Properties

Label 418.2.a.g.1.2
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $3$
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] = [418,2,Mod(1,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("418.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.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: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.523976 q^{3} +1.00000 q^{4} +2.72545 q^{5} +0.523976 q^{6} -4.67750 q^{7} -1.00000 q^{8} -2.72545 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.523976 q^{3} +1.00000 q^{4} +2.72545 q^{5} +0.523976 q^{6} -4.67750 q^{7} -1.00000 q^{8} -2.72545 q^{9} -2.72545 q^{10} -1.00000 q^{11} -0.523976 q^{12} -2.67750 q^{13} +4.67750 q^{14} -1.42807 q^{15} +1.00000 q^{16} +0.201472 q^{17} +2.72545 q^{18} -1.00000 q^{19} +2.72545 q^{20} +2.45090 q^{21} +1.00000 q^{22} -1.79853 q^{23} +0.523976 q^{24} +2.42807 q^{25} +2.67750 q^{26} +3.00000 q^{27} -4.67750 q^{28} -4.92692 q^{29} +1.42807 q^{30} -2.57193 q^{31} -1.00000 q^{32} +0.523976 q^{33} -0.201472 q^{34} -12.7483 q^{35} -2.72545 q^{36} +4.40294 q^{37} +1.00000 q^{38} +1.40294 q^{39} -2.72545 q^{40} +4.97487 q^{41} -2.45090 q^{42} -11.7734 q^{43} -1.00000 q^{44} -7.42807 q^{45} +1.79853 q^{46} -12.4989 q^{47} -0.523976 q^{48} +14.8790 q^{49} -2.42807 q^{50} -0.105567 q^{51} -2.67750 q^{52} -4.10557 q^{53} -3.00000 q^{54} -2.72545 q^{55} +4.67750 q^{56} +0.523976 q^{57} +4.92692 q^{58} -5.24943 q^{59} -1.42807 q^{60} +1.59706 q^{61} +2.57193 q^{62} +12.7483 q^{63} +1.00000 q^{64} -7.29738 q^{65} -0.523976 q^{66} +9.08044 q^{67} +0.201472 q^{68} +0.942386 q^{69} +12.7483 q^{70} +12.8214 q^{71} +2.72545 q^{72} +2.20147 q^{73} -4.40294 q^{74} -1.27225 q^{75} -1.00000 q^{76} +4.67750 q^{77} -1.40294 q^{78} +11.8538 q^{79} +2.72545 q^{80} +6.60442 q^{81} -4.97487 q^{82} -13.6775 q^{83} +2.45090 q^{84} +0.549103 q^{85} +11.7734 q^{86} +2.58159 q^{87} +1.00000 q^{88} -5.45090 q^{89} +7.42807 q^{90} +12.5240 q^{91} -1.79853 q^{92} +1.34763 q^{93} +12.4989 q^{94} -2.72545 q^{95} +0.523976 q^{96} +16.8059 q^{97} -14.8790 q^{98} +2.72545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25} + 9 q^{27} - 6 q^{28} + 6 q^{29} + 9 q^{30} - 3 q^{31} - 3 q^{32} + 9 q^{34} + 3 q^{36} - 6 q^{37} + 3 q^{38} - 15 q^{39} + 3 q^{40} - 9 q^{41} + 15 q^{42} - 21 q^{43} - 3 q^{44} - 27 q^{45} + 15 q^{46} - 12 q^{47} + 27 q^{49} - 12 q^{50} + 3 q^{51} - 9 q^{53} - 9 q^{54} + 3 q^{55} + 6 q^{56} - 6 q^{58} - 3 q^{59} - 9 q^{60} + 24 q^{61} + 3 q^{62} + 3 q^{64} - 6 q^{65} - 9 q^{68} + 3 q^{69} + 21 q^{71} - 3 q^{72} - 3 q^{73} + 6 q^{74} + 36 q^{75} - 3 q^{76} + 6 q^{77} + 15 q^{78} - 6 q^{79} - 3 q^{80} - 9 q^{81} + 9 q^{82} - 33 q^{83} - 15 q^{84} + 24 q^{85} + 21 q^{86} + 6 q^{87} + 3 q^{88} + 6 q^{89} + 27 q^{90} + 36 q^{91} - 15 q^{92} + 36 q^{93} + 12 q^{94} + 3 q^{95} + 12 q^{97} - 27 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −0.523976 −0.302518 −0.151259 0.988494i \(-0.548333\pi\)
−0.151259 + 0.988494i \(0.548333\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.72545 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(6\) 0.523976 0.213912
\(7\) −4.67750 −1.76793 −0.883964 0.467556i \(-0.845135\pi\)
−0.883964 + 0.467556i \(0.845135\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.72545 −0.908483
\(10\) −2.72545 −0.861863
\(11\) −1.00000 −0.301511
\(12\) −0.523976 −0.151259
\(13\) −2.67750 −0.742604 −0.371302 0.928512i \(-0.621089\pi\)
−0.371302 + 0.928512i \(0.621089\pi\)
\(14\) 4.67750 1.25011
\(15\) −1.42807 −0.368726
\(16\) 1.00000 0.250000
\(17\) 0.201472 0.0488642 0.0244321 0.999701i \(-0.492222\pi\)
0.0244321 + 0.999701i \(0.492222\pi\)
\(18\) 2.72545 0.642394
\(19\) −1.00000 −0.229416
\(20\) 2.72545 0.609429
\(21\) 2.45090 0.534830
\(22\) 1.00000 0.213201
\(23\) −1.79853 −0.375019 −0.187509 0.982263i \(-0.560042\pi\)
−0.187509 + 0.982263i \(0.560042\pi\)
\(24\) 0.523976 0.106956
\(25\) 2.42807 0.485614
\(26\) 2.67750 0.525100
\(27\) 3.00000 0.577350
\(28\) −4.67750 −0.883964
\(29\) −4.92692 −0.914906 −0.457453 0.889234i \(-0.651238\pi\)
−0.457453 + 0.889234i \(0.651238\pi\)
\(30\) 1.42807 0.260729
\(31\) −2.57193 −0.461932 −0.230966 0.972962i \(-0.574189\pi\)
−0.230966 + 0.972962i \(0.574189\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.523976 0.0912126
\(34\) −0.201472 −0.0345522
\(35\) −12.7483 −2.15485
\(36\) −2.72545 −0.454241
\(37\) 4.40294 0.723840 0.361920 0.932209i \(-0.382121\pi\)
0.361920 + 0.932209i \(0.382121\pi\)
\(38\) 1.00000 0.162221
\(39\) 1.40294 0.224651
\(40\) −2.72545 −0.430931
\(41\) 4.97487 0.776945 0.388472 0.921460i \(-0.373003\pi\)
0.388472 + 0.921460i \(0.373003\pi\)
\(42\) −2.45090 −0.378182
\(43\) −11.7734 −1.79543 −0.897713 0.440580i \(-0.854773\pi\)
−0.897713 + 0.440580i \(0.854773\pi\)
\(44\) −1.00000 −0.150756
\(45\) −7.42807 −1.10731
\(46\) 1.79853 0.265178
\(47\) −12.4989 −1.82314 −0.911572 0.411140i \(-0.865131\pi\)
−0.911572 + 0.411140i \(0.865131\pi\)
\(48\) −0.523976 −0.0756295
\(49\) 14.8790 2.12557
\(50\) −2.42807 −0.343381
\(51\) −0.105567 −0.0147823
\(52\) −2.67750 −0.371302
\(53\) −4.10557 −0.563943 −0.281971 0.959423i \(-0.590988\pi\)
−0.281971 + 0.959423i \(0.590988\pi\)
\(54\) −3.00000 −0.408248
\(55\) −2.72545 −0.367499
\(56\) 4.67750 0.625057
\(57\) 0.523976 0.0694024
\(58\) 4.92692 0.646936
\(59\) −5.24943 −0.683417 −0.341708 0.939806i \(-0.611006\pi\)
−0.341708 + 0.939806i \(0.611006\pi\)
\(60\) −1.42807 −0.184363
\(61\) 1.59706 0.204482 0.102241 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) 2.57193 0.326635
\(63\) 12.7483 1.60613
\(64\) 1.00000 0.125000
\(65\) −7.29738 −0.905128
\(66\) −0.523976 −0.0644970
\(67\) 9.08044 1.10935 0.554676 0.832066i \(-0.312842\pi\)
0.554676 + 0.832066i \(0.312842\pi\)
\(68\) 0.201472 0.0244321
\(69\) 0.942386 0.113450
\(70\) 12.7483 1.52371
\(71\) 12.8214 1.52161 0.760807 0.648978i \(-0.224803\pi\)
0.760807 + 0.648978i \(0.224803\pi\)
\(72\) 2.72545 0.321197
\(73\) 2.20147 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) −4.40294 −0.511832
\(75\) −1.27225 −0.146907
\(76\) −1.00000 −0.114708
\(77\) 4.67750 0.533050
\(78\) −1.40294 −0.158852
\(79\) 11.8538 1.33366 0.666831 0.745209i \(-0.267650\pi\)
0.666831 + 0.745209i \(0.267650\pi\)
\(80\) 2.72545 0.304714
\(81\) 6.60442 0.733824
\(82\) −4.97487 −0.549383
\(83\) −13.6775 −1.50130 −0.750650 0.660700i \(-0.770260\pi\)
−0.750650 + 0.660700i \(0.770260\pi\)
\(84\) 2.45090 0.267415
\(85\) 0.549103 0.0595585
\(86\) 11.7734 1.26956
\(87\) 2.58159 0.276776
\(88\) 1.00000 0.106600
\(89\) −5.45090 −0.577794 −0.288897 0.957360i \(-0.593289\pi\)
−0.288897 + 0.957360i \(0.593289\pi\)
\(90\) 7.42807 0.782987
\(91\) 12.5240 1.31287
\(92\) −1.79853 −0.187509
\(93\) 1.34763 0.139743
\(94\) 12.4989 1.28916
\(95\) −2.72545 −0.279625
\(96\) 0.523976 0.0534781
\(97\) 16.8059 1.70638 0.853190 0.521601i \(-0.174665\pi\)
0.853190 + 0.521601i \(0.174665\pi\)
\(98\) −14.8790 −1.50300
\(99\) 2.72545 0.273918
\(100\) 2.42807 0.242807
\(101\) 7.45090 0.741392 0.370696 0.928754i \(-0.379119\pi\)
0.370696 + 0.928754i \(0.379119\pi\)
\(102\) 0.105567 0.0104527
\(103\) −14.4834 −1.42709 −0.713545 0.700609i \(-0.752912\pi\)
−0.713545 + 0.700609i \(0.752912\pi\)
\(104\) 2.67750 0.262550
\(105\) 6.67980 0.651881
\(106\) 4.10557 0.398768
\(107\) 1.15352 0.111515 0.0557575 0.998444i \(-0.482243\pi\)
0.0557575 + 0.998444i \(0.482243\pi\)
\(108\) 3.00000 0.288675
\(109\) −9.55646 −0.915343 −0.457672 0.889121i \(-0.651316\pi\)
−0.457672 + 0.889121i \(0.651316\pi\)
\(110\) 2.72545 0.259861
\(111\) −2.30704 −0.218974
\(112\) −4.67750 −0.441982
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) −0.523976 −0.0490749
\(115\) −4.90179 −0.457095
\(116\) −4.92692 −0.457453
\(117\) 7.29738 0.674643
\(118\) 5.24943 0.483249
\(119\) −0.942386 −0.0863884
\(120\) 1.42807 0.130364
\(121\) 1.00000 0.0909091
\(122\) −1.59706 −0.144591
\(123\) −2.60672 −0.235040
\(124\) −2.57193 −0.230966
\(125\) −7.00966 −0.626963
\(126\) −12.7483 −1.13571
\(127\) 5.75794 0.510934 0.255467 0.966818i \(-0.417771\pi\)
0.255467 + 0.966818i \(0.417771\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.16898 0.543149
\(130\) 7.29738 0.640022
\(131\) 1.42807 0.124771 0.0623856 0.998052i \(-0.480129\pi\)
0.0623856 + 0.998052i \(0.480129\pi\)
\(132\) 0.523976 0.0456063
\(133\) 4.67750 0.405590
\(134\) −9.08044 −0.784431
\(135\) 8.17635 0.703708
\(136\) −0.201472 −0.0172761
\(137\) −14.5313 −1.24150 −0.620748 0.784010i \(-0.713171\pi\)
−0.620748 + 0.784010i \(0.713171\pi\)
\(138\) −0.942386 −0.0802212
\(139\) 16.1763 1.37206 0.686030 0.727573i \(-0.259352\pi\)
0.686030 + 0.727573i \(0.259352\pi\)
\(140\) −12.7483 −1.07743
\(141\) 6.54910 0.551534
\(142\) −12.8214 −1.07594
\(143\) 2.67750 0.223903
\(144\) −2.72545 −0.227121
\(145\) −13.4281 −1.11514
\(146\) −2.20147 −0.182195
\(147\) −7.79623 −0.643022
\(148\) 4.40294 0.361920
\(149\) 6.30704 0.516693 0.258346 0.966052i \(-0.416822\pi\)
0.258346 + 0.966052i \(0.416822\pi\)
\(150\) 1.27225 0.103879
\(151\) 0.498850 0.0405959 0.0202979 0.999794i \(-0.493539\pi\)
0.0202979 + 0.999794i \(0.493539\pi\)
\(152\) 1.00000 0.0811107
\(153\) −0.549103 −0.0443923
\(154\) −4.67750 −0.376923
\(155\) −7.00966 −0.563030
\(156\) 1.40294 0.112325
\(157\) −13.6272 −1.08757 −0.543786 0.839224i \(-0.683010\pi\)
−0.543786 + 0.839224i \(0.683010\pi\)
\(158\) −11.8538 −0.943041
\(159\) 2.15122 0.170603
\(160\) −2.72545 −0.215466
\(161\) 8.41261 0.663006
\(162\) −6.60442 −0.518892
\(163\) 15.9497 1.24928 0.624640 0.780913i \(-0.285246\pi\)
0.624640 + 0.780913i \(0.285246\pi\)
\(164\) 4.97487 0.388472
\(165\) 1.42807 0.111175
\(166\) 13.6775 1.06158
\(167\) 18.8059 1.45524 0.727622 0.685979i \(-0.240626\pi\)
0.727622 + 0.685979i \(0.240626\pi\)
\(168\) −2.45090 −0.189091
\(169\) −5.83102 −0.448540
\(170\) −0.549103 −0.0421142
\(171\) 2.72545 0.208420
\(172\) −11.7734 −0.897713
\(173\) −3.22430 −0.245139 −0.122569 0.992460i \(-0.539113\pi\)
−0.122569 + 0.992460i \(0.539113\pi\)
\(174\) −2.58159 −0.195710
\(175\) −11.3573 −0.858531
\(176\) −1.00000 −0.0753778
\(177\) 2.75057 0.206746
\(178\) 5.45090 0.408562
\(179\) 13.3778 0.999905 0.499953 0.866053i \(-0.333351\pi\)
0.499953 + 0.866053i \(0.333351\pi\)
\(180\) −7.42807 −0.553656
\(181\) 15.9497 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(182\) −12.5240 −0.928339
\(183\) −0.836819 −0.0618595
\(184\) 1.79853 0.132589
\(185\) 12.0000 0.882258
\(186\) −1.34763 −0.0988130
\(187\) −0.201472 −0.0147331
\(188\) −12.4989 −0.911572
\(189\) −14.0325 −1.02071
\(190\) 2.72545 0.197725
\(191\) −24.0553 −1.74058 −0.870291 0.492538i \(-0.836069\pi\)
−0.870291 + 0.492538i \(0.836069\pi\)
\(192\) −0.523976 −0.0378147
\(193\) 13.0325 0.938099 0.469050 0.883172i \(-0.344597\pi\)
0.469050 + 0.883172i \(0.344597\pi\)
\(194\) −16.8059 −1.20659
\(195\) 3.82365 0.273818
\(196\) 14.8790 1.06278
\(197\) −5.59706 −0.398774 −0.199387 0.979921i \(-0.563895\pi\)
−0.199387 + 0.979921i \(0.563895\pi\)
\(198\) −2.72545 −0.193689
\(199\) −7.39558 −0.524259 −0.262129 0.965033i \(-0.584425\pi\)
−0.262129 + 0.965033i \(0.584425\pi\)
\(200\) −2.42807 −0.171691
\(201\) −4.75794 −0.335599
\(202\) −7.45090 −0.524243
\(203\) 23.0457 1.61749
\(204\) −0.105567 −0.00739115
\(205\) 13.5588 0.946985
\(206\) 14.4834 1.00911
\(207\) 4.90179 0.340698
\(208\) −2.67750 −0.185651
\(209\) 1.00000 0.0691714
\(210\) −6.67980 −0.460950
\(211\) −14.2974 −0.984272 −0.492136 0.870518i \(-0.663784\pi\)
−0.492136 + 0.870518i \(0.663784\pi\)
\(212\) −4.10557 −0.281971
\(213\) −6.71809 −0.460316
\(214\) −1.15352 −0.0788530
\(215\) −32.0878 −2.18837
\(216\) −3.00000 −0.204124
\(217\) 12.0302 0.816662
\(218\) 9.55646 0.647245
\(219\) −1.15352 −0.0779476
\(220\) −2.72545 −0.183750
\(221\) −0.539441 −0.0362868
\(222\) 2.30704 0.154838
\(223\) −0.402945 −0.0269832 −0.0134916 0.999909i \(-0.504295\pi\)
−0.0134916 + 0.999909i \(0.504295\pi\)
\(224\) 4.67750 0.312528
\(225\) −6.61758 −0.441172
\(226\) 8.00000 0.532152
\(227\) −21.0074 −1.39431 −0.697154 0.716922i \(-0.745551\pi\)
−0.697154 + 0.716922i \(0.745551\pi\)
\(228\) 0.523976 0.0347012
\(229\) −14.3299 −0.946944 −0.473472 0.880809i \(-0.657000\pi\)
−0.473472 + 0.880809i \(0.657000\pi\)
\(230\) 4.90179 0.323215
\(231\) −2.45090 −0.161257
\(232\) 4.92692 0.323468
\(233\) −24.4029 −1.59869 −0.799345 0.600872i \(-0.794820\pi\)
−0.799345 + 0.600872i \(0.794820\pi\)
\(234\) −7.29738 −0.477045
\(235\) −34.0650 −2.22215
\(236\) −5.24943 −0.341708
\(237\) −6.21113 −0.403456
\(238\) 0.942386 0.0610858
\(239\) −12.6849 −0.820515 −0.410258 0.911970i \(-0.634561\pi\)
−0.410258 + 0.911970i \(0.634561\pi\)
\(240\) −1.42807 −0.0921816
\(241\) −23.9343 −1.54174 −0.770871 0.636991i \(-0.780179\pi\)
−0.770871 + 0.636991i \(0.780179\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −12.4606 −0.799345
\(244\) 1.59706 0.102241
\(245\) 40.5519 2.59076
\(246\) 2.60672 0.166198
\(247\) 2.67750 0.170365
\(248\) 2.57193 0.163318
\(249\) 7.16669 0.454170
\(250\) 7.00966 0.443330
\(251\) 18.4989 1.16764 0.583819 0.811884i \(-0.301558\pi\)
0.583819 + 0.811884i \(0.301558\pi\)
\(252\) 12.7483 0.803066
\(253\) 1.79853 0.113072
\(254\) −5.75794 −0.361285
\(255\) −0.287717 −0.0180175
\(256\) 1.00000 0.0625000
\(257\) 27.7579 1.73149 0.865746 0.500483i \(-0.166844\pi\)
0.865746 + 0.500483i \(0.166844\pi\)
\(258\) −6.16898 −0.384064
\(259\) −20.5948 −1.27970
\(260\) −7.29738 −0.452564
\(261\) 13.4281 0.831177
\(262\) −1.42807 −0.0882265
\(263\) 20.2723 1.25004 0.625020 0.780608i \(-0.285091\pi\)
0.625020 + 0.780608i \(0.285091\pi\)
\(264\) −0.523976 −0.0322485
\(265\) −11.1895 −0.687366
\(266\) −4.67750 −0.286796
\(267\) 2.85614 0.174793
\(268\) 9.08044 0.554676
\(269\) −15.6427 −0.953753 −0.476876 0.878970i \(-0.658231\pi\)
−0.476876 + 0.878970i \(0.658231\pi\)
\(270\) −8.17635 −0.497597
\(271\) −12.3395 −0.749573 −0.374786 0.927111i \(-0.622284\pi\)
−0.374786 + 0.927111i \(0.622284\pi\)
\(272\) 0.201472 0.0122161
\(273\) −6.56227 −0.397167
\(274\) 14.5313 0.877870
\(275\) −2.42807 −0.146418
\(276\) 0.942386 0.0567250
\(277\) 8.49885 0.510646 0.255323 0.966856i \(-0.417818\pi\)
0.255323 + 0.966856i \(0.417818\pi\)
\(278\) −16.1763 −0.970193
\(279\) 7.00966 0.419657
\(280\) 12.7483 0.761855
\(281\) −7.12839 −0.425244 −0.212622 0.977134i \(-0.568200\pi\)
−0.212622 + 0.977134i \(0.568200\pi\)
\(282\) −6.54910 −0.389993
\(283\) 11.5240 0.685029 0.342515 0.939512i \(-0.388721\pi\)
0.342515 + 0.939512i \(0.388721\pi\)
\(284\) 12.8214 0.760807
\(285\) 1.42807 0.0845916
\(286\) −2.67750 −0.158324
\(287\) −23.2700 −1.37358
\(288\) 2.72545 0.160599
\(289\) −16.9594 −0.997612
\(290\) 13.4281 0.788523
\(291\) −8.80589 −0.516210
\(292\) 2.20147 0.128831
\(293\) 10.6775 0.623786 0.311893 0.950117i \(-0.399037\pi\)
0.311893 + 0.950117i \(0.399037\pi\)
\(294\) 7.79623 0.454685
\(295\) −14.3070 −0.832988
\(296\) −4.40294 −0.255916
\(297\) −3.00000 −0.174078
\(298\) −6.30704 −0.365357
\(299\) 4.81555 0.278490
\(300\) −1.27225 −0.0734535
\(301\) 55.0700 3.17418
\(302\) −0.498850 −0.0287056
\(303\) −3.90409 −0.224284
\(304\) −1.00000 −0.0573539
\(305\) 4.35269 0.249234
\(306\) 0.549103 0.0313901
\(307\) −30.2568 −1.72685 −0.863423 0.504481i \(-0.831684\pi\)
−0.863423 + 0.504481i \(0.831684\pi\)
\(308\) 4.67750 0.266525
\(309\) 7.58895 0.431720
\(310\) 7.00966 0.398122
\(311\) 8.79623 0.498788 0.249394 0.968402i \(-0.419769\pi\)
0.249394 + 0.968402i \(0.419769\pi\)
\(312\) −1.40294 −0.0794261
\(313\) 10.1860 0.575747 0.287874 0.957668i \(-0.407052\pi\)
0.287874 + 0.957668i \(0.407052\pi\)
\(314\) 13.6272 0.769030
\(315\) 34.7448 1.95765
\(316\) 11.8538 0.666831
\(317\) −12.0097 −0.674530 −0.337265 0.941410i \(-0.609502\pi\)
−0.337265 + 0.941410i \(0.609502\pi\)
\(318\) −2.15122 −0.120634
\(319\) 4.92692 0.275855
\(320\) 2.72545 0.152357
\(321\) −0.604417 −0.0337353
\(322\) −8.41261 −0.468816
\(323\) −0.201472 −0.0112102
\(324\) 6.60442 0.366912
\(325\) −6.50115 −0.360619
\(326\) −15.9497 −0.883375
\(327\) 5.00736 0.276908
\(328\) −4.97487 −0.274691
\(329\) 58.4633 3.22319
\(330\) −1.42807 −0.0786127
\(331\) 26.1667 1.43825 0.719126 0.694880i \(-0.244543\pi\)
0.719126 + 0.694880i \(0.244543\pi\)
\(332\) −13.6775 −0.750650
\(333\) −12.0000 −0.657596
\(334\) −18.8059 −1.02901
\(335\) 24.7483 1.35214
\(336\) 2.45090 0.133707
\(337\) 17.3778 0.946630 0.473315 0.880893i \(-0.343057\pi\)
0.473315 + 0.880893i \(0.343057\pi\)
\(338\) 5.83102 0.317165
\(339\) 4.19181 0.227668
\(340\) 0.549103 0.0297793
\(341\) 2.57193 0.139278
\(342\) −2.72545 −0.147375
\(343\) −36.8538 −1.98992
\(344\) 11.7734 0.634779
\(345\) 2.56842 0.138279
\(346\) 3.22430 0.173339
\(347\) −12.4029 −0.665825 −0.332912 0.942958i \(-0.608031\pi\)
−0.332912 + 0.942958i \(0.608031\pi\)
\(348\) 2.58159 0.138388
\(349\) −23.8036 −1.27418 −0.637088 0.770791i \(-0.719861\pi\)
−0.637088 + 0.770791i \(0.719861\pi\)
\(350\) 11.3573 0.607073
\(351\) −8.03249 −0.428742
\(352\) 1.00000 0.0533002
\(353\) −24.2664 −1.29157 −0.645786 0.763518i \(-0.723470\pi\)
−0.645786 + 0.763518i \(0.723470\pi\)
\(354\) −2.75057 −0.146191
\(355\) 34.9439 1.85463
\(356\) −5.45090 −0.288897
\(357\) 0.493788 0.0261340
\(358\) −13.3778 −0.707040
\(359\) 20.3395 1.07348 0.536740 0.843748i \(-0.319656\pi\)
0.536740 + 0.843748i \(0.319656\pi\)
\(360\) 7.42807 0.391494
\(361\) 1.00000 0.0526316
\(362\) −15.9497 −0.838300
\(363\) −0.523976 −0.0275016
\(364\) 12.5240 0.656435
\(365\) 6.00000 0.314054
\(366\) 0.836819 0.0437412
\(367\) −23.0170 −1.20148 −0.600739 0.799445i \(-0.705127\pi\)
−0.600739 + 0.799445i \(0.705127\pi\)
\(368\) −1.79853 −0.0937547
\(369\) −13.5588 −0.705841
\(370\) −12.0000 −0.623850
\(371\) 19.2038 0.997010
\(372\) 1.34763 0.0698714
\(373\) −18.5890 −0.962499 −0.481250 0.876584i \(-0.659817\pi\)
−0.481250 + 0.876584i \(0.659817\pi\)
\(374\) 0.201472 0.0104179
\(375\) 3.67290 0.189668
\(376\) 12.4989 0.644579
\(377\) 13.1918 0.679413
\(378\) 14.0325 0.721753
\(379\) −10.5816 −0.543540 −0.271770 0.962362i \(-0.587609\pi\)
−0.271770 + 0.962362i \(0.587609\pi\)
\(380\) −2.72545 −0.139813
\(381\) −3.01702 −0.154567
\(382\) 24.0553 1.23078
\(383\) −26.0878 −1.33302 −0.666512 0.745494i \(-0.732214\pi\)
−0.666512 + 0.745494i \(0.732214\pi\)
\(384\) 0.523976 0.0267391
\(385\) 12.7483 0.649712
\(386\) −13.0325 −0.663336
\(387\) 32.0878 1.63111
\(388\) 16.8059 0.853190
\(389\) −9.46636 −0.479964 −0.239982 0.970777i \(-0.577141\pi\)
−0.239982 + 0.970777i \(0.577141\pi\)
\(390\) −3.82365 −0.193618
\(391\) −0.362354 −0.0183250
\(392\) −14.8790 −0.751501
\(393\) −0.748275 −0.0377455
\(394\) 5.59706 0.281976
\(395\) 32.3070 1.62554
\(396\) 2.72545 0.136959
\(397\) −25.1667 −1.26308 −0.631540 0.775343i \(-0.717577\pi\)
−0.631540 + 0.775343i \(0.717577\pi\)
\(398\) 7.39558 0.370707
\(399\) −2.45090 −0.122698
\(400\) 2.42807 0.121404
\(401\) −16.0650 −0.802247 −0.401123 0.916024i \(-0.631380\pi\)
−0.401123 + 0.916024i \(0.631380\pi\)
\(402\) 4.75794 0.237304
\(403\) 6.88633 0.343033
\(404\) 7.45090 0.370696
\(405\) 18.0000 0.894427
\(406\) −23.0457 −1.14374
\(407\) −4.40294 −0.218246
\(408\) 0.105567 0.00522633
\(409\) −19.5313 −0.965763 −0.482881 0.875686i \(-0.660410\pi\)
−0.482881 + 0.875686i \(0.660410\pi\)
\(410\) −13.5588 −0.669620
\(411\) 7.61408 0.375575
\(412\) −14.4834 −0.713545
\(413\) 24.5542 1.20823
\(414\) −4.90179 −0.240910
\(415\) −37.2773 −1.82987
\(416\) 2.67750 0.131275
\(417\) −8.47602 −0.415073
\(418\) −1.00000 −0.0489116
\(419\) 24.9018 1.21653 0.608266 0.793733i \(-0.291865\pi\)
0.608266 + 0.793733i \(0.291865\pi\)
\(420\) 6.67980 0.325941
\(421\) 26.6044 1.29662 0.648310 0.761377i \(-0.275476\pi\)
0.648310 + 0.761377i \(0.275476\pi\)
\(422\) 14.2974 0.695985
\(423\) 34.0650 1.65630
\(424\) 4.10557 0.199384
\(425\) 0.489189 0.0237292
\(426\) 6.71809 0.325492
\(427\) −7.47022 −0.361509
\(428\) 1.15352 0.0557575
\(429\) −1.40294 −0.0677348
\(430\) 32.0878 1.54741
\(431\) −32.5948 −1.57003 −0.785017 0.619474i \(-0.787346\pi\)
−0.785017 + 0.619474i \(0.787346\pi\)
\(432\) 3.00000 0.144338
\(433\) 19.7386 0.948577 0.474289 0.880369i \(-0.342705\pi\)
0.474289 + 0.880369i \(0.342705\pi\)
\(434\) −12.0302 −0.577468
\(435\) 7.03599 0.337350
\(436\) −9.55646 −0.457672
\(437\) 1.79853 0.0860352
\(438\) 1.15352 0.0551173
\(439\) −24.4029 −1.16469 −0.582345 0.812942i \(-0.697865\pi\)
−0.582345 + 0.812942i \(0.697865\pi\)
\(440\) 2.72545 0.129931
\(441\) −40.5519 −1.93104
\(442\) 0.539441 0.0256586
\(443\) −16.5948 −0.788441 −0.394220 0.919016i \(-0.628985\pi\)
−0.394220 + 0.919016i \(0.628985\pi\)
\(444\) −2.30704 −0.109487
\(445\) −14.8561 −0.704249
\(446\) 0.402945 0.0190800
\(447\) −3.30474 −0.156309
\(448\) −4.67750 −0.220991
\(449\) −20.9520 −0.988788 −0.494394 0.869238i \(-0.664610\pi\)
−0.494394 + 0.869238i \(0.664610\pi\)
\(450\) 6.61758 0.311956
\(451\) −4.97487 −0.234258
\(452\) −8.00000 −0.376288
\(453\) −0.261386 −0.0122810
\(454\) 21.0074 0.985924
\(455\) 34.1335 1.60020
\(456\) −0.523976 −0.0245374
\(457\) 23.3144 1.09060 0.545301 0.838240i \(-0.316415\pi\)
0.545301 + 0.838240i \(0.316415\pi\)
\(458\) 14.3299 0.669591
\(459\) 0.604417 0.0282118
\(460\) −4.90179 −0.228547
\(461\) 20.8059 0.969027 0.484513 0.874784i \(-0.338997\pi\)
0.484513 + 0.874784i \(0.338997\pi\)
\(462\) 2.45090 0.114026
\(463\) −9.29002 −0.431744 −0.215872 0.976422i \(-0.569259\pi\)
−0.215872 + 0.976422i \(0.569259\pi\)
\(464\) −4.92692 −0.228727
\(465\) 3.67290 0.170327
\(466\) 24.4029 1.13044
\(467\) 12.0457 0.557406 0.278703 0.960377i \(-0.410095\pi\)
0.278703 + 0.960377i \(0.410095\pi\)
\(468\) 7.29738 0.337321
\(469\) −42.4737 −1.96125
\(470\) 34.0650 1.57130
\(471\) 7.14035 0.329010
\(472\) 5.24943 0.241624
\(473\) 11.7734 0.541342
\(474\) 6.21113 0.285287
\(475\) −2.42807 −0.111408
\(476\) −0.942386 −0.0431942
\(477\) 11.1895 0.512333
\(478\) 12.6849 0.580192
\(479\) 0.188307 0.00860396 0.00430198 0.999991i \(-0.498631\pi\)
0.00430198 + 0.999991i \(0.498631\pi\)
\(480\) 1.42807 0.0651822
\(481\) −11.7889 −0.537526
\(482\) 23.9343 1.09018
\(483\) −4.40801 −0.200571
\(484\) 1.00000 0.0454545
\(485\) 45.8036 2.07983
\(486\) 12.4606 0.565222
\(487\) −33.3852 −1.51283 −0.756413 0.654094i \(-0.773050\pi\)
−0.756413 + 0.654094i \(0.773050\pi\)
\(488\) −1.59706 −0.0722953
\(489\) −8.35729 −0.377930
\(490\) −40.5519 −1.83195
\(491\) 12.5217 0.565095 0.282548 0.959253i \(-0.408820\pi\)
0.282548 + 0.959253i \(0.408820\pi\)
\(492\) −2.60672 −0.117520
\(493\) −0.992638 −0.0447062
\(494\) −2.67750 −0.120466
\(495\) 7.42807 0.333867
\(496\) −2.57193 −0.115483
\(497\) −59.9718 −2.69010
\(498\) −7.16669 −0.321147
\(499\) 16.3070 0.730003 0.365002 0.931007i \(-0.381068\pi\)
0.365002 + 0.931007i \(0.381068\pi\)
\(500\) −7.00966 −0.313482
\(501\) −9.85384 −0.440237
\(502\) −18.4989 −0.825644
\(503\) −18.9726 −0.845945 −0.422973 0.906142i \(-0.639013\pi\)
−0.422973 + 0.906142i \(0.639013\pi\)
\(504\) −12.7483 −0.567853
\(505\) 20.3070 0.903651
\(506\) −1.79853 −0.0799543
\(507\) 3.05531 0.135691
\(508\) 5.75794 0.255467
\(509\) −30.6907 −1.36034 −0.680170 0.733055i \(-0.738094\pi\)
−0.680170 + 0.733055i \(0.738094\pi\)
\(510\) 0.287717 0.0127403
\(511\) −10.2974 −0.455529
\(512\) −1.00000 −0.0441942
\(513\) −3.00000 −0.132453
\(514\) −27.7579 −1.22435
\(515\) −39.4737 −1.73942
\(516\) 6.16898 0.271574
\(517\) 12.4989 0.549699
\(518\) 20.5948 0.904882
\(519\) 1.68946 0.0741589
\(520\) 7.29738 0.320011
\(521\) −42.1106 −1.84490 −0.922450 0.386116i \(-0.873816\pi\)
−0.922450 + 0.386116i \(0.873816\pi\)
\(522\) −13.4281 −0.587731
\(523\) 37.8133 1.65346 0.826729 0.562600i \(-0.190199\pi\)
0.826729 + 0.562600i \(0.190199\pi\)
\(524\) 1.42807 0.0623856
\(525\) 5.95095 0.259721
\(526\) −20.2723 −0.883912
\(527\) −0.518173 −0.0225720
\(528\) 0.523976 0.0228031
\(529\) −19.7653 −0.859361
\(530\) 11.1895 0.486041
\(531\) 14.3070 0.620873
\(532\) 4.67750 0.202795
\(533\) −13.3202 −0.576962
\(534\) −2.85614 −0.123597
\(535\) 3.14386 0.135921
\(536\) −9.08044 −0.392215
\(537\) −7.00966 −0.302489
\(538\) 15.6427 0.674405
\(539\) −14.8790 −0.640883
\(540\) 8.17635 0.351854
\(541\) 13.5661 0.583253 0.291627 0.956532i \(-0.405804\pi\)
0.291627 + 0.956532i \(0.405804\pi\)
\(542\) 12.3395 0.530028
\(543\) −8.35729 −0.358646
\(544\) −0.201472 −0.00863806
\(545\) −26.0457 −1.11567
\(546\) 6.56227 0.280839
\(547\) 46.4177 1.98468 0.992338 0.123552i \(-0.0394286\pi\)
0.992338 + 0.123552i \(0.0394286\pi\)
\(548\) −14.5313 −0.620748
\(549\) −4.35269 −0.185768
\(550\) 2.42807 0.103533
\(551\) 4.92692 0.209894
\(552\) −0.942386 −0.0401106
\(553\) −55.4463 −2.35782
\(554\) −8.49885 −0.361082
\(555\) −6.28772 −0.266899
\(556\) 16.1763 0.686030
\(557\) −38.2065 −1.61886 −0.809431 0.587214i \(-0.800225\pi\)
−0.809431 + 0.587214i \(0.800225\pi\)
\(558\) −7.00966 −0.296743
\(559\) 31.5232 1.33329
\(560\) −12.7483 −0.538713
\(561\) 0.105567 0.00445703
\(562\) 7.12839 0.300693
\(563\) −7.01702 −0.295732 −0.147866 0.989007i \(-0.547240\pi\)
−0.147866 + 0.989007i \(0.547240\pi\)
\(564\) 6.54910 0.275767
\(565\) −21.8036 −0.917284
\(566\) −11.5240 −0.484389
\(567\) −30.8921 −1.29735
\(568\) −12.8214 −0.537972
\(569\) −41.4235 −1.73656 −0.868281 0.496072i \(-0.834775\pi\)
−0.868281 + 0.496072i \(0.834775\pi\)
\(570\) −1.42807 −0.0598153
\(571\) 12.1381 0.507962 0.253981 0.967209i \(-0.418260\pi\)
0.253981 + 0.967209i \(0.418260\pi\)
\(572\) 2.67750 0.111952
\(573\) 12.6044 0.526557
\(574\) 23.2700 0.971269
\(575\) −4.36695 −0.182115
\(576\) −2.72545 −0.113560
\(577\) 12.8384 0.534469 0.267234 0.963632i \(-0.413890\pi\)
0.267234 + 0.963632i \(0.413890\pi\)
\(578\) 16.9594 0.705418
\(579\) −6.82872 −0.283792
\(580\) −13.4281 −0.557570
\(581\) 63.9764 2.65419
\(582\) 8.80589 0.365016
\(583\) 4.10557 0.170035
\(584\) −2.20147 −0.0910976
\(585\) 19.8886 0.822294
\(586\) −10.6775 −0.441083
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −7.79623 −0.321511
\(589\) 2.57193 0.105974
\(590\) 14.3070 0.589011
\(591\) 2.93272 0.120636
\(592\) 4.40294 0.180960
\(593\) 40.4177 1.65975 0.829877 0.557946i \(-0.188410\pi\)
0.829877 + 0.557946i \(0.188410\pi\)
\(594\) 3.00000 0.123091
\(595\) −2.56842 −0.105295
\(596\) 6.30704 0.258346
\(597\) 3.87511 0.158598
\(598\) −4.81555 −0.196923
\(599\) 0.679795 0.0277757 0.0138878 0.999904i \(-0.495579\pi\)
0.0138878 + 0.999904i \(0.495579\pi\)
\(600\) 1.27225 0.0519395
\(601\) 21.5240 0.877981 0.438991 0.898492i \(-0.355336\pi\)
0.438991 + 0.898492i \(0.355336\pi\)
\(602\) −55.0700 −2.24449
\(603\) −24.7483 −1.00783
\(604\) 0.498850 0.0202979
\(605\) 2.72545 0.110805
\(606\) 3.90409 0.158593
\(607\) 13.3550 0.542062 0.271031 0.962571i \(-0.412635\pi\)
0.271031 + 0.962571i \(0.412635\pi\)
\(608\) 1.00000 0.0405554
\(609\) −12.0754 −0.489319
\(610\) −4.35269 −0.176235
\(611\) 33.4656 1.35387
\(612\) −0.549103 −0.0221962
\(613\) 9.11293 0.368068 0.184034 0.982920i \(-0.441084\pi\)
0.184034 + 0.982920i \(0.441084\pi\)
\(614\) 30.2568 1.22106
\(615\) −7.10447 −0.286480
\(616\) −4.67750 −0.188462
\(617\) 0.329866 0.0132799 0.00663995 0.999978i \(-0.497886\pi\)
0.00663995 + 0.999978i \(0.497886\pi\)
\(618\) −7.58895 −0.305272
\(619\) 8.92112 0.358570 0.179285 0.983797i \(-0.442622\pi\)
0.179285 + 0.983797i \(0.442622\pi\)
\(620\) −7.00966 −0.281515
\(621\) −5.39558 −0.216517
\(622\) −8.79623 −0.352697
\(623\) 25.4966 1.02150
\(624\) 1.40294 0.0561627
\(625\) −31.2448 −1.24979
\(626\) −10.1860 −0.407115
\(627\) −0.523976 −0.0209256
\(628\) −13.6272 −0.543786
\(629\) 0.887072 0.0353699
\(630\) −34.7448 −1.38426
\(631\) 44.6597 1.77788 0.888938 0.458028i \(-0.151444\pi\)
0.888938 + 0.458028i \(0.151444\pi\)
\(632\) −11.8538 −0.471521
\(633\) 7.49149 0.297760
\(634\) 12.0097 0.476965
\(635\) 15.6930 0.622756
\(636\) 2.15122 0.0853014
\(637\) −39.8384 −1.57845
\(638\) −4.92692 −0.195059
\(639\) −34.9439 −1.38236
\(640\) −2.72545 −0.107733
\(641\) −6.93272 −0.273826 −0.136913 0.990583i \(-0.543718\pi\)
−0.136913 + 0.990583i \(0.543718\pi\)
\(642\) 0.604417 0.0238544
\(643\) 23.4966 0.926614 0.463307 0.886198i \(-0.346663\pi\)
0.463307 + 0.886198i \(0.346663\pi\)
\(644\) 8.41261 0.331503
\(645\) 16.8133 0.662021
\(646\) 0.201472 0.00792682
\(647\) 20.9880 0.825125 0.412562 0.910929i \(-0.364634\pi\)
0.412562 + 0.910929i \(0.364634\pi\)
\(648\) −6.60442 −0.259446
\(649\) 5.24943 0.206058
\(650\) 6.50115 0.254996
\(651\) −6.30353 −0.247055
\(652\) 15.9497 0.624640
\(653\) −44.1335 −1.72708 −0.863538 0.504284i \(-0.831756\pi\)
−0.863538 + 0.504284i \(0.831756\pi\)
\(654\) −5.00736 −0.195803
\(655\) 3.89213 0.152078
\(656\) 4.97487 0.194236
\(657\) −6.00000 −0.234082
\(658\) −58.4633 −2.27914
\(659\) 24.0862 0.938267 0.469133 0.883127i \(-0.344566\pi\)
0.469133 + 0.883127i \(0.344566\pi\)
\(660\) 1.42807 0.0555876
\(661\) −9.65237 −0.375434 −0.187717 0.982223i \(-0.560109\pi\)
−0.187717 + 0.982223i \(0.560109\pi\)
\(662\) −26.1667 −1.01700
\(663\) 0.282655 0.0109774
\(664\) 13.6775 0.530790
\(665\) 12.7483 0.494357
\(666\) 12.0000 0.464991
\(667\) 8.86120 0.343107
\(668\) 18.8059 0.727622
\(669\) 0.211133 0.00816289
\(670\) −24.7483 −0.956109
\(671\) −1.59706 −0.0616536
\(672\) −2.45090 −0.0945454
\(673\) 24.6176 0.948938 0.474469 0.880272i \(-0.342640\pi\)
0.474469 + 0.880272i \(0.342640\pi\)
\(674\) −17.3778 −0.669369
\(675\) 7.28421 0.280369
\(676\) −5.83102 −0.224270
\(677\) 21.5719 0.829077 0.414538 0.910032i \(-0.363943\pi\)
0.414538 + 0.910032i \(0.363943\pi\)
\(678\) −4.19181 −0.160986
\(679\) −78.6095 −3.01675
\(680\) −0.549103 −0.0210571
\(681\) 11.0074 0.421803
\(682\) −2.57193 −0.0984843
\(683\) −0.997701 −0.0381759 −0.0190880 0.999818i \(-0.506076\pi\)
−0.0190880 + 0.999818i \(0.506076\pi\)
\(684\) 2.72545 0.104210
\(685\) −39.6044 −1.51321
\(686\) 36.8538 1.40709
\(687\) 7.50851 0.286468
\(688\) −11.7734 −0.448857
\(689\) 10.9926 0.418786
\(690\) −2.56842 −0.0977783
\(691\) −9.00230 −0.342464 −0.171232 0.985231i \(-0.554775\pi\)
−0.171232 + 0.985231i \(0.554775\pi\)
\(692\) −3.22430 −0.122569
\(693\) −12.7483 −0.484267
\(694\) 12.4029 0.470809
\(695\) 44.0878 1.67235
\(696\) −2.58159 −0.0978549
\(697\) 1.00230 0.0379648
\(698\) 23.8036 0.900979
\(699\) 12.7866 0.483632
\(700\) −11.3573 −0.429265
\(701\) −15.0936 −0.570078 −0.285039 0.958516i \(-0.592006\pi\)
−0.285039 + 0.958516i \(0.592006\pi\)
\(702\) 8.03249 0.303167
\(703\) −4.40294 −0.166060
\(704\) −1.00000 −0.0376889
\(705\) 17.8492 0.672241
\(706\) 24.2664 0.913280
\(707\) −34.8515 −1.31073
\(708\) 2.75057 0.103373
\(709\) −22.9246 −0.860952 −0.430476 0.902602i \(-0.641654\pi\)
−0.430476 + 0.902602i \(0.641654\pi\)
\(710\) −34.9439 −1.31142
\(711\) −32.3070 −1.21161
\(712\) 5.45090 0.204281
\(713\) 4.62569 0.173233
\(714\) −0.493788 −0.0184796
\(715\) 7.29738 0.272906
\(716\) 13.3778 0.499953
\(717\) 6.64657 0.248221
\(718\) −20.3395 −0.759064
\(719\) 6.70032 0.249880 0.124940 0.992164i \(-0.460126\pi\)
0.124940 + 0.992164i \(0.460126\pi\)
\(720\) −7.42807 −0.276828
\(721\) 67.7460 2.52299
\(722\) −1.00000 −0.0372161
\(723\) 12.5410 0.466405
\(724\) 15.9497 0.592768
\(725\) −11.9629 −0.444291
\(726\) 0.523976 0.0194466
\(727\) −18.4583 −0.684579 −0.342289 0.939595i \(-0.611202\pi\)
−0.342289 + 0.939595i \(0.611202\pi\)
\(728\) −12.5240 −0.464169
\(729\) −13.2842 −0.492008
\(730\) −6.00000 −0.222070
\(731\) −2.37201 −0.0877321
\(732\) −0.836819 −0.0309297
\(733\) 41.8995 1.54759 0.773797 0.633434i \(-0.218355\pi\)
0.773797 + 0.633434i \(0.218355\pi\)
\(734\) 23.0170 0.849574
\(735\) −21.2482 −0.783752
\(736\) 1.79853 0.0662946
\(737\) −9.08044 −0.334482
\(738\) 13.5588 0.499105
\(739\) −34.8361 −1.28147 −0.640733 0.767764i \(-0.721369\pi\)
−0.640733 + 0.767764i \(0.721369\pi\)
\(740\) 12.0000 0.441129
\(741\) −1.40294 −0.0515385
\(742\) −19.2038 −0.704993
\(743\) −25.7889 −0.946102 −0.473051 0.881035i \(-0.656847\pi\)
−0.473051 + 0.881035i \(0.656847\pi\)
\(744\) −1.34763 −0.0494065
\(745\) 17.1895 0.629775
\(746\) 18.5890 0.680590
\(747\) 37.2773 1.36391
\(748\) −0.201472 −0.00736656
\(749\) −5.39558 −0.197150
\(750\) −3.67290 −0.134115
\(751\) −33.5011 −1.22247 −0.611237 0.791447i \(-0.709328\pi\)
−0.611237 + 0.791447i \(0.709328\pi\)
\(752\) −12.4989 −0.455786
\(753\) −9.69296 −0.353231
\(754\) −13.1918 −0.480417
\(755\) 1.35959 0.0494806
\(756\) −14.0325 −0.510357
\(757\) 7.53134 0.273731 0.136866 0.990590i \(-0.456297\pi\)
0.136866 + 0.990590i \(0.456297\pi\)
\(758\) 10.5816 0.384341
\(759\) −0.942386 −0.0342064
\(760\) 2.72545 0.0988624
\(761\) 38.1969 1.38464 0.692318 0.721593i \(-0.256590\pi\)
0.692318 + 0.721593i \(0.256590\pi\)
\(762\) 3.01702 0.109295
\(763\) 44.7003 1.61826
\(764\) −24.0553 −0.870291
\(765\) −1.49655 −0.0541079
\(766\) 26.0878 0.942591
\(767\) 14.0553 0.507508
\(768\) −0.523976 −0.0189074
\(769\) 16.6501 0.600417 0.300208 0.953874i \(-0.402944\pi\)
0.300208 + 0.953874i \(0.402944\pi\)
\(770\) −12.7483 −0.459416
\(771\) −14.5445 −0.523808
\(772\) 13.0325 0.469050
\(773\) −23.6330 −0.850022 −0.425011 0.905188i \(-0.639730\pi\)
−0.425011 + 0.905188i \(0.639730\pi\)
\(774\) −32.0878 −1.15337
\(775\) −6.24483 −0.224321
\(776\) −16.8059 −0.603296
\(777\) 10.7912 0.387131
\(778\) 9.46636 0.339386
\(779\) −4.97487 −0.178243
\(780\) 3.82365 0.136909
\(781\) −12.8214 −0.458784
\(782\) 0.362354 0.0129577
\(783\) −14.7808 −0.528221
\(784\) 14.8790 0.531392
\(785\) −37.1404 −1.32560
\(786\) 0.748275 0.0266901
\(787\) −20.3933 −0.726942 −0.363471 0.931606i \(-0.618408\pi\)
−0.363471 + 0.931606i \(0.618408\pi\)
\(788\) −5.59706 −0.199387
\(789\) −10.6222 −0.378160
\(790\) −32.3070 −1.14943
\(791\) 37.4200 1.33050
\(792\) −2.72545 −0.0968446
\(793\) −4.27611 −0.151849
\(794\) 25.1667 0.893132
\(795\) 5.86304 0.207941
\(796\) −7.39558 −0.262129
\(797\) −9.70262 −0.343685 −0.171842 0.985124i \(-0.554972\pi\)
−0.171842 + 0.985124i \(0.554972\pi\)
\(798\) 2.45090 0.0867608
\(799\) −2.51817 −0.0890865
\(800\) −2.42807 −0.0858453
\(801\) 14.8561 0.524916
\(802\) 16.0650 0.567274
\(803\) −2.20147 −0.0776883
\(804\) −4.75794 −0.167799
\(805\) 22.9281 0.808110
\(806\) −6.88633 −0.242561
\(807\) 8.19641 0.288527
\(808\) −7.45090 −0.262122
\(809\) −27.0723 −0.951813 −0.475906 0.879496i \(-0.657880\pi\)
−0.475906 + 0.879496i \(0.657880\pi\)
\(810\) −18.0000 −0.632456
\(811\) 51.9548 1.82438 0.912190 0.409767i \(-0.134390\pi\)
0.912190 + 0.409767i \(0.134390\pi\)
\(812\) 23.0457 0.808744
\(813\) 6.46562 0.226759
\(814\) 4.40294 0.154323
\(815\) 43.4702 1.52270
\(816\) −0.105567 −0.00369558
\(817\) 11.7734 0.411899
\(818\) 19.5313 0.682897
\(819\) −34.1335 −1.19272
\(820\) 13.5588 0.473493
\(821\) −25.6574 −0.895451 −0.447725 0.894171i \(-0.647766\pi\)
−0.447725 + 0.894171i \(0.647766\pi\)
\(822\) −7.61408 −0.265571
\(823\) −41.8635 −1.45927 −0.729635 0.683837i \(-0.760310\pi\)
−0.729635 + 0.683837i \(0.760310\pi\)
\(824\) 14.4834 0.504553
\(825\) 1.27225 0.0442941
\(826\) −24.5542 −0.854349
\(827\) 6.15582 0.214059 0.107029 0.994256i \(-0.465866\pi\)
0.107029 + 0.994256i \(0.465866\pi\)
\(828\) 4.90179 0.170349
\(829\) −0.247126 −0.00858303 −0.00429151 0.999991i \(-0.501366\pi\)
−0.00429151 + 0.999991i \(0.501366\pi\)
\(830\) 37.2773 1.29391
\(831\) −4.45320 −0.154480
\(832\) −2.67750 −0.0928255
\(833\) 2.99770 0.103864
\(834\) 8.47602 0.293501
\(835\) 51.2545 1.77373
\(836\) 1.00000 0.0345857
\(837\) −7.71579 −0.266697
\(838\) −24.9018 −0.860218
\(839\) 16.9320 0.584557 0.292278 0.956333i \(-0.405587\pi\)
0.292278 + 0.956333i \(0.405587\pi\)
\(840\) −6.67980 −0.230475
\(841\) −4.72545 −0.162947
\(842\) −26.6044 −0.916849
\(843\) 3.73511 0.128644
\(844\) −14.2974 −0.492136
\(845\) −15.8921 −0.546706
\(846\) −34.0650 −1.17118
\(847\) −4.67750 −0.160721
\(848\) −4.10557 −0.140986
\(849\) −6.03829 −0.207234
\(850\) −0.489189 −0.0167790
\(851\) −7.91882 −0.271454
\(852\) −6.71809 −0.230158
\(853\) 21.6930 0.742753 0.371376 0.928482i \(-0.378886\pi\)
0.371376 + 0.928482i \(0.378886\pi\)
\(854\) 7.47022 0.255626
\(855\) 7.42807 0.254035
\(856\) −1.15352 −0.0394265
\(857\) 34.7328 1.18645 0.593225 0.805037i \(-0.297854\pi\)
0.593225 + 0.805037i \(0.297854\pi\)
\(858\) 1.40294 0.0478957
\(859\) −15.3047 −0.522191 −0.261095 0.965313i \(-0.584084\pi\)
−0.261095 + 0.965313i \(0.584084\pi\)
\(860\) −32.0878 −1.09418
\(861\) 12.1929 0.415533
\(862\) 32.5948 1.11018
\(863\) 35.8457 1.22020 0.610102 0.792323i \(-0.291129\pi\)
0.610102 + 0.792323i \(0.291129\pi\)
\(864\) −3.00000 −0.102062
\(865\) −8.78766 −0.298789
\(866\) −19.7386 −0.670745
\(867\) 8.88633 0.301796
\(868\) 12.0302 0.408331
\(869\) −11.8538 −0.402114
\(870\) −7.03599 −0.238542
\(871\) −24.3128 −0.823809
\(872\) 9.55646 0.323623
\(873\) −45.8036 −1.55022
\(874\) −1.79853 −0.0608361
\(875\) 32.7877 1.10843
\(876\) −1.15352 −0.0389738
\(877\) −45.4907 −1.53611 −0.768057 0.640382i \(-0.778776\pi\)
−0.768057 + 0.640382i \(0.778776\pi\)
\(878\) 24.4029 0.823559
\(879\) −5.59476 −0.188706
\(880\) −2.72545 −0.0918749
\(881\) 31.3322 1.05561 0.527804 0.849366i \(-0.323016\pi\)
0.527804 + 0.849366i \(0.323016\pi\)
\(882\) 40.5519 1.36545
\(883\) 35.3550 1.18979 0.594895 0.803803i \(-0.297194\pi\)
0.594895 + 0.803803i \(0.297194\pi\)
\(884\) −0.539441 −0.0181434
\(885\) 7.49655 0.251994
\(886\) 16.5948 0.557512
\(887\) −16.6141 −0.557846 −0.278923 0.960313i \(-0.589977\pi\)
−0.278923 + 0.960313i \(0.589977\pi\)
\(888\) 2.30704 0.0774192
\(889\) −26.9327 −0.903295
\(890\) 14.8561 0.497979
\(891\) −6.60442 −0.221256
\(892\) −0.402945 −0.0134916
\(893\) 12.4989 0.418258
\(894\) 3.30474 0.110527
\(895\) 36.4606 1.21874
\(896\) 4.67750 0.156264
\(897\) −2.52323 −0.0842484
\(898\) 20.9520 0.699179
\(899\) 12.6717 0.422625
\(900\) −6.61758 −0.220586
\(901\) −0.827158 −0.0275566
\(902\) 4.97487 0.165645
\(903\) −28.8554 −0.960248
\(904\) 8.00000 0.266076
\(905\) 43.4702 1.44500
\(906\) 0.261386 0.00868396
\(907\) 54.8612 1.82164 0.910818 0.412808i \(-0.135452\pi\)
0.910818 + 0.412808i \(0.135452\pi\)
\(908\) −21.0074 −0.697154
\(909\) −20.3070 −0.673542
\(910\) −34.1335 −1.13151
\(911\) 2.98298 0.0988304 0.0494152 0.998778i \(-0.484264\pi\)
0.0494152 + 0.998778i \(0.484264\pi\)
\(912\) 0.523976 0.0173506
\(913\) 13.6775 0.452659
\(914\) −23.3144 −0.771172
\(915\) −2.28071 −0.0753979
\(916\) −14.3299 −0.473472
\(917\) −6.67980 −0.220586
\(918\) −0.604417 −0.0199487
\(919\) −26.6701 −0.879767 −0.439883 0.898055i \(-0.644980\pi\)
−0.439883 + 0.898055i \(0.644980\pi\)
\(920\) 4.90179 0.161607
\(921\) 15.8538 0.522402
\(922\) −20.8059 −0.685205
\(923\) −34.3291 −1.12996
\(924\) −2.45090 −0.0806286
\(925\) 10.6907 0.351507
\(926\) 9.29002 0.305289
\(927\) 39.4737 1.29649
\(928\) 4.92692 0.161734
\(929\) 43.7595 1.43570 0.717851 0.696197i \(-0.245126\pi\)
0.717851 + 0.696197i \(0.245126\pi\)
\(930\) −3.67290 −0.120439
\(931\) −14.8790 −0.487638
\(932\) −24.4029 −0.799345
\(933\) −4.60902 −0.150892
\(934\) −12.0457 −0.394146
\(935\) −0.549103 −0.0179576
\(936\) −7.29738 −0.238522
\(937\) 47.7823 1.56098 0.780490 0.625168i \(-0.214970\pi\)
0.780490 + 0.625168i \(0.214970\pi\)
\(938\) 42.4737 1.38682
\(939\) −5.33723 −0.174174
\(940\) −34.0650 −1.11108
\(941\) 36.2664 1.18225 0.591126 0.806579i \(-0.298684\pi\)
0.591126 + 0.806579i \(0.298684\pi\)
\(942\) −7.14035 −0.232645
\(943\) −8.94745 −0.291369
\(944\) −5.24943 −0.170854
\(945\) −38.2448 −1.24410
\(946\) −11.7734 −0.382786
\(947\) 35.5468 1.15512 0.577558 0.816350i \(-0.304006\pi\)
0.577558 + 0.816350i \(0.304006\pi\)
\(948\) −6.21113 −0.201728
\(949\) −5.89443 −0.191341
\(950\) 2.42807 0.0787770
\(951\) 6.29278 0.204057
\(952\) 0.942386 0.0305429
\(953\) 23.2088 0.751808 0.375904 0.926659i \(-0.377332\pi\)
0.375904 + 0.926659i \(0.377332\pi\)
\(954\) −11.1895 −0.362274
\(955\) −65.5615 −2.12152
\(956\) −12.6849 −0.410258
\(957\) −2.58159 −0.0834510
\(958\) −0.188307 −0.00608392
\(959\) 67.9703 2.19487
\(960\) −1.42807 −0.0460908
\(961\) −24.3852 −0.786619
\(962\) 11.7889 0.380088
\(963\) −3.14386 −0.101309
\(964\) −23.9343 −0.770871
\(965\) 35.5194 1.14341
\(966\) 4.40801 0.141825
\(967\) −6.09591 −0.196031 −0.0980156 0.995185i \(-0.531249\pi\)
−0.0980156 + 0.995185i \(0.531249\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0.105567 0.00339129
\(970\) −45.8036 −1.47066
\(971\) 34.9822 1.12263 0.561317 0.827601i \(-0.310295\pi\)
0.561317 + 0.827601i \(0.310295\pi\)
\(972\) −12.4606 −0.399673
\(973\) −75.6648 −2.42570
\(974\) 33.3852 1.06973
\(975\) 3.40645 0.109094
\(976\) 1.59706 0.0511205
\(977\) −10.9018 −0.348779 −0.174390 0.984677i \(-0.555795\pi\)
−0.174390 + 0.984677i \(0.555795\pi\)
\(978\) 8.35729 0.267237
\(979\) 5.45090 0.174211
\(980\) 40.5519 1.29538
\(981\) 26.0457 0.831574
\(982\) −12.5217 −0.399583
\(983\) −25.4281 −0.811030 −0.405515 0.914089i \(-0.632908\pi\)
−0.405515 + 0.914089i \(0.632908\pi\)
\(984\) 2.60672 0.0830991
\(985\) −15.2545 −0.486048
\(986\) 0.992638 0.0316120
\(987\) −30.6334 −0.975072
\(988\) 2.67750 0.0851825
\(989\) 21.1748 0.673319
\(990\) −7.42807 −0.236080
\(991\) 49.3925 1.56901 0.784503 0.620125i \(-0.212918\pi\)
0.784503 + 0.620125i \(0.212918\pi\)
\(992\) 2.57193 0.0816588
\(993\) −13.7107 −0.435097
\(994\) 59.9718 1.90219
\(995\) −20.1563 −0.638997
\(996\) 7.16669 0.227085
\(997\) 20.7409 0.656871 0.328436 0.944526i \(-0.393479\pi\)
0.328436 + 0.944526i \(0.393479\pi\)
\(998\) −16.3070 −0.516190
\(999\) 13.2088 0.417909
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 418.2.a.g.1.2 3
3.2 odd 2 3762.2.a.bg.1.1 3
4.3 odd 2 3344.2.a.q.1.2 3
11.10 odd 2 4598.2.a.bo.1.2 3
19.18 odd 2 7942.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 3 1.1 even 1 trivial
3344.2.a.q.1.2 3 4.3 odd 2
3762.2.a.bg.1.1 3 3.2 odd 2
4598.2.a.bo.1.2 3 11.10 odd 2
7942.2.a.bi.1.2 3 19.18 odd 2