Properties

Label 418.2.a.f.1.2
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 \(-1.79129\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.79129 q^{3} +1.00000 q^{4} +3.79129 q^{5} +1.79129 q^{6} -4.79129 q^{7} +1.00000 q^{8} +0.208712 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.79129 q^{3} +1.00000 q^{4} +3.79129 q^{5} +1.79129 q^{6} -4.79129 q^{7} +1.00000 q^{8} +0.208712 q^{9} +3.79129 q^{10} +1.00000 q^{11} +1.79129 q^{12} +1.20871 q^{13} -4.79129 q^{14} +6.79129 q^{15} +1.00000 q^{16} -7.58258 q^{17} +0.208712 q^{18} +1.00000 q^{19} +3.79129 q^{20} -8.58258 q^{21} +1.00000 q^{22} -1.58258 q^{23} +1.79129 q^{24} +9.37386 q^{25} +1.20871 q^{26} -5.00000 q^{27} -4.79129 q^{28} +2.20871 q^{29} +6.79129 q^{30} -10.7913 q^{31} +1.00000 q^{32} +1.79129 q^{33} -7.58258 q^{34} -18.1652 q^{35} +0.208712 q^{36} +8.00000 q^{37} +1.00000 q^{38} +2.16515 q^{39} +3.79129 q^{40} -0.791288 q^{41} -8.58258 q^{42} +7.37386 q^{43} +1.00000 q^{44} +0.791288 q^{45} -1.58258 q^{46} +9.16515 q^{47} +1.79129 q^{48} +15.9564 q^{49} +9.37386 q^{50} -13.5826 q^{51} +1.20871 q^{52} -1.58258 q^{53} -5.00000 q^{54} +3.79129 q^{55} -4.79129 q^{56} +1.79129 q^{57} +2.20871 q^{58} +6.79129 q^{60} +2.00000 q^{61} -10.7913 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.58258 q^{65} +1.79129 q^{66} -4.79129 q^{67} -7.58258 q^{68} -2.83485 q^{69} -18.1652 q^{70} -11.3739 q^{71} +0.208712 q^{72} +9.58258 q^{73} +8.00000 q^{74} +16.7913 q^{75} +1.00000 q^{76} -4.79129 q^{77} +2.16515 q^{78} -5.58258 q^{79} +3.79129 q^{80} -9.58258 q^{81} -0.791288 q^{82} -3.79129 q^{83} -8.58258 q^{84} -28.7477 q^{85} +7.37386 q^{86} +3.95644 q^{87} +1.00000 q^{88} -4.41742 q^{89} +0.791288 q^{90} -5.79129 q^{91} -1.58258 q^{92} -19.3303 q^{93} +9.16515 q^{94} +3.79129 q^{95} +1.79129 q^{96} +8.00000 q^{97} +15.9564 q^{98} +0.208712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} + 5 q^{9} + 3 q^{10} + 2 q^{11} - q^{12} + 7 q^{13} - 5 q^{14} + 9 q^{15} + 2 q^{16} - 6 q^{17} + 5 q^{18} + 2 q^{19} + 3 q^{20} - 8 q^{21} + 2 q^{22} + 6 q^{23} - q^{24} + 5 q^{25} + 7 q^{26} - 10 q^{27} - 5 q^{28} + 9 q^{29} + 9 q^{30} - 17 q^{31} + 2 q^{32} - q^{33} - 6 q^{34} - 18 q^{35} + 5 q^{36} + 16 q^{37} + 2 q^{38} - 14 q^{39} + 3 q^{40} + 3 q^{41} - 8 q^{42} + q^{43} + 2 q^{44} - 3 q^{45} + 6 q^{46} - q^{48} + 9 q^{49} + 5 q^{50} - 18 q^{51} + 7 q^{52} + 6 q^{53} - 10 q^{54} + 3 q^{55} - 5 q^{56} - q^{57} + 9 q^{58} + 9 q^{60} + 4 q^{61} - 17 q^{62} - 2 q^{63} + 2 q^{64} - q^{66} - 5 q^{67} - 6 q^{68} - 24 q^{69} - 18 q^{70} - 9 q^{71} + 5 q^{72} + 10 q^{73} + 16 q^{74} + 29 q^{75} + 2 q^{76} - 5 q^{77} - 14 q^{78} - 2 q^{79} + 3 q^{80} - 10 q^{81} + 3 q^{82} - 3 q^{83} - 8 q^{84} - 30 q^{85} + q^{86} - 15 q^{87} + 2 q^{88} - 18 q^{89} - 3 q^{90} - 7 q^{91} + 6 q^{92} - 2 q^{93} + 3 q^{95} - q^{96} + 16 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.79129 1.03420 0.517100 0.855925i \(-0.327011\pi\)
0.517100 + 0.855925i \(0.327011\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.79129 1.69552 0.847758 0.530384i \(-0.177952\pi\)
0.847758 + 0.530384i \(0.177952\pi\)
\(6\) 1.79129 0.731290
\(7\) −4.79129 −1.81094 −0.905468 0.424414i \(-0.860480\pi\)
−0.905468 + 0.424414i \(0.860480\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.208712 0.0695707
\(10\) 3.79129 1.19891
\(11\) 1.00000 0.301511
\(12\) 1.79129 0.517100
\(13\) 1.20871 0.335236 0.167618 0.985852i \(-0.446392\pi\)
0.167618 + 0.985852i \(0.446392\pi\)
\(14\) −4.79129 −1.28053
\(15\) 6.79129 1.75350
\(16\) 1.00000 0.250000
\(17\) −7.58258 −1.83904 −0.919522 0.393038i \(-0.871424\pi\)
−0.919522 + 0.393038i \(0.871424\pi\)
\(18\) 0.208712 0.0491939
\(19\) 1.00000 0.229416
\(20\) 3.79129 0.847758
\(21\) −8.58258 −1.87287
\(22\) 1.00000 0.213201
\(23\) −1.58258 −0.329990 −0.164995 0.986294i \(-0.552761\pi\)
−0.164995 + 0.986294i \(0.552761\pi\)
\(24\) 1.79129 0.365645
\(25\) 9.37386 1.87477
\(26\) 1.20871 0.237048
\(27\) −5.00000 −0.962250
\(28\) −4.79129 −0.905468
\(29\) 2.20871 0.410148 0.205074 0.978747i \(-0.434257\pi\)
0.205074 + 0.978747i \(0.434257\pi\)
\(30\) 6.79129 1.23991
\(31\) −10.7913 −1.93817 −0.969086 0.246722i \(-0.920646\pi\)
−0.969086 + 0.246722i \(0.920646\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.79129 0.311823
\(34\) −7.58258 −1.30040
\(35\) −18.1652 −3.07047
\(36\) 0.208712 0.0347854
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.16515 0.346702
\(40\) 3.79129 0.599455
\(41\) −0.791288 −0.123578 −0.0617892 0.998089i \(-0.519681\pi\)
−0.0617892 + 0.998089i \(0.519681\pi\)
\(42\) −8.58258 −1.32432
\(43\) 7.37386 1.12450 0.562252 0.826966i \(-0.309935\pi\)
0.562252 + 0.826966i \(0.309935\pi\)
\(44\) 1.00000 0.150756
\(45\) 0.791288 0.117958
\(46\) −1.58258 −0.233338
\(47\) 9.16515 1.33687 0.668437 0.743768i \(-0.266963\pi\)
0.668437 + 0.743768i \(0.266963\pi\)
\(48\) 1.79129 0.258550
\(49\) 15.9564 2.27949
\(50\) 9.37386 1.32566
\(51\) −13.5826 −1.90194
\(52\) 1.20871 0.167618
\(53\) −1.58258 −0.217383 −0.108692 0.994076i \(-0.534666\pi\)
−0.108692 + 0.994076i \(0.534666\pi\)
\(54\) −5.00000 −0.680414
\(55\) 3.79129 0.511217
\(56\) −4.79129 −0.640263
\(57\) 1.79129 0.237262
\(58\) 2.20871 0.290018
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 6.79129 0.876751
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.7913 −1.37049
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 4.58258 0.568399
\(66\) 1.79129 0.220492
\(67\) −4.79129 −0.585349 −0.292674 0.956212i \(-0.594545\pi\)
−0.292674 + 0.956212i \(0.594545\pi\)
\(68\) −7.58258 −0.919522
\(69\) −2.83485 −0.341276
\(70\) −18.1652 −2.17115
\(71\) −11.3739 −1.34983 −0.674915 0.737896i \(-0.735820\pi\)
−0.674915 + 0.737896i \(0.735820\pi\)
\(72\) 0.208712 0.0245970
\(73\) 9.58258 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(74\) 8.00000 0.929981
\(75\) 16.7913 1.93889
\(76\) 1.00000 0.114708
\(77\) −4.79129 −0.546018
\(78\) 2.16515 0.245155
\(79\) −5.58258 −0.628089 −0.314044 0.949408i \(-0.601684\pi\)
−0.314044 + 0.949408i \(0.601684\pi\)
\(80\) 3.79129 0.423879
\(81\) −9.58258 −1.06473
\(82\) −0.791288 −0.0873831
\(83\) −3.79129 −0.416148 −0.208074 0.978113i \(-0.566719\pi\)
−0.208074 + 0.978113i \(0.566719\pi\)
\(84\) −8.58258 −0.936436
\(85\) −28.7477 −3.11813
\(86\) 7.37386 0.795144
\(87\) 3.95644 0.424175
\(88\) 1.00000 0.106600
\(89\) −4.41742 −0.468246 −0.234123 0.972207i \(-0.575222\pi\)
−0.234123 + 0.972207i \(0.575222\pi\)
\(90\) 0.791288 0.0834091
\(91\) −5.79129 −0.607092
\(92\) −1.58258 −0.164995
\(93\) −19.3303 −2.00446
\(94\) 9.16515 0.945313
\(95\) 3.79129 0.388978
\(96\) 1.79129 0.182823
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 15.9564 1.61184
\(99\) 0.208712 0.0209764
\(100\) 9.37386 0.937386
\(101\) −13.5826 −1.35152 −0.675758 0.737123i \(-0.736184\pi\)
−0.675758 + 0.737123i \(0.736184\pi\)
\(102\) −13.5826 −1.34488
\(103\) 2.95644 0.291307 0.145653 0.989336i \(-0.453472\pi\)
0.145653 + 0.989336i \(0.453472\pi\)
\(104\) 1.20871 0.118524
\(105\) −32.5390 −3.17548
\(106\) −1.58258 −0.153713
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −5.00000 −0.481125
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 3.79129 0.361485
\(111\) 14.3303 1.36017
\(112\) −4.79129 −0.452734
\(113\) 15.1652 1.42662 0.713309 0.700850i \(-0.247196\pi\)
0.713309 + 0.700850i \(0.247196\pi\)
\(114\) 1.79129 0.167769
\(115\) −6.00000 −0.559503
\(116\) 2.20871 0.205074
\(117\) 0.252273 0.0233226
\(118\) 0 0
\(119\) 36.3303 3.33039
\(120\) 6.79129 0.619957
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −1.41742 −0.127805
\(124\) −10.7913 −0.969086
\(125\) 16.5826 1.48319
\(126\) −1.00000 −0.0890871
\(127\) −8.74773 −0.776235 −0.388118 0.921610i \(-0.626875\pi\)
−0.388118 + 0.921610i \(0.626875\pi\)
\(128\) 1.00000 0.0883883
\(129\) 13.2087 1.16296
\(130\) 4.58258 0.401918
\(131\) 3.62614 0.316817 0.158409 0.987374i \(-0.449364\pi\)
0.158409 + 0.987374i \(0.449364\pi\)
\(132\) 1.79129 0.155912
\(133\) −4.79129 −0.415457
\(134\) −4.79129 −0.413904
\(135\) −18.9564 −1.63151
\(136\) −7.58258 −0.650201
\(137\) 5.20871 0.445010 0.222505 0.974932i \(-0.428577\pi\)
0.222505 + 0.974932i \(0.428577\pi\)
\(138\) −2.83485 −0.241318
\(139\) −12.2087 −1.03553 −0.517765 0.855523i \(-0.673236\pi\)
−0.517765 + 0.855523i \(0.673236\pi\)
\(140\) −18.1652 −1.53524
\(141\) 16.4174 1.38260
\(142\) −11.3739 −0.954473
\(143\) 1.20871 0.101078
\(144\) 0.208712 0.0173927
\(145\) 8.37386 0.695412
\(146\) 9.58258 0.793060
\(147\) 28.5826 2.35745
\(148\) 8.00000 0.657596
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 16.7913 1.37100
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 1.00000 0.0811107
\(153\) −1.58258 −0.127944
\(154\) −4.79129 −0.386093
\(155\) −40.9129 −3.28620
\(156\) 2.16515 0.173351
\(157\) −10.9564 −0.874419 −0.437210 0.899360i \(-0.644033\pi\)
−0.437210 + 0.899360i \(0.644033\pi\)
\(158\) −5.58258 −0.444126
\(159\) −2.83485 −0.224818
\(160\) 3.79129 0.299728
\(161\) 7.58258 0.597591
\(162\) −9.58258 −0.752878
\(163\) −11.5826 −0.907217 −0.453609 0.891201i \(-0.649864\pi\)
−0.453609 + 0.891201i \(0.649864\pi\)
\(164\) −0.791288 −0.0617892
\(165\) 6.79129 0.528701
\(166\) −3.79129 −0.294261
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −8.58258 −0.662160
\(169\) −11.5390 −0.887617
\(170\) −28.7477 −2.20485
\(171\) 0.208712 0.0159606
\(172\) 7.37386 0.562252
\(173\) 12.9564 0.985060 0.492530 0.870296i \(-0.336072\pi\)
0.492530 + 0.870296i \(0.336072\pi\)
\(174\) 3.95644 0.299937
\(175\) −44.9129 −3.39509
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −4.41742 −0.331100
\(179\) 3.95644 0.295718 0.147859 0.989008i \(-0.452762\pi\)
0.147859 + 0.989008i \(0.452762\pi\)
\(180\) 0.791288 0.0589791
\(181\) 18.7477 1.39351 0.696754 0.717310i \(-0.254627\pi\)
0.696754 + 0.717310i \(0.254627\pi\)
\(182\) −5.79129 −0.429279
\(183\) 3.58258 0.264832
\(184\) −1.58258 −0.116669
\(185\) 30.3303 2.22993
\(186\) −19.3303 −1.41737
\(187\) −7.58258 −0.554493
\(188\) 9.16515 0.668437
\(189\) 23.9564 1.74257
\(190\) 3.79129 0.275049
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.79129 0.129275
\(193\) 7.37386 0.530782 0.265391 0.964141i \(-0.414499\pi\)
0.265391 + 0.964141i \(0.414499\pi\)
\(194\) 8.00000 0.574367
\(195\) 8.20871 0.587838
\(196\) 15.9564 1.13975
\(197\) 9.16515 0.652990 0.326495 0.945199i \(-0.394132\pi\)
0.326495 + 0.945199i \(0.394132\pi\)
\(198\) 0.208712 0.0148325
\(199\) 6.74773 0.478334 0.239167 0.970978i \(-0.423126\pi\)
0.239167 + 0.970978i \(0.423126\pi\)
\(200\) 9.37386 0.662832
\(201\) −8.58258 −0.605368
\(202\) −13.5826 −0.955667
\(203\) −10.5826 −0.742751
\(204\) −13.5826 −0.950971
\(205\) −3.00000 −0.209529
\(206\) 2.95644 0.205985
\(207\) −0.330303 −0.0229576
\(208\) 1.20871 0.0838091
\(209\) 1.00000 0.0691714
\(210\) −32.5390 −2.24541
\(211\) 18.7477 1.29065 0.645323 0.763910i \(-0.276723\pi\)
0.645323 + 0.763910i \(0.276723\pi\)
\(212\) −1.58258 −0.108692
\(213\) −20.3739 −1.39599
\(214\) 18.0000 1.23045
\(215\) 27.9564 1.90661
\(216\) −5.00000 −0.340207
\(217\) 51.7042 3.50991
\(218\) −10.0000 −0.677285
\(219\) 17.1652 1.15991
\(220\) 3.79129 0.255609
\(221\) −9.16515 −0.616515
\(222\) 14.3303 0.961787
\(223\) −7.16515 −0.479814 −0.239907 0.970796i \(-0.577117\pi\)
−0.239907 + 0.970796i \(0.577117\pi\)
\(224\) −4.79129 −0.320131
\(225\) 1.95644 0.130429
\(226\) 15.1652 1.00877
\(227\) −7.58258 −0.503273 −0.251637 0.967822i \(-0.580969\pi\)
−0.251637 + 0.967822i \(0.580969\pi\)
\(228\) 1.79129 0.118631
\(229\) −18.3739 −1.21418 −0.607090 0.794633i \(-0.707663\pi\)
−0.607090 + 0.794633i \(0.707663\pi\)
\(230\) −6.00000 −0.395628
\(231\) −8.58258 −0.564692
\(232\) 2.20871 0.145009
\(233\) 15.1652 0.993502 0.496751 0.867893i \(-0.334526\pi\)
0.496751 + 0.867893i \(0.334526\pi\)
\(234\) 0.252273 0.0164916
\(235\) 34.7477 2.26669
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 36.3303 2.35494
\(239\) −2.20871 −0.142870 −0.0714349 0.997445i \(-0.522758\pi\)
−0.0714349 + 0.997445i \(0.522758\pi\)
\(240\) 6.79129 0.438376
\(241\) −0.208712 −0.0134443 −0.00672217 0.999977i \(-0.502140\pi\)
−0.00672217 + 0.999977i \(0.502140\pi\)
\(242\) 1.00000 0.0642824
\(243\) −2.16515 −0.138895
\(244\) 2.00000 0.128037
\(245\) 60.4955 3.86491
\(246\) −1.41742 −0.0903717
\(247\) 1.20871 0.0769085
\(248\) −10.7913 −0.685247
\(249\) −6.79129 −0.430380
\(250\) 16.5826 1.04877
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −1.58258 −0.0994957
\(254\) −8.74773 −0.548881
\(255\) −51.4955 −3.22477
\(256\) 1.00000 0.0625000
\(257\) −7.58258 −0.472988 −0.236494 0.971633i \(-0.575998\pi\)
−0.236494 + 0.971633i \(0.575998\pi\)
\(258\) 13.2087 0.822338
\(259\) −38.3303 −2.38173
\(260\) 4.58258 0.284199
\(261\) 0.460985 0.0285343
\(262\) 3.62614 0.224023
\(263\) 12.9564 0.798928 0.399464 0.916749i \(-0.369196\pi\)
0.399464 + 0.916749i \(0.369196\pi\)
\(264\) 1.79129 0.110246
\(265\) −6.00000 −0.368577
\(266\) −4.79129 −0.293773
\(267\) −7.91288 −0.484260
\(268\) −4.79129 −0.292674
\(269\) −16.7477 −1.02113 −0.510563 0.859840i \(-0.670563\pi\)
−0.510563 + 0.859840i \(0.670563\pi\)
\(270\) −18.9564 −1.15365
\(271\) −6.37386 −0.387185 −0.193592 0.981082i \(-0.562014\pi\)
−0.193592 + 0.981082i \(0.562014\pi\)
\(272\) −7.58258 −0.459761
\(273\) −10.3739 −0.627855
\(274\) 5.20871 0.314670
\(275\) 9.37386 0.565265
\(276\) −2.83485 −0.170638
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −12.2087 −0.732230
\(279\) −2.25227 −0.134840
\(280\) −18.1652 −1.08558
\(281\) 22.1216 1.31966 0.659832 0.751413i \(-0.270628\pi\)
0.659832 + 0.751413i \(0.270628\pi\)
\(282\) 16.4174 0.977643
\(283\) 9.12159 0.542222 0.271111 0.962548i \(-0.412609\pi\)
0.271111 + 0.962548i \(0.412609\pi\)
\(284\) −11.3739 −0.674915
\(285\) 6.79129 0.402281
\(286\) 1.20871 0.0714726
\(287\) 3.79129 0.223793
\(288\) 0.208712 0.0122985
\(289\) 40.4955 2.38209
\(290\) 8.37386 0.491730
\(291\) 14.3303 0.840057
\(292\) 9.58258 0.560778
\(293\) 6.79129 0.396751 0.198376 0.980126i \(-0.436433\pi\)
0.198376 + 0.980126i \(0.436433\pi\)
\(294\) 28.5826 1.66697
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −1.91288 −0.110625
\(300\) 16.7913 0.969445
\(301\) −35.3303 −2.03640
\(302\) 2.00000 0.115087
\(303\) −24.3303 −1.39774
\(304\) 1.00000 0.0573539
\(305\) 7.58258 0.434177
\(306\) −1.58258 −0.0904698
\(307\) −5.25227 −0.299763 −0.149882 0.988704i \(-0.547889\pi\)
−0.149882 + 0.988704i \(0.547889\pi\)
\(308\) −4.79129 −0.273009
\(309\) 5.29583 0.301269
\(310\) −40.9129 −2.32370
\(311\) −25.9129 −1.46938 −0.734692 0.678401i \(-0.762673\pi\)
−0.734692 + 0.678401i \(0.762673\pi\)
\(312\) 2.16515 0.122578
\(313\) −9.37386 −0.529842 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(314\) −10.9564 −0.618308
\(315\) −3.79129 −0.213615
\(316\) −5.58258 −0.314044
\(317\) 1.25227 0.0703347 0.0351673 0.999381i \(-0.488804\pi\)
0.0351673 + 0.999381i \(0.488804\pi\)
\(318\) −2.83485 −0.158970
\(319\) 2.20871 0.123664
\(320\) 3.79129 0.211939
\(321\) 32.2432 1.79964
\(322\) 7.58258 0.422560
\(323\) −7.58258 −0.421906
\(324\) −9.58258 −0.532365
\(325\) 11.3303 0.628492
\(326\) −11.5826 −0.641500
\(327\) −17.9129 −0.990584
\(328\) −0.791288 −0.0436916
\(329\) −43.9129 −2.42100
\(330\) 6.79129 0.373848
\(331\) 29.7913 1.63748 0.818739 0.574166i \(-0.194674\pi\)
0.818739 + 0.574166i \(0.194674\pi\)
\(332\) −3.79129 −0.208074
\(333\) 1.66970 0.0914988
\(334\) 18.0000 0.984916
\(335\) −18.1652 −0.992468
\(336\) −8.58258 −0.468218
\(337\) −13.6261 −0.742263 −0.371131 0.928580i \(-0.621030\pi\)
−0.371131 + 0.928580i \(0.621030\pi\)
\(338\) −11.5390 −0.627640
\(339\) 27.1652 1.47541
\(340\) −28.7477 −1.55906
\(341\) −10.7913 −0.584381
\(342\) 0.208712 0.0112859
\(343\) −42.9129 −2.31708
\(344\) 7.37386 0.397572
\(345\) −10.7477 −0.578638
\(346\) 12.9564 0.696542
\(347\) −3.16515 −0.169914 −0.0849571 0.996385i \(-0.527075\pi\)
−0.0849571 + 0.996385i \(0.527075\pi\)
\(348\) 3.95644 0.212087
\(349\) 23.4955 1.25768 0.628841 0.777534i \(-0.283529\pi\)
0.628841 + 0.777534i \(0.283529\pi\)
\(350\) −44.9129 −2.40069
\(351\) −6.04356 −0.322581
\(352\) 1.00000 0.0533002
\(353\) −14.8348 −0.789579 −0.394790 0.918772i \(-0.629183\pi\)
−0.394790 + 0.918772i \(0.629183\pi\)
\(354\) 0 0
\(355\) −43.1216 −2.28866
\(356\) −4.41742 −0.234123
\(357\) 65.0780 3.44429
\(358\) 3.95644 0.209104
\(359\) 20.7042 1.09272 0.546362 0.837549i \(-0.316012\pi\)
0.546362 + 0.837549i \(0.316012\pi\)
\(360\) 0.791288 0.0417045
\(361\) 1.00000 0.0526316
\(362\) 18.7477 0.985359
\(363\) 1.79129 0.0940182
\(364\) −5.79129 −0.303546
\(365\) 36.3303 1.90161
\(366\) 3.58258 0.187264
\(367\) 20.3303 1.06123 0.530617 0.847612i \(-0.321960\pi\)
0.530617 + 0.847612i \(0.321960\pi\)
\(368\) −1.58258 −0.0824975
\(369\) −0.165151 −0.00859744
\(370\) 30.3303 1.57680
\(371\) 7.58258 0.393668
\(372\) −19.3303 −1.00223
\(373\) −9.37386 −0.485360 −0.242680 0.970106i \(-0.578027\pi\)
−0.242680 + 0.970106i \(0.578027\pi\)
\(374\) −7.58258 −0.392086
\(375\) 29.7042 1.53392
\(376\) 9.16515 0.472657
\(377\) 2.66970 0.137496
\(378\) 23.9564 1.23219
\(379\) 32.7913 1.68438 0.842188 0.539185i \(-0.181267\pi\)
0.842188 + 0.539185i \(0.181267\pi\)
\(380\) 3.79129 0.194489
\(381\) −15.6697 −0.802783
\(382\) −18.0000 −0.920960
\(383\) −9.95644 −0.508750 −0.254375 0.967106i \(-0.581870\pi\)
−0.254375 + 0.967106i \(0.581870\pi\)
\(384\) 1.79129 0.0914113
\(385\) −18.1652 −0.925782
\(386\) 7.37386 0.375320
\(387\) 1.53901 0.0782325
\(388\) 8.00000 0.406138
\(389\) 10.1216 0.513185 0.256593 0.966520i \(-0.417400\pi\)
0.256593 + 0.966520i \(0.417400\pi\)
\(390\) 8.20871 0.415664
\(391\) 12.0000 0.606866
\(392\) 15.9564 0.805922
\(393\) 6.49545 0.327652
\(394\) 9.16515 0.461734
\(395\) −21.1652 −1.06493
\(396\) 0.208712 0.0104882
\(397\) 4.04356 0.202940 0.101470 0.994839i \(-0.467645\pi\)
0.101470 + 0.994839i \(0.467645\pi\)
\(398\) 6.74773 0.338233
\(399\) −8.58258 −0.429666
\(400\) 9.37386 0.468693
\(401\) −13.5826 −0.678281 −0.339141 0.940736i \(-0.610136\pi\)
−0.339141 + 0.940736i \(0.610136\pi\)
\(402\) −8.58258 −0.428060
\(403\) −13.0436 −0.649746
\(404\) −13.5826 −0.675758
\(405\) −36.3303 −1.80527
\(406\) −10.5826 −0.525204
\(407\) 8.00000 0.396545
\(408\) −13.5826 −0.672438
\(409\) −3.37386 −0.166827 −0.0834134 0.996515i \(-0.526582\pi\)
−0.0834134 + 0.996515i \(0.526582\pi\)
\(410\) −3.00000 −0.148159
\(411\) 9.33030 0.460230
\(412\) 2.95644 0.145653
\(413\) 0 0
\(414\) −0.330303 −0.0162335
\(415\) −14.3739 −0.705585
\(416\) 1.20871 0.0592620
\(417\) −21.8693 −1.07095
\(418\) 1.00000 0.0489116
\(419\) 12.3303 0.602375 0.301187 0.953565i \(-0.402617\pi\)
0.301187 + 0.953565i \(0.402617\pi\)
\(420\) −32.5390 −1.58774
\(421\) −11.2523 −0.548402 −0.274201 0.961672i \(-0.588413\pi\)
−0.274201 + 0.961672i \(0.588413\pi\)
\(422\) 18.7477 0.912625
\(423\) 1.91288 0.0930073
\(424\) −1.58258 −0.0768567
\(425\) −71.0780 −3.44779
\(426\) −20.3739 −0.987117
\(427\) −9.58258 −0.463733
\(428\) 18.0000 0.870063
\(429\) 2.16515 0.104534
\(430\) 27.9564 1.34818
\(431\) −21.4955 −1.03540 −0.517700 0.855562i \(-0.673212\pi\)
−0.517700 + 0.855562i \(0.673212\pi\)
\(432\) −5.00000 −0.240563
\(433\) −11.5826 −0.556623 −0.278312 0.960491i \(-0.589775\pi\)
−0.278312 + 0.960491i \(0.589775\pi\)
\(434\) 51.7042 2.48188
\(435\) 15.0000 0.719195
\(436\) −10.0000 −0.478913
\(437\) −1.58258 −0.0757049
\(438\) 17.1652 0.820183
\(439\) −31.1652 −1.48743 −0.743716 0.668496i \(-0.766938\pi\)
−0.743716 + 0.668496i \(0.766938\pi\)
\(440\) 3.79129 0.180743
\(441\) 3.33030 0.158586
\(442\) −9.16515 −0.435942
\(443\) −18.3303 −0.870899 −0.435449 0.900213i \(-0.643411\pi\)
−0.435449 + 0.900213i \(0.643411\pi\)
\(444\) 14.3303 0.680086
\(445\) −16.7477 −0.793918
\(446\) −7.16515 −0.339280
\(447\) 0 0
\(448\) −4.79129 −0.226367
\(449\) 16.7477 0.790374 0.395187 0.918601i \(-0.370680\pi\)
0.395187 + 0.918601i \(0.370680\pi\)
\(450\) 1.95644 0.0922274
\(451\) −0.791288 −0.0372603
\(452\) 15.1652 0.713309
\(453\) 3.58258 0.168324
\(454\) −7.58258 −0.355868
\(455\) −21.9564 −1.02933
\(456\) 1.79129 0.0838847
\(457\) −5.25227 −0.245691 −0.122845 0.992426i \(-0.539202\pi\)
−0.122845 + 0.992426i \(0.539202\pi\)
\(458\) −18.3739 −0.858554
\(459\) 37.9129 1.76962
\(460\) −6.00000 −0.279751
\(461\) −30.3303 −1.41262 −0.706312 0.707901i \(-0.749642\pi\)
−0.706312 + 0.707901i \(0.749642\pi\)
\(462\) −8.58258 −0.399298
\(463\) −40.6606 −1.88966 −0.944829 0.327563i \(-0.893773\pi\)
−0.944829 + 0.327563i \(0.893773\pi\)
\(464\) 2.20871 0.102537
\(465\) −73.2867 −3.39859
\(466\) 15.1652 0.702512
\(467\) 13.5826 0.628527 0.314263 0.949336i \(-0.398243\pi\)
0.314263 + 0.949336i \(0.398243\pi\)
\(468\) 0.252273 0.0116613
\(469\) 22.9564 1.06003
\(470\) 34.7477 1.60279
\(471\) −19.6261 −0.904325
\(472\) 0 0
\(473\) 7.37386 0.339051
\(474\) −10.0000 −0.459315
\(475\) 9.37386 0.430102
\(476\) 36.3303 1.66520
\(477\) −0.330303 −0.0151235
\(478\) −2.20871 −0.101024
\(479\) −12.7913 −0.584449 −0.292224 0.956350i \(-0.594395\pi\)
−0.292224 + 0.956350i \(0.594395\pi\)
\(480\) 6.79129 0.309978
\(481\) 9.66970 0.440900
\(482\) −0.208712 −0.00950658
\(483\) 13.5826 0.618029
\(484\) 1.00000 0.0454545
\(485\) 30.3303 1.37723
\(486\) −2.16515 −0.0982133
\(487\) 18.1216 0.821168 0.410584 0.911823i \(-0.365325\pi\)
0.410584 + 0.911823i \(0.365325\pi\)
\(488\) 2.00000 0.0905357
\(489\) −20.7477 −0.938245
\(490\) 60.4955 2.73291
\(491\) −29.8693 −1.34798 −0.673992 0.738739i \(-0.735422\pi\)
−0.673992 + 0.738739i \(0.735422\pi\)
\(492\) −1.41742 −0.0639024
\(493\) −16.7477 −0.754280
\(494\) 1.20871 0.0543825
\(495\) 0.791288 0.0355657
\(496\) −10.7913 −0.484543
\(497\) 54.4955 2.44446
\(498\) −6.79129 −0.304325
\(499\) −1.16515 −0.0521593 −0.0260797 0.999660i \(-0.508302\pi\)
−0.0260797 + 0.999660i \(0.508302\pi\)
\(500\) 16.5826 0.741595
\(501\) 32.2432 1.44052
\(502\) 12.0000 0.535586
\(503\) 12.9564 0.577699 0.288850 0.957375i \(-0.406727\pi\)
0.288850 + 0.957375i \(0.406727\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −51.4955 −2.29152
\(506\) −1.58258 −0.0703541
\(507\) −20.6697 −0.917973
\(508\) −8.74773 −0.388118
\(509\) −8.83485 −0.391598 −0.195799 0.980644i \(-0.562730\pi\)
−0.195799 + 0.980644i \(0.562730\pi\)
\(510\) −51.4955 −2.28026
\(511\) −45.9129 −2.03107
\(512\) 1.00000 0.0441942
\(513\) −5.00000 −0.220755
\(514\) −7.58258 −0.334453
\(515\) 11.2087 0.493915
\(516\) 13.2087 0.581481
\(517\) 9.16515 0.403083
\(518\) −38.3303 −1.68414
\(519\) 23.2087 1.01875
\(520\) 4.58258 0.200959
\(521\) 33.1652 1.45299 0.726496 0.687171i \(-0.241148\pi\)
0.726496 + 0.687171i \(0.241148\pi\)
\(522\) 0.460985 0.0201768
\(523\) −2.74773 −0.120150 −0.0600749 0.998194i \(-0.519134\pi\)
−0.0600749 + 0.998194i \(0.519134\pi\)
\(524\) 3.62614 0.158409
\(525\) −80.4519 −3.51121
\(526\) 12.9564 0.564928
\(527\) 81.8258 3.56439
\(528\) 1.79129 0.0779558
\(529\) −20.4955 −0.891107
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −4.79129 −0.207729
\(533\) −0.956439 −0.0414280
\(534\) −7.91288 −0.342424
\(535\) 68.2432 2.95041
\(536\) −4.79129 −0.206952
\(537\) 7.08712 0.305832
\(538\) −16.7477 −0.722046
\(539\) 15.9564 0.687292
\(540\) −18.9564 −0.815755
\(541\) −35.9129 −1.54402 −0.772008 0.635613i \(-0.780747\pi\)
−0.772008 + 0.635613i \(0.780747\pi\)
\(542\) −6.37386 −0.273781
\(543\) 33.5826 1.44117
\(544\) −7.58258 −0.325100
\(545\) −37.9129 −1.62401
\(546\) −10.3739 −0.443960
\(547\) −43.1652 −1.84561 −0.922804 0.385269i \(-0.874109\pi\)
−0.922804 + 0.385269i \(0.874109\pi\)
\(548\) 5.20871 0.222505
\(549\) 0.417424 0.0178152
\(550\) 9.37386 0.399703
\(551\) 2.20871 0.0940943
\(552\) −2.83485 −0.120659
\(553\) 26.7477 1.13743
\(554\) −22.0000 −0.934690
\(555\) 54.3303 2.30619
\(556\) −12.2087 −0.517765
\(557\) −20.8348 −0.882801 −0.441400 0.897310i \(-0.645518\pi\)
−0.441400 + 0.897310i \(0.645518\pi\)
\(558\) −2.25227 −0.0953463
\(559\) 8.91288 0.376975
\(560\) −18.1652 −0.767618
\(561\) −13.5826 −0.573457
\(562\) 22.1216 0.933143
\(563\) −30.6606 −1.29219 −0.646095 0.763257i \(-0.723599\pi\)
−0.646095 + 0.763257i \(0.723599\pi\)
\(564\) 16.4174 0.691298
\(565\) 57.4955 2.41885
\(566\) 9.12159 0.383409
\(567\) 45.9129 1.92816
\(568\) −11.3739 −0.477237
\(569\) −3.95644 −0.165863 −0.0829313 0.996555i \(-0.526428\pi\)
−0.0829313 + 0.996555i \(0.526428\pi\)
\(570\) 6.79129 0.284456
\(571\) 14.7913 0.618996 0.309498 0.950900i \(-0.399839\pi\)
0.309498 + 0.950900i \(0.399839\pi\)
\(572\) 1.20871 0.0505388
\(573\) −32.2432 −1.34698
\(574\) 3.79129 0.158245
\(575\) −14.8348 −0.618656
\(576\) 0.208712 0.00869634
\(577\) −33.8693 −1.41000 −0.704999 0.709208i \(-0.749053\pi\)
−0.704999 + 0.709208i \(0.749053\pi\)
\(578\) 40.4955 1.68439
\(579\) 13.2087 0.548935
\(580\) 8.37386 0.347706
\(581\) 18.1652 0.753617
\(582\) 14.3303 0.594010
\(583\) −1.58258 −0.0655436
\(584\) 9.58258 0.396530
\(585\) 0.956439 0.0395439
\(586\) 6.79129 0.280546
\(587\) −3.16515 −0.130640 −0.0653199 0.997864i \(-0.520807\pi\)
−0.0653199 + 0.997864i \(0.520807\pi\)
\(588\) 28.5826 1.17873
\(589\) −10.7913 −0.444647
\(590\) 0 0
\(591\) 16.4174 0.675323
\(592\) 8.00000 0.328798
\(593\) 2.83485 0.116413 0.0582066 0.998305i \(-0.481462\pi\)
0.0582066 + 0.998305i \(0.481462\pi\)
\(594\) −5.00000 −0.205152
\(595\) 137.739 5.64673
\(596\) 0 0
\(597\) 12.0871 0.494693
\(598\) −1.91288 −0.0782234
\(599\) −18.9564 −0.774539 −0.387270 0.921967i \(-0.626582\pi\)
−0.387270 + 0.921967i \(0.626582\pi\)
\(600\) 16.7913 0.685501
\(601\) −15.2087 −0.620376 −0.310188 0.950675i \(-0.600392\pi\)
−0.310188 + 0.950675i \(0.600392\pi\)
\(602\) −35.3303 −1.43996
\(603\) −1.00000 −0.0407231
\(604\) 2.00000 0.0813788
\(605\) 3.79129 0.154138
\(606\) −24.3303 −0.988351
\(607\) 15.9129 0.645884 0.322942 0.946419i \(-0.395328\pi\)
0.322942 + 0.946419i \(0.395328\pi\)
\(608\) 1.00000 0.0405554
\(609\) −18.9564 −0.768154
\(610\) 7.58258 0.307010
\(611\) 11.0780 0.448169
\(612\) −1.58258 −0.0639718
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −5.25227 −0.211964
\(615\) −5.37386 −0.216695
\(616\) −4.79129 −0.193046
\(617\) 5.20871 0.209695 0.104847 0.994488i \(-0.466565\pi\)
0.104847 + 0.994488i \(0.466565\pi\)
\(618\) 5.29583 0.213030
\(619\) −1.16515 −0.0468314 −0.0234157 0.999726i \(-0.507454\pi\)
−0.0234157 + 0.999726i \(0.507454\pi\)
\(620\) −40.9129 −1.64310
\(621\) 7.91288 0.317533
\(622\) −25.9129 −1.03901
\(623\) 21.1652 0.847964
\(624\) 2.16515 0.0866754
\(625\) 16.0000 0.640000
\(626\) −9.37386 −0.374655
\(627\) 1.79129 0.0715371
\(628\) −10.9564 −0.437210
\(629\) −60.6606 −2.41870
\(630\) −3.79129 −0.151049
\(631\) −35.9129 −1.42967 −0.714835 0.699294i \(-0.753498\pi\)
−0.714835 + 0.699294i \(0.753498\pi\)
\(632\) −5.58258 −0.222063
\(633\) 33.5826 1.33479
\(634\) 1.25227 0.0497341
\(635\) −33.1652 −1.31612
\(636\) −2.83485 −0.112409
\(637\) 19.2867 0.764169
\(638\) 2.20871 0.0874438
\(639\) −2.37386 −0.0939086
\(640\) 3.79129 0.149864
\(641\) 16.4174 0.648449 0.324225 0.945980i \(-0.394897\pi\)
0.324225 + 0.945980i \(0.394897\pi\)
\(642\) 32.2432 1.27254
\(643\) 26.3303 1.03837 0.519183 0.854663i \(-0.326236\pi\)
0.519183 + 0.854663i \(0.326236\pi\)
\(644\) 7.58258 0.298795
\(645\) 50.0780 1.97182
\(646\) −7.58258 −0.298332
\(647\) 25.9129 1.01874 0.509370 0.860548i \(-0.329878\pi\)
0.509370 + 0.860548i \(0.329878\pi\)
\(648\) −9.58258 −0.376439
\(649\) 0 0
\(650\) 11.3303 0.444411
\(651\) 92.6170 3.62995
\(652\) −11.5826 −0.453609
\(653\) 10.2867 0.402551 0.201276 0.979535i \(-0.435491\pi\)
0.201276 + 0.979535i \(0.435491\pi\)
\(654\) −17.9129 −0.700449
\(655\) 13.7477 0.537168
\(656\) −0.791288 −0.0308946
\(657\) 2.00000 0.0780274
\(658\) −43.9129 −1.71190
\(659\) 7.91288 0.308242 0.154121 0.988052i \(-0.450745\pi\)
0.154121 + 0.988052i \(0.450745\pi\)
\(660\) 6.79129 0.264351
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 29.7913 1.15787
\(663\) −16.4174 −0.637600
\(664\) −3.79129 −0.147131
\(665\) −18.1652 −0.704414
\(666\) 1.66970 0.0646995
\(667\) −3.49545 −0.135345
\(668\) 18.0000 0.696441
\(669\) −12.8348 −0.496224
\(670\) −18.1652 −0.701781
\(671\) 2.00000 0.0772091
\(672\) −8.58258 −0.331080
\(673\) −12.0436 −0.464245 −0.232123 0.972687i \(-0.574567\pi\)
−0.232123 + 0.972687i \(0.574567\pi\)
\(674\) −13.6261 −0.524859
\(675\) −46.8693 −1.80400
\(676\) −11.5390 −0.443808
\(677\) 32.5390 1.25058 0.625288 0.780394i \(-0.284982\pi\)
0.625288 + 0.780394i \(0.284982\pi\)
\(678\) 27.1652 1.04327
\(679\) −38.3303 −1.47098
\(680\) −28.7477 −1.10243
\(681\) −13.5826 −0.520485
\(682\) −10.7913 −0.413220
\(683\) 48.6606 1.86195 0.930973 0.365088i \(-0.118961\pi\)
0.930973 + 0.365088i \(0.118961\pi\)
\(684\) 0.208712 0.00798031
\(685\) 19.7477 0.754522
\(686\) −42.9129 −1.63842
\(687\) −32.9129 −1.25570
\(688\) 7.37386 0.281126
\(689\) −1.91288 −0.0728749
\(690\) −10.7477 −0.409159
\(691\) −31.4955 −1.19814 −0.599072 0.800695i \(-0.704464\pi\)
−0.599072 + 0.800695i \(0.704464\pi\)
\(692\) 12.9564 0.492530
\(693\) −1.00000 −0.0379869
\(694\) −3.16515 −0.120148
\(695\) −46.2867 −1.75576
\(696\) 3.95644 0.149968
\(697\) 6.00000 0.227266
\(698\) 23.4955 0.889316
\(699\) 27.1652 1.02748
\(700\) −44.9129 −1.69755
\(701\) −5.66970 −0.214142 −0.107071 0.994251i \(-0.534147\pi\)
−0.107071 + 0.994251i \(0.534147\pi\)
\(702\) −6.04356 −0.228100
\(703\) 8.00000 0.301726
\(704\) 1.00000 0.0376889
\(705\) 62.2432 2.34421
\(706\) −14.8348 −0.558317
\(707\) 65.0780 2.44751
\(708\) 0 0
\(709\) 32.7913 1.23150 0.615751 0.787941i \(-0.288853\pi\)
0.615751 + 0.787941i \(0.288853\pi\)
\(710\) −43.1216 −1.61832
\(711\) −1.16515 −0.0436966
\(712\) −4.41742 −0.165550
\(713\) 17.0780 0.639577
\(714\) 65.0780 2.43548
\(715\) 4.58258 0.171379
\(716\) 3.95644 0.147859
\(717\) −3.95644 −0.147756
\(718\) 20.7042 0.772673
\(719\) −29.0780 −1.08443 −0.542214 0.840241i \(-0.682414\pi\)
−0.542214 + 0.840241i \(0.682414\pi\)
\(720\) 0.791288 0.0294896
\(721\) −14.1652 −0.527538
\(722\) 1.00000 0.0372161
\(723\) −0.373864 −0.0139041
\(724\) 18.7477 0.696754
\(725\) 20.7042 0.768933
\(726\) 1.79129 0.0664809
\(727\) −25.4955 −0.945574 −0.472787 0.881177i \(-0.656752\pi\)
−0.472787 + 0.881177i \(0.656752\pi\)
\(728\) −5.79129 −0.214639
\(729\) 24.8693 0.921086
\(730\) 36.3303 1.34464
\(731\) −55.9129 −2.06801
\(732\) 3.58258 0.132416
\(733\) −19.4955 −0.720081 −0.360041 0.932937i \(-0.617237\pi\)
−0.360041 + 0.932937i \(0.617237\pi\)
\(734\) 20.3303 0.750405
\(735\) 108.365 3.99709
\(736\) −1.58258 −0.0583345
\(737\) −4.79129 −0.176489
\(738\) −0.165151 −0.00607931
\(739\) 38.9564 1.43304 0.716518 0.697569i \(-0.245735\pi\)
0.716518 + 0.697569i \(0.245735\pi\)
\(740\) 30.3303 1.11496
\(741\) 2.16515 0.0795388
\(742\) 7.58258 0.278365
\(743\) 45.1652 1.65695 0.828474 0.560027i \(-0.189209\pi\)
0.828474 + 0.560027i \(0.189209\pi\)
\(744\) −19.3303 −0.708683
\(745\) 0 0
\(746\) −9.37386 −0.343202
\(747\) −0.791288 −0.0289517
\(748\) −7.58258 −0.277246
\(749\) −86.2432 −3.15126
\(750\) 29.7042 1.08464
\(751\) −10.3303 −0.376958 −0.188479 0.982077i \(-0.560356\pi\)
−0.188479 + 0.982077i \(0.560356\pi\)
\(752\) 9.16515 0.334219
\(753\) 21.4955 0.783338
\(754\) 2.66970 0.0972246
\(755\) 7.58258 0.275958
\(756\) 23.9564 0.871287
\(757\) −24.2087 −0.879881 −0.439940 0.898027i \(-0.645000\pi\)
−0.439940 + 0.898027i \(0.645000\pi\)
\(758\) 32.7913 1.19103
\(759\) −2.83485 −0.102898
\(760\) 3.79129 0.137524
\(761\) 49.9129 1.80934 0.904670 0.426112i \(-0.140117\pi\)
0.904670 + 0.426112i \(0.140117\pi\)
\(762\) −15.6697 −0.567653
\(763\) 47.9129 1.73456
\(764\) −18.0000 −0.651217
\(765\) −6.00000 −0.216930
\(766\) −9.95644 −0.359741
\(767\) 0 0
\(768\) 1.79129 0.0646375
\(769\) 7.66970 0.276576 0.138288 0.990392i \(-0.455840\pi\)
0.138288 + 0.990392i \(0.455840\pi\)
\(770\) −18.1652 −0.654627
\(771\) −13.5826 −0.489165
\(772\) 7.37386 0.265391
\(773\) −39.4955 −1.42055 −0.710276 0.703923i \(-0.751430\pi\)
−0.710276 + 0.703923i \(0.751430\pi\)
\(774\) 1.53901 0.0553187
\(775\) −101.156 −3.63363
\(776\) 8.00000 0.287183
\(777\) −68.6606 −2.46319
\(778\) 10.1216 0.362877
\(779\) −0.791288 −0.0283508
\(780\) 8.20871 0.293919
\(781\) −11.3739 −0.406989
\(782\) 12.0000 0.429119
\(783\) −11.0436 −0.394665
\(784\) 15.9564 0.569873
\(785\) −41.5390 −1.48259
\(786\) 6.49545 0.231685
\(787\) 3.58258 0.127705 0.0638525 0.997959i \(-0.479661\pi\)
0.0638525 + 0.997959i \(0.479661\pi\)
\(788\) 9.16515 0.326495
\(789\) 23.2087 0.826252
\(790\) −21.1652 −0.753022
\(791\) −72.6606 −2.58351
\(792\) 0.208712 0.00741626
\(793\) 2.41742 0.0858453
\(794\) 4.04356 0.143501
\(795\) −10.7477 −0.381183
\(796\) 6.74773 0.239167
\(797\) −32.2432 −1.14211 −0.571056 0.820911i \(-0.693466\pi\)
−0.571056 + 0.820911i \(0.693466\pi\)
\(798\) −8.58258 −0.303820
\(799\) −69.4955 −2.45857
\(800\) 9.37386 0.331416
\(801\) −0.921970 −0.0325762
\(802\) −13.5826 −0.479617
\(803\) 9.58258 0.338162
\(804\) −8.58258 −0.302684
\(805\) 28.7477 1.01322
\(806\) −13.0436 −0.459440
\(807\) −30.0000 −1.05605
\(808\) −13.5826 −0.477833
\(809\) −42.3303 −1.48825 −0.744127 0.668038i \(-0.767134\pi\)
−0.744127 + 0.668038i \(0.767134\pi\)
\(810\) −36.3303 −1.27652
\(811\) −31.4955 −1.10595 −0.552977 0.833196i \(-0.686508\pi\)
−0.552977 + 0.833196i \(0.686508\pi\)
\(812\) −10.5826 −0.371376
\(813\) −11.4174 −0.400427
\(814\) 8.00000 0.280400
\(815\) −43.9129 −1.53820
\(816\) −13.5826 −0.475485
\(817\) 7.37386 0.257979
\(818\) −3.37386 −0.117964
\(819\) −1.20871 −0.0422358
\(820\) −3.00000 −0.104765
\(821\) −47.0780 −1.64303 −0.821517 0.570184i \(-0.806872\pi\)
−0.821517 + 0.570184i \(0.806872\pi\)
\(822\) 9.33030 0.325432
\(823\) 17.4955 0.609853 0.304927 0.952376i \(-0.401368\pi\)
0.304927 + 0.952376i \(0.401368\pi\)
\(824\) 2.95644 0.102992
\(825\) 16.7913 0.584598
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −0.330303 −0.0114788
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −14.3739 −0.498924
\(831\) −39.4083 −1.36706
\(832\) 1.20871 0.0419046
\(833\) −120.991 −4.19209
\(834\) −21.8693 −0.757273
\(835\) 68.2432 2.36165
\(836\) 1.00000 0.0345857
\(837\) 53.9564 1.86501
\(838\) 12.3303 0.425943
\(839\) 18.9564 0.654449 0.327224 0.944947i \(-0.393887\pi\)
0.327224 + 0.944947i \(0.393887\pi\)
\(840\) −32.5390 −1.12270
\(841\) −24.1216 −0.831779
\(842\) −11.2523 −0.387779
\(843\) 39.6261 1.36480
\(844\) 18.7477 0.645323
\(845\) −43.7477 −1.50497
\(846\) 1.91288 0.0657661
\(847\) −4.79129 −0.164631
\(848\) −1.58258 −0.0543459
\(849\) 16.3394 0.560767
\(850\) −71.0780 −2.43796
\(851\) −12.6606 −0.434000
\(852\) −20.3739 −0.697997
\(853\) −7.16515 −0.245330 −0.122665 0.992448i \(-0.539144\pi\)
−0.122665 + 0.992448i \(0.539144\pi\)
\(854\) −9.58258 −0.327909
\(855\) 0.791288 0.0270615
\(856\) 18.0000 0.615227
\(857\) 34.2867 1.17121 0.585606 0.810596i \(-0.300856\pi\)
0.585606 + 0.810596i \(0.300856\pi\)
\(858\) 2.16515 0.0739170
\(859\) −13.4955 −0.460459 −0.230229 0.973136i \(-0.573948\pi\)
−0.230229 + 0.973136i \(0.573948\pi\)
\(860\) 27.9564 0.953307
\(861\) 6.79129 0.231446
\(862\) −21.4955 −0.732138
\(863\) 23.5390 0.801277 0.400639 0.916236i \(-0.368788\pi\)
0.400639 + 0.916236i \(0.368788\pi\)
\(864\) −5.00000 −0.170103
\(865\) 49.1216 1.67018
\(866\) −11.5826 −0.393592
\(867\) 72.5390 2.46355
\(868\) 51.7042 1.75495
\(869\) −5.58258 −0.189376
\(870\) 15.0000 0.508548
\(871\) −5.79129 −0.196230
\(872\) −10.0000 −0.338643
\(873\) 1.66970 0.0565107
\(874\) −1.58258 −0.0535314
\(875\) −79.4519 −2.68596
\(876\) 17.1652 0.579957
\(877\) −11.8784 −0.401105 −0.200553 0.979683i \(-0.564274\pi\)
−0.200553 + 0.979683i \(0.564274\pi\)
\(878\) −31.1652 −1.05177
\(879\) 12.1652 0.410320
\(880\) 3.79129 0.127804
\(881\) −13.1216 −0.442078 −0.221039 0.975265i \(-0.570945\pi\)
−0.221039 + 0.975265i \(0.570945\pi\)
\(882\) 3.33030 0.112137
\(883\) 21.9129 0.737427 0.368714 0.929543i \(-0.379798\pi\)
0.368714 + 0.929543i \(0.379798\pi\)
\(884\) −9.16515 −0.308257
\(885\) 0 0
\(886\) −18.3303 −0.615819
\(887\) 21.4955 0.721747 0.360873 0.932615i \(-0.382479\pi\)
0.360873 + 0.932615i \(0.382479\pi\)
\(888\) 14.3303 0.480893
\(889\) 41.9129 1.40571
\(890\) −16.7477 −0.561385
\(891\) −9.58258 −0.321028
\(892\) −7.16515 −0.239907
\(893\) 9.16515 0.306700
\(894\) 0 0
\(895\) 15.0000 0.501395
\(896\) −4.79129 −0.160066
\(897\) −3.42652 −0.114408
\(898\) 16.7477 0.558879
\(899\) −23.8348 −0.794937
\(900\) 1.95644 0.0652146
\(901\) 12.0000 0.399778
\(902\) −0.791288 −0.0263470
\(903\) −63.2867 −2.10605
\(904\) 15.1652 0.504385
\(905\) 71.0780 2.36271
\(906\) 3.58258 0.119023
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) −7.58258 −0.251637
\(909\) −2.83485 −0.0940260
\(910\) −21.9564 −0.727849
\(911\) −39.1652 −1.29760 −0.648800 0.760959i \(-0.724729\pi\)
−0.648800 + 0.760959i \(0.724729\pi\)
\(912\) 1.79129 0.0593155
\(913\) −3.79129 −0.125473
\(914\) −5.25227 −0.173730
\(915\) 13.5826 0.449026
\(916\) −18.3739 −0.607090
\(917\) −17.3739 −0.573736
\(918\) 37.9129 1.25131
\(919\) 51.2867 1.69179 0.845897 0.533347i \(-0.179066\pi\)
0.845897 + 0.533347i \(0.179066\pi\)
\(920\) −6.00000 −0.197814
\(921\) −9.40833 −0.310015
\(922\) −30.3303 −0.998875
\(923\) −13.7477 −0.452512
\(924\) −8.58258 −0.282346
\(925\) 74.9909 2.46569
\(926\) −40.6606 −1.33619
\(927\) 0.617045 0.0202664
\(928\) 2.20871 0.0725045
\(929\) −17.2087 −0.564600 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(930\) −73.2867 −2.40317
\(931\) 15.9564 0.522951
\(932\) 15.1652 0.496751
\(933\) −46.4174 −1.51964
\(934\) 13.5826 0.444435
\(935\) −28.7477 −0.940151
\(936\) 0.252273 0.00824580
\(937\) −37.8258 −1.23571 −0.617857 0.786291i \(-0.711999\pi\)
−0.617857 + 0.786291i \(0.711999\pi\)
\(938\) 22.9564 0.749554
\(939\) −16.7913 −0.547963
\(940\) 34.7477 1.13335
\(941\) −26.8348 −0.874791 −0.437396 0.899269i \(-0.644099\pi\)
−0.437396 + 0.899269i \(0.644099\pi\)
\(942\) −19.6261 −0.639454
\(943\) 1.25227 0.0407796
\(944\) 0 0
\(945\) 90.8258 2.95456
\(946\) 7.37386 0.239745
\(947\) 17.0780 0.554961 0.277481 0.960731i \(-0.410501\pi\)
0.277481 + 0.960731i \(0.410501\pi\)
\(948\) −10.0000 −0.324785
\(949\) 11.5826 0.375986
\(950\) 9.37386 0.304128
\(951\) 2.24318 0.0727401
\(952\) 36.3303 1.17747
\(953\) −14.8348 −0.480548 −0.240274 0.970705i \(-0.577237\pi\)
−0.240274 + 0.970705i \(0.577237\pi\)
\(954\) −0.330303 −0.0106939
\(955\) −68.2432 −2.20830
\(956\) −2.20871 −0.0714349
\(957\) 3.95644 0.127894
\(958\) −12.7913 −0.413268
\(959\) −24.9564 −0.805885
\(960\) 6.79129 0.219188
\(961\) 85.4519 2.75651
\(962\) 9.66970 0.311764
\(963\) 3.75682 0.121062
\(964\) −0.208712 −0.00672217
\(965\) 27.9564 0.899950
\(966\) 13.5826 0.437012
\(967\) 41.4955 1.33440 0.667202 0.744877i \(-0.267492\pi\)
0.667202 + 0.744877i \(0.267492\pi\)
\(968\) 1.00000 0.0321412
\(969\) −13.5826 −0.436335
\(970\) 30.3303 0.973847
\(971\) −19.2867 −0.618941 −0.309471 0.950909i \(-0.600152\pi\)
−0.309471 + 0.950909i \(0.600152\pi\)
\(972\) −2.16515 −0.0694473
\(973\) 58.4955 1.87528
\(974\) 18.1216 0.580653
\(975\) 20.2958 0.649987
\(976\) 2.00000 0.0640184
\(977\) −20.8348 −0.666566 −0.333283 0.942827i \(-0.608156\pi\)
−0.333283 + 0.942827i \(0.608156\pi\)
\(978\) −20.7477 −0.663439
\(979\) −4.41742 −0.141181
\(980\) 60.4955 1.93246
\(981\) −2.08712 −0.0666367
\(982\) −29.8693 −0.953168
\(983\) 6.79129 0.216608 0.108304 0.994118i \(-0.465458\pi\)
0.108304 + 0.994118i \(0.465458\pi\)
\(984\) −1.41742 −0.0451858
\(985\) 34.7477 1.10715
\(986\) −16.7477 −0.533356
\(987\) −78.6606 −2.50379
\(988\) 1.20871 0.0384543
\(989\) −11.6697 −0.371075
\(990\) 0.791288 0.0251488
\(991\) 5.95644 0.189213 0.0946063 0.995515i \(-0.469841\pi\)
0.0946063 + 0.995515i \(0.469841\pi\)
\(992\) −10.7913 −0.342624
\(993\) 53.3648 1.69348
\(994\) 54.4955 1.72849
\(995\) 25.5826 0.811022
\(996\) −6.79129 −0.215190
\(997\) −59.9129 −1.89746 −0.948730 0.316088i \(-0.897631\pi\)
−0.948730 + 0.316088i \(0.897631\pi\)
\(998\) −1.16515 −0.0368822
\(999\) −40.0000 −1.26554
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.f.1.2 2
3.2 odd 2 3762.2.a.s.1.1 2
4.3 odd 2 3344.2.a.l.1.1 2
11.10 odd 2 4598.2.a.y.1.2 2
19.18 odd 2 7942.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.f.1.2 2 1.1 even 1 trivial
3344.2.a.l.1.1 2 4.3 odd 2
3762.2.a.s.1.1 2 3.2 odd 2
4598.2.a.y.1.2 2 11.10 odd 2
7942.2.a.w.1.1 2 19.18 odd 2