Properties

Label 8038.2.a.c.1.15
Level $8038$
Weight $2$
Character 8038.1
Self dual yes
Analytic conductor $64.184$
Analytic rank $1$
Dimension $84$
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] = [8038,2,Mod(1,8038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8038, 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("8038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8038.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: \(64.1837531447\)
Analytic rank: \(1\)
Dimension: \(84\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 8038.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.53354 q^{3} +1.00000 q^{4} +1.39163 q^{5} +2.53354 q^{6} +4.71879 q^{7} -1.00000 q^{8} +3.41884 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.53354 q^{3} +1.00000 q^{4} +1.39163 q^{5} +2.53354 q^{6} +4.71879 q^{7} -1.00000 q^{8} +3.41884 q^{9} -1.39163 q^{10} -2.76601 q^{11} -2.53354 q^{12} -2.19021 q^{13} -4.71879 q^{14} -3.52576 q^{15} +1.00000 q^{16} +5.13908 q^{17} -3.41884 q^{18} -6.17334 q^{19} +1.39163 q^{20} -11.9552 q^{21} +2.76601 q^{22} +6.53406 q^{23} +2.53354 q^{24} -3.06336 q^{25} +2.19021 q^{26} -1.06116 q^{27} +4.71879 q^{28} -3.72122 q^{29} +3.52576 q^{30} +0.253263 q^{31} -1.00000 q^{32} +7.00781 q^{33} -5.13908 q^{34} +6.56682 q^{35} +3.41884 q^{36} +2.29806 q^{37} +6.17334 q^{38} +5.54899 q^{39} -1.39163 q^{40} -11.3204 q^{41} +11.9552 q^{42} +2.00522 q^{43} -2.76601 q^{44} +4.75778 q^{45} -6.53406 q^{46} +7.02008 q^{47} -2.53354 q^{48} +15.2669 q^{49} +3.06336 q^{50} -13.0201 q^{51} -2.19021 q^{52} -0.800200 q^{53} +1.06116 q^{54} -3.84927 q^{55} -4.71879 q^{56} +15.6404 q^{57} +3.72122 q^{58} +4.01802 q^{59} -3.52576 q^{60} -3.35908 q^{61} -0.253263 q^{62} +16.1328 q^{63} +1.00000 q^{64} -3.04797 q^{65} -7.00781 q^{66} -7.50851 q^{67} +5.13908 q^{68} -16.5543 q^{69} -6.56682 q^{70} -13.6153 q^{71} -3.41884 q^{72} +14.0968 q^{73} -2.29806 q^{74} +7.76115 q^{75} -6.17334 q^{76} -13.0522 q^{77} -5.54899 q^{78} -16.6002 q^{79} +1.39163 q^{80} -7.56804 q^{81} +11.3204 q^{82} -17.3391 q^{83} -11.9552 q^{84} +7.15172 q^{85} -2.00522 q^{86} +9.42786 q^{87} +2.76601 q^{88} +14.1295 q^{89} -4.75778 q^{90} -10.3351 q^{91} +6.53406 q^{92} -0.641654 q^{93} -7.02008 q^{94} -8.59102 q^{95} +2.53354 q^{96} -10.0057 q^{97} -15.2669 q^{98} -9.45656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 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
\(3\) −2.53354 −1.46274 −0.731371 0.681980i \(-0.761119\pi\)
−0.731371 + 0.681980i \(0.761119\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.39163 0.622357 0.311179 0.950351i \(-0.399276\pi\)
0.311179 + 0.950351i \(0.399276\pi\)
\(6\) 2.53354 1.03431
\(7\) 4.71879 1.78353 0.891767 0.452496i \(-0.149466\pi\)
0.891767 + 0.452496i \(0.149466\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.41884 1.13961
\(10\) −1.39163 −0.440073
\(11\) −2.76601 −0.833984 −0.416992 0.908910i \(-0.636916\pi\)
−0.416992 + 0.908910i \(0.636916\pi\)
\(12\) −2.53354 −0.731371
\(13\) −2.19021 −0.607455 −0.303727 0.952759i \(-0.598231\pi\)
−0.303727 + 0.952759i \(0.598231\pi\)
\(14\) −4.71879 −1.26115
\(15\) −3.52576 −0.910348
\(16\) 1.00000 0.250000
\(17\) 5.13908 1.24641 0.623205 0.782058i \(-0.285830\pi\)
0.623205 + 0.782058i \(0.285830\pi\)
\(18\) −3.41884 −0.805829
\(19\) −6.17334 −1.41626 −0.708130 0.706082i \(-0.750461\pi\)
−0.708130 + 0.706082i \(0.750461\pi\)
\(20\) 1.39163 0.311179
\(21\) −11.9552 −2.60885
\(22\) 2.76601 0.589716
\(23\) 6.53406 1.36245 0.681223 0.732076i \(-0.261448\pi\)
0.681223 + 0.732076i \(0.261448\pi\)
\(24\) 2.53354 0.517157
\(25\) −3.06336 −0.612671
\(26\) 2.19021 0.429536
\(27\) −1.06116 −0.204220
\(28\) 4.71879 0.891767
\(29\) −3.72122 −0.691012 −0.345506 0.938416i \(-0.612293\pi\)
−0.345506 + 0.938416i \(0.612293\pi\)
\(30\) 3.52576 0.643713
\(31\) 0.253263 0.0454874 0.0227437 0.999741i \(-0.492760\pi\)
0.0227437 + 0.999741i \(0.492760\pi\)
\(32\) −1.00000 −0.176777
\(33\) 7.00781 1.21990
\(34\) −5.13908 −0.881345
\(35\) 6.56682 1.10999
\(36\) 3.41884 0.569807
\(37\) 2.29806 0.377798 0.188899 0.981996i \(-0.439508\pi\)
0.188899 + 0.981996i \(0.439508\pi\)
\(38\) 6.17334 1.00145
\(39\) 5.54899 0.888550
\(40\) −1.39163 −0.220037
\(41\) −11.3204 −1.76795 −0.883974 0.467536i \(-0.845142\pi\)
−0.883974 + 0.467536i \(0.845142\pi\)
\(42\) 11.9552 1.84474
\(43\) 2.00522 0.305793 0.152897 0.988242i \(-0.451140\pi\)
0.152897 + 0.988242i \(0.451140\pi\)
\(44\) −2.76601 −0.416992
\(45\) 4.75778 0.709247
\(46\) −6.53406 −0.963395
\(47\) 7.02008 1.02398 0.511992 0.858990i \(-0.328908\pi\)
0.511992 + 0.858990i \(0.328908\pi\)
\(48\) −2.53354 −0.365686
\(49\) 15.2669 2.18099
\(50\) 3.06336 0.433224
\(51\) −13.0201 −1.82318
\(52\) −2.19021 −0.303727
\(53\) −0.800200 −0.109916 −0.0549580 0.998489i \(-0.517502\pi\)
−0.0549580 + 0.998489i \(0.517502\pi\)
\(54\) 1.06116 0.144405
\(55\) −3.84927 −0.519036
\(56\) −4.71879 −0.630574
\(57\) 15.6404 2.07162
\(58\) 3.72122 0.488619
\(59\) 4.01802 0.523102 0.261551 0.965190i \(-0.415766\pi\)
0.261551 + 0.965190i \(0.415766\pi\)
\(60\) −3.52576 −0.455174
\(61\) −3.35908 −0.430086 −0.215043 0.976605i \(-0.568989\pi\)
−0.215043 + 0.976605i \(0.568989\pi\)
\(62\) −0.253263 −0.0321645
\(63\) 16.1328 2.03254
\(64\) 1.00000 0.125000
\(65\) −3.04797 −0.378054
\(66\) −7.00781 −0.862602
\(67\) −7.50851 −0.917310 −0.458655 0.888614i \(-0.651669\pi\)
−0.458655 + 0.888614i \(0.651669\pi\)
\(68\) 5.13908 0.623205
\(69\) −16.5543 −1.99291
\(70\) −6.56682 −0.784885
\(71\) −13.6153 −1.61584 −0.807918 0.589295i \(-0.799405\pi\)
−0.807918 + 0.589295i \(0.799405\pi\)
\(72\) −3.41884 −0.402915
\(73\) 14.0968 1.64991 0.824953 0.565201i \(-0.191201\pi\)
0.824953 + 0.565201i \(0.191201\pi\)
\(74\) −2.29806 −0.267144
\(75\) 7.76115 0.896180
\(76\) −6.17334 −0.708130
\(77\) −13.0522 −1.48744
\(78\) −5.54899 −0.628300
\(79\) −16.6002 −1.86766 −0.933832 0.357713i \(-0.883557\pi\)
−0.933832 + 0.357713i \(0.883557\pi\)
\(80\) 1.39163 0.155589
\(81\) −7.56804 −0.840893
\(82\) 11.3204 1.25013
\(83\) −17.3391 −1.90322 −0.951608 0.307314i \(-0.900570\pi\)
−0.951608 + 0.307314i \(0.900570\pi\)
\(84\) −11.9552 −1.30442
\(85\) 7.15172 0.775713
\(86\) −2.00522 −0.216228
\(87\) 9.42786 1.01077
\(88\) 2.76601 0.294858
\(89\) 14.1295 1.49773 0.748863 0.662725i \(-0.230600\pi\)
0.748863 + 0.662725i \(0.230600\pi\)
\(90\) −4.75778 −0.501514
\(91\) −10.3351 −1.08342
\(92\) 6.53406 0.681223
\(93\) −0.641654 −0.0665364
\(94\) −7.02008 −0.724066
\(95\) −8.59102 −0.881420
\(96\) 2.53354 0.258579
\(97\) −10.0057 −1.01592 −0.507961 0.861380i \(-0.669600\pi\)
−0.507961 + 0.861380i \(0.669600\pi\)
\(98\) −15.2669 −1.54219
\(99\) −9.45656 −0.950420
\(100\) −3.06336 −0.306336
\(101\) −18.5334 −1.84414 −0.922072 0.387017i \(-0.873505\pi\)
−0.922072 + 0.387017i \(0.873505\pi\)
\(102\) 13.0201 1.28918
\(103\) −2.45893 −0.242285 −0.121143 0.992635i \(-0.538656\pi\)
−0.121143 + 0.992635i \(0.538656\pi\)
\(104\) 2.19021 0.214768
\(105\) −16.6373 −1.62364
\(106\) 0.800200 0.0777223
\(107\) −0.166982 −0.0161428 −0.00807140 0.999967i \(-0.502569\pi\)
−0.00807140 + 0.999967i \(0.502569\pi\)
\(108\) −1.06116 −0.102110
\(109\) 9.02543 0.864479 0.432240 0.901759i \(-0.357723\pi\)
0.432240 + 0.901759i \(0.357723\pi\)
\(110\) 3.84927 0.367014
\(111\) −5.82223 −0.552622
\(112\) 4.71879 0.445883
\(113\) −15.1876 −1.42872 −0.714362 0.699776i \(-0.753283\pi\)
−0.714362 + 0.699776i \(0.753283\pi\)
\(114\) −15.6404 −1.46486
\(115\) 9.09301 0.847928
\(116\) −3.72122 −0.345506
\(117\) −7.48799 −0.692264
\(118\) −4.01802 −0.369889
\(119\) 24.2502 2.22301
\(120\) 3.52576 0.321857
\(121\) −3.34918 −0.304471
\(122\) 3.35908 0.304117
\(123\) 28.6807 2.58605
\(124\) 0.253263 0.0227437
\(125\) −11.2212 −1.00366
\(126\) −16.1328 −1.43722
\(127\) 11.5985 1.02920 0.514602 0.857429i \(-0.327940\pi\)
0.514602 + 0.857429i \(0.327940\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.08031 −0.447296
\(130\) 3.04797 0.267325
\(131\) 4.18442 0.365594 0.182797 0.983151i \(-0.441485\pi\)
0.182797 + 0.983151i \(0.441485\pi\)
\(132\) 7.00781 0.609952
\(133\) −29.1307 −2.52595
\(134\) 7.50851 0.648636
\(135\) −1.47674 −0.127098
\(136\) −5.13908 −0.440673
\(137\) 8.88632 0.759210 0.379605 0.925149i \(-0.376060\pi\)
0.379605 + 0.925149i \(0.376060\pi\)
\(138\) 16.5543 1.40920
\(139\) 4.84928 0.411311 0.205655 0.978624i \(-0.434067\pi\)
0.205655 + 0.978624i \(0.434067\pi\)
\(140\) 6.56682 0.554997
\(141\) −17.7857 −1.49783
\(142\) 13.6153 1.14257
\(143\) 6.05815 0.506608
\(144\) 3.41884 0.284904
\(145\) −5.17857 −0.430057
\(146\) −14.0968 −1.16666
\(147\) −38.6795 −3.19023
\(148\) 2.29806 0.188899
\(149\) 1.12039 0.0917856 0.0458928 0.998946i \(-0.485387\pi\)
0.0458928 + 0.998946i \(0.485387\pi\)
\(150\) −7.76115 −0.633695
\(151\) −14.2465 −1.15937 −0.579683 0.814842i \(-0.696824\pi\)
−0.579683 + 0.814842i \(0.696824\pi\)
\(152\) 6.17334 0.500724
\(153\) 17.5697 1.42043
\(154\) 13.0522 1.05178
\(155\) 0.352450 0.0283094
\(156\) 5.54899 0.444275
\(157\) 5.69937 0.454859 0.227430 0.973795i \(-0.426968\pi\)
0.227430 + 0.973795i \(0.426968\pi\)
\(158\) 16.6002 1.32064
\(159\) 2.02734 0.160779
\(160\) −1.39163 −0.110018
\(161\) 30.8328 2.42997
\(162\) 7.56804 0.594601
\(163\) 21.1843 1.65928 0.829640 0.558299i \(-0.188546\pi\)
0.829640 + 0.558299i \(0.188546\pi\)
\(164\) −11.3204 −0.883974
\(165\) 9.75230 0.759216
\(166\) 17.3391 1.34578
\(167\) −3.61223 −0.279523 −0.139761 0.990185i \(-0.544634\pi\)
−0.139761 + 0.990185i \(0.544634\pi\)
\(168\) 11.9552 0.922368
\(169\) −8.20298 −0.630998
\(170\) −7.15172 −0.548512
\(171\) −21.1057 −1.61399
\(172\) 2.00522 0.152897
\(173\) −5.78462 −0.439797 −0.219898 0.975523i \(-0.570573\pi\)
−0.219898 + 0.975523i \(0.570573\pi\)
\(174\) −9.42786 −0.714724
\(175\) −14.4553 −1.09272
\(176\) −2.76601 −0.208496
\(177\) −10.1798 −0.765163
\(178\) −14.1295 −1.05905
\(179\) 12.0434 0.900164 0.450082 0.892987i \(-0.351395\pi\)
0.450082 + 0.892987i \(0.351395\pi\)
\(180\) 4.75778 0.354624
\(181\) −17.4236 −1.29509 −0.647544 0.762028i \(-0.724204\pi\)
−0.647544 + 0.762028i \(0.724204\pi\)
\(182\) 10.3351 0.766091
\(183\) 8.51037 0.629105
\(184\) −6.53406 −0.481697
\(185\) 3.19805 0.235126
\(186\) 0.641654 0.0470483
\(187\) −14.2148 −1.03949
\(188\) 7.02008 0.511992
\(189\) −5.00738 −0.364233
\(190\) 8.59102 0.623258
\(191\) 1.21193 0.0876924 0.0438462 0.999038i \(-0.486039\pi\)
0.0438462 + 0.999038i \(0.486039\pi\)
\(192\) −2.53354 −0.182843
\(193\) 0.959405 0.0690595 0.0345298 0.999404i \(-0.489007\pi\)
0.0345298 + 0.999404i \(0.489007\pi\)
\(194\) 10.0057 0.718365
\(195\) 7.72216 0.552996
\(196\) 15.2669 1.09050
\(197\) −8.73442 −0.622302 −0.311151 0.950360i \(-0.600714\pi\)
−0.311151 + 0.950360i \(0.600714\pi\)
\(198\) 9.45656 0.672049
\(199\) 15.2750 1.08281 0.541407 0.840760i \(-0.317892\pi\)
0.541407 + 0.840760i \(0.317892\pi\)
\(200\) 3.06336 0.216612
\(201\) 19.0231 1.34179
\(202\) 18.5334 1.30401
\(203\) −17.5596 −1.23244
\(204\) −13.0201 −0.911588
\(205\) −15.7538 −1.10030
\(206\) 2.45893 0.171322
\(207\) 22.3389 1.55266
\(208\) −2.19021 −0.151864
\(209\) 17.0755 1.18114
\(210\) 16.6373 1.14808
\(211\) −8.91288 −0.613588 −0.306794 0.951776i \(-0.599256\pi\)
−0.306794 + 0.951776i \(0.599256\pi\)
\(212\) −0.800200 −0.0549580
\(213\) 34.4949 2.36355
\(214\) 0.166982 0.0114147
\(215\) 2.79053 0.190313
\(216\) 1.06116 0.0722027
\(217\) 1.19510 0.0811284
\(218\) −9.02543 −0.611279
\(219\) −35.7149 −2.41339
\(220\) −3.84927 −0.259518
\(221\) −11.2557 −0.757138
\(222\) 5.82223 0.390763
\(223\) 6.57988 0.440621 0.220311 0.975430i \(-0.429293\pi\)
0.220311 + 0.975430i \(0.429293\pi\)
\(224\) −4.71879 −0.315287
\(225\) −10.4731 −0.698209
\(226\) 15.1876 1.01026
\(227\) 6.55570 0.435117 0.217558 0.976047i \(-0.430191\pi\)
0.217558 + 0.976047i \(0.430191\pi\)
\(228\) 15.6404 1.03581
\(229\) 8.51214 0.562498 0.281249 0.959635i \(-0.409251\pi\)
0.281249 + 0.959635i \(0.409251\pi\)
\(230\) −9.09301 −0.599576
\(231\) 33.0684 2.17574
\(232\) 3.72122 0.244310
\(233\) −17.4576 −1.14368 −0.571842 0.820364i \(-0.693771\pi\)
−0.571842 + 0.820364i \(0.693771\pi\)
\(234\) 7.48799 0.489505
\(235\) 9.76938 0.637284
\(236\) 4.01802 0.261551
\(237\) 42.0572 2.73191
\(238\) −24.2502 −1.57191
\(239\) −25.5963 −1.65569 −0.827845 0.560958i \(-0.810433\pi\)
−0.827845 + 0.560958i \(0.810433\pi\)
\(240\) −3.52576 −0.227587
\(241\) −21.5827 −1.39026 −0.695132 0.718882i \(-0.744654\pi\)
−0.695132 + 0.718882i \(0.744654\pi\)
\(242\) 3.34918 0.215293
\(243\) 22.3574 1.43423
\(244\) −3.35908 −0.215043
\(245\) 21.2460 1.35736
\(246\) −28.6807 −1.82861
\(247\) 13.5209 0.860314
\(248\) −0.253263 −0.0160822
\(249\) 43.9294 2.78391
\(250\) 11.2212 0.709693
\(251\) 0.556712 0.0351394 0.0175697 0.999846i \(-0.494407\pi\)
0.0175697 + 0.999846i \(0.494407\pi\)
\(252\) 16.1328 1.01627
\(253\) −18.0733 −1.13626
\(254\) −11.5985 −0.727757
\(255\) −18.1192 −1.13467
\(256\) 1.00000 0.0625000
\(257\) 31.1754 1.94467 0.972333 0.233599i \(-0.0750503\pi\)
0.972333 + 0.233599i \(0.0750503\pi\)
\(258\) 5.08031 0.316286
\(259\) 10.8440 0.673816
\(260\) −3.04797 −0.189027
\(261\) −12.7223 −0.787488
\(262\) −4.18442 −0.258514
\(263\) 12.2728 0.756773 0.378387 0.925648i \(-0.376479\pi\)
0.378387 + 0.925648i \(0.376479\pi\)
\(264\) −7.00781 −0.431301
\(265\) −1.11358 −0.0684070
\(266\) 29.1307 1.78611
\(267\) −35.7978 −2.19079
\(268\) −7.50851 −0.458655
\(269\) −23.1202 −1.40966 −0.704832 0.709374i \(-0.748978\pi\)
−0.704832 + 0.709374i \(0.748978\pi\)
\(270\) 1.47674 0.0898717
\(271\) 23.9570 1.45528 0.727642 0.685957i \(-0.240616\pi\)
0.727642 + 0.685957i \(0.240616\pi\)
\(272\) 5.13908 0.311603
\(273\) 26.1845 1.58476
\(274\) −8.88632 −0.536842
\(275\) 8.47328 0.510958
\(276\) −16.5543 −0.996453
\(277\) −11.6360 −0.699138 −0.349569 0.936911i \(-0.613672\pi\)
−0.349569 + 0.936911i \(0.613672\pi\)
\(278\) −4.84928 −0.290841
\(279\) 0.865868 0.0518381
\(280\) −6.56682 −0.392442
\(281\) 16.6857 0.995384 0.497692 0.867354i \(-0.334181\pi\)
0.497692 + 0.867354i \(0.334181\pi\)
\(282\) 17.7857 1.05912
\(283\) −6.71441 −0.399130 −0.199565 0.979885i \(-0.563953\pi\)
−0.199565 + 0.979885i \(0.563953\pi\)
\(284\) −13.6153 −0.807918
\(285\) 21.7657 1.28929
\(286\) −6.05815 −0.358226
\(287\) −53.4185 −3.15319
\(288\) −3.41884 −0.201457
\(289\) 9.41016 0.553539
\(290\) 5.17857 0.304096
\(291\) 25.3498 1.48603
\(292\) 14.0968 0.824953
\(293\) 5.47976 0.320131 0.160066 0.987106i \(-0.448829\pi\)
0.160066 + 0.987106i \(0.448829\pi\)
\(294\) 38.6795 2.25583
\(295\) 5.59161 0.325556
\(296\) −2.29806 −0.133572
\(297\) 2.93518 0.170316
\(298\) −1.12039 −0.0649022
\(299\) −14.3110 −0.827624
\(300\) 7.76115 0.448090
\(301\) 9.46220 0.545392
\(302\) 14.2465 0.819796
\(303\) 46.9552 2.69751
\(304\) −6.17334 −0.354065
\(305\) −4.67460 −0.267667
\(306\) −17.5697 −1.00439
\(307\) 27.5631 1.57311 0.786556 0.617519i \(-0.211862\pi\)
0.786556 + 0.617519i \(0.211862\pi\)
\(308\) −13.0522 −0.743719
\(309\) 6.22980 0.354401
\(310\) −0.352450 −0.0200178
\(311\) −6.31777 −0.358248 −0.179124 0.983827i \(-0.557326\pi\)
−0.179124 + 0.983827i \(0.557326\pi\)
\(312\) −5.54899 −0.314150
\(313\) −8.46763 −0.478619 −0.239309 0.970943i \(-0.576921\pi\)
−0.239309 + 0.970943i \(0.576921\pi\)
\(314\) −5.69937 −0.321634
\(315\) 22.4509 1.26497
\(316\) −16.6002 −0.933832
\(317\) 0.996946 0.0559940 0.0279970 0.999608i \(-0.491087\pi\)
0.0279970 + 0.999608i \(0.491087\pi\)
\(318\) −2.02734 −0.113688
\(319\) 10.2929 0.576293
\(320\) 1.39163 0.0777947
\(321\) 0.423057 0.0236128
\(322\) −30.8328 −1.71825
\(323\) −31.7253 −1.76524
\(324\) −7.56804 −0.420447
\(325\) 6.70940 0.372170
\(326\) −21.1843 −1.17329
\(327\) −22.8663 −1.26451
\(328\) 11.3204 0.625064
\(329\) 33.1263 1.82631
\(330\) −9.75230 −0.536847
\(331\) −14.4797 −0.795878 −0.397939 0.917412i \(-0.630274\pi\)
−0.397939 + 0.917412i \(0.630274\pi\)
\(332\) −17.3391 −0.951608
\(333\) 7.85670 0.430545
\(334\) 3.61223 0.197653
\(335\) −10.4491 −0.570895
\(336\) −11.9552 −0.652212
\(337\) 18.8737 1.02812 0.514058 0.857755i \(-0.328142\pi\)
0.514058 + 0.857755i \(0.328142\pi\)
\(338\) 8.20298 0.446183
\(339\) 38.4783 2.08986
\(340\) 7.15172 0.387856
\(341\) −0.700529 −0.0379358
\(342\) 21.1057 1.14126
\(343\) 39.0099 2.10634
\(344\) −2.00522 −0.108114
\(345\) −23.0375 −1.24030
\(346\) 5.78462 0.310983
\(347\) 19.9968 1.07348 0.536742 0.843747i \(-0.319655\pi\)
0.536742 + 0.843747i \(0.319655\pi\)
\(348\) 9.42786 0.505386
\(349\) −4.49839 −0.240793 −0.120397 0.992726i \(-0.538417\pi\)
−0.120397 + 0.992726i \(0.538417\pi\)
\(350\) 14.4553 0.772670
\(351\) 2.32416 0.124054
\(352\) 2.76601 0.147429
\(353\) −34.2822 −1.82466 −0.912328 0.409461i \(-0.865717\pi\)
−0.912328 + 0.409461i \(0.865717\pi\)
\(354\) 10.1798 0.541052
\(355\) −18.9475 −1.00563
\(356\) 14.1295 0.748863
\(357\) −61.4390 −3.25170
\(358\) −12.0434 −0.636512
\(359\) −4.56154 −0.240749 −0.120374 0.992729i \(-0.538410\pi\)
−0.120374 + 0.992729i \(0.538410\pi\)
\(360\) −4.75778 −0.250757
\(361\) 19.1101 1.00579
\(362\) 17.4236 0.915766
\(363\) 8.48529 0.445362
\(364\) −10.3351 −0.541708
\(365\) 19.6176 1.02683
\(366\) −8.51037 −0.444844
\(367\) 11.4321 0.596748 0.298374 0.954449i \(-0.403556\pi\)
0.298374 + 0.954449i \(0.403556\pi\)
\(368\) 6.53406 0.340611
\(369\) −38.7026 −2.01478
\(370\) −3.19805 −0.166259
\(371\) −3.77597 −0.196039
\(372\) −0.641654 −0.0332682
\(373\) −5.21755 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(374\) 14.2148 0.735028
\(375\) 28.4295 1.46809
\(376\) −7.02008 −0.362033
\(377\) 8.15024 0.419759
\(378\) 5.00738 0.257552
\(379\) 27.1167 1.39289 0.696445 0.717611i \(-0.254764\pi\)
0.696445 + 0.717611i \(0.254764\pi\)
\(380\) −8.59102 −0.440710
\(381\) −29.3854 −1.50546
\(382\) −1.21193 −0.0620079
\(383\) −21.8116 −1.11452 −0.557261 0.830337i \(-0.688148\pi\)
−0.557261 + 0.830337i \(0.688148\pi\)
\(384\) 2.53354 0.129289
\(385\) −18.1639 −0.925718
\(386\) −0.959405 −0.0488325
\(387\) 6.85553 0.348486
\(388\) −10.0057 −0.507961
\(389\) −1.21853 −0.0617821 −0.0308910 0.999523i \(-0.509834\pi\)
−0.0308910 + 0.999523i \(0.509834\pi\)
\(390\) −7.72216 −0.391027
\(391\) 33.5791 1.69817
\(392\) −15.2669 −0.771097
\(393\) −10.6014 −0.534770
\(394\) 8.73442 0.440034
\(395\) −23.1013 −1.16235
\(396\) −9.45656 −0.475210
\(397\) −15.7252 −0.789225 −0.394612 0.918848i \(-0.629121\pi\)
−0.394612 + 0.918848i \(0.629121\pi\)
\(398\) −15.2750 −0.765666
\(399\) 73.8038 3.69481
\(400\) −3.06336 −0.153168
\(401\) −35.7473 −1.78514 −0.892568 0.450914i \(-0.851098\pi\)
−0.892568 + 0.450914i \(0.851098\pi\)
\(402\) −19.0231 −0.948788
\(403\) −0.554700 −0.0276316
\(404\) −18.5334 −0.922072
\(405\) −10.5319 −0.523336
\(406\) 17.5596 0.871469
\(407\) −6.35646 −0.315078
\(408\) 13.0201 0.644590
\(409\) 1.14419 0.0565765 0.0282882 0.999600i \(-0.490994\pi\)
0.0282882 + 0.999600i \(0.490994\pi\)
\(410\) 15.7538 0.778026
\(411\) −22.5139 −1.11053
\(412\) −2.45893 −0.121143
\(413\) 18.9602 0.932970
\(414\) −22.3389 −1.09790
\(415\) −24.1297 −1.18448
\(416\) 2.19021 0.107384
\(417\) −12.2859 −0.601642
\(418\) −17.0755 −0.835191
\(419\) −33.2960 −1.62662 −0.813308 0.581833i \(-0.802336\pi\)
−0.813308 + 0.581833i \(0.802336\pi\)
\(420\) −16.6373 −0.811818
\(421\) −27.8954 −1.35954 −0.679768 0.733427i \(-0.737920\pi\)
−0.679768 + 0.733427i \(0.737920\pi\)
\(422\) 8.91288 0.433872
\(423\) 24.0006 1.16695
\(424\) 0.800200 0.0388611
\(425\) −15.7428 −0.763640
\(426\) −34.4949 −1.67128
\(427\) −15.8508 −0.767072
\(428\) −0.166982 −0.00807140
\(429\) −15.3486 −0.741036
\(430\) −2.79053 −0.134571
\(431\) 1.78701 0.0860771 0.0430385 0.999073i \(-0.486296\pi\)
0.0430385 + 0.999073i \(0.486296\pi\)
\(432\) −1.06116 −0.0510550
\(433\) 5.92092 0.284541 0.142271 0.989828i \(-0.454560\pi\)
0.142271 + 0.989828i \(0.454560\pi\)
\(434\) −1.19510 −0.0573664
\(435\) 13.1201 0.629062
\(436\) 9.02543 0.432240
\(437\) −40.3369 −1.92958
\(438\) 35.7149 1.70652
\(439\) 26.8472 1.28135 0.640674 0.767813i \(-0.278655\pi\)
0.640674 + 0.767813i \(0.278655\pi\)
\(440\) 3.84927 0.183507
\(441\) 52.1953 2.48549
\(442\) 11.2557 0.535378
\(443\) −4.13134 −0.196286 −0.0981430 0.995172i \(-0.531290\pi\)
−0.0981430 + 0.995172i \(0.531290\pi\)
\(444\) −5.82223 −0.276311
\(445\) 19.6631 0.932121
\(446\) −6.57988 −0.311566
\(447\) −2.83855 −0.134259
\(448\) 4.71879 0.222942
\(449\) 17.7201 0.836261 0.418131 0.908387i \(-0.362685\pi\)
0.418131 + 0.908387i \(0.362685\pi\)
\(450\) 10.4731 0.493708
\(451\) 31.3123 1.47444
\(452\) −15.1876 −0.714362
\(453\) 36.0942 1.69585
\(454\) −6.55570 −0.307674
\(455\) −14.3827 −0.674272
\(456\) −15.6404 −0.732430
\(457\) 35.9463 1.68150 0.840748 0.541426i \(-0.182115\pi\)
0.840748 + 0.541426i \(0.182115\pi\)
\(458\) −8.51214 −0.397746
\(459\) −5.45338 −0.254542
\(460\) 9.09301 0.423964
\(461\) −31.0691 −1.44703 −0.723516 0.690307i \(-0.757475\pi\)
−0.723516 + 0.690307i \(0.757475\pi\)
\(462\) −33.0684 −1.53848
\(463\) 23.6141 1.09744 0.548721 0.836005i \(-0.315115\pi\)
0.548721 + 0.836005i \(0.315115\pi\)
\(464\) −3.72122 −0.172753
\(465\) −0.892947 −0.0414094
\(466\) 17.4576 0.808707
\(467\) 15.3694 0.711211 0.355606 0.934636i \(-0.384275\pi\)
0.355606 + 0.934636i \(0.384275\pi\)
\(468\) −7.48799 −0.346132
\(469\) −35.4310 −1.63605
\(470\) −9.76938 −0.450628
\(471\) −14.4396 −0.665342
\(472\) −4.01802 −0.184944
\(473\) −5.54646 −0.255026
\(474\) −42.0572 −1.93175
\(475\) 18.9111 0.867702
\(476\) 24.2502 1.11151
\(477\) −2.73576 −0.125262
\(478\) 25.5963 1.17075
\(479\) −40.1793 −1.83584 −0.917919 0.396768i \(-0.870132\pi\)
−0.917919 + 0.396768i \(0.870132\pi\)
\(480\) 3.52576 0.160928
\(481\) −5.03323 −0.229496
\(482\) 21.5827 0.983065
\(483\) −78.1163 −3.55442
\(484\) −3.34918 −0.152235
\(485\) −13.9242 −0.632266
\(486\) −22.3574 −1.01415
\(487\) −38.6875 −1.75310 −0.876549 0.481313i \(-0.840160\pi\)
−0.876549 + 0.481313i \(0.840160\pi\)
\(488\) 3.35908 0.152058
\(489\) −53.6712 −2.42710
\(490\) −21.2460 −0.959795
\(491\) 42.2643 1.90736 0.953682 0.300818i \(-0.0972595\pi\)
0.953682 + 0.300818i \(0.0972595\pi\)
\(492\) 28.6807 1.29303
\(493\) −19.1236 −0.861285
\(494\) −13.5209 −0.608334
\(495\) −13.1601 −0.591501
\(496\) 0.253263 0.0113719
\(497\) −64.2475 −2.88190
\(498\) −43.9294 −1.96852
\(499\) 2.54463 0.113913 0.0569567 0.998377i \(-0.481860\pi\)
0.0569567 + 0.998377i \(0.481860\pi\)
\(500\) −11.2212 −0.501829
\(501\) 9.15175 0.408870
\(502\) −0.556712 −0.0248473
\(503\) 0.501645 0.0223673 0.0111836 0.999937i \(-0.496440\pi\)
0.0111836 + 0.999937i \(0.496440\pi\)
\(504\) −16.1328 −0.718612
\(505\) −25.7917 −1.14772
\(506\) 18.0733 0.803456
\(507\) 20.7826 0.922988
\(508\) 11.5985 0.514602
\(509\) −34.0775 −1.51046 −0.755229 0.655461i \(-0.772474\pi\)
−0.755229 + 0.655461i \(0.772474\pi\)
\(510\) 18.1192 0.802331
\(511\) 66.5198 2.94266
\(512\) −1.00000 −0.0441942
\(513\) 6.55089 0.289229
\(514\) −31.1754 −1.37509
\(515\) −3.42192 −0.150788
\(516\) −5.08031 −0.223648
\(517\) −19.4176 −0.853987
\(518\) −10.8440 −0.476460
\(519\) 14.6556 0.643309
\(520\) 3.04797 0.133662
\(521\) −10.9875 −0.481372 −0.240686 0.970603i \(-0.577372\pi\)
−0.240686 + 0.970603i \(0.577372\pi\)
\(522\) 12.7223 0.556838
\(523\) −9.13391 −0.399398 −0.199699 0.979857i \(-0.563996\pi\)
−0.199699 + 0.979857i \(0.563996\pi\)
\(524\) 4.18442 0.182797
\(525\) 36.6232 1.59837
\(526\) −12.2728 −0.535119
\(527\) 1.30154 0.0566960
\(528\) 7.00781 0.304976
\(529\) 19.6939 0.856258
\(530\) 1.11358 0.0483710
\(531\) 13.7370 0.596134
\(532\) −29.1307 −1.26297
\(533\) 24.7940 1.07395
\(534\) 35.7978 1.54912
\(535\) −0.232378 −0.0100466
\(536\) 7.50851 0.324318
\(537\) −30.5124 −1.31671
\(538\) 23.1202 0.996783
\(539\) −42.2285 −1.81891
\(540\) −1.47674 −0.0635489
\(541\) 1.53054 0.0658029 0.0329015 0.999459i \(-0.489525\pi\)
0.0329015 + 0.999459i \(0.489525\pi\)
\(542\) −23.9570 −1.02904
\(543\) 44.1435 1.89438
\(544\) −5.13908 −0.220336
\(545\) 12.5601 0.538015
\(546\) −26.1845 −1.12059
\(547\) 20.9634 0.896331 0.448166 0.893951i \(-0.352077\pi\)
0.448166 + 0.893951i \(0.352077\pi\)
\(548\) 8.88632 0.379605
\(549\) −11.4842 −0.490132
\(550\) −8.47328 −0.361302
\(551\) 22.9723 0.978653
\(552\) 16.5543 0.704599
\(553\) −78.3326 −3.33104
\(554\) 11.6360 0.494365
\(555\) −8.10241 −0.343928
\(556\) 4.84928 0.205655
\(557\) 32.3469 1.37058 0.685290 0.728270i \(-0.259675\pi\)
0.685290 + 0.728270i \(0.259675\pi\)
\(558\) −0.865868 −0.0366551
\(559\) −4.39185 −0.185756
\(560\) 6.56682 0.277499
\(561\) 36.0137 1.52050
\(562\) −16.6857 −0.703843
\(563\) −32.5250 −1.37076 −0.685382 0.728184i \(-0.740365\pi\)
−0.685382 + 0.728184i \(0.740365\pi\)
\(564\) −17.7857 −0.748913
\(565\) −21.1355 −0.889177
\(566\) 6.71441 0.282228
\(567\) −35.7120 −1.49976
\(568\) 13.6153 0.571284
\(569\) −17.5976 −0.737731 −0.368865 0.929483i \(-0.620254\pi\)
−0.368865 + 0.929483i \(0.620254\pi\)
\(570\) −21.7657 −0.911666
\(571\) 1.19428 0.0499790 0.0249895 0.999688i \(-0.492045\pi\)
0.0249895 + 0.999688i \(0.492045\pi\)
\(572\) 6.05815 0.253304
\(573\) −3.07048 −0.128271
\(574\) 53.4185 2.22964
\(575\) −20.0162 −0.834732
\(576\) 3.41884 0.142452
\(577\) 10.3514 0.430936 0.215468 0.976511i \(-0.430872\pi\)
0.215468 + 0.976511i \(0.430872\pi\)
\(578\) −9.41016 −0.391411
\(579\) −2.43070 −0.101016
\(580\) −5.17857 −0.215028
\(581\) −81.8196 −3.39445
\(582\) −25.3498 −1.05078
\(583\) 2.21336 0.0916681
\(584\) −14.0968 −0.583330
\(585\) −10.4205 −0.430836
\(586\) −5.47976 −0.226367
\(587\) −4.78956 −0.197686 −0.0988432 0.995103i \(-0.531514\pi\)
−0.0988432 + 0.995103i \(0.531514\pi\)
\(588\) −38.6795 −1.59511
\(589\) −1.56348 −0.0644221
\(590\) −5.59161 −0.230203
\(591\) 22.1290 0.910267
\(592\) 2.29806 0.0944496
\(593\) 23.4098 0.961323 0.480662 0.876906i \(-0.340397\pi\)
0.480662 + 0.876906i \(0.340397\pi\)
\(594\) −2.93518 −0.120432
\(595\) 33.7474 1.38351
\(596\) 1.12039 0.0458928
\(597\) −38.6998 −1.58388
\(598\) 14.3110 0.585219
\(599\) −0.809170 −0.0330618 −0.0165309 0.999863i \(-0.505262\pi\)
−0.0165309 + 0.999863i \(0.505262\pi\)
\(600\) −7.76115 −0.316848
\(601\) 19.0585 0.777411 0.388706 0.921362i \(-0.372922\pi\)
0.388706 + 0.921362i \(0.372922\pi\)
\(602\) −9.46220 −0.385650
\(603\) −25.6704 −1.04538
\(604\) −14.2465 −0.579683
\(605\) −4.66083 −0.189490
\(606\) −46.9552 −1.90743
\(607\) −4.20146 −0.170532 −0.0852660 0.996358i \(-0.527174\pi\)
−0.0852660 + 0.996358i \(0.527174\pi\)
\(608\) 6.17334 0.250362
\(609\) 44.4881 1.80275
\(610\) 4.67460 0.189269
\(611\) −15.3755 −0.622024
\(612\) 17.5697 0.710214
\(613\) −6.18511 −0.249814 −0.124907 0.992168i \(-0.539863\pi\)
−0.124907 + 0.992168i \(0.539863\pi\)
\(614\) −27.5631 −1.11236
\(615\) 39.9130 1.60945
\(616\) 13.0522 0.525889
\(617\) −45.1203 −1.81648 −0.908238 0.418455i \(-0.862572\pi\)
−0.908238 + 0.418455i \(0.862572\pi\)
\(618\) −6.22980 −0.250599
\(619\) 23.2152 0.933099 0.466549 0.884495i \(-0.345497\pi\)
0.466549 + 0.884495i \(0.345497\pi\)
\(620\) 0.352450 0.0141547
\(621\) −6.93367 −0.278239
\(622\) 6.31777 0.253320
\(623\) 66.6742 2.67125
\(624\) 5.54899 0.222137
\(625\) −0.299058 −0.0119623
\(626\) 8.46763 0.338435
\(627\) −43.2616 −1.72770
\(628\) 5.69937 0.227430
\(629\) 11.8099 0.470892
\(630\) −22.4509 −0.894466
\(631\) 0.147010 0.00585236 0.00292618 0.999996i \(-0.499069\pi\)
0.00292618 + 0.999996i \(0.499069\pi\)
\(632\) 16.6002 0.660319
\(633\) 22.5812 0.897521
\(634\) −0.996946 −0.0395938
\(635\) 16.1409 0.640533
\(636\) 2.02734 0.0803893
\(637\) −33.4378 −1.32485
\(638\) −10.2929 −0.407501
\(639\) −46.5485 −1.84143
\(640\) −1.39163 −0.0550091
\(641\) 2.86159 0.113026 0.0565130 0.998402i \(-0.482002\pi\)
0.0565130 + 0.998402i \(0.482002\pi\)
\(642\) −0.423057 −0.0166967
\(643\) −44.7087 −1.76314 −0.881569 0.472056i \(-0.843512\pi\)
−0.881569 + 0.472056i \(0.843512\pi\)
\(644\) 30.8328 1.21498
\(645\) −7.06993 −0.278378
\(646\) 31.7253 1.24821
\(647\) −20.1785 −0.793297 −0.396649 0.917971i \(-0.629827\pi\)
−0.396649 + 0.917971i \(0.629827\pi\)
\(648\) 7.56804 0.297301
\(649\) −11.1139 −0.436259
\(650\) −6.70940 −0.263164
\(651\) −3.02783 −0.118670
\(652\) 21.1843 0.829640
\(653\) −31.7629 −1.24298 −0.621489 0.783423i \(-0.713472\pi\)
−0.621489 + 0.783423i \(0.713472\pi\)
\(654\) 22.8663 0.894144
\(655\) 5.82317 0.227530
\(656\) −11.3204 −0.441987
\(657\) 48.1948 1.88026
\(658\) −33.1263 −1.29140
\(659\) −31.0390 −1.20911 −0.604553 0.796565i \(-0.706648\pi\)
−0.604553 + 0.796565i \(0.706648\pi\)
\(660\) 9.75230 0.379608
\(661\) −19.5650 −0.760991 −0.380496 0.924783i \(-0.624247\pi\)
−0.380496 + 0.924783i \(0.624247\pi\)
\(662\) 14.4797 0.562771
\(663\) 28.5167 1.10750
\(664\) 17.3391 0.672889
\(665\) −40.5392 −1.57204
\(666\) −7.85670 −0.304441
\(667\) −24.3146 −0.941467
\(668\) −3.61223 −0.139761
\(669\) −16.6704 −0.644515
\(670\) 10.4491 0.403683
\(671\) 9.29125 0.358685
\(672\) 11.9552 0.461184
\(673\) −22.9077 −0.883028 −0.441514 0.897254i \(-0.645558\pi\)
−0.441514 + 0.897254i \(0.645558\pi\)
\(674\) −18.8737 −0.726988
\(675\) 3.25071 0.125120
\(676\) −8.20298 −0.315499
\(677\) −39.5681 −1.52073 −0.760364 0.649498i \(-0.774979\pi\)
−0.760364 + 0.649498i \(0.774979\pi\)
\(678\) −38.4783 −1.47775
\(679\) −47.2146 −1.81193
\(680\) −7.15172 −0.274256
\(681\) −16.6091 −0.636464
\(682\) 0.700529 0.0268247
\(683\) −38.2437 −1.46336 −0.731678 0.681651i \(-0.761262\pi\)
−0.731678 + 0.681651i \(0.761262\pi\)
\(684\) −21.1057 −0.806995
\(685\) 12.3665 0.472500
\(686\) −39.0099 −1.48941
\(687\) −21.5659 −0.822789
\(688\) 2.00522 0.0764483
\(689\) 1.75261 0.0667690
\(690\) 23.0375 0.877024
\(691\) 28.1325 1.07021 0.535105 0.844786i \(-0.320272\pi\)
0.535105 + 0.844786i \(0.320272\pi\)
\(692\) −5.78462 −0.219898
\(693\) −44.6235 −1.69511
\(694\) −19.9968 −0.759068
\(695\) 6.74842 0.255982
\(696\) −9.42786 −0.357362
\(697\) −58.1764 −2.20359
\(698\) 4.49839 0.170267
\(699\) 44.2296 1.67292
\(700\) −14.4553 −0.546360
\(701\) −21.1791 −0.799924 −0.399962 0.916532i \(-0.630977\pi\)
−0.399962 + 0.916532i \(0.630977\pi\)
\(702\) −2.32416 −0.0877197
\(703\) −14.1867 −0.535061
\(704\) −2.76601 −0.104248
\(705\) −24.7512 −0.932182
\(706\) 34.2822 1.29023
\(707\) −87.4553 −3.28909
\(708\) −10.1798 −0.382582
\(709\) −51.3793 −1.92959 −0.964794 0.263005i \(-0.915286\pi\)
−0.964794 + 0.263005i \(0.915286\pi\)
\(710\) 18.9475 0.711086
\(711\) −56.7533 −2.12842
\(712\) −14.1295 −0.529526
\(713\) 1.65484 0.0619742
\(714\) 61.4390 2.29930
\(715\) 8.43072 0.315291
\(716\) 12.0434 0.450082
\(717\) 64.8494 2.42185
\(718\) 4.56154 0.170235
\(719\) 11.6311 0.433768 0.216884 0.976197i \(-0.430411\pi\)
0.216884 + 0.976197i \(0.430411\pi\)
\(720\) 4.75778 0.177312
\(721\) −11.6031 −0.432124
\(722\) −19.1101 −0.711204
\(723\) 54.6807 2.03360
\(724\) −17.4236 −0.647544
\(725\) 11.3994 0.423363
\(726\) −8.48529 −0.314919
\(727\) −12.9139 −0.478950 −0.239475 0.970903i \(-0.576975\pi\)
−0.239475 + 0.970903i \(0.576975\pi\)
\(728\) 10.3351 0.383045
\(729\) −33.9394 −1.25702
\(730\) −19.6176 −0.726080
\(731\) 10.3050 0.381144
\(732\) 8.51037 0.314552
\(733\) −4.29538 −0.158653 −0.0793267 0.996849i \(-0.525277\pi\)
−0.0793267 + 0.996849i \(0.525277\pi\)
\(734\) −11.4321 −0.421965
\(735\) −53.8276 −1.98546
\(736\) −6.53406 −0.240849
\(737\) 20.7686 0.765022
\(738\) 38.7026 1.42466
\(739\) −22.6438 −0.832967 −0.416484 0.909143i \(-0.636738\pi\)
−0.416484 + 0.909143i \(0.636738\pi\)
\(740\) 3.19805 0.117563
\(741\) −34.2558 −1.25842
\(742\) 3.77597 0.138620
\(743\) 30.5830 1.12198 0.560991 0.827822i \(-0.310420\pi\)
0.560991 + 0.827822i \(0.310420\pi\)
\(744\) 0.641654 0.0235242
\(745\) 1.55917 0.0571234
\(746\) 5.21755 0.191028
\(747\) −59.2798 −2.16893
\(748\) −14.2148 −0.519743
\(749\) −0.787955 −0.0287912
\(750\) −28.4295 −1.03810
\(751\) 36.4686 1.33076 0.665380 0.746505i \(-0.268270\pi\)
0.665380 + 0.746505i \(0.268270\pi\)
\(752\) 7.02008 0.255996
\(753\) −1.41045 −0.0513998
\(754\) −8.15024 −0.296814
\(755\) −19.8259 −0.721540
\(756\) −5.00738 −0.182117
\(757\) 14.7458 0.535947 0.267973 0.963426i \(-0.413646\pi\)
0.267973 + 0.963426i \(0.413646\pi\)
\(758\) −27.1167 −0.984921
\(759\) 45.7895 1.66205
\(760\) 8.59102 0.311629
\(761\) 41.5565 1.50642 0.753211 0.657779i \(-0.228504\pi\)
0.753211 + 0.657779i \(0.228504\pi\)
\(762\) 29.3854 1.06452
\(763\) 42.5891 1.54183
\(764\) 1.21193 0.0438462
\(765\) 24.4506 0.884013
\(766\) 21.8116 0.788086
\(767\) −8.80031 −0.317761
\(768\) −2.53354 −0.0914214
\(769\) 18.9906 0.684818 0.342409 0.939551i \(-0.388757\pi\)
0.342409 + 0.939551i \(0.388757\pi\)
\(770\) 18.1639 0.654581
\(771\) −78.9842 −2.84454
\(772\) 0.959405 0.0345298
\(773\) 18.4500 0.663600 0.331800 0.943350i \(-0.392344\pi\)
0.331800 + 0.943350i \(0.392344\pi\)
\(774\) −6.85553 −0.246417
\(775\) −0.775836 −0.0278689
\(776\) 10.0057 0.359182
\(777\) −27.4739 −0.985619
\(778\) 1.21853 0.0436865
\(779\) 69.8846 2.50387
\(780\) 7.72216 0.276498
\(781\) 37.6600 1.34758
\(782\) −33.5791 −1.20078
\(783\) 3.94880 0.141119
\(784\) 15.2669 0.545248
\(785\) 7.93143 0.283085
\(786\) 10.6014 0.378140
\(787\) −25.7800 −0.918958 −0.459479 0.888189i \(-0.651964\pi\)
−0.459479 + 0.888189i \(0.651964\pi\)
\(788\) −8.73442 −0.311151
\(789\) −31.0937 −1.10696
\(790\) 23.1013 0.821908
\(791\) −71.6668 −2.54818
\(792\) 9.45656 0.336024
\(793\) 7.35709 0.261258
\(794\) 15.7252 0.558066
\(795\) 2.82132 0.100062
\(796\) 15.2750 0.541407
\(797\) 26.4267 0.936081 0.468041 0.883707i \(-0.344960\pi\)
0.468041 + 0.883707i \(0.344960\pi\)
\(798\) −73.8038 −2.61263
\(799\) 36.0768 1.27630
\(800\) 3.06336 0.108306
\(801\) 48.3066 1.70683
\(802\) 35.7473 1.26228
\(803\) −38.9919 −1.37600
\(804\) 19.0231 0.670894
\(805\) 42.9080 1.51231
\(806\) 0.554700 0.0195385
\(807\) 58.5761 2.06198
\(808\) 18.5334 0.652004
\(809\) 35.2271 1.23852 0.619260 0.785186i \(-0.287433\pi\)
0.619260 + 0.785186i \(0.287433\pi\)
\(810\) 10.5319 0.370054
\(811\) 25.6753 0.901581 0.450791 0.892630i \(-0.351142\pi\)
0.450791 + 0.892630i \(0.351142\pi\)
\(812\) −17.5596 −0.616222
\(813\) −60.6961 −2.12870
\(814\) 6.35646 0.222794
\(815\) 29.4807 1.03266
\(816\) −13.0201 −0.455794
\(817\) −12.3789 −0.433083
\(818\) −1.14419 −0.0400056
\(819\) −35.3342 −1.23468
\(820\) −15.7538 −0.550148
\(821\) −7.63648 −0.266515 −0.133258 0.991081i \(-0.542544\pi\)
−0.133258 + 0.991081i \(0.542544\pi\)
\(822\) 22.5139 0.785262
\(823\) −30.6017 −1.06671 −0.533355 0.845892i \(-0.679069\pi\)
−0.533355 + 0.845892i \(0.679069\pi\)
\(824\) 2.45893 0.0856608
\(825\) −21.4674 −0.747400
\(826\) −18.9602 −0.659709
\(827\) 13.0639 0.454275 0.227138 0.973863i \(-0.427063\pi\)
0.227138 + 0.973863i \(0.427063\pi\)
\(828\) 22.3389 0.776331
\(829\) 42.4264 1.47353 0.736765 0.676149i \(-0.236353\pi\)
0.736765 + 0.676149i \(0.236353\pi\)
\(830\) 24.1297 0.837554
\(831\) 29.4803 1.02266
\(832\) −2.19021 −0.0759319
\(833\) 78.4580 2.71841
\(834\) 12.2859 0.425425
\(835\) −5.02690 −0.173963
\(836\) 17.0755 0.590569
\(837\) −0.268752 −0.00928944
\(838\) 33.2960 1.15019
\(839\) −20.2524 −0.699191 −0.349596 0.936901i \(-0.613681\pi\)
−0.349596 + 0.936901i \(0.613681\pi\)
\(840\) 16.6373 0.574042
\(841\) −15.1526 −0.522502
\(842\) 27.8954 0.961337
\(843\) −42.2739 −1.45599
\(844\) −8.91288 −0.306794
\(845\) −11.4155 −0.392706
\(846\) −24.0006 −0.825156
\(847\) −15.8041 −0.543034
\(848\) −0.800200 −0.0274790
\(849\) 17.0113 0.583825
\(850\) 15.7428 0.539975
\(851\) 15.0157 0.514730
\(852\) 34.4949 1.18178
\(853\) −31.3101 −1.07204 −0.536018 0.844206i \(-0.680072\pi\)
−0.536018 + 0.844206i \(0.680072\pi\)
\(854\) 15.8508 0.542402
\(855\) −29.3714 −1.00448
\(856\) 0.166982 0.00570734
\(857\) −46.2106 −1.57852 −0.789261 0.614057i \(-0.789536\pi\)
−0.789261 + 0.614057i \(0.789536\pi\)
\(858\) 15.3486 0.523992
\(859\) 11.6321 0.396881 0.198441 0.980113i \(-0.436412\pi\)
0.198441 + 0.980113i \(0.436412\pi\)
\(860\) 2.79053 0.0951563
\(861\) 135.338 4.61231
\(862\) −1.78701 −0.0608657
\(863\) −36.7764 −1.25188 −0.625941 0.779871i \(-0.715285\pi\)
−0.625941 + 0.779871i \(0.715285\pi\)
\(864\) 1.06116 0.0361013
\(865\) −8.05008 −0.273711
\(866\) −5.92092 −0.201201
\(867\) −23.8410 −0.809684
\(868\) 1.19510 0.0405642
\(869\) 45.9162 1.55760
\(870\) −13.1201 −0.444814
\(871\) 16.4452 0.557225
\(872\) −9.02543 −0.305640
\(873\) −34.2078 −1.15776
\(874\) 40.3369 1.36442
\(875\) −52.9506 −1.79006
\(876\) −35.7149 −1.20669
\(877\) −2.78864 −0.0941655 −0.0470828 0.998891i \(-0.514992\pi\)
−0.0470828 + 0.998891i \(0.514992\pi\)
\(878\) −26.8472 −0.906050
\(879\) −13.8832 −0.468269
\(880\) −3.84927 −0.129759
\(881\) −41.4499 −1.39648 −0.698241 0.715862i \(-0.746034\pi\)
−0.698241 + 0.715862i \(0.746034\pi\)
\(882\) −52.1953 −1.75751
\(883\) −29.0427 −0.977364 −0.488682 0.872462i \(-0.662522\pi\)
−0.488682 + 0.872462i \(0.662522\pi\)
\(884\) −11.2557 −0.378569
\(885\) −14.1666 −0.476205
\(886\) 4.13134 0.138795
\(887\) −20.0790 −0.674187 −0.337093 0.941471i \(-0.609444\pi\)
−0.337093 + 0.941471i \(0.609444\pi\)
\(888\) 5.82223 0.195381
\(889\) 54.7310 1.83562
\(890\) −19.6631 −0.659109
\(891\) 20.9333 0.701292
\(892\) 6.57988 0.220311
\(893\) −43.3373 −1.45023
\(894\) 2.83855 0.0949352
\(895\) 16.7600 0.560224
\(896\) −4.71879 −0.157644
\(897\) 36.2574 1.21060
\(898\) −17.7201 −0.591326
\(899\) −0.942447 −0.0314324
\(900\) −10.4731 −0.349105
\(901\) −4.11229 −0.137000
\(902\) −31.3123 −1.04259
\(903\) −23.9729 −0.797768
\(904\) 15.1876 0.505130
\(905\) −24.2473 −0.806008
\(906\) −36.0942 −1.19915
\(907\) −23.6438 −0.785080 −0.392540 0.919735i \(-0.628404\pi\)
−0.392540 + 0.919735i \(0.628404\pi\)
\(908\) 6.55570 0.217558
\(909\) −63.3629 −2.10161
\(910\) 14.3827 0.476782
\(911\) 25.2248 0.835735 0.417867 0.908508i \(-0.362778\pi\)
0.417867 + 0.908508i \(0.362778\pi\)
\(912\) 15.6404 0.517906
\(913\) 47.9602 1.58725
\(914\) −35.9463 −1.18900
\(915\) 11.8433 0.391528
\(916\) 8.51214 0.281249
\(917\) 19.7454 0.652049
\(918\) 5.45338 0.179988
\(919\) 21.3841 0.705398 0.352699 0.935737i \(-0.385264\pi\)
0.352699 + 0.935737i \(0.385264\pi\)
\(920\) −9.09301 −0.299788
\(921\) −69.8324 −2.30106
\(922\) 31.0691 1.02321
\(923\) 29.8203 0.981547
\(924\) 33.0684 1.08787
\(925\) −7.03977 −0.231466
\(926\) −23.6141 −0.776009
\(927\) −8.40669 −0.276112
\(928\) 3.72122 0.122155
\(929\) 6.05803 0.198758 0.0993788 0.995050i \(-0.468314\pi\)
0.0993788 + 0.995050i \(0.468314\pi\)
\(930\) 0.892947 0.0292809
\(931\) −94.2479 −3.08885
\(932\) −17.4576 −0.571842
\(933\) 16.0064 0.524024
\(934\) −15.3694 −0.502902
\(935\) −19.7817 −0.646932
\(936\) 7.48799 0.244752
\(937\) −40.3196 −1.31718 −0.658592 0.752500i \(-0.728848\pi\)
−0.658592 + 0.752500i \(0.728848\pi\)
\(938\) 35.4310 1.15686
\(939\) 21.4531 0.700096
\(940\) 9.76938 0.318642
\(941\) −33.3535 −1.08729 −0.543647 0.839314i \(-0.682957\pi\)
−0.543647 + 0.839314i \(0.682957\pi\)
\(942\) 14.4396 0.470468
\(943\) −73.9681 −2.40873
\(944\) 4.01802 0.130775
\(945\) −6.96843 −0.226683
\(946\) 5.54646 0.180331
\(947\) −37.9584 −1.23348 −0.616741 0.787166i \(-0.711547\pi\)
−0.616741 + 0.787166i \(0.711547\pi\)
\(948\) 42.0572 1.36595
\(949\) −30.8750 −1.00224
\(950\) −18.9111 −0.613558
\(951\) −2.52581 −0.0819049
\(952\) −24.2502 −0.785954
\(953\) −20.8214 −0.674472 −0.337236 0.941420i \(-0.609492\pi\)
−0.337236 + 0.941420i \(0.609492\pi\)
\(954\) 2.73576 0.0885734
\(955\) 1.68657 0.0545760
\(956\) −25.5963 −0.827845
\(957\) −26.0776 −0.842968
\(958\) 40.1793 1.29813
\(959\) 41.9327 1.35408
\(960\) −3.52576 −0.113794
\(961\) −30.9359 −0.997931
\(962\) 5.03323 0.162278
\(963\) −0.570887 −0.0183966
\(964\) −21.5827 −0.695132
\(965\) 1.33514 0.0429797
\(966\) 78.1163 2.51335
\(967\) 44.6289 1.43517 0.717585 0.696471i \(-0.245248\pi\)
0.717585 + 0.696471i \(0.245248\pi\)
\(968\) 3.34918 0.107647
\(969\) 80.3774 2.58209
\(970\) 13.9242 0.447080
\(971\) 54.7566 1.75722 0.878612 0.477537i \(-0.158470\pi\)
0.878612 + 0.477537i \(0.158470\pi\)
\(972\) 22.3574 0.717115
\(973\) 22.8827 0.733587
\(974\) 38.6875 1.23963
\(975\) −16.9985 −0.544389
\(976\) −3.35908 −0.107521
\(977\) 23.3875 0.748234 0.374117 0.927382i \(-0.377946\pi\)
0.374117 + 0.927382i \(0.377946\pi\)
\(978\) 53.6712 1.71622
\(979\) −39.0824 −1.24908
\(980\) 21.2460 0.678678
\(981\) 30.8565 0.985173
\(982\) −42.2643 −1.34871
\(983\) 48.1613 1.53611 0.768053 0.640387i \(-0.221226\pi\)
0.768053 + 0.640387i \(0.221226\pi\)
\(984\) −28.6807 −0.914307
\(985\) −12.1551 −0.387294
\(986\) 19.1236 0.609020
\(987\) −83.9269 −2.67142
\(988\) 13.5209 0.430157
\(989\) 13.1022 0.416626
\(990\) 13.1601 0.418254
\(991\) 39.1554 1.24381 0.621906 0.783092i \(-0.286358\pi\)
0.621906 + 0.783092i \(0.286358\pi\)
\(992\) −0.253263 −0.00804112
\(993\) 36.6850 1.16416
\(994\) 64.2475 2.03781
\(995\) 21.2572 0.673898
\(996\) 43.9294 1.39196
\(997\) 26.3336 0.833992 0.416996 0.908908i \(-0.363083\pi\)
0.416996 + 0.908908i \(0.363083\pi\)
\(998\) −2.54463 −0.0805490
\(999\) −2.43860 −0.0771540
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 8038.2.a.c.1.15 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8038.2.a.c.1.15 84 1.1 even 1 trivial