.690596 0.0590016 0.0295008 0.999565i \(-0.490608\pi\)
0.0295008 + 0.999565i \(0.490608\pi\)
\(138\) 6.27908 0.534511
\(139\) −15.5473 −1.31871 −0.659353 0.751834i \(-0.729169\pi\)
−0.659353 + 0.751834i \(0.729169\pi\)
\(140\) −7.57961 −0.640594
\(141\) −26.6201 −2.24182
\(142\) −15.1780 −1.27371
\(143\) −1.43618 −0.120100
\(144\) 5.56701 0.463917
\(145\) −0.955763 −0.0793718
\(146\) 2.71306 0.224534
\(147\) −6.90788 −0.569752
\(148\) −8.31027 −0.683100
\(149\) 8.77859 0.719170 0.359585 0.933112i \(-0.382918\pi\)
0.359585 + 0.933112i \(0.382918\pi\)
\(150\) −21.6062 −1.76414
\(151\) 6.59271 0.536507 0.268254 0.963348i \(-0.413553\pi\)
0.268254 + 0.963348i \(0.413553\pi\)
\(152\) −6.03206 −0.489265
\(153\) 16.3890 1.32497
\(154\) −3.04674 −0.245513
\(155\) 17.8572 1.43433
\(156\) 2.97197 0.237948
\(157\) 17.0219 1.35850 0.679249 0.733908i \(-0.262306\pi\)
0.679249 + 0.733908i \(0.262306\pi\)
\(158\) 4.28520 0.340912
\(159\) 22.4150 1.77763
\(160\) −3.51878 −0.278184
\(161\) 4.62100 0.364186
\(162\) −5.29054 −0.415664
\(163\) −1.29657 −0.101555 −0.0507775 0.998710i \(-0.516170\pi\)
−0.0507775 + 0.998710i \(0.516170\pi\)
\(164\) 9.15654 0.715006
\(165\) −14.5676 −1.13409
\(166\) 11.1344 0.864194
\(167\) −3.97341 −0.307472 −0.153736 0.988112i \(-0.549130\pi\)
−0.153736 + 0.988112i \(0.549130\pi\)
\(168\) 6.30477 0.486423
\(169\) −11.9690 −0.920692
\(170\) −10.3591 −0.794511
\(171\) 33.5805 2.56797
\(172\) 5.82028 0.443792
\(173\) 15.9429 1.21212 0.606060 0.795419i \(-0.292749\pi\)
0.606060 + 0.795419i \(0.292749\pi\)
\(174\) 0.795009 0.0602695
\(175\) −15.9008 −1.20199
\(176\) −1.41443 −0.106616
\(177\) 17.9073 1.34599
\(178\) 15.1343 1.13437
\(179\) 10.0601 0.751924 0.375962 0.926635i \(-0.377312\pi\)
0.375962 + 0.926635i \(0.377312\pi\)
\(180\) 19.5891 1.46008
\(181\) 5.63218 0.418636 0.209318 0.977848i \(-0.432876\pi\)
0.209318 + 0.977848i \(0.432876\pi\)
\(182\) 2.18718 0.162124
\(183\) −5.81860 −0.430124
\(184\) 2.14527 0.158151
\(185\) −29.2420 −2.14992
\(186\) −14.8537 −1.08913
\(187\) −4.16401 −0.304503
\(188\) −9.09485 −0.663310
\(189\) −16.1844 −1.17724
\(190\) −21.2255 −1.53986
\(191\) −17.7971 −1.28776 −0.643878 0.765128i \(-0.722676\pi\)
−0.643878 + 0.765128i \(0.722676\pi\)
\(192\) 2.92694 0.211234
\(193\) 16.1480 1.16236 0.581179 0.813776i \(-0.302591\pi\)
0.581179 + 0.813776i \(0.302591\pi\)
\(194\) 15.5772 1.11838
\(195\) 10.4577 0.748893
\(196\) −2.36010 −0.168578
\(197\) 5.37830 0.383188 0.191594 0.981474i \(-0.438634\pi\)
0.191594 + 0.981474i \(0.438634\pi\)
\(198\) 7.87412 0.559590
\(199\) −12.2368 −0.867447 −0.433724 0.901046i \(-0.642801\pi\)
−0.433724 + 0.901046i \(0.642801\pi\)
\(200\) −7.38184 −0.521975
\(201\) 12.0654 0.851026
\(202\) 0.540634 0.0380389
\(203\) 0.585076 0.0410643
\(204\) 8.61680 0.603297
\(205\) 32.2199 2.25034
\(206\) −6.79230 −0.473242
\(207\) −11.9427 −0.830076
\(208\) 1.01538 0.0704041
\(209\) −8.53191 −0.590165
\(210\) 22.1851 1.53092
\(211\) −9.56427 −0.658431 −0.329216 0.944255i \(-0.606784\pi\)
−0.329216 + 0.944255i \(0.606784\pi\)
\(212\) 7.65817 0.525965
\(213\) 44.4251 3.04396
\(214\) −13.4644 −0.920407
\(215\) 20.4803 1.39674
\(216\) −7.51348 −0.511228
\(217\) −10.9314 −0.742072
\(218\) −6.72962 −0.455787
\(219\) −7.94097 −0.536601
\(220\) −4.97706 −0.335554
\(221\) 2.98924 0.201078
\(222\) 24.3237 1.63250
\(223\) 20.9492 1.40286 0.701430 0.712739i \(-0.252545\pi\)
0.701430 + 0.712739i \(0.252545\pi\)
\(224\) 2.15404 0.143923
\(225\) 41.0947 2.73965
\(226\) 10.2145 0.679458
\(227\) 25.0539 1.66288 0.831442 0.555611i \(-0.187516\pi\)
0.831442 + 0.555611i \(0.187516\pi\)
\(228\) 17.6555 1.16926
\(229\) 11.8010 0.779829 0.389914 0.920851i \(-0.372505\pi\)
0.389914 + 0.920851i \(0.372505\pi\)
\(230\) 7.54873 0.497749
\(231\) 8.91763 0.586737
\(232\) 0.271618 0.0178326
\(233\) −16.7353 −1.09636 −0.548182 0.836359i \(-0.684680\pi\)
−0.548182 + 0.836359i \(0.684680\pi\)
\(234\) −5.65264 −0.369525
\(235\) −32.0028 −2.08763
\(236\) 6.11808 0.398253
\(237\) −12.5425 −0.814725
\(238\) 6.34141 0.411053
\(239\) −18.5731 −1.20140 −0.600698 0.799476i \(-0.705111\pi\)
−0.600698 + 0.799476i \(0.705111\pi\)
\(240\) 10.2993 0.664816
\(241\) −5.35589 −0.345003 −0.172502 0.985009i \(-0.555185\pi\)
−0.172502 + 0.985009i \(0.555185\pi\)
\(242\) 8.99940 0.578503
\(243\) −7.05534 −0.452600
\(244\) −1.98794 −0.127265
\(245\) −8.30467 −0.530566
\(246\) −26.8007 −1.70875
\(247\) 6.12485 0.389715
\(248\) −5.07483 −0.322252
\(249\) −32.5897 −2.06529
\(250\) −8.38118 −0.530072
\(251\) −16.9179 −1.06785 −0.533923 0.845533i \(-0.679283\pi\)
−0.533923 + 0.845533i \(0.679283\pi\)
\(252\) −11.9916 −0.755398
\(253\) 3.03432 0.190766
\(254\) −8.20947 −0.515108
\(255\) 30.3207 1.89875
\(256\) 1.00000 0.0625000
\(257\) −5.30216 −0.330740 −0.165370 0.986232i \(-0.552882\pi\)
−0.165370 + 0.986232i \(0.552882\pi\)
\(258\) −17.0356 −1.06059
\(259\) 17.9007 1.11229
\(260\) 3.57291 0.221583
\(261\) −1.51210 −0.0935964
\(262\) 12.4641 0.770037
\(263\) 27.9487 1.72339 0.861697 0.507424i \(-0.169402\pi\)
0.861697 + 0.507424i \(0.169402\pi\)
\(264\) 4.13995 0.254796
\(265\) 26.9474 1.65537
\(266\) 12.9933 0.796672
\(267\) −44.2974 −2.71096
\(268\) 4.12217 0.251802
\(269\) 6.57533 0.400905 0.200452 0.979703i \(-0.435759\pi\)
0.200452 + 0.979703i \(0.435759\pi\)
\(270\) −26.4383 −1.60899
\(271\) 29.0271 1.76327 0.881634 0.471934i \(-0.156444\pi\)
0.881634 + 0.471934i \(0.156444\pi\)
\(272\) 2.94396 0.178504
\(273\) −6.40175 −0.387452
\(274\) −0.690596 −0.0417204
\(275\) −10.4411 −0.629620
\(276\) −6.27908 −0.377956
\(277\) 31.1579 1.87209 0.936047 0.351874i \(-0.114456\pi\)
0.936047 + 0.351874i \(0.114456\pi\)
\(278\) 15.5473 0.932465
\(279\) 28.2516 1.69138
\(280\) 7.57961 0.452968
\(281\) −10.7216 −0.639599 −0.319800 0.947485i \(-0.603616\pi\)
−0.319800 + 0.947485i \(0.603616\pi\)
\(282\) 26.6201 1.58521
\(283\) 27.0020 1.60510 0.802550 0.596584i \(-0.203476\pi\)
0.802550 + 0.596584i \(0.203476\pi\)
\(284\) 15.1780 0.900647
\(285\) 62.1259 3.68002
\(286\) 1.43618 0.0849234
\(287\) −19.7236 −1.16425
\(288\) −5.56701 −0.328039
\(289\) −8.33312 −0.490183
\(290\) 0.955763 0.0561244
\(291\) −45.5937 −2.67275
\(292\) −2.71306 −0.158770
\(293\) 1.63516 0.0955270 0.0477635 0.998859i \(-0.484791\pi\)
0.0477635 + 0.998859i \(0.484791\pi\)
\(294\) 6.90788 0.402876
\(295\) 21.5282 1.25342
\(296\) 8.31027 0.483025
\(297\) −10.6273 −0.616657
\(298\) −8.77859 −0.508530
\(299\) −2.17827 −0.125972
\(300\) 21.6062 1.24744
\(301\) −12.5371 −0.722628
\(302\) −6.59271 −0.379368
\(303\) −1.58241 −0.0909068
\(304\) 6.03206 0.345962
\(305\) −6.99515 −0.400541
\(306\) −16.3890 −0.936898
\(307\) 14.1830 0.809464 0.404732 0.914435i \(-0.367365\pi\)
0.404732 + 0.914435i \(0.367365\pi\)
\(308\) 3.04674 0.173604
\(309\) 19.8807 1.13097
\(310\) −17.8572 −1.01422
\(311\) 1.76971 0.100351 0.0501755 0.998740i \(-0.484022\pi\)
0.0501755 + 0.998740i \(0.484022\pi\)
\(312\) −2.97197 −0.168255
\(313\) −5.95556 −0.336628 −0.168314 0.985733i \(-0.553832\pi\)
−0.168314 + 0.985733i \(0.553832\pi\)
\(314\) −17.0219 −0.960603
\(315\) −42.1957 −2.37746
\(316\) −4.28520 −0.241061
\(317\) 1.51891 0.0853103 0.0426552 0.999090i \(-0.486418\pi\)
0.0426552 + 0.999090i \(0.486418\pi\)
\(318\) −22.4150 −1.25697
\(319\) 0.384183 0.0215101
\(320\) 3.51878 0.196706
\(321\) 39.4095 2.19963
\(322\) −4.62100 −0.257518
\(323\) 17.7581 0.988089
\(324\) 5.29054 0.293919
\(325\) 7.49539 0.415770
\(326\) 1.29657 0.0718102
\(327\) 19.6972 1.08926
\(328\) −9.15654 −0.505586
\(329\) 19.5907 1.08007
\(330\) 14.5676 0.801919
\(331\) −26.2245 −1.44143 −0.720713 0.693233i \(-0.756186\pi\)
−0.720713 + 0.693233i \(0.756186\pi\)
\(332\) −11.1344 −0.611077
\(333\) −46.2633 −2.53521
\(334\) 3.97341 0.217415
\(335\) 14.5050 0.792495
\(336\) −6.30477 −0.343953
\(337\) −16.2124 −0.883146 −0.441573 0.897225i \(-0.645579\pi\)
−0.441573 + 0.897225i \(0.645579\pi\)
\(338\) 11.9690 0.651028
\(339\) −29.8973 −1.62380
\(340\) 10.3591 0.561804
\(341\) −7.17797 −0.388709
\(342\) −33.5805 −1.81583
\(343\) 20.1621 1.08865
\(344\) −5.82028 −0.313808
\(345\) −22.0947 −1.18954
\(346\) −15.9429 −0.857098
\(347\) −12.6302 −0.678024 −0.339012 0.940782i \(-0.610093\pi\)
−0.339012 + 0.940782i \(0.610093\pi\)
\(348\) −0.795009 −0.0426170
\(349\) −22.2269 −1.18978 −0.594888 0.803809i \(-0.702804\pi\)
−0.594888 + 0.803809i \(0.702804\pi\)
\(350\) 15.9008 0.849934
\(351\) 7.62906 0.407209
\(352\) 1.41443 0.0753892
\(353\) 7.44590 0.396305 0.198153 0.980171i \(-0.436506\pi\)
0.198153 + 0.980171i \(0.436506\pi\)
\(354\) −17.9073 −0.951762
\(355\) 53.4080 2.83460
\(356\) −15.1343 −0.802119
\(357\) −18.5610 −0.982350
\(358\) −10.0601 −0.531691
\(359\) 12.3914 0.653993 0.326996 0.945026i \(-0.393964\pi\)
0.326996 + 0.945026i \(0.393964\pi\)
\(360\) −19.5891 −1.03244
\(361\) 17.3858 0.915040
\(362\) −5.63218 −0.296021
\(363\) −26.3407 −1.38253
\(364\) −2.18718 −0.114639
\(365\) −9.54666 −0.499695
\(366\) 5.81860 0.304143
\(367\) −13.6207 −0.710994 −0.355497 0.934677i \(-0.615688\pi\)
−0.355497 + 0.934677i \(0.615688\pi\)
\(368\) −2.14527 −0.111830
\(369\) 50.9745 2.65363
\(370\) 29.2420 1.52022
\(371\) −16.4960 −0.856431
\(372\) 14.8537 0.770131
\(373\) 9.21697 0.477236 0.238618 0.971113i \(-0.423306\pi\)
0.238618 + 0.971113i \(0.423306\pi\)
\(374\) 4.16401 0.215316
\(375\) 24.5312 1.26679
\(376\) 9.09485 0.469031
\(377\) −0.275796 −0.0142042
\(378\) 16.1844 0.832434
\(379\) −5.15285 −0.264684 −0.132342 0.991204i \(-0.542250\pi\)
−0.132342 + 0.991204i \(0.542250\pi\)
\(380\) 21.2255 1.08885
\(381\) 24.0287 1.23103
\(382\) 17.7971 0.910581
\(383\) 25.9053 1.32370 0.661850 0.749636i \(-0.269772\pi\)
0.661850 + 0.749636i \(0.269772\pi\)
\(384\) −2.92694 −0.149365
\(385\) 10.7208 0.546383
\(386\) −16.1480 −0.821912
\(387\) 32.4015 1.64706
\(388\) −15.5772 −0.790813
\(389\) −14.2715 −0.723595 −0.361798 0.932257i \(-0.617837\pi\)
−0.361798 + 0.932257i \(0.617837\pi\)
\(390\) −10.4577 −0.529547
\(391\) −6.31558 −0.319392
\(392\) 2.36010 0.119203
\(393\) −36.4818 −1.84027
\(394\) −5.37830 −0.270955
\(395\) −15.0787 −0.758691
\(396\) −7.87412 −0.395690
\(397\) −28.3844 −1.42457 −0.712286 0.701890i \(-0.752340\pi\)
−0.712286 + 0.701890i \(0.752340\pi\)
\(398\) 12.2368 0.613378
\(399\) −38.0307 −1.90392
\(400\) 7.38184 0.369092
\(401\) −38.1003 −1.90264 −0.951320 0.308206i \(-0.900271\pi\)
−0.951320 + 0.308206i \(0.900271\pi\)
\(402\) −12.0654 −0.601766
\(403\) 5.15289 0.256684
\(404\) −0.540634 −0.0268975
\(405\) 18.6163 0.925049
\(406\) −0.585076 −0.0290368
\(407\) 11.7543 0.582638
\(408\) −8.61680 −0.426595
\(409\) 20.3927 1.00836 0.504178 0.863600i \(-0.331796\pi\)
0.504178 + 0.863600i \(0.331796\pi\)
\(410\) −32.2199 −1.59123
\(411\) 2.02134 0.0997052
\(412\) 6.79230 0.334632
\(413\) −13.1786 −0.648477
\(414\) 11.9427 0.586953
\(415\) −39.1794 −1.92324
\(416\) −1.01538 −0.0497832
\(417\) −45.5061 −2.22844
\(418\) 8.53191 0.417309
\(419\) 40.3433 1.97090 0.985450 0.169968i \(-0.0543664\pi\)
0.985450 + 0.169968i \(0.0543664\pi\)
\(420\) −22.1851 −1.08252
\(421\) 14.5045 0.706906 0.353453 0.935452i \(-0.385007\pi\)
0.353453 + 0.935452i \(0.385007\pi\)
\(422\) 9.56427 0.465581
\(423\) −50.6311 −2.46177
\(424\) −7.65817 −0.371914
\(425\) 21.7318 1.05415
\(426\) −44.4251 −2.15240
\(427\) 4.28212 0.207226
\(428\) 13.4644 0.650826
\(429\) −4.20363 −0.202953
\(430\) −20.4803 −0.987647
\(431\) −25.4001 −1.22348 −0.611739 0.791060i \(-0.709530\pi\)
−0.611739 + 0.791060i \(0.709530\pi\)
\(432\) 7.51348 0.361493
\(433\) −25.0402 −1.20336 −0.601678 0.798738i \(-0.705501\pi\)
−0.601678 + 0.798738i \(0.705501\pi\)
\(434\) 10.9314 0.524724
\(435\) −2.79747 −0.134128
\(436\) 6.72962 0.322290
\(437\) −12.9404 −0.619023
\(438\) 7.94097 0.379434
\(439\) −13.4799 −0.643361 −0.321681 0.946848i \(-0.604248\pi\)
−0.321681 + 0.946848i \(0.604248\pi\)
\(440\) 4.97706 0.237272
\(441\) −13.1387 −0.625651
\(442\) −2.98924 −0.142184
\(443\) 11.5523 0.548867 0.274433 0.961606i \(-0.411510\pi\)
0.274433 + 0.961606i \(0.411510\pi\)
\(444\) −24.3237 −1.15435
\(445\) −53.2545 −2.52450
\(446\) −20.9492 −0.991971
\(447\) 25.6944 1.21531
\(448\) −2.15404 −0.101769
\(449\) −5.54771 −0.261813 −0.130906 0.991395i \(-0.541789\pi\)
−0.130906 + 0.991395i \(0.541789\pi\)
\(450\) −41.0947 −1.93722
\(451\) −12.9513 −0.609851
\(452\) −10.2145 −0.480450
\(453\) 19.2965 0.906629
\(454\) −25.0539 −1.17584
\(455\) −7.69621 −0.360804
\(456\) −17.6555 −0.826795
\(457\) −11.9046 −0.556876 −0.278438 0.960454i \(-0.589817\pi\)
−0.278438 + 0.960454i \(0.589817\pi\)
\(458\) −11.8010 −0.551422
\(459\) 22.1194 1.03244
\(460\) −7.54873 −0.351961
\(461\) −33.4774 −1.55920 −0.779599 0.626279i \(-0.784577\pi\)
−0.779599 + 0.626279i \(0.784577\pi\)
\(462\) −8.91763 −0.414886
\(463\) −4.15271 −0.192993 −0.0964964 0.995333i \(-0.530764\pi\)
−0.0964964 + 0.995333i \(0.530764\pi\)
\(464\) −0.271618 −0.0126095
\(465\) 52.2671 2.42383
\(466\) 16.7353 0.775246
\(467\) 24.1830 1.11906 0.559528 0.828811i \(-0.310982\pi\)
0.559528 + 0.828811i \(0.310982\pi\)
\(468\) 5.65264 0.261293
\(469\) −8.87934 −0.410010
\(470\) 32.0028 1.47618
\(471\) 49.8222 2.29569
\(472\) −6.11808 −0.281608
\(473\) −8.23236 −0.378524
\(474\) 12.5425 0.576098
\(475\) 44.5277 2.04307
\(476\) −6.34141 −0.290658
\(477\) 42.6331 1.95203
\(478\) 18.5731 0.849516
\(479\) 27.9934 1.27905 0.639526 0.768769i \(-0.279131\pi\)
0.639526 + 0.768769i \(0.279131\pi\)
\(480\) −10.2993 −0.470096
\(481\) −8.43810 −0.384744
\(482\) 5.35589 0.243954
\(483\) 13.5254 0.615427
\(484\) −8.99940 −0.409063
\(485\) −54.8129 −2.48892
\(486\) 7.05534 0.320037
\(487\) 6.83469 0.309709 0.154855 0.987937i \(-0.450509\pi\)
0.154855 + 0.987937i \(0.450509\pi\)
\(488\) 1.98794 0.0899900
\(489\) −3.79498 −0.171615
\(490\) 8.30467 0.375167
\(491\) −3.00393 −0.135565 −0.0677826 0.997700i \(-0.521592\pi\)
−0.0677826 + 0.997700i \(0.521592\pi\)
\(492\) 26.8007 1.20827
\(493\) −0.799630 −0.0360135
\(494\) −6.12485 −0.275570
\(495\) −27.7073 −1.24535
\(496\) 5.07483 0.227867
\(497\) −32.6940 −1.46653
\(498\) 32.5897 1.46038
\(499\) −11.0943 −0.496649 −0.248325 0.968677i \(-0.579880\pi\)
−0.248325 + 0.968677i \(0.579880\pi\)
\(500\) 8.38118 0.374818
\(501\) −11.6300 −0.519588
\(502\) 16.9179 0.755081
\(503\) −30.2152 −1.34723 −0.673615 0.739082i \(-0.735260\pi\)
−0.673615 + 0.739082i \(0.735260\pi\)
\(504\) 11.9916 0.534147
\(505\) −1.90237 −0.0846545
\(506\) −3.03432 −0.134892
\(507\) −35.0326 −1.55585
\(508\) 8.20947 0.364236
\(509\) 14.0591 0.623158 0.311579 0.950220i \(-0.399142\pi\)
0.311579 + 0.950220i \(0.399142\pi\)
\(510\) −30.3207 −1.34262
\(511\) 5.84404 0.258525
\(512\) −1.00000 −0.0441942
\(513\) 45.3218 2.00101
\(514\) 5.30216 0.233868
\(515\) 23.9006 1.05319
\(516\) 17.0356 0.749952
\(517\) 12.8640 0.565758
\(518\) −17.9007 −0.786511
\(519\) 46.6641 2.04833
\(520\) −3.57291 −0.156683
\(521\) 11.2641 0.493489 0.246745 0.969081i \(-0.420639\pi\)
0.246745 + 0.969081i \(0.420639\pi\)
\(522\) 1.51210 0.0661827
\(523\) −0.483039 −0.0211218 −0.0105609 0.999944i \(-0.503362\pi\)
−0.0105609 + 0.999944i \(0.503362\pi\)
\(524\) −12.4641 −0.544498
\(525\) −46.5408 −2.03121
\(526\) −27.9487 −1.21862
\(527\) 14.9401 0.650800
\(528\) −4.13995 −0.180168
\(529\) −18.3978 −0.799906
\(530\) −26.9474 −1.17052
\(531\) 34.0594 1.47805
\(532\) −12.9933 −0.563332
\(533\) 9.29739 0.402715
\(534\) 44.2974 1.91694
\(535\) 47.3783 2.04834
\(536\) −4.12217 −0.178051
\(537\) 29.4452 1.27066
\(538\) −6.57533 −0.283482
\(539\) 3.33819 0.143786
\(540\) 26.4383 1.13772
\(541\) 25.4837 1.09563 0.547814 0.836600i \(-0.315460\pi\)
0.547814 + 0.836600i \(0.315460\pi\)
\(542\) −29.0271 −1.24682
\(543\) 16.4851 0.707442
\(544\) −2.94396 −0.126221
\(545\) 23.6801 1.01434
\(546\) 6.40175 0.273970
\(547\) 8.31873 0.355683 0.177842 0.984059i \(-0.443089\pi\)
0.177842 + 0.984059i \(0.443089\pi\)
\(548\) 0.690596 0.0295008
\(549\) −11.0669 −0.472324
\(550\) 10.4411 0.445209
\(551\) −1.63841 −0.0697988
\(552\) 6.27908 0.267255
\(553\) 9.23050 0.392521
\(554\) −31.1579 −1.32377
\(555\) −85.5898 −3.63309
\(556\) −15.5473 −0.659353
\(557\) −16.7477 −0.709622 −0.354811 0.934938i \(-0.615455\pi\)
−0.354811 + 0.934938i \(0.615455\pi\)
\(558\) −28.2516 −1.19599
\(559\) 5.90981 0.249958
\(560\) −7.57961 −0.320297
\(561\) −12.1878 −0.514571
\(562\) 10.7216 0.452265
\(563\) −32.9048 −1.38677 −0.693386 0.720566i \(-0.743882\pi\)
−0.693386 + 0.720566i \(0.743882\pi\)
\(564\) −26.6201 −1.12091
\(565\) −35.9426 −1.51212
\(566\) −27.0020 −1.13498
\(567\) −11.3960 −0.478589
\(568\) −15.1780 −0.636853
\(569\) −26.5208 −1.11181 −0.555905 0.831246i \(-0.687628\pi\)
−0.555905 + 0.831246i \(0.687628\pi\)
\(570\) −62.1259 −2.60217
\(571\) −1.48419 −0.0621114 −0.0310557 0.999518i \(-0.509887\pi\)
−0.0310557 + 0.999518i \(0.509887\pi\)
\(572\) −1.43618 −0.0600499
\(573\) −52.0913 −2.17614
\(574\) 19.7236 0.823247
\(575\) −15.8360 −0.660408
\(576\) 5.56701 0.231959
\(577\) 15.9449 0.663794 0.331897 0.943316i \(-0.392311\pi\)
0.331897 + 0.943316i \(0.392311\pi\)
\(578\) 8.33312 0.346612
\(579\) 47.2643 1.96424
\(580\) −0.955763 −0.0396859
\(581\) 23.9839 0.995020
\(582\) 45.5937 1.88992
\(583\) −10.8319 −0.448612
\(584\) 2.71306 0.112267
\(585\) 19.8904 0.822368
\(586\) −1.63516 −0.0675478
\(587\) −15.7778 −0.651219 −0.325609 0.945504i \(-0.605569\pi\)
−0.325609 + 0.945504i \(0.605569\pi\)
\(588\) −6.90788 −0.284876
\(589\) 30.6117 1.26133
\(590\) −21.5282 −0.886302
\(591\) 15.7420 0.647539
\(592\) −8.31027 −0.341550
\(593\) −6.60690 −0.271313 −0.135656 0.990756i \(-0.543314\pi\)
−0.135656 + 0.990756i \(0.543314\pi\)
\(594\) 10.6273 0.436042
\(595\) −22.3141 −0.914787
\(596\) 8.77859 0.359585
\(597\) −35.8166 −1.46587
\(598\) 2.17827 0.0890760
\(599\) 2.89962 0.118475 0.0592376 0.998244i \(-0.481133\pi\)
0.0592376 + 0.998244i \(0.481133\pi\)
\(600\) −21.6062 −0.882071
\(601\) −8.56656 −0.349437 −0.174719 0.984618i \(-0.555902\pi\)
−0.174719 + 0.984618i \(0.555902\pi\)
\(602\) 12.5371 0.510975
\(603\) 22.9482 0.934521
\(604\) 6.59271 0.268254
\(605\) −31.6669 −1.28744
\(606\) 1.58241 0.0642808
\(607\) −11.4992 −0.466737 −0.233368 0.972388i \(-0.574975\pi\)
−0.233368 + 0.972388i \(0.574975\pi\)
\(608\) −6.03206 −0.244632
\(609\) 1.71248 0.0693934
\(610\) 6.99515 0.283225
\(611\) −9.23475 −0.373598
\(612\) 16.3890 0.662487
\(613\) −36.6162 −1.47892 −0.739458 0.673203i \(-0.764918\pi\)
−0.739458 + 0.673203i \(0.764918\pi\)
\(614\) −14.1830 −0.572378
\(615\) 94.3059 3.80278
\(616\) −3.04674 −0.122757
\(617\) −4.97994 −0.200485 −0.100242 0.994963i \(-0.531962\pi\)
−0.100242 + 0.994963i \(0.531962\pi\)
\(618\) −19.8807 −0.799718
\(619\) −16.3902 −0.658780 −0.329390 0.944194i \(-0.606843\pi\)
−0.329390 + 0.944194i \(0.606843\pi\)
\(620\) 17.8572 0.717163
\(621\) −16.1184 −0.646811
\(622\) −1.76971 −0.0709589
\(623\) 32.6000 1.30609
\(624\) 2.97197 0.118974
\(625\) −7.41764 −0.296706
\(626\) 5.95556 0.238032
\(627\) −24.9724 −0.997303
\(628\) 17.0219 0.679249
\(629\) −24.4651 −0.975487
\(630\) 42.1957 1.68112
\(631\) −28.5874 −1.13805 −0.569023 0.822321i \(-0.692679\pi\)
−0.569023 + 0.822321i \(0.692679\pi\)
\(632\) 4.28520 0.170456
\(633\) −27.9941 −1.11266
\(634\) −1.51891 −0.0603235
\(635\) 28.8874 1.14636
\(636\) 22.4150 0.888814
\(637\) −2.39640 −0.0949489
\(638\) −0.384183 −0.0152100
\(639\) 84.4958 3.34260
\(640\) −3.51878 −0.139092
\(641\) −37.5347 −1.48253 −0.741265 0.671212i \(-0.765774\pi\)
−0.741265 + 0.671212i \(0.765774\pi\)
\(642\) −39.4095 −1.55537
\(643\) −6.96859 −0.274814 −0.137407 0.990515i \(-0.543877\pi\)
−0.137407 + 0.990515i \(0.543877\pi\)
\(644\) 4.62100 0.182093
\(645\) 59.9447 2.36032
\(646\) −17.7581 −0.698684
\(647\) 31.5656 1.24097 0.620485 0.784218i \(-0.286936\pi\)
0.620485 + 0.784218i \(0.286936\pi\)
\(648\) −5.29054 −0.207832
\(649\) −8.65358 −0.339683
\(650\) −7.49539 −0.293993
\(651\) −31.9956 −1.25401
\(652\) −1.29657 −0.0507775
\(653\) −35.5943 −1.39291 −0.696457 0.717599i \(-0.745241\pi\)
−0.696457 + 0.717599i \(0.745241\pi\)
\(654\) −19.6972 −0.770222
\(655\) −43.8586 −1.71370
\(656\) 9.15654 0.357503
\(657\) −15.1036 −0.589248
\(658\) −19.5907 −0.763725
\(659\) 16.9929 0.661950 0.330975 0.943640i \(-0.392622\pi\)
0.330975 + 0.943640i \(0.392622\pi\)
\(660\) −14.5676 −0.567043
\(661\) −24.8625 −0.967039 −0.483520 0.875333i \(-0.660642\pi\)
−0.483520 + 0.875333i \(0.660642\pi\)
\(662\) 26.2245 1.01924
\(663\) 8.74935 0.339797
\(664\) 11.1344 0.432097
\(665\) −45.7207 −1.77297
\(666\) 46.2633 1.79267
\(667\) 0.582692 0.0225619
\(668\) −3.97341 −0.153736
\(669\) 61.3170 2.37065
\(670\) −14.5050 −0.560378
\(671\) 2.81180 0.108548
\(672\) 6.30477 0.243212
\(673\) 6.76622 0.260819 0.130409 0.991460i \(-0.458371\pi\)
0.130409 + 0.991460i \(0.458371\pi\)
\(674\) 16.2124 0.624479
\(675\) 55.4633 2.13478
\(676\) −11.9690 −0.460346
\(677\) 30.6185 1.17677 0.588383 0.808583i \(-0.299765\pi\)
0.588383 + 0.808583i \(0.299765\pi\)
\(678\) 29.8973 1.14820
\(679\) 33.5540 1.28768
\(680\) −10.3591 −0.397255
\(681\) 73.3313 2.81006
\(682\) 7.17797 0.274859
\(683\) −11.4771 −0.439158 −0.219579 0.975595i \(-0.570468\pi\)
−0.219579 + 0.975595i \(0.570468\pi\)
\(684\) 33.5805 1.28398
\(685\) 2.43006 0.0928477
\(686\) −20.1621 −0.769791
\(687\) 34.5407 1.31781
\(688\) 5.82028 0.221896
\(689\) 7.77597 0.296241
\(690\) 22.0947 0.841132
\(691\) −44.1455 −1.67937 −0.839687 0.543070i \(-0.817262\pi\)
−0.839687 + 0.543070i \(0.817262\pi\)
\(692\) 15.9429 0.606060
\(693\) 16.9612 0.644303
\(694\) 12.6302 0.479435
\(695\) −54.7076 −2.07518
\(696\) 0.795009 0.0301348
\(697\) 26.9565 1.02105
\(698\) 22.2269 0.841299
\(699\) −48.9832 −1.85271
\(700\) −15.9008 −0.600994
\(701\) −39.5282 −1.49296 −0.746480 0.665408i \(-0.768258\pi\)
−0.746480 + 0.665408i \(0.768258\pi\)
\(702\) −7.62906 −0.287940
\(703\) −50.1281 −1.89062
\(704\) −1.41443 −0.0533082
\(705\) −93.6704 −3.52783
\(706\) −7.44590 −0.280230
\(707\) 1.16455 0.0437974
\(708\) 17.9073 0.672997
\(709\) 5.77479 0.216877 0.108438 0.994103i \(-0.465415\pi\)
0.108438 + 0.994103i \(0.465415\pi\)
\(710\) −53.4080 −2.00437
\(711\) −23.8557 −0.894659
\(712\) 15.1343 0.567184
\(713\) −10.8869 −0.407716
\(714\) 18.5610 0.694627
\(715\) −5.05362 −0.188995
\(716\) 10.0601 0.375962
\(717\) −54.3626 −2.03021
\(718\) −12.3914 −0.462443
\(719\) −43.1721 −1.61005 −0.805025 0.593241i \(-0.797848\pi\)
−0.805025 + 0.593241i \(0.797848\pi\)
\(720\) 19.5891 0.730042
\(721\) −14.6309 −0.544883
\(722\) −17.3858 −0.647031
\(723\) −15.6764 −0.583012
\(724\) 5.63218 0.209318
\(725\) −2.00504 −0.0744652
\(726\) 26.3407 0.977596
\(727\) −5.30795 −0.196861 −0.0984304 0.995144i \(-0.531382\pi\)
−0.0984304 + 0.995144i \(0.531382\pi\)
\(728\) 2.18718 0.0810622
\(729\) −36.5222 −1.35267
\(730\) 9.54666 0.353338
\(731\) 17.1346 0.633748
\(732\) −5.81860 −0.215062
\(733\) 14.1569 0.522896 0.261448 0.965218i \(-0.415800\pi\)
0.261448 + 0.965218i \(0.415800\pi\)
\(734\) 13.6207 0.502749
\(735\) −24.3073 −0.896589
\(736\) 2.14527 0.0790756
\(737\) −5.83051 −0.214770
\(738\) −50.9745 −1.87640
\(739\) 40.8528 1.50279 0.751397 0.659850i \(-0.229380\pi\)
0.751397 + 0.659850i \(0.229380\pi\)
\(740\) −29.2420 −1.07496
\(741\) 17.9271 0.658568
\(742\) 16.4960 0.605588
\(743\) 30.1773 1.10710 0.553549 0.832817i \(-0.313273\pi\)
0.553549 + 0.832817i \(0.313273\pi\)
\(744\) −14.8537 −0.544565
\(745\) 30.8900 1.13172
\(746\) −9.21697 −0.337457
\(747\) −61.9850 −2.26791
\(748\) −4.16401 −0.152251
\(749\) −29.0029 −1.05974
\(750\) −24.5312 −0.895755
\(751\) −16.3337 −0.596025 −0.298012 0.954562i \(-0.596324\pi\)
−0.298012 + 0.954562i \(0.596324\pi\)
\(752\) −9.09485 −0.331655
\(753\) −49.5177 −1.80452
\(754\) 0.275796 0.0100439
\(755\) 23.1983 0.844273
\(756\) −16.1844 −0.588620
\(757\) −10.7404 −0.390367 −0.195184 0.980767i \(-0.562530\pi\)
−0.195184 + 0.980767i \(0.562530\pi\)
\(758\) 5.15285 0.187160
\(759\) 8.88130 0.322371
\(760\) −21.2255 −0.769930
\(761\) −47.6840 −1.72854 −0.864272 0.503025i \(-0.832220\pi\)
−0.864272 + 0.503025i \(0.832220\pi\)
\(762\) −24.0287 −0.870467
\(763\) −14.4959 −0.524786
\(764\) −17.7971 −0.643878
\(765\) 57.6694 2.08504
\(766\) −25.9053 −0.935997
\(767\) 6.21219 0.224309
\(768\) 2.92694 0.105617
\(769\) 5.92408 0.213628 0.106814 0.994279i \(-0.465935\pi\)
0.106814 + 0.994279i \(0.465935\pi\)
\(770\) −10.7208 −0.386351
\(771\) −15.5191 −0.558908
\(772\) 16.1480 0.581179
\(773\) 17.2054 0.618837 0.309418 0.950926i \(-0.399866\pi\)
0.309418 + 0.950926i \(0.399866\pi\)
\(774\) −32.4015 −1.16465
\(775\) 37.4616 1.34566
\(776\) 15.5772 0.559189
\(777\) 52.3943 1.87964
\(778\) 14.2715 0.511659
\(779\) 55.2328 1.97892
\(780\) 10.4577 0.374446
\(781\) −21.4681 −0.768190
\(782\) 6.31558 0.225845
\(783\) −2.04079 −0.0729320
\(784\) −2.36010 −0.0842892
\(785\) 59.8965 2.13780
\(786\) 36.4818 1.30126
\(787\) 31.0946 1.10840 0.554201 0.832383i \(-0.313024\pi\)
0.554201 + 0.832383i \(0.313024\pi\)
\(788\) 5.37830 0.191594
\(789\) 81.8044 2.91232
\(790\) 15.0787 0.536475
\(791\) 22.0025 0.782318
\(792\) 7.87412 0.279795
\(793\) −2.01852 −0.0716799
\(794\) 28.3844 1.00732
\(795\) 78.8737 2.79736
\(796\) −12.2368 −0.433724
\(797\) −18.6722 −0.661404 −0.330702 0.943735i \(-0.607286\pi\)
−0.330702 + 0.943735i \(0.607286\pi\)
\(798\) 38.0307 1.34627
\(799\) −26.7748 −0.947226
\(800\) −7.38184 −0.260987
\(801\) −84.2530 −2.97693
\(802\) 38.1003 1.34537
\(803\) 3.83742 0.135420
\(804\) 12.0654 0.425513
\(805\) 16.2603 0.573100
\(806\) −5.15289 −0.181503
\(807\) 19.2456 0.677478
\(808\) 0.540634 0.0190194
\(809\) 21.5480 0.757588 0.378794 0.925481i \(-0.376339\pi\)
0.378794 + 0.925481i \(0.376339\pi\)
\(810\) −18.6163 −0.654109
\(811\) −17.4411 −0.612439 −0.306219 0.951961i \(-0.599064\pi\)
−0.306219 + 0.951961i \(0.599064\pi\)
\(812\) 0.585076 0.0205321
\(813\) 84.9606 2.97970
\(814\) −11.7543 −0.411987
\(815\) −4.56234 −0.159812
\(816\) 8.61680 0.301648
\(817\) 35.1083 1.22828
\(818\) −20.3927 −0.713015
\(819\) −12.1760 −0.425465
\(820\) 32.2199 1.12517
\(821\) −49.2282 −1.71807 −0.859037 0.511913i \(-0.828937\pi\)
−0.859037 + 0.511913i \(0.828937\pi\)
\(822\) −2.02134 −0.0705022
\(823\) −39.4215 −1.37415 −0.687073 0.726589i \(-0.741105\pi\)
−0.687073 + 0.726589i \(0.741105\pi\)
\(824\) −6.79230 −0.236621
\(825\) −30.5604 −1.06398
\(826\) 13.1786 0.458543
\(827\) 9.99263 0.347478 0.173739 0.984792i \(-0.444415\pi\)
0.173739 + 0.984792i \(0.444415\pi\)
\(828\) −11.9427 −0.415038
\(829\) 12.4418 0.432120 0.216060 0.976380i \(-0.430679\pi\)
0.216060 + 0.976380i \(0.430679\pi\)
\(830\) 39.1794 1.35994
\(831\) 91.1973 3.16360
\(832\) 1.01538 0.0352021
\(833\) −6.94803 −0.240735
\(834\) 45.5061 1.57575
\(835\) −13.9816 −0.483852
\(836\) −8.53191 −0.295082
\(837\) 38.1296 1.31795
\(838\) −40.3433 −1.39364
\(839\) 40.2755 1.39047 0.695233 0.718785i \(-0.255301\pi\)
0.695233 + 0.718785i \(0.255301\pi\)
\(840\) 22.1851 0.765459
\(841\) −28.9262 −0.997456
\(842\) −14.5045 −0.499858
\(843\) −31.3817 −1.08084
\(844\) −9.56427 −0.329216
\(845\) −42.1163 −1.44885
\(846\) 50.6311 1.74073
\(847\) 19.3851 0.666080
\(848\) 7.65817 0.262983
\(849\) 79.0333 2.71242
\(850\) −21.7318 −0.745395
\(851\) 17.8278 0.611127
\(852\) 44.4251 1.52198
\(853\) 36.3548 1.24476 0.622382 0.782714i \(-0.286165\pi\)
0.622382 + 0.782714i \(0.286165\pi\)
\(854\) −4.28212 −0.146531
\(855\) 118.163 4.04108
\(856\) −13.4644 −0.460204
\(857\) −23.1125 −0.789509 −0.394755 0.918787i \(-0.629170\pi\)
−0.394755 + 0.918787i \(0.629170\pi\)
\(858\) 4.20363 0.143510
\(859\) −22.4839 −0.767141 −0.383570 0.923512i \(-0.625306\pi\)
−0.383570 + 0.923512i \(0.625306\pi\)
\(860\) 20.4803 0.698372
\(861\) −57.7299 −1.96743
\(862\) 25.4001 0.865130
\(863\) 18.5867 0.632698 0.316349 0.948643i \(-0.397543\pi\)
0.316349 + 0.948643i \(0.397543\pi\)
\(864\) −7.51348 −0.255614
\(865\) 56.0998 1.90745
\(866\) 25.0402 0.850902
\(867\) −24.3906 −0.828347
\(868\) −10.9314 −0.371036
\(869\) 6.06110 0.205609
\(870\) 2.79747 0.0948430
\(871\) 4.18558 0.141823
\(872\) −6.72962 −0.227894
\(873\) −86.7185 −2.93497
\(874\) 12.9404 0.437715
\(875\) −18.0534 −0.610317
\(876\) −7.94097 −0.268301
\(877\) 20.6282 0.696564 0.348282 0.937390i \(-0.386765\pi\)
0.348282 + 0.937390i \(0.386765\pi\)
\(878\) 13.4799 0.454925
\(879\) 4.78602 0.161429
\(880\) −4.97706 −0.167777
\(881\) −14.9572 −0.503920 −0.251960 0.967738i \(-0.581075\pi\)
−0.251960 + 0.967738i \(0.581075\pi\)
\(882\) 13.1387 0.442402
\(883\) −21.2829 −0.716228 −0.358114 0.933678i \(-0.616580\pi\)
−0.358114 + 0.933678i \(0.616580\pi\)
\(884\) 2.98924 0.100539
\(885\) 63.0119 2.11812
\(886\) −11.5523 −0.388107
\(887\) 14.2275 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(888\) 24.3237 0.816250
\(889\) −17.6836 −0.593088
\(890\) 53.2545 1.78509
\(891\) −7.48308 −0.250693
\(892\) 20.9492 0.701430
\(893\) −54.8607 −1.83584
\(894\) −25.6944 −0.859351
\(895\) 35.3992 1.18326
\(896\) 2.15404 0.0719615
\(897\) −6.37567 −0.212877
\(898\) 5.54771 0.185130
\(899\) −1.37841 −0.0459726
\(900\) 41.0947 1.36982
\(901\) 22.5453 0.751093
\(902\) 12.9513 0.431230
\(903\) −36.6955 −1.22115
\(904\) 10.2145 0.339729
\(905\) 19.8184 0.658786
\(906\) −19.2965 −0.641083
\(907\) 34.2535 1.13737 0.568684 0.822556i \(-0.307453\pi\)
0.568684 + 0.822556i \(0.307453\pi\)
\(908\) 25.0539 0.831442
\(909\) −3.00971 −0.0998259
\(910\) 7.69621 0.255127
\(911\) 1.47381 0.0488297 0.0244148 0.999702i \(-0.492228\pi\)
0.0244148 + 0.999702i \(0.492228\pi\)
\(912\) 17.6555 0.584632
\(913\) 15.7487 0.521207
\(914\) 11.9046 0.393771
\(915\) −20.4744 −0.676863
\(916\) 11.8010 0.389914
\(917\) 26.8483 0.886609
\(918\) −22.1194 −0.730048
\(919\) 18.9914 0.626467 0.313234 0.949676i \(-0.398588\pi\)
0.313234 + 0.949676i \(0.398588\pi\)
\(920\) 7.54873 0.248874
\(921\) 41.5127 1.36789
\(922\) 33.4774 1.10252
\(923\) 15.4114 0.507274
\(924\) 8.91763 0.293369
\(925\) −61.3451 −2.01701
\(926\) 4.15271 0.136467
\(927\) 37.8127 1.24193
\(928\) 0.271618 0.00891628
\(929\) 29.6480 0.972719 0.486359 0.873759i \(-0.338325\pi\)
0.486359 + 0.873759i \(0.338325\pi\)
\(930\) −52.2671 −1.71391
\(931\) −14.2363 −0.466574
\(932\) −16.7353 −0.548182
\(933\) 5.17984 0.169580
\(934\) −24.1830 −0.791292
\(935\) −14.6523 −0.479180
\(936\) −5.65264 −0.184762
\(937\) −32.0794 −1.04799 −0.523993 0.851722i \(-0.675558\pi\)
−0.523993 + 0.851722i \(0.675558\pi\)
\(938\) 8.87934 0.289921
\(939\) −17.4316 −0.568859
\(940\) −32.0028 −1.04382
\(941\) −13.1020 −0.427113 −0.213556 0.976931i \(-0.568505\pi\)
−0.213556 + 0.976931i \(0.568505\pi\)
\(942\) −49.8222 −1.62330
\(943\) −19.6432 −0.639672
\(944\) 6.11808 0.199127
\(945\) −56.9493 −1.85256
\(946\) 8.23236 0.267657
\(947\) 54.0459 1.75625 0.878127 0.478427i \(-0.158793\pi\)
0.878127 + 0.478427i \(0.158793\pi\)
\(948\) −12.5425 −0.407363
\(949\) −2.75479 −0.0894243
\(950\) −44.5277 −1.44467
\(951\) 4.44576 0.144164
\(952\) 6.34141 0.205526
\(953\) −3.23242 −0.104708 −0.0523541 0.998629i \(-0.516672\pi\)
−0.0523541 + 0.998629i \(0.516672\pi\)
\(954\) −42.6331 −1.38030
\(955\) −62.6243 −2.02648
\(956\) −18.5731 −0.600698
\(957\) 1.12448 0.0363494
\(958\) −27.9934 −0.904427
\(959\) −1.48757 −0.0480363
\(960\) 10.2993 0.332408
\(961\) −5.24612 −0.169230
\(962\) 8.43810 0.272055
\(963\) 74.9564 2.41544
\(964\) −5.35589 −0.172502
\(965\) 56.8213 1.82914
\(966\) −13.5254 −0.435173
\(967\) 41.9691 1.34963 0.674817 0.737985i \(-0.264222\pi\)
0.674817 + 0.737985i \(0.264222\pi\)
\(968\) 8.99940 0.289252
\(969\) 51.9771 1.66974
\(970\) 54.8129 1.75993
\(971\) 11.3915 0.365571 0.182785 0.983153i \(-0.441489\pi\)
0.182785 + 0.983153i \(0.441489\pi\)
\(972\) −7.05534 −0.226300
\(973\) 33.4896 1.07363
\(974\) −6.83469 −0.218998
\(975\) 21.9386 0.702597
\(976\) −1.98794 −0.0636326
\(977\) 4.91695 0.157307 0.0786536 0.996902i \(-0.474938\pi\)
0.0786536 + 0.996902i \(0.474938\pi\)
\(978\) 3.79498 0.121350
\(979\) 21.4064 0.684152
\(980\) −8.30467 −0.265283
\(981\) 37.4638 1.19613
\(982\) 3.00393 0.0958591
\(983\) −11.2907 −0.360119 −0.180059 0.983656i \(-0.557629\pi\)
−0.180059 + 0.983656i \(0.557629\pi\)
\(984\) −26.8007 −0.854375
\(985\) 18.9251 0.603003
\(986\) 0.799630 0.0254654
\(987\) 57.3409 1.82518
\(988\) 6.12485 0.194857
\(989\) −12.4860 −0.397033
\(990\) 27.7073 0.880597
\(991\) −43.8862 −1.39409 −0.697046 0.717026i \(-0.745503\pi\)
−0.697046 + 0.717026i \(0.745503\pi\)
\(992\) −5.07483 −0.161126
\(993\) −76.7576 −2.43583
\(994\) 32.6940 1.03699
\(995\) −43.0588 −1.36506
\(996\) −32.5897 −1.03264
\(997\) −37.8524 −1.19880 −0.599399 0.800450i \(-0.704594\pi\)
−0.599399 + 0.800450i \(0.704594\pi\)
\(998\) 11.0943 0.351184
\(999\) −62.4391 −1.97549
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 8002.2.a.e.1.72 77
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.e.1.72 77 1.1 even 1 trivial