Properties

Label 231.2.a.b.1.1
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
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] = [231,2,Mod(1,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.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: \(1.84454428669\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} +3.00000 q^{5} +2.79129 q^{6} +1.00000 q^{7} -10.5826 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} +3.00000 q^{5} +2.79129 q^{6} +1.00000 q^{7} -10.5826 q^{8} +1.00000 q^{9} -8.37386 q^{10} -1.00000 q^{11} -5.79129 q^{12} +1.00000 q^{13} -2.79129 q^{14} -3.00000 q^{15} +17.9564 q^{16} -1.58258 q^{17} -2.79129 q^{18} +2.58258 q^{19} +17.3739 q^{20} -1.00000 q^{21} +2.79129 q^{22} +3.58258 q^{23} +10.5826 q^{24} +4.00000 q^{25} -2.79129 q^{26} -1.00000 q^{27} +5.79129 q^{28} +10.1652 q^{29} +8.37386 q^{30} -5.58258 q^{31} -28.9564 q^{32} +1.00000 q^{33} +4.41742 q^{34} +3.00000 q^{35} +5.79129 q^{36} +1.00000 q^{37} -7.20871 q^{38} -1.00000 q^{39} -31.7477 q^{40} +7.16515 q^{41} +2.79129 q^{42} -7.58258 q^{43} -5.79129 q^{44} +3.00000 q^{45} -10.0000 q^{46} +10.5826 q^{47} -17.9564 q^{48} +1.00000 q^{49} -11.1652 q^{50} +1.58258 q^{51} +5.79129 q^{52} -0.417424 q^{53} +2.79129 q^{54} -3.00000 q^{55} -10.5826 q^{56} -2.58258 q^{57} -28.3739 q^{58} -4.58258 q^{59} -17.3739 q^{60} +10.0000 q^{61} +15.5826 q^{62} +1.00000 q^{63} +44.9129 q^{64} +3.00000 q^{65} -2.79129 q^{66} -0.582576 q^{67} -9.16515 q^{68} -3.58258 q^{69} -8.37386 q^{70} -7.16515 q^{71} -10.5826 q^{72} +7.00000 q^{73} -2.79129 q^{74} -4.00000 q^{75} +14.9564 q^{76} -1.00000 q^{77} +2.79129 q^{78} -11.1652 q^{79} +53.8693 q^{80} +1.00000 q^{81} -20.0000 q^{82} -2.41742 q^{83} -5.79129 q^{84} -4.74773 q^{85} +21.1652 q^{86} -10.1652 q^{87} +10.5826 q^{88} -9.16515 q^{89} -8.37386 q^{90} +1.00000 q^{91} +20.7477 q^{92} +5.58258 q^{93} -29.5390 q^{94} +7.74773 q^{95} +28.9564 q^{96} -11.5826 q^{97} -2.79129 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} - 3 q^{10} - 2 q^{11} - 7 q^{12} + 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + 21 q^{20} - 2 q^{21} + q^{22} - 2 q^{23} + 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} + 7 q^{28} + 2 q^{29} + 3 q^{30} - 2 q^{31} - 35 q^{32} + 2 q^{33} + 18 q^{34} + 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} - 2 q^{39} - 36 q^{40} - 4 q^{41} + q^{42} - 6 q^{43} - 7 q^{44} + 6 q^{45} - 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} - 4 q^{50} - 6 q^{51} + 7 q^{52} - 10 q^{53} + q^{54} - 6 q^{55} - 12 q^{56} + 4 q^{57} - 43 q^{58} - 21 q^{60} + 20 q^{61} + 22 q^{62} + 2 q^{63} + 44 q^{64} + 6 q^{65} - q^{66} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} - 12 q^{72} + 14 q^{73} - q^{74} - 8 q^{75} + 7 q^{76} - 2 q^{77} + q^{78} - 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} - 14 q^{83} - 7 q^{84} + 18 q^{85} + 24 q^{86} - 2 q^{87} + 12 q^{88} - 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} - 27 q^{94} - 12 q^{95} + 35 q^{96} - 14 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.79129 −1.97374 −0.986869 0.161521i \(-0.948360\pi\)
−0.986869 + 0.161521i \(0.948360\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.79129 2.89564
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 2.79129 1.13954
\(7\) 1.00000 0.377964
\(8\) −10.5826 −3.74151
\(9\) 1.00000 0.333333
\(10\) −8.37386 −2.64805
\(11\) −1.00000 −0.301511
\(12\) −5.79129 −1.67180
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −2.79129 −0.746003
\(15\) −3.00000 −0.774597
\(16\) 17.9564 4.48911
\(17\) −1.58258 −0.383831 −0.191915 0.981411i \(-0.561470\pi\)
−0.191915 + 0.981411i \(0.561470\pi\)
\(18\) −2.79129 −0.657913
\(19\) 2.58258 0.592483 0.296242 0.955113i \(-0.404267\pi\)
0.296242 + 0.955113i \(0.404267\pi\)
\(20\) 17.3739 3.88491
\(21\) −1.00000 −0.218218
\(22\) 2.79129 0.595105
\(23\) 3.58258 0.747019 0.373509 0.927626i \(-0.378154\pi\)
0.373509 + 0.927626i \(0.378154\pi\)
\(24\) 10.5826 2.16016
\(25\) 4.00000 0.800000
\(26\) −2.79129 −0.547417
\(27\) −1.00000 −0.192450
\(28\) 5.79129 1.09445
\(29\) 10.1652 1.88762 0.943811 0.330487i \(-0.107213\pi\)
0.943811 + 0.330487i \(0.107213\pi\)
\(30\) 8.37386 1.52885
\(31\) −5.58258 −1.00266 −0.501330 0.865256i \(-0.667156\pi\)
−0.501330 + 0.865256i \(0.667156\pi\)
\(32\) −28.9564 −5.11882
\(33\) 1.00000 0.174078
\(34\) 4.41742 0.757582
\(35\) 3.00000 0.507093
\(36\) 5.79129 0.965215
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) −7.20871 −1.16941
\(39\) −1.00000 −0.160128
\(40\) −31.7477 −5.01976
\(41\) 7.16515 1.11901 0.559504 0.828827i \(-0.310992\pi\)
0.559504 + 0.828827i \(0.310992\pi\)
\(42\) 2.79129 0.430705
\(43\) −7.58258 −1.15633 −0.578166 0.815919i \(-0.696231\pi\)
−0.578166 + 0.815919i \(0.696231\pi\)
\(44\) −5.79129 −0.873069
\(45\) 3.00000 0.447214
\(46\) −10.0000 −1.47442
\(47\) 10.5826 1.54363 0.771814 0.635849i \(-0.219350\pi\)
0.771814 + 0.635849i \(0.219350\pi\)
\(48\) −17.9564 −2.59179
\(49\) 1.00000 0.142857
\(50\) −11.1652 −1.57899
\(51\) 1.58258 0.221605
\(52\) 5.79129 0.803107
\(53\) −0.417424 −0.0573376 −0.0286688 0.999589i \(-0.509127\pi\)
−0.0286688 + 0.999589i \(0.509127\pi\)
\(54\) 2.79129 0.379846
\(55\) −3.00000 −0.404520
\(56\) −10.5826 −1.41416
\(57\) −2.58258 −0.342071
\(58\) −28.3739 −3.72567
\(59\) −4.58258 −0.596601 −0.298300 0.954472i \(-0.596420\pi\)
−0.298300 + 0.954472i \(0.596420\pi\)
\(60\) −17.3739 −2.24296
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 15.5826 1.97899
\(63\) 1.00000 0.125988
\(64\) 44.9129 5.61411
\(65\) 3.00000 0.372104
\(66\) −2.79129 −0.343584
\(67\) −0.582576 −0.0711729 −0.0355865 0.999367i \(-0.511330\pi\)
−0.0355865 + 0.999367i \(0.511330\pi\)
\(68\) −9.16515 −1.11144
\(69\) −3.58258 −0.431291
\(70\) −8.37386 −1.00087
\(71\) −7.16515 −0.850347 −0.425174 0.905112i \(-0.639787\pi\)
−0.425174 + 0.905112i \(0.639787\pi\)
\(72\) −10.5826 −1.24717
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) −2.79129 −0.324481
\(75\) −4.00000 −0.461880
\(76\) 14.9564 1.71562
\(77\) −1.00000 −0.113961
\(78\) 2.79129 0.316051
\(79\) −11.1652 −1.25618 −0.628089 0.778142i \(-0.716163\pi\)
−0.628089 + 0.778142i \(0.716163\pi\)
\(80\) 53.8693 6.02277
\(81\) 1.00000 0.111111
\(82\) −20.0000 −2.20863
\(83\) −2.41742 −0.265347 −0.132673 0.991160i \(-0.542356\pi\)
−0.132673 + 0.991160i \(0.542356\pi\)
\(84\) −5.79129 −0.631881
\(85\) −4.74773 −0.514963
\(86\) 21.1652 2.28230
\(87\) −10.1652 −1.08982
\(88\) 10.5826 1.12811
\(89\) −9.16515 −0.971504 −0.485752 0.874097i \(-0.661454\pi\)
−0.485752 + 0.874097i \(0.661454\pi\)
\(90\) −8.37386 −0.882683
\(91\) 1.00000 0.104828
\(92\) 20.7477 2.16310
\(93\) 5.58258 0.578886
\(94\) −29.5390 −3.04672
\(95\) 7.74773 0.794900
\(96\) 28.9564 2.95535
\(97\) −11.5826 −1.17603 −0.588016 0.808849i \(-0.700091\pi\)
−0.588016 + 0.808849i \(0.700091\pi\)
\(98\) −2.79129 −0.281963
\(99\) −1.00000 −0.100504
\(100\) 23.1652 2.31652
\(101\) 2.41742 0.240543 0.120271 0.992741i \(-0.461624\pi\)
0.120271 + 0.992741i \(0.461624\pi\)
\(102\) −4.41742 −0.437390
\(103\) 17.1652 1.69133 0.845666 0.533712i \(-0.179203\pi\)
0.845666 + 0.533712i \(0.179203\pi\)
\(104\) −10.5826 −1.03771
\(105\) −3.00000 −0.292770
\(106\) 1.16515 0.113170
\(107\) −3.41742 −0.330375 −0.165187 0.986262i \(-0.552823\pi\)
−0.165187 + 0.986262i \(0.552823\pi\)
\(108\) −5.79129 −0.557267
\(109\) −5.58258 −0.534714 −0.267357 0.963598i \(-0.586150\pi\)
−0.267357 + 0.963598i \(0.586150\pi\)
\(110\) 8.37386 0.798417
\(111\) −1.00000 −0.0949158
\(112\) 17.9564 1.69672
\(113\) −9.16515 −0.862185 −0.431092 0.902308i \(-0.641872\pi\)
−0.431092 + 0.902308i \(0.641872\pi\)
\(114\) 7.20871 0.675158
\(115\) 10.7477 1.00223
\(116\) 58.8693 5.46588
\(117\) 1.00000 0.0924500
\(118\) 12.7913 1.17753
\(119\) −1.58258 −0.145074
\(120\) 31.7477 2.89816
\(121\) 1.00000 0.0909091
\(122\) −27.9129 −2.52711
\(123\) −7.16515 −0.646060
\(124\) −32.3303 −2.90335
\(125\) −3.00000 −0.268328
\(126\) −2.79129 −0.248668
\(127\) −2.41742 −0.214512 −0.107256 0.994231i \(-0.534206\pi\)
−0.107256 + 0.994231i \(0.534206\pi\)
\(128\) −67.4519 −5.96196
\(129\) 7.58258 0.667609
\(130\) −8.37386 −0.734436
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 5.79129 0.504067
\(133\) 2.58258 0.223938
\(134\) 1.62614 0.140477
\(135\) −3.00000 −0.258199
\(136\) 16.7477 1.43611
\(137\) −2.41742 −0.206534 −0.103267 0.994654i \(-0.532930\pi\)
−0.103267 + 0.994654i \(0.532930\pi\)
\(138\) 10.0000 0.851257
\(139\) −7.16515 −0.607740 −0.303870 0.952713i \(-0.598279\pi\)
−0.303870 + 0.952713i \(0.598279\pi\)
\(140\) 17.3739 1.46836
\(141\) −10.5826 −0.891214
\(142\) 20.0000 1.67836
\(143\) −1.00000 −0.0836242
\(144\) 17.9564 1.49637
\(145\) 30.4955 2.53251
\(146\) −19.5390 −1.61706
\(147\) −1.00000 −0.0824786
\(148\) 5.79129 0.476041
\(149\) −12.1652 −0.996608 −0.498304 0.867002i \(-0.666044\pi\)
−0.498304 + 0.867002i \(0.666044\pi\)
\(150\) 11.1652 0.911631
\(151\) −5.58258 −0.454304 −0.227152 0.973859i \(-0.572941\pi\)
−0.227152 + 0.973859i \(0.572941\pi\)
\(152\) −27.3303 −2.21678
\(153\) −1.58258 −0.127944
\(154\) 2.79129 0.224928
\(155\) −16.7477 −1.34521
\(156\) −5.79129 −0.463674
\(157\) 0.834849 0.0666282 0.0333141 0.999445i \(-0.489394\pi\)
0.0333141 + 0.999445i \(0.489394\pi\)
\(158\) 31.1652 2.47937
\(159\) 0.417424 0.0331039
\(160\) −86.8693 −6.86762
\(161\) 3.58258 0.282347
\(162\) −2.79129 −0.219304
\(163\) −0.582576 −0.0456309 −0.0228154 0.999740i \(-0.507263\pi\)
−0.0228154 + 0.999740i \(0.507263\pi\)
\(164\) 41.4955 3.24025
\(165\) 3.00000 0.233550
\(166\) 6.74773 0.523725
\(167\) 22.7477 1.76027 0.880136 0.474722i \(-0.157451\pi\)
0.880136 + 0.474722i \(0.157451\pi\)
\(168\) 10.5826 0.816463
\(169\) −12.0000 −0.923077
\(170\) 13.2523 1.01640
\(171\) 2.58258 0.197494
\(172\) −43.9129 −3.34833
\(173\) 11.1652 0.848871 0.424435 0.905458i \(-0.360473\pi\)
0.424435 + 0.905458i \(0.360473\pi\)
\(174\) 28.3739 2.15102
\(175\) 4.00000 0.302372
\(176\) −17.9564 −1.35352
\(177\) 4.58258 0.344447
\(178\) 25.5826 1.91750
\(179\) −22.3303 −1.66905 −0.834523 0.550974i \(-0.814256\pi\)
−0.834523 + 0.550974i \(0.814256\pi\)
\(180\) 17.3739 1.29497
\(181\) 3.58258 0.266291 0.133145 0.991097i \(-0.457492\pi\)
0.133145 + 0.991097i \(0.457492\pi\)
\(182\) −2.79129 −0.206904
\(183\) −10.0000 −0.739221
\(184\) −37.9129 −2.79497
\(185\) 3.00000 0.220564
\(186\) −15.5826 −1.14257
\(187\) 1.58258 0.115729
\(188\) 61.2867 4.46980
\(189\) −1.00000 −0.0727393
\(190\) −21.6261 −1.56892
\(191\) 2.41742 0.174919 0.0874593 0.996168i \(-0.472125\pi\)
0.0874593 + 0.996168i \(0.472125\pi\)
\(192\) −44.9129 −3.24131
\(193\) −11.5826 −0.833732 −0.416866 0.908968i \(-0.636872\pi\)
−0.416866 + 0.908968i \(0.636872\pi\)
\(194\) 32.3303 2.32118
\(195\) −3.00000 −0.214834
\(196\) 5.79129 0.413663
\(197\) −13.1652 −0.937978 −0.468989 0.883204i \(-0.655382\pi\)
−0.468989 + 0.883204i \(0.655382\pi\)
\(198\) 2.79129 0.198368
\(199\) −0.417424 −0.0295904 −0.0147952 0.999891i \(-0.504710\pi\)
−0.0147952 + 0.999891i \(0.504710\pi\)
\(200\) −42.3303 −2.99320
\(201\) 0.582576 0.0410917
\(202\) −6.74773 −0.474768
\(203\) 10.1652 0.713454
\(204\) 9.16515 0.641689
\(205\) 21.4955 1.50131
\(206\) −47.9129 −3.33825
\(207\) 3.58258 0.249006
\(208\) 17.9564 1.24506
\(209\) −2.58258 −0.178640
\(210\) 8.37386 0.577851
\(211\) 5.16515 0.355584 0.177792 0.984068i \(-0.443105\pi\)
0.177792 + 0.984068i \(0.443105\pi\)
\(212\) −2.41742 −0.166029
\(213\) 7.16515 0.490948
\(214\) 9.53901 0.652074
\(215\) −22.7477 −1.55138
\(216\) 10.5826 0.720053
\(217\) −5.58258 −0.378970
\(218\) 15.5826 1.05539
\(219\) −7.00000 −0.473016
\(220\) −17.3739 −1.17135
\(221\) −1.58258 −0.106456
\(222\) 2.79129 0.187339
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −28.9564 −1.93473
\(225\) 4.00000 0.266667
\(226\) 25.5826 1.70173
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) −14.9564 −0.990514
\(229\) −26.7477 −1.76754 −0.883770 0.467922i \(-0.845003\pi\)
−0.883770 + 0.467922i \(0.845003\pi\)
\(230\) −30.0000 −1.97814
\(231\) 1.00000 0.0657952
\(232\) −107.573 −7.06255
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −2.79129 −0.182472
\(235\) 31.7477 2.07099
\(236\) −26.5390 −1.72754
\(237\) 11.1652 0.725255
\(238\) 4.41742 0.286339
\(239\) 7.41742 0.479793 0.239897 0.970798i \(-0.422886\pi\)
0.239897 + 0.970798i \(0.422886\pi\)
\(240\) −53.8693 −3.47725
\(241\) 8.16515 0.525964 0.262982 0.964801i \(-0.415294\pi\)
0.262982 + 0.964801i \(0.415294\pi\)
\(242\) −2.79129 −0.179431
\(243\) −1.00000 −0.0641500
\(244\) 57.9129 3.70749
\(245\) 3.00000 0.191663
\(246\) 20.0000 1.27515
\(247\) 2.58258 0.164325
\(248\) 59.0780 3.75146
\(249\) 2.41742 0.153198
\(250\) 8.37386 0.529610
\(251\) 16.5826 1.04668 0.523341 0.852123i \(-0.324685\pi\)
0.523341 + 0.852123i \(0.324685\pi\)
\(252\) 5.79129 0.364817
\(253\) −3.58258 −0.225235
\(254\) 6.74773 0.423390
\(255\) 4.74773 0.297314
\(256\) 98.4519 6.15324
\(257\) 19.0000 1.18519 0.592594 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) −21.1652 −1.31768
\(259\) 1.00000 0.0621370
\(260\) 17.3739 1.07748
\(261\) 10.1652 0.629207
\(262\) 44.6606 2.75914
\(263\) 22.9129 1.41287 0.706434 0.707779i \(-0.250303\pi\)
0.706434 + 0.707779i \(0.250303\pi\)
\(264\) −10.5826 −0.651313
\(265\) −1.25227 −0.0769265
\(266\) −7.20871 −0.441995
\(267\) 9.16515 0.560898
\(268\) −3.37386 −0.206092
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 8.37386 0.509617
\(271\) 14.5826 0.885828 0.442914 0.896564i \(-0.353945\pi\)
0.442914 + 0.896564i \(0.353945\pi\)
\(272\) −28.4174 −1.72306
\(273\) −1.00000 −0.0605228
\(274\) 6.74773 0.407645
\(275\) −4.00000 −0.241209
\(276\) −20.7477 −1.24887
\(277\) −0.834849 −0.0501612 −0.0250806 0.999685i \(-0.507984\pi\)
−0.0250806 + 0.999685i \(0.507984\pi\)
\(278\) 20.0000 1.19952
\(279\) −5.58258 −0.334220
\(280\) −31.7477 −1.89729
\(281\) 9.33030 0.556599 0.278300 0.960494i \(-0.410229\pi\)
0.278300 + 0.960494i \(0.410229\pi\)
\(282\) 29.5390 1.75902
\(283\) 0.252273 0.0149961 0.00749803 0.999972i \(-0.497613\pi\)
0.00749803 + 0.999972i \(0.497613\pi\)
\(284\) −41.4955 −2.46230
\(285\) −7.74773 −0.458936
\(286\) 2.79129 0.165052
\(287\) 7.16515 0.422946
\(288\) −28.9564 −1.70627
\(289\) −14.4955 −0.852674
\(290\) −85.1216 −4.99851
\(291\) 11.5826 0.678983
\(292\) 40.5390 2.37237
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 2.79129 0.162791
\(295\) −13.7477 −0.800424
\(296\) −10.5826 −0.615100
\(297\) 1.00000 0.0580259
\(298\) 33.9564 1.96704
\(299\) 3.58258 0.207186
\(300\) −23.1652 −1.33744
\(301\) −7.58258 −0.437052
\(302\) 15.5826 0.896676
\(303\) −2.41742 −0.138877
\(304\) 46.3739 2.65972
\(305\) 30.0000 1.71780
\(306\) 4.41742 0.252527
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −5.79129 −0.329989
\(309\) −17.1652 −0.976491
\(310\) 46.7477 2.65509
\(311\) −22.3303 −1.26624 −0.633118 0.774056i \(-0.718225\pi\)
−0.633118 + 0.774056i \(0.718225\pi\)
\(312\) 10.5826 0.599120
\(313\) 10.4174 0.588828 0.294414 0.955678i \(-0.404876\pi\)
0.294414 + 0.955678i \(0.404876\pi\)
\(314\) −2.33030 −0.131507
\(315\) 3.00000 0.169031
\(316\) −64.6606 −3.63744
\(317\) −31.5826 −1.77385 −0.886927 0.461909i \(-0.847165\pi\)
−0.886927 + 0.461909i \(0.847165\pi\)
\(318\) −1.16515 −0.0653384
\(319\) −10.1652 −0.569139
\(320\) 134.739 7.53212
\(321\) 3.41742 0.190742
\(322\) −10.0000 −0.557278
\(323\) −4.08712 −0.227414
\(324\) 5.79129 0.321738
\(325\) 4.00000 0.221880
\(326\) 1.62614 0.0900634
\(327\) 5.58258 0.308717
\(328\) −75.8258 −4.18678
\(329\) 10.5826 0.583436
\(330\) −8.37386 −0.460966
\(331\) 15.1652 0.833552 0.416776 0.909009i \(-0.363160\pi\)
0.416776 + 0.909009i \(0.363160\pi\)
\(332\) −14.0000 −0.768350
\(333\) 1.00000 0.0547997
\(334\) −63.4955 −3.47432
\(335\) −1.74773 −0.0954885
\(336\) −17.9564 −0.979604
\(337\) 8.41742 0.458526 0.229263 0.973364i \(-0.426368\pi\)
0.229263 + 0.973364i \(0.426368\pi\)
\(338\) 33.4955 1.82191
\(339\) 9.16515 0.497783
\(340\) −27.4955 −1.49115
\(341\) 5.58258 0.302313
\(342\) −7.20871 −0.389803
\(343\) 1.00000 0.0539949
\(344\) 80.2432 4.32642
\(345\) −10.7477 −0.578638
\(346\) −31.1652 −1.67545
\(347\) −10.3303 −0.554560 −0.277280 0.960789i \(-0.589433\pi\)
−0.277280 + 0.960789i \(0.589433\pi\)
\(348\) −58.8693 −3.15573
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) −11.1652 −0.596802
\(351\) −1.00000 −0.0533761
\(352\) 28.9564 1.54338
\(353\) 5.83485 0.310558 0.155279 0.987871i \(-0.450372\pi\)
0.155279 + 0.987871i \(0.450372\pi\)
\(354\) −12.7913 −0.679849
\(355\) −21.4955 −1.14086
\(356\) −53.0780 −2.81313
\(357\) 1.58258 0.0837588
\(358\) 62.3303 3.29426
\(359\) −27.1652 −1.43372 −0.716861 0.697216i \(-0.754422\pi\)
−0.716861 + 0.697216i \(0.754422\pi\)
\(360\) −31.7477 −1.67325
\(361\) −12.3303 −0.648963
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 5.79129 0.303546
\(365\) 21.0000 1.09919
\(366\) 27.9129 1.45903
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 64.3303 3.35345
\(369\) 7.16515 0.373003
\(370\) −8.37386 −0.435336
\(371\) −0.417424 −0.0216716
\(372\) 32.3303 1.67625
\(373\) −7.25227 −0.375508 −0.187754 0.982216i \(-0.560121\pi\)
−0.187754 + 0.982216i \(0.560121\pi\)
\(374\) −4.41742 −0.228420
\(375\) 3.00000 0.154919
\(376\) −111.991 −5.77549
\(377\) 10.1652 0.523532
\(378\) 2.79129 0.143568
\(379\) −3.41742 −0.175541 −0.0877706 0.996141i \(-0.527974\pi\)
−0.0877706 + 0.996141i \(0.527974\pi\)
\(380\) 44.8693 2.30175
\(381\) 2.41742 0.123848
\(382\) −6.74773 −0.345244
\(383\) −26.3303 −1.34542 −0.672708 0.739908i \(-0.734869\pi\)
−0.672708 + 0.739908i \(0.734869\pi\)
\(384\) 67.4519 3.44214
\(385\) −3.00000 −0.152894
\(386\) 32.3303 1.64557
\(387\) −7.58258 −0.385444
\(388\) −67.0780 −3.40537
\(389\) 10.3303 0.523767 0.261884 0.965099i \(-0.415656\pi\)
0.261884 + 0.965099i \(0.415656\pi\)
\(390\) 8.37386 0.424027
\(391\) −5.66970 −0.286729
\(392\) −10.5826 −0.534501
\(393\) 16.0000 0.807093
\(394\) 36.7477 1.85132
\(395\) −33.4955 −1.68534
\(396\) −5.79129 −0.291023
\(397\) −22.4174 −1.12510 −0.562549 0.826764i \(-0.690179\pi\)
−0.562549 + 0.826764i \(0.690179\pi\)
\(398\) 1.16515 0.0584038
\(399\) −2.58258 −0.129290
\(400\) 71.8258 3.59129
\(401\) −13.9129 −0.694776 −0.347388 0.937721i \(-0.612931\pi\)
−0.347388 + 0.937721i \(0.612931\pi\)
\(402\) −1.62614 −0.0811043
\(403\) −5.58258 −0.278088
\(404\) 14.0000 0.696526
\(405\) 3.00000 0.149071
\(406\) −28.3739 −1.40817
\(407\) −1.00000 −0.0495682
\(408\) −16.7477 −0.829136
\(409\) −28.3303 −1.40084 −0.700422 0.713729i \(-0.747005\pi\)
−0.700422 + 0.713729i \(0.747005\pi\)
\(410\) −60.0000 −2.96319
\(411\) 2.41742 0.119243
\(412\) 99.4083 4.89750
\(413\) −4.58258 −0.225494
\(414\) −10.0000 −0.491473
\(415\) −7.25227 −0.356000
\(416\) −28.9564 −1.41971
\(417\) 7.16515 0.350879
\(418\) 7.20871 0.352590
\(419\) −6.58258 −0.321580 −0.160790 0.986989i \(-0.551404\pi\)
−0.160790 + 0.986989i \(0.551404\pi\)
\(420\) −17.3739 −0.847758
\(421\) 39.6606 1.93294 0.966470 0.256780i \(-0.0826617\pi\)
0.966470 + 0.256780i \(0.0826617\pi\)
\(422\) −14.4174 −0.701829
\(423\) 10.5826 0.514542
\(424\) 4.41742 0.214529
\(425\) −6.33030 −0.307065
\(426\) −20.0000 −0.969003
\(427\) 10.0000 0.483934
\(428\) −19.7913 −0.956648
\(429\) 1.00000 0.0482805
\(430\) 63.4955 3.06202
\(431\) −9.74773 −0.469531 −0.234766 0.972052i \(-0.575432\pi\)
−0.234766 + 0.972052i \(0.575432\pi\)
\(432\) −17.9564 −0.863930
\(433\) 7.16515 0.344335 0.172168 0.985068i \(-0.444923\pi\)
0.172168 + 0.985068i \(0.444923\pi\)
\(434\) 15.5826 0.747988
\(435\) −30.4955 −1.46215
\(436\) −32.3303 −1.54834
\(437\) 9.25227 0.442596
\(438\) 19.5390 0.933610
\(439\) 26.5826 1.26872 0.634359 0.773039i \(-0.281264\pi\)
0.634359 + 0.773039i \(0.281264\pi\)
\(440\) 31.7477 1.51351
\(441\) 1.00000 0.0476190
\(442\) 4.41742 0.210115
\(443\) −4.83485 −0.229711 −0.114855 0.993382i \(-0.536640\pi\)
−0.114855 + 0.993382i \(0.536640\pi\)
\(444\) −5.79129 −0.274842
\(445\) −27.4955 −1.30341
\(446\) −16.7477 −0.793028
\(447\) 12.1652 0.575392
\(448\) 44.9129 2.12193
\(449\) 18.3303 0.865060 0.432530 0.901619i \(-0.357621\pi\)
0.432530 + 0.901619i \(0.357621\pi\)
\(450\) −11.1652 −0.526330
\(451\) −7.16515 −0.337394
\(452\) −53.0780 −2.49658
\(453\) 5.58258 0.262292
\(454\) 61.4083 2.88204
\(455\) 3.00000 0.140642
\(456\) 27.3303 1.27986
\(457\) 25.9129 1.21215 0.606077 0.795406i \(-0.292742\pi\)
0.606077 + 0.795406i \(0.292742\pi\)
\(458\) 74.6606 3.48866
\(459\) 1.58258 0.0738683
\(460\) 62.2432 2.90210
\(461\) −18.3303 −0.853727 −0.426864 0.904316i \(-0.640382\pi\)
−0.426864 + 0.904316i \(0.640382\pi\)
\(462\) −2.79129 −0.129862
\(463\) −0.582576 −0.0270746 −0.0135373 0.999908i \(-0.504309\pi\)
−0.0135373 + 0.999908i \(0.504309\pi\)
\(464\) 182.530 8.47374
\(465\) 16.7477 0.776657
\(466\) 39.0780 1.81025
\(467\) −29.4174 −1.36128 −0.680638 0.732620i \(-0.738297\pi\)
−0.680638 + 0.732620i \(0.738297\pi\)
\(468\) 5.79129 0.267702
\(469\) −0.582576 −0.0269008
\(470\) −88.6170 −4.08760
\(471\) −0.834849 −0.0384678
\(472\) 48.4955 2.23218
\(473\) 7.58258 0.348647
\(474\) −31.1652 −1.43146
\(475\) 10.3303 0.473987
\(476\) −9.16515 −0.420084
\(477\) −0.417424 −0.0191125
\(478\) −20.7042 −0.946987
\(479\) −6.41742 −0.293220 −0.146610 0.989194i \(-0.546836\pi\)
−0.146610 + 0.989194i \(0.546836\pi\)
\(480\) 86.8693 3.96502
\(481\) 1.00000 0.0455961
\(482\) −22.7913 −1.03811
\(483\) −3.58258 −0.163013
\(484\) 5.79129 0.263240
\(485\) −34.7477 −1.57781
\(486\) 2.79129 0.126615
\(487\) −26.3303 −1.19314 −0.596570 0.802561i \(-0.703470\pi\)
−0.596570 + 0.802561i \(0.703470\pi\)
\(488\) −105.826 −4.79051
\(489\) 0.582576 0.0263450
\(490\) −8.37386 −0.378293
\(491\) 22.9129 1.03404 0.517022 0.855972i \(-0.327041\pi\)
0.517022 + 0.855972i \(0.327041\pi\)
\(492\) −41.4955 −1.87076
\(493\) −16.0871 −0.724528
\(494\) −7.20871 −0.324335
\(495\) −3.00000 −0.134840
\(496\) −100.243 −4.50105
\(497\) −7.16515 −0.321401
\(498\) −6.74773 −0.302373
\(499\) 14.2523 0.638019 0.319010 0.947751i \(-0.396650\pi\)
0.319010 + 0.947751i \(0.396650\pi\)
\(500\) −17.3739 −0.776983
\(501\) −22.7477 −1.01629
\(502\) −46.2867 −2.06588
\(503\) 26.7477 1.19262 0.596311 0.802753i \(-0.296632\pi\)
0.596311 + 0.802753i \(0.296632\pi\)
\(504\) −10.5826 −0.471385
\(505\) 7.25227 0.322722
\(506\) 10.0000 0.444554
\(507\) 12.0000 0.532939
\(508\) −14.0000 −0.621150
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −13.2523 −0.586821
\(511\) 7.00000 0.309662
\(512\) −139.904 −6.18293
\(513\) −2.58258 −0.114024
\(514\) −53.0345 −2.33925
\(515\) 51.4955 2.26916
\(516\) 43.9129 1.93316
\(517\) −10.5826 −0.465421
\(518\) −2.79129 −0.122642
\(519\) −11.1652 −0.490096
\(520\) −31.7477 −1.39223
\(521\) 34.1652 1.49680 0.748401 0.663246i \(-0.230822\pi\)
0.748401 + 0.663246i \(0.230822\pi\)
\(522\) −28.3739 −1.24189
\(523\) 24.5826 1.07492 0.537460 0.843289i \(-0.319384\pi\)
0.537460 + 0.843289i \(0.319384\pi\)
\(524\) −92.6606 −4.04790
\(525\) −4.00000 −0.174574
\(526\) −63.9564 −2.78863
\(527\) 8.83485 0.384852
\(528\) 17.9564 0.781454
\(529\) −10.1652 −0.441963
\(530\) 3.49545 0.151833
\(531\) −4.58258 −0.198867
\(532\) 14.9564 0.648444
\(533\) 7.16515 0.310357
\(534\) −25.5826 −1.10707
\(535\) −10.2523 −0.443244
\(536\) 6.16515 0.266294
\(537\) 22.3303 0.963624
\(538\) −27.9129 −1.20341
\(539\) −1.00000 −0.0430730
\(540\) −17.3739 −0.747652
\(541\) −18.3303 −0.788081 −0.394041 0.919093i \(-0.628923\pi\)
−0.394041 + 0.919093i \(0.628923\pi\)
\(542\) −40.7042 −1.74839
\(543\) −3.58258 −0.153743
\(544\) 45.8258 1.96476
\(545\) −16.7477 −0.717394
\(546\) 2.79129 0.119456
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −14.0000 −0.598050
\(549\) 10.0000 0.426790
\(550\) 11.1652 0.476084
\(551\) 26.2523 1.11838
\(552\) 37.9129 1.61368
\(553\) −11.1652 −0.474791
\(554\) 2.33030 0.0990051
\(555\) −3.00000 −0.127343
\(556\) −41.4955 −1.75980
\(557\) −27.3303 −1.15802 −0.579011 0.815320i \(-0.696561\pi\)
−0.579011 + 0.815320i \(0.696561\pi\)
\(558\) 15.5826 0.659663
\(559\) −7.58258 −0.320709
\(560\) 53.8693 2.27639
\(561\) −1.58258 −0.0668164
\(562\) −26.0436 −1.09858
\(563\) −28.4174 −1.19765 −0.598826 0.800879i \(-0.704366\pi\)
−0.598826 + 0.800879i \(0.704366\pi\)
\(564\) −61.2867 −2.58064
\(565\) −27.4955 −1.15674
\(566\) −0.704166 −0.0295983
\(567\) 1.00000 0.0419961
\(568\) 75.8258 3.18158
\(569\) 46.6606 1.95611 0.978057 0.208337i \(-0.0668050\pi\)
0.978057 + 0.208337i \(0.0668050\pi\)
\(570\) 21.6261 0.905819
\(571\) 47.1652 1.97380 0.986900 0.161333i \(-0.0515792\pi\)
0.986900 + 0.161333i \(0.0515792\pi\)
\(572\) −5.79129 −0.242146
\(573\) −2.41742 −0.100989
\(574\) −20.0000 −0.834784
\(575\) 14.3303 0.597615
\(576\) 44.9129 1.87137
\(577\) 23.9129 0.995506 0.497753 0.867319i \(-0.334159\pi\)
0.497753 + 0.867319i \(0.334159\pi\)
\(578\) 40.4610 1.68296
\(579\) 11.5826 0.481355
\(580\) 176.608 7.33325
\(581\) −2.41742 −0.100292
\(582\) −32.3303 −1.34013
\(583\) 0.417424 0.0172879
\(584\) −74.0780 −3.06537
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) 10.2523 0.423157 0.211578 0.977361i \(-0.432140\pi\)
0.211578 + 0.977361i \(0.432140\pi\)
\(588\) −5.79129 −0.238829
\(589\) −14.4174 −0.594060
\(590\) 38.3739 1.57983
\(591\) 13.1652 0.541542
\(592\) 17.9564 0.738005
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) −2.79129 −0.114528
\(595\) −4.74773 −0.194638
\(596\) −70.4519 −2.88582
\(597\) 0.417424 0.0170840
\(598\) −10.0000 −0.408930
\(599\) −11.1652 −0.456196 −0.228098 0.973638i \(-0.573251\pi\)
−0.228098 + 0.973638i \(0.573251\pi\)
\(600\) 42.3303 1.72813
\(601\) −30.4955 −1.24394 −0.621968 0.783043i \(-0.713667\pi\)
−0.621968 + 0.783043i \(0.713667\pi\)
\(602\) 21.1652 0.862627
\(603\) −0.582576 −0.0237243
\(604\) −32.3303 −1.31550
\(605\) 3.00000 0.121967
\(606\) 6.74773 0.274108
\(607\) 5.74773 0.233293 0.116647 0.993173i \(-0.462786\pi\)
0.116647 + 0.993173i \(0.462786\pi\)
\(608\) −74.7822 −3.03282
\(609\) −10.1652 −0.411913
\(610\) −83.7386 −3.39048
\(611\) 10.5826 0.428125
\(612\) −9.16515 −0.370479
\(613\) 0.747727 0.0302004 0.0151002 0.999886i \(-0.495193\pi\)
0.0151002 + 0.999886i \(0.495193\pi\)
\(614\) 0 0
\(615\) −21.4955 −0.866780
\(616\) 10.5826 0.426384
\(617\) 21.1652 0.852077 0.426038 0.904705i \(-0.359909\pi\)
0.426038 + 0.904705i \(0.359909\pi\)
\(618\) 47.9129 1.92734
\(619\) 35.0780 1.40991 0.704953 0.709254i \(-0.250968\pi\)
0.704953 + 0.709254i \(0.250968\pi\)
\(620\) −96.9909 −3.89525
\(621\) −3.58258 −0.143764
\(622\) 62.3303 2.49922
\(623\) −9.16515 −0.367194
\(624\) −17.9564 −0.718833
\(625\) −29.0000 −1.16000
\(626\) −29.0780 −1.16219
\(627\) 2.58258 0.103138
\(628\) 4.83485 0.192931
\(629\) −1.58258 −0.0631014
\(630\) −8.37386 −0.333623
\(631\) 4.83485 0.192472 0.0962361 0.995359i \(-0.469320\pi\)
0.0962361 + 0.995359i \(0.469320\pi\)
\(632\) 118.156 4.70000
\(633\) −5.16515 −0.205296
\(634\) 88.1561 3.50112
\(635\) −7.25227 −0.287798
\(636\) 2.41742 0.0958571
\(637\) 1.00000 0.0396214
\(638\) 28.3739 1.12333
\(639\) −7.16515 −0.283449
\(640\) −202.356 −7.99881
\(641\) 34.4174 1.35941 0.679703 0.733487i \(-0.262109\pi\)
0.679703 + 0.733487i \(0.262109\pi\)
\(642\) −9.53901 −0.376475
\(643\) −44.2432 −1.74478 −0.872390 0.488810i \(-0.837431\pi\)
−0.872390 + 0.488810i \(0.837431\pi\)
\(644\) 20.7477 0.817575
\(645\) 22.7477 0.895691
\(646\) 11.4083 0.448855
\(647\) 34.9129 1.37257 0.686283 0.727334i \(-0.259241\pi\)
0.686283 + 0.727334i \(0.259241\pi\)
\(648\) −10.5826 −0.415723
\(649\) 4.58258 0.179882
\(650\) −11.1652 −0.437933
\(651\) 5.58258 0.218798
\(652\) −3.37386 −0.132131
\(653\) 6.33030 0.247724 0.123862 0.992299i \(-0.460472\pi\)
0.123862 + 0.992299i \(0.460472\pi\)
\(654\) −15.5826 −0.609327
\(655\) −48.0000 −1.87552
\(656\) 128.661 5.02335
\(657\) 7.00000 0.273096
\(658\) −29.5390 −1.15155
\(659\) −19.4174 −0.756395 −0.378198 0.925725i \(-0.623456\pi\)
−0.378198 + 0.925725i \(0.623456\pi\)
\(660\) 17.3739 0.676277
\(661\) 25.0780 0.975422 0.487711 0.873005i \(-0.337832\pi\)
0.487711 + 0.873005i \(0.337832\pi\)
\(662\) −42.3303 −1.64521
\(663\) 1.58258 0.0614621
\(664\) 25.5826 0.992796
\(665\) 7.74773 0.300444
\(666\) −2.79129 −0.108160
\(667\) 36.4174 1.41009
\(668\) 131.739 5.09712
\(669\) −6.00000 −0.231973
\(670\) 4.87841 0.188469
\(671\) −10.0000 −0.386046
\(672\) 28.9564 1.11702
\(673\) 38.7477 1.49362 0.746808 0.665040i \(-0.231586\pi\)
0.746808 + 0.665040i \(0.231586\pi\)
\(674\) −23.4955 −0.905011
\(675\) −4.00000 −0.153960
\(676\) −69.4955 −2.67290
\(677\) −26.8348 −1.03135 −0.515674 0.856785i \(-0.672458\pi\)
−0.515674 + 0.856785i \(0.672458\pi\)
\(678\) −25.5826 −0.982493
\(679\) −11.5826 −0.444498
\(680\) 50.2432 1.92674
\(681\) 22.0000 0.843042
\(682\) −15.5826 −0.596688
\(683\) 31.0780 1.18917 0.594584 0.804034i \(-0.297317\pi\)
0.594584 + 0.804034i \(0.297317\pi\)
\(684\) 14.9564 0.571874
\(685\) −7.25227 −0.277095
\(686\) −2.79129 −0.106572
\(687\) 26.7477 1.02049
\(688\) −136.156 −5.19090
\(689\) −0.417424 −0.0159026
\(690\) 30.0000 1.14208
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 64.6606 2.45803
\(693\) −1.00000 −0.0379869
\(694\) 28.8348 1.09456
\(695\) −21.4955 −0.815369
\(696\) 107.573 4.07756
\(697\) −11.3394 −0.429510
\(698\) 41.8693 1.58478
\(699\) 14.0000 0.529529
\(700\) 23.1652 0.875560
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 2.79129 0.105350
\(703\) 2.58258 0.0974037
\(704\) −44.9129 −1.69272
\(705\) −31.7477 −1.19569
\(706\) −16.2867 −0.612960
\(707\) 2.41742 0.0909166
\(708\) 26.5390 0.997397
\(709\) −45.6606 −1.71482 −0.857410 0.514634i \(-0.827928\pi\)
−0.857410 + 0.514634i \(0.827928\pi\)
\(710\) 60.0000 2.25176
\(711\) −11.1652 −0.418726
\(712\) 96.9909 3.63489
\(713\) −20.0000 −0.749006
\(714\) −4.41742 −0.165318
\(715\) −3.00000 −0.112194
\(716\) −129.321 −4.83296
\(717\) −7.41742 −0.277009
\(718\) 75.8258 2.82979
\(719\) 50.0780 1.86760 0.933798 0.357801i \(-0.116474\pi\)
0.933798 + 0.357801i \(0.116474\pi\)
\(720\) 53.8693 2.00759
\(721\) 17.1652 0.639264
\(722\) 34.4174 1.28088
\(723\) −8.16515 −0.303665
\(724\) 20.7477 0.771083
\(725\) 40.6606 1.51010
\(726\) 2.79129 0.103594
\(727\) 29.9129 1.10941 0.554704 0.832048i \(-0.312832\pi\)
0.554704 + 0.832048i \(0.312832\pi\)
\(728\) −10.5826 −0.392216
\(729\) 1.00000 0.0370370
\(730\) −58.6170 −2.16951
\(731\) 12.0000 0.443836
\(732\) −57.9129 −2.14052
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −61.4083 −2.26662
\(735\) −3.00000 −0.110657
\(736\) −103.739 −3.82386
\(737\) 0.582576 0.0214595
\(738\) −20.0000 −0.736210
\(739\) 13.9129 0.511794 0.255897 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(740\) 17.3739 0.638676
\(741\) −2.58258 −0.0948733
\(742\) 1.16515 0.0427741
\(743\) 29.2432 1.07283 0.536414 0.843955i \(-0.319779\pi\)
0.536414 + 0.843955i \(0.319779\pi\)
\(744\) −59.0780 −2.16591
\(745\) −36.4955 −1.33709
\(746\) 20.2432 0.741156
\(747\) −2.41742 −0.0884489
\(748\) 9.16515 0.335111
\(749\) −3.41742 −0.124870
\(750\) −8.37386 −0.305770
\(751\) 36.9129 1.34697 0.673485 0.739201i \(-0.264797\pi\)
0.673485 + 0.739201i \(0.264797\pi\)
\(752\) 190.025 6.92951
\(753\) −16.5826 −0.604303
\(754\) −28.3739 −1.03332
\(755\) −16.7477 −0.609512
\(756\) −5.79129 −0.210627
\(757\) −9.33030 −0.339116 −0.169558 0.985520i \(-0.554234\pi\)
−0.169558 + 0.985520i \(0.554234\pi\)
\(758\) 9.53901 0.346473
\(759\) 3.58258 0.130039
\(760\) −81.9909 −2.97412
\(761\) 5.66970 0.205526 0.102763 0.994706i \(-0.467232\pi\)
0.102763 + 0.994706i \(0.467232\pi\)
\(762\) −6.74773 −0.244444
\(763\) −5.58258 −0.202103
\(764\) 14.0000 0.506502
\(765\) −4.74773 −0.171654
\(766\) 73.4955 2.65550
\(767\) −4.58258 −0.165467
\(768\) −98.4519 −3.55258
\(769\) −48.4955 −1.74879 −0.874395 0.485214i \(-0.838742\pi\)
−0.874395 + 0.485214i \(0.838742\pi\)
\(770\) 8.37386 0.301773
\(771\) −19.0000 −0.684268
\(772\) −67.0780 −2.41419
\(773\) −12.1652 −0.437550 −0.218775 0.975775i \(-0.570206\pi\)
−0.218775 + 0.975775i \(0.570206\pi\)
\(774\) 21.1652 0.760766
\(775\) −22.3303 −0.802128
\(776\) 122.573 4.40013
\(777\) −1.00000 −0.0358748
\(778\) −28.8348 −1.03378
\(779\) 18.5045 0.662994
\(780\) −17.3739 −0.622084
\(781\) 7.16515 0.256389
\(782\) 15.8258 0.565928
\(783\) −10.1652 −0.363273
\(784\) 17.9564 0.641301
\(785\) 2.50455 0.0893911
\(786\) −44.6606 −1.59299
\(787\) 29.4174 1.04862 0.524309 0.851528i \(-0.324324\pi\)
0.524309 + 0.851528i \(0.324324\pi\)
\(788\) −76.2432 −2.71605
\(789\) −22.9129 −0.815720
\(790\) 93.4955 3.32642
\(791\) −9.16515 −0.325875
\(792\) 10.5826 0.376035
\(793\) 10.0000 0.355110
\(794\) 62.5735 2.22065
\(795\) 1.25227 0.0444135
\(796\) −2.41742 −0.0856833
\(797\) 2.49545 0.0883935 0.0441968 0.999023i \(-0.485927\pi\)
0.0441968 + 0.999023i \(0.485927\pi\)
\(798\) 7.20871 0.255186
\(799\) −16.7477 −0.592492
\(800\) −115.826 −4.09506
\(801\) −9.16515 −0.323835
\(802\) 38.8348 1.37131
\(803\) −7.00000 −0.247025
\(804\) 3.37386 0.118987
\(805\) 10.7477 0.378808
\(806\) 15.5826 0.548873
\(807\) −10.0000 −0.352017
\(808\) −25.5826 −0.899992
\(809\) −27.3303 −0.960882 −0.480441 0.877027i \(-0.659523\pi\)
−0.480441 + 0.877027i \(0.659523\pi\)
\(810\) −8.37386 −0.294228
\(811\) −29.7477 −1.04458 −0.522292 0.852767i \(-0.674923\pi\)
−0.522292 + 0.852767i \(0.674923\pi\)
\(812\) 58.8693 2.06591
\(813\) −14.5826 −0.511433
\(814\) 2.79129 0.0978346
\(815\) −1.74773 −0.0612202
\(816\) 28.4174 0.994809
\(817\) −19.5826 −0.685108
\(818\) 79.0780 2.76490
\(819\) 1.00000 0.0349428
\(820\) 124.486 4.34725
\(821\) −47.0000 −1.64031 −0.820156 0.572140i \(-0.806113\pi\)
−0.820156 + 0.572140i \(0.806113\pi\)
\(822\) −6.74773 −0.235354
\(823\) −21.4174 −0.746564 −0.373282 0.927718i \(-0.621768\pi\)
−0.373282 + 0.927718i \(0.621768\pi\)
\(824\) −181.652 −6.32813
\(825\) 4.00000 0.139262
\(826\) 12.7913 0.445066
\(827\) −36.9129 −1.28359 −0.641793 0.766878i \(-0.721809\pi\)
−0.641793 + 0.766878i \(0.721809\pi\)
\(828\) 20.7477 0.721033
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 20.2432 0.702651
\(831\) 0.834849 0.0289606
\(832\) 44.9129 1.55707
\(833\) −1.58258 −0.0548330
\(834\) −20.0000 −0.692543
\(835\) 68.2432 2.36165
\(836\) −14.9564 −0.517279
\(837\) 5.58258 0.192962
\(838\) 18.3739 0.634715
\(839\) 52.9129 1.82676 0.913378 0.407113i \(-0.133465\pi\)
0.913378 + 0.407113i \(0.133465\pi\)
\(840\) 31.7477 1.09540
\(841\) 74.3303 2.56311
\(842\) −110.704 −3.81512
\(843\) −9.33030 −0.321353
\(844\) 29.9129 1.02964
\(845\) −36.0000 −1.23844
\(846\) −29.5390 −1.01557
\(847\) 1.00000 0.0343604
\(848\) −7.49545 −0.257395
\(849\) −0.252273 −0.00865798
\(850\) 17.6697 0.606066
\(851\) 3.58258 0.122809
\(852\) 41.4955 1.42161
\(853\) −1.16515 −0.0398940 −0.0199470 0.999801i \(-0.506350\pi\)
−0.0199470 + 0.999801i \(0.506350\pi\)
\(854\) −27.9129 −0.955159
\(855\) 7.74773 0.264967
\(856\) 36.1652 1.23610
\(857\) −44.3303 −1.51429 −0.757147 0.653244i \(-0.773407\pi\)
−0.757147 + 0.653244i \(0.773407\pi\)
\(858\) −2.79129 −0.0952930
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) −131.739 −4.49225
\(861\) −7.16515 −0.244188
\(862\) 27.2087 0.926732
\(863\) 23.5826 0.802760 0.401380 0.915912i \(-0.368531\pi\)
0.401380 + 0.915912i \(0.368531\pi\)
\(864\) 28.9564 0.985118
\(865\) 33.4955 1.13888
\(866\) −20.0000 −0.679628
\(867\) 14.4955 0.492291
\(868\) −32.3303 −1.09736
\(869\) 11.1652 0.378752
\(870\) 85.1216 2.88589
\(871\) −0.582576 −0.0197398
\(872\) 59.0780 2.00063
\(873\) −11.5826 −0.392011
\(874\) −25.8258 −0.873569
\(875\) −3.00000 −0.101419
\(876\) −40.5390 −1.36969
\(877\) −45.4955 −1.53627 −0.768136 0.640287i \(-0.778816\pi\)
−0.768136 + 0.640287i \(0.778816\pi\)
\(878\) −74.1996 −2.50412
\(879\) 0 0
\(880\) −53.8693 −1.81593
\(881\) 35.6606 1.20144 0.600718 0.799461i \(-0.294881\pi\)
0.600718 + 0.799461i \(0.294881\pi\)
\(882\) −2.79129 −0.0939876
\(883\) −28.2523 −0.950765 −0.475382 0.879779i \(-0.657690\pi\)
−0.475382 + 0.879779i \(0.657690\pi\)
\(884\) −9.16515 −0.308257
\(885\) 13.7477 0.462125
\(886\) 13.4955 0.453389
\(887\) 11.2523 0.377814 0.188907 0.981995i \(-0.439505\pi\)
0.188907 + 0.981995i \(0.439505\pi\)
\(888\) 10.5826 0.355128
\(889\) −2.41742 −0.0810778
\(890\) 76.7477 2.57259
\(891\) −1.00000 −0.0335013
\(892\) 34.7477 1.16344
\(893\) 27.3303 0.914574
\(894\) −33.9564 −1.13567
\(895\) −66.9909 −2.23926
\(896\) −67.4519 −2.25341
\(897\) −3.58258 −0.119619
\(898\) −51.1652 −1.70740
\(899\) −56.7477 −1.89264
\(900\) 23.1652 0.772172
\(901\) 0.660606 0.0220080
\(902\) 20.0000 0.665927
\(903\) 7.58258 0.252332
\(904\) 96.9909 3.22587
\(905\) 10.7477 0.357267
\(906\) −15.5826 −0.517696
\(907\) 5.66970 0.188259 0.0941296 0.995560i \(-0.469993\pi\)
0.0941296 + 0.995560i \(0.469993\pi\)
\(908\) −127.408 −4.22819
\(909\) 2.41742 0.0801809
\(910\) −8.37386 −0.277591
\(911\) 51.4955 1.70612 0.853060 0.521812i \(-0.174744\pi\)
0.853060 + 0.521812i \(0.174744\pi\)
\(912\) −46.3739 −1.53559
\(913\) 2.41742 0.0800051
\(914\) −72.3303 −2.39247
\(915\) −30.0000 −0.991769
\(916\) −154.904 −5.11817
\(917\) −16.0000 −0.528367
\(918\) −4.41742 −0.145797
\(919\) −53.9129 −1.77842 −0.889211 0.457498i \(-0.848746\pi\)
−0.889211 + 0.457498i \(0.848746\pi\)
\(920\) −113.739 −3.74985
\(921\) 0 0
\(922\) 51.1652 1.68503
\(923\) −7.16515 −0.235844
\(924\) 5.79129 0.190519
\(925\) 4.00000 0.131519
\(926\) 1.62614 0.0534382
\(927\) 17.1652 0.563778
\(928\) −294.347 −9.66240
\(929\) 15.3303 0.502971 0.251485 0.967861i \(-0.419081\pi\)
0.251485 + 0.967861i \(0.419081\pi\)
\(930\) −46.7477 −1.53292
\(931\) 2.58258 0.0846405
\(932\) −81.0780 −2.65580
\(933\) 22.3303 0.731061
\(934\) 82.1125 2.68680
\(935\) 4.74773 0.155267
\(936\) −10.5826 −0.345902
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 1.62614 0.0530952
\(939\) −10.4174 −0.339960
\(940\) 183.860 5.99686
\(941\) 16.8348 0.548800 0.274400 0.961616i \(-0.411521\pi\)
0.274400 + 0.961616i \(0.411521\pi\)
\(942\) 2.33030 0.0759254
\(943\) 25.6697 0.835920
\(944\) −82.2867 −2.67821
\(945\) −3.00000 −0.0975900
\(946\) −21.1652 −0.688138
\(947\) 26.8348 0.872015 0.436008 0.899943i \(-0.356392\pi\)
0.436008 + 0.899943i \(0.356392\pi\)
\(948\) 64.6606 2.10008
\(949\) 7.00000 0.227230
\(950\) −28.8348 −0.935526
\(951\) 31.5826 1.02414
\(952\) 16.7477 0.542797
\(953\) −9.83485 −0.318582 −0.159291 0.987232i \(-0.550921\pi\)
−0.159291 + 0.987232i \(0.550921\pi\)
\(954\) 1.16515 0.0377232
\(955\) 7.25227 0.234678
\(956\) 42.9564 1.38931
\(957\) 10.1652 0.328593
\(958\) 17.9129 0.578739
\(959\) −2.41742 −0.0780627
\(960\) −134.739 −4.34867
\(961\) 0.165151 0.00532746
\(962\) −2.79129 −0.0899947
\(963\) −3.41742 −0.110125
\(964\) 47.2867 1.52300
\(965\) −34.7477 −1.11857
\(966\) 10.0000 0.321745
\(967\) 5.16515 0.166100 0.0830500 0.996545i \(-0.473534\pi\)
0.0830500 + 0.996545i \(0.473534\pi\)
\(968\) −10.5826 −0.340137
\(969\) 4.08712 0.131297
\(970\) 96.9909 3.11419
\(971\) −3.41742 −0.109670 −0.0548352 0.998495i \(-0.517463\pi\)
−0.0548352 + 0.998495i \(0.517463\pi\)
\(972\) −5.79129 −0.185756
\(973\) −7.16515 −0.229704
\(974\) 73.4955 2.35495
\(975\) −4.00000 −0.128103
\(976\) 179.564 5.74772
\(977\) 50.7477 1.62356 0.811782 0.583961i \(-0.198498\pi\)
0.811782 + 0.583961i \(0.198498\pi\)
\(978\) −1.62614 −0.0519981
\(979\) 9.16515 0.292920
\(980\) 17.3739 0.554988
\(981\) −5.58258 −0.178238
\(982\) −63.9564 −2.04093
\(983\) −23.1652 −0.738854 −0.369427 0.929260i \(-0.620446\pi\)
−0.369427 + 0.929260i \(0.620446\pi\)
\(984\) 75.8258 2.41724
\(985\) −39.4955 −1.25843
\(986\) 44.9038 1.43003
\(987\) −10.5826 −0.336847
\(988\) 14.9564 0.475828
\(989\) −27.1652 −0.863802
\(990\) 8.37386 0.266139
\(991\) −47.7477 −1.51676 −0.758378 0.651815i \(-0.774008\pi\)
−0.758378 + 0.651815i \(0.774008\pi\)
\(992\) 161.652 5.13244
\(993\) −15.1652 −0.481252
\(994\) 20.0000 0.634361
\(995\) −1.25227 −0.0396997
\(996\) 14.0000 0.443607
\(997\) −62.6606 −1.98448 −0.992241 0.124332i \(-0.960321\pi\)
−0.992241 + 0.124332i \(0.960321\pi\)
\(998\) −39.7822 −1.25928
\(999\) −1.00000 −0.0316386
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 231.2.a.b.1.1 2
3.2 odd 2 693.2.a.j.1.2 2
4.3 odd 2 3696.2.a.bl.1.1 2
5.4 even 2 5775.2.a.bn.1.2 2
7.6 odd 2 1617.2.a.o.1.1 2
11.10 odd 2 2541.2.a.z.1.2 2
21.20 even 2 4851.2.a.ba.1.2 2
33.32 even 2 7623.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.1 2 1.1 even 1 trivial
693.2.a.j.1.2 2 3.2 odd 2
1617.2.a.o.1.1 2 7.6 odd 2
2541.2.a.z.1.2 2 11.10 odd 2
3696.2.a.bl.1.1 2 4.3 odd 2
4851.2.a.ba.1.2 2 21.20 even 2
5775.2.a.bn.1.2 2 5.4 even 2
7623.2.a.bf.1.1 2 33.32 even 2