Properties

Label 538.2.a.d.1.3
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $1$
Dimension $7$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.30699\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.20937 q^{3} +1.00000 q^{4} -2.40321 q^{5} +1.20937 q^{6} +1.52017 q^{7} -1.00000 q^{8} -1.53742 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.20937 q^{3} +1.00000 q^{4} -2.40321 q^{5} +1.20937 q^{6} +1.52017 q^{7} -1.00000 q^{8} -1.53742 q^{9} +2.40321 q^{10} +5.07638 q^{11} -1.20937 q^{12} +5.11021 q^{13} -1.52017 q^{14} +2.90637 q^{15} +1.00000 q^{16} -6.65764 q^{17} +1.53742 q^{18} -5.71222 q^{19} -2.40321 q^{20} -1.83846 q^{21} -5.07638 q^{22} -6.36685 q^{23} +1.20937 q^{24} +0.775401 q^{25} -5.11021 q^{26} +5.48743 q^{27} +1.52017 q^{28} -6.00511 q^{29} -2.90637 q^{30} +2.77822 q^{31} -1.00000 q^{32} -6.13924 q^{33} +6.65764 q^{34} -3.65329 q^{35} -1.53742 q^{36} -1.59410 q^{37} +5.71222 q^{38} -6.18015 q^{39} +2.40321 q^{40} +9.69619 q^{41} +1.83846 q^{42} -9.26120 q^{43} +5.07638 q^{44} +3.69473 q^{45} +6.36685 q^{46} -7.55235 q^{47} -1.20937 q^{48} -4.68907 q^{49} -0.775401 q^{50} +8.05157 q^{51} +5.11021 q^{52} -9.42058 q^{53} -5.48743 q^{54} -12.1996 q^{55} -1.52017 q^{56} +6.90820 q^{57} +6.00511 q^{58} -6.31332 q^{59} +2.90637 q^{60} -7.97918 q^{61} -2.77822 q^{62} -2.33714 q^{63} +1.00000 q^{64} -12.2809 q^{65} +6.13924 q^{66} -2.54885 q^{67} -6.65764 q^{68} +7.69990 q^{69} +3.65329 q^{70} +1.98957 q^{71} +1.53742 q^{72} +12.7523 q^{73} +1.59410 q^{74} -0.937750 q^{75} -5.71222 q^{76} +7.71699 q^{77} +6.18015 q^{78} +3.64283 q^{79} -2.40321 q^{80} -2.02411 q^{81} -9.69619 q^{82} -0.689504 q^{83} -1.83846 q^{84} +15.9997 q^{85} +9.26120 q^{86} +7.26242 q^{87} -5.07638 q^{88} +2.78972 q^{89} -3.69473 q^{90} +7.76841 q^{91} -6.36685 q^{92} -3.35991 q^{93} +7.55235 q^{94} +13.7276 q^{95} +1.20937 q^{96} +11.9363 q^{97} +4.68907 q^{98} -7.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20} - 5 q^{21} + 12 q^{22} - 22 q^{23} + 4 q^{24} + 9 q^{25} - 3 q^{26} - 22 q^{27} - 3 q^{28} - 7 q^{29} + 6 q^{30} - 3 q^{31} - 7 q^{32} - 19 q^{33} + 8 q^{34} - 25 q^{35} + 7 q^{36} - 13 q^{37} + 7 q^{38} + q^{39} + 6 q^{40} - 11 q^{41} + 5 q^{42} - 14 q^{43} - 12 q^{44} - 17 q^{45} + 22 q^{46} - 23 q^{47} - 4 q^{48} + 6 q^{49} - 9 q^{50} - 21 q^{51} + 3 q^{52} - 21 q^{53} + 22 q^{54} + 15 q^{55} + 3 q^{56} - 14 q^{57} + 7 q^{58} - 20 q^{59} - 6 q^{60} + 13 q^{61} + 3 q^{62} - 12 q^{63} + 7 q^{64} - 29 q^{65} + 19 q^{66} - 30 q^{67} - 8 q^{68} + 13 q^{69} + 25 q^{70} - 23 q^{71} - 7 q^{72} + 9 q^{73} + 13 q^{74} + 3 q^{75} - 7 q^{76} + 8 q^{77} - q^{78} + 11 q^{79} - 6 q^{80} + 43 q^{81} + 11 q^{82} - 3 q^{83} - 5 q^{84} + 21 q^{85} + 14 q^{86} + 28 q^{87} + 12 q^{88} + 16 q^{89} + 17 q^{90} - 10 q^{91} - 22 q^{92} - 13 q^{93} + 23 q^{94} + 4 q^{96} + 21 q^{97} - 6 q^{98} - 9 q^{99}+O(q^{100}) \) 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
\(3\) −1.20937 −0.698232 −0.349116 0.937079i \(-0.613518\pi\)
−0.349116 + 0.937079i \(0.613518\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.40321 −1.07475 −0.537373 0.843344i \(-0.680583\pi\)
−0.537373 + 0.843344i \(0.680583\pi\)
\(6\) 1.20937 0.493725
\(7\) 1.52017 0.574572 0.287286 0.957845i \(-0.407247\pi\)
0.287286 + 0.957845i \(0.407247\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.53742 −0.512472
\(10\) 2.40321 0.759961
\(11\) 5.07638 1.53059 0.765294 0.643681i \(-0.222594\pi\)
0.765294 + 0.643681i \(0.222594\pi\)
\(12\) −1.20937 −0.349116
\(13\) 5.11021 1.41732 0.708659 0.705552i \(-0.249301\pi\)
0.708659 + 0.705552i \(0.249301\pi\)
\(14\) −1.52017 −0.406284
\(15\) 2.90637 0.750423
\(16\) 1.00000 0.250000
\(17\) −6.65764 −1.61471 −0.807357 0.590063i \(-0.799103\pi\)
−0.807357 + 0.590063i \(0.799103\pi\)
\(18\) 1.53742 0.362372
\(19\) −5.71222 −1.31047 −0.655236 0.755424i \(-0.727431\pi\)
−0.655236 + 0.755424i \(0.727431\pi\)
\(20\) −2.40321 −0.537373
\(21\) −1.83846 −0.401184
\(22\) −5.07638 −1.08229
\(23\) −6.36685 −1.32758 −0.663790 0.747919i \(-0.731053\pi\)
−0.663790 + 0.747919i \(0.731053\pi\)
\(24\) 1.20937 0.246862
\(25\) 0.775401 0.155080
\(26\) −5.11021 −1.00219
\(27\) 5.48743 1.05606
\(28\) 1.52017 0.287286
\(29\) −6.00511 −1.11512 −0.557560 0.830137i \(-0.688262\pi\)
−0.557560 + 0.830137i \(0.688262\pi\)
\(30\) −2.90637 −0.530629
\(31\) 2.77822 0.498984 0.249492 0.968377i \(-0.419736\pi\)
0.249492 + 0.968377i \(0.419736\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.13924 −1.06871
\(34\) 6.65764 1.14178
\(35\) −3.65329 −0.617519
\(36\) −1.53742 −0.256236
\(37\) −1.59410 −0.262069 −0.131034 0.991378i \(-0.541830\pi\)
−0.131034 + 0.991378i \(0.541830\pi\)
\(38\) 5.71222 0.926644
\(39\) −6.18015 −0.989616
\(40\) 2.40321 0.379980
\(41\) 9.69619 1.51429 0.757146 0.653246i \(-0.226593\pi\)
0.757146 + 0.653246i \(0.226593\pi\)
\(42\) 1.83846 0.283680
\(43\) −9.26120 −1.41232 −0.706160 0.708053i \(-0.749574\pi\)
−0.706160 + 0.708053i \(0.749574\pi\)
\(44\) 5.07638 0.765294
\(45\) 3.69473 0.550777
\(46\) 6.36685 0.938741
\(47\) −7.55235 −1.10162 −0.550811 0.834630i \(-0.685682\pi\)
−0.550811 + 0.834630i \(0.685682\pi\)
\(48\) −1.20937 −0.174558
\(49\) −4.68907 −0.669867
\(50\) −0.775401 −0.109658
\(51\) 8.05157 1.12745
\(52\) 5.11021 0.708659
\(53\) −9.42058 −1.29402 −0.647008 0.762483i \(-0.723980\pi\)
−0.647008 + 0.762483i \(0.723980\pi\)
\(54\) −5.48743 −0.746745
\(55\) −12.1996 −1.64499
\(56\) −1.52017 −0.203142
\(57\) 6.90820 0.915014
\(58\) 6.00511 0.788509
\(59\) −6.31332 −0.821924 −0.410962 0.911652i \(-0.634807\pi\)
−0.410962 + 0.911652i \(0.634807\pi\)
\(60\) 2.90637 0.375211
\(61\) −7.97918 −1.02163 −0.510815 0.859691i \(-0.670656\pi\)
−0.510815 + 0.859691i \(0.670656\pi\)
\(62\) −2.77822 −0.352835
\(63\) −2.33714 −0.294452
\(64\) 1.00000 0.125000
\(65\) −12.2809 −1.52326
\(66\) 6.13924 0.755689
\(67\) −2.54885 −0.311392 −0.155696 0.987805i \(-0.549762\pi\)
−0.155696 + 0.987805i \(0.549762\pi\)
\(68\) −6.65764 −0.807357
\(69\) 7.69990 0.926959
\(70\) 3.65329 0.436652
\(71\) 1.98957 0.236118 0.118059 0.993007i \(-0.462333\pi\)
0.118059 + 0.993007i \(0.462333\pi\)
\(72\) 1.53742 0.181186
\(73\) 12.7523 1.49254 0.746269 0.665644i \(-0.231843\pi\)
0.746269 + 0.665644i \(0.231843\pi\)
\(74\) 1.59410 0.185311
\(75\) −0.937750 −0.108282
\(76\) −5.71222 −0.655236
\(77\) 7.71699 0.879432
\(78\) 6.18015 0.699764
\(79\) 3.64283 0.409851 0.204925 0.978778i \(-0.434305\pi\)
0.204925 + 0.978778i \(0.434305\pi\)
\(80\) −2.40321 −0.268687
\(81\) −2.02411 −0.224901
\(82\) −9.69619 −1.07077
\(83\) −0.689504 −0.0756829 −0.0378415 0.999284i \(-0.512048\pi\)
−0.0378415 + 0.999284i \(0.512048\pi\)
\(84\) −1.83846 −0.200592
\(85\) 15.9997 1.73541
\(86\) 9.26120 0.998660
\(87\) 7.26242 0.778613
\(88\) −5.07638 −0.541144
\(89\) 2.78972 0.295709 0.147855 0.989009i \(-0.452763\pi\)
0.147855 + 0.989009i \(0.452763\pi\)
\(90\) −3.69473 −0.389458
\(91\) 7.76841 0.814350
\(92\) −6.36685 −0.663790
\(93\) −3.35991 −0.348407
\(94\) 7.55235 0.778965
\(95\) 13.7276 1.40843
\(96\) 1.20937 0.123431
\(97\) 11.9363 1.21194 0.605972 0.795486i \(-0.292784\pi\)
0.605972 + 0.795486i \(0.292784\pi\)
\(98\) 4.68907 0.473668
\(99\) −7.80451 −0.784383
\(100\) 0.775401 0.0775401
\(101\) −14.5273 −1.44552 −0.722759 0.691100i \(-0.757127\pi\)
−0.722759 + 0.691100i \(0.757127\pi\)
\(102\) −8.05157 −0.797224
\(103\) −10.1048 −0.995656 −0.497828 0.867276i \(-0.665869\pi\)
−0.497828 + 0.867276i \(0.665869\pi\)
\(104\) −5.11021 −0.501097
\(105\) 4.41819 0.431172
\(106\) 9.42058 0.915007
\(107\) −12.2725 −1.18643 −0.593214 0.805045i \(-0.702141\pi\)
−0.593214 + 0.805045i \(0.702141\pi\)
\(108\) 5.48743 0.528028
\(109\) 14.2163 1.36168 0.680838 0.732434i \(-0.261616\pi\)
0.680838 + 0.732434i \(0.261616\pi\)
\(110\) 12.1996 1.16319
\(111\) 1.92786 0.182985
\(112\) 1.52017 0.143643
\(113\) 12.7024 1.19494 0.597471 0.801891i \(-0.296172\pi\)
0.597471 + 0.801891i \(0.296172\pi\)
\(114\) −6.90820 −0.647013
\(115\) 15.3009 1.42681
\(116\) −6.00511 −0.557560
\(117\) −7.85652 −0.726335
\(118\) 6.31332 0.581188
\(119\) −10.1208 −0.927769
\(120\) −2.90637 −0.265314
\(121\) 14.7697 1.34270
\(122\) 7.97918 0.722401
\(123\) −11.7263 −1.05733
\(124\) 2.77822 0.249492
\(125\) 10.1526 0.908075
\(126\) 2.33714 0.208209
\(127\) −17.1800 −1.52448 −0.762238 0.647297i \(-0.775899\pi\)
−0.762238 + 0.647297i \(0.775899\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.2002 0.986127
\(130\) 12.2809 1.07711
\(131\) 5.22919 0.456877 0.228438 0.973558i \(-0.426638\pi\)
0.228438 + 0.973558i \(0.426638\pi\)
\(132\) −6.13924 −0.534353
\(133\) −8.68356 −0.752960
\(134\) 2.54885 0.220187
\(135\) −13.1874 −1.13499
\(136\) 6.65764 0.570888
\(137\) −12.9664 −1.10779 −0.553895 0.832586i \(-0.686859\pi\)
−0.553895 + 0.832586i \(0.686859\pi\)
\(138\) −7.69990 −0.655459
\(139\) −0.692646 −0.0587495 −0.0293747 0.999568i \(-0.509352\pi\)
−0.0293747 + 0.999568i \(0.509352\pi\)
\(140\) −3.65329 −0.308759
\(141\) 9.13361 0.769189
\(142\) −1.98957 −0.166961
\(143\) 25.9414 2.16933
\(144\) −1.53742 −0.128118
\(145\) 14.4315 1.19847
\(146\) −12.7523 −1.05538
\(147\) 5.67084 0.467723
\(148\) −1.59410 −0.131034
\(149\) 11.5986 0.950197 0.475099 0.879933i \(-0.342412\pi\)
0.475099 + 0.879933i \(0.342412\pi\)
\(150\) 0.937750 0.0765669
\(151\) 1.14291 0.0930089 0.0465045 0.998918i \(-0.485192\pi\)
0.0465045 + 0.998918i \(0.485192\pi\)
\(152\) 5.71222 0.463322
\(153\) 10.2356 0.827496
\(154\) −7.71699 −0.621852
\(155\) −6.67665 −0.536281
\(156\) −6.18015 −0.494808
\(157\) −12.2350 −0.976460 −0.488230 0.872715i \(-0.662357\pi\)
−0.488230 + 0.872715i \(0.662357\pi\)
\(158\) −3.64283 −0.289808
\(159\) 11.3930 0.903523
\(160\) 2.40321 0.189990
\(161\) −9.67872 −0.762790
\(162\) 2.02411 0.159029
\(163\) −23.6815 −1.85488 −0.927438 0.373978i \(-0.877994\pi\)
−0.927438 + 0.373978i \(0.877994\pi\)
\(164\) 9.69619 0.757146
\(165\) 14.7539 1.14859
\(166\) 0.689504 0.0535159
\(167\) −3.54694 −0.274470 −0.137235 0.990538i \(-0.543822\pi\)
−0.137235 + 0.990538i \(0.543822\pi\)
\(168\) 1.83846 0.141840
\(169\) 13.1142 1.00879
\(170\) −15.9997 −1.22712
\(171\) 8.78205 0.671580
\(172\) −9.26120 −0.706160
\(173\) 20.8927 1.58844 0.794220 0.607630i \(-0.207880\pi\)
0.794220 + 0.607630i \(0.207880\pi\)
\(174\) −7.26242 −0.550562
\(175\) 1.17874 0.0891047
\(176\) 5.07638 0.382647
\(177\) 7.63516 0.573894
\(178\) −2.78972 −0.209098
\(179\) 2.13073 0.159258 0.0796290 0.996825i \(-0.474626\pi\)
0.0796290 + 0.996825i \(0.474626\pi\)
\(180\) 3.69473 0.275389
\(181\) 18.5589 1.37948 0.689738 0.724059i \(-0.257726\pi\)
0.689738 + 0.724059i \(0.257726\pi\)
\(182\) −7.76841 −0.575833
\(183\) 9.64981 0.713334
\(184\) 6.36685 0.469371
\(185\) 3.83095 0.281657
\(186\) 3.35991 0.246361
\(187\) −33.7967 −2.47146
\(188\) −7.55235 −0.550811
\(189\) 8.34185 0.606780
\(190\) −13.7276 −0.995907
\(191\) 15.6213 1.13032 0.565160 0.824982i \(-0.308815\pi\)
0.565160 + 0.824982i \(0.308815\pi\)
\(192\) −1.20937 −0.0872790
\(193\) −18.5417 −1.33466 −0.667330 0.744762i \(-0.732563\pi\)
−0.667330 + 0.744762i \(0.732563\pi\)
\(194\) −11.9363 −0.856974
\(195\) 14.8522 1.06359
\(196\) −4.68907 −0.334934
\(197\) −9.98285 −0.711249 −0.355624 0.934629i \(-0.615732\pi\)
−0.355624 + 0.934629i \(0.615732\pi\)
\(198\) 7.80451 0.554643
\(199\) 5.33431 0.378139 0.189070 0.981964i \(-0.439453\pi\)
0.189070 + 0.981964i \(0.439453\pi\)
\(200\) −0.775401 −0.0548292
\(201\) 3.08251 0.217424
\(202\) 14.5273 1.02214
\(203\) −9.12880 −0.640716
\(204\) 8.05157 0.563723
\(205\) −23.3020 −1.62748
\(206\) 10.1048 0.704035
\(207\) 9.78850 0.680348
\(208\) 5.11021 0.354329
\(209\) −28.9974 −2.00579
\(210\) −4.41819 −0.304884
\(211\) −19.7945 −1.36271 −0.681355 0.731953i \(-0.738609\pi\)
−0.681355 + 0.731953i \(0.738609\pi\)
\(212\) −9.42058 −0.647008
\(213\) −2.40613 −0.164865
\(214\) 12.2725 0.838931
\(215\) 22.2566 1.51789
\(216\) −5.48743 −0.373372
\(217\) 4.22338 0.286702
\(218\) −14.2163 −0.962850
\(219\) −15.4222 −1.04214
\(220\) −12.1996 −0.822497
\(221\) −34.0219 −2.28856
\(222\) −1.92786 −0.129390
\(223\) 29.2824 1.96089 0.980447 0.196783i \(-0.0630495\pi\)
0.980447 + 0.196783i \(0.0630495\pi\)
\(224\) −1.52017 −0.101571
\(225\) −1.19211 −0.0794743
\(226\) −12.7024 −0.844951
\(227\) −5.90479 −0.391915 −0.195957 0.980612i \(-0.562781\pi\)
−0.195957 + 0.980612i \(0.562781\pi\)
\(228\) 6.90820 0.457507
\(229\) 14.2761 0.943391 0.471695 0.881762i \(-0.343642\pi\)
0.471695 + 0.881762i \(0.343642\pi\)
\(230\) −15.3009 −1.00891
\(231\) −9.33272 −0.614048
\(232\) 6.00511 0.394254
\(233\) −27.0850 −1.77440 −0.887198 0.461389i \(-0.847351\pi\)
−0.887198 + 0.461389i \(0.847351\pi\)
\(234\) 7.85652 0.513597
\(235\) 18.1498 1.18397
\(236\) −6.31332 −0.410962
\(237\) −4.40555 −0.286171
\(238\) 10.1208 0.656032
\(239\) 7.97200 0.515666 0.257833 0.966190i \(-0.416992\pi\)
0.257833 + 0.966190i \(0.416992\pi\)
\(240\) 2.90637 0.187606
\(241\) −2.36516 −0.152353 −0.0761767 0.997094i \(-0.524271\pi\)
−0.0761767 + 0.997094i \(0.524271\pi\)
\(242\) −14.7697 −0.949430
\(243\) −14.0144 −0.899024
\(244\) −7.97918 −0.510815
\(245\) 11.2688 0.719938
\(246\) 11.7263 0.747643
\(247\) −29.1906 −1.85735
\(248\) −2.77822 −0.176417
\(249\) 0.833868 0.0528443
\(250\) −10.1526 −0.642106
\(251\) −23.0435 −1.45449 −0.727246 0.686376i \(-0.759200\pi\)
−0.727246 + 0.686376i \(0.759200\pi\)
\(252\) −2.33714 −0.147226
\(253\) −32.3206 −2.03198
\(254\) 17.1800 1.07797
\(255\) −19.3496 −1.21172
\(256\) 1.00000 0.0625000
\(257\) 23.2436 1.44990 0.724948 0.688803i \(-0.241864\pi\)
0.724948 + 0.688803i \(0.241864\pi\)
\(258\) −11.2002 −0.697297
\(259\) −2.42331 −0.150577
\(260\) −12.2809 −0.761628
\(261\) 9.23234 0.571468
\(262\) −5.22919 −0.323061
\(263\) 4.62308 0.285072 0.142536 0.989790i \(-0.454474\pi\)
0.142536 + 0.989790i \(0.454474\pi\)
\(264\) 6.13924 0.377844
\(265\) 22.6396 1.39074
\(266\) 8.68356 0.532423
\(267\) −3.37381 −0.206474
\(268\) −2.54885 −0.155696
\(269\) −1.00000 −0.0609711
\(270\) 13.1874 0.802561
\(271\) 12.4311 0.755135 0.377568 0.925982i \(-0.376761\pi\)
0.377568 + 0.925982i \(0.376761\pi\)
\(272\) −6.65764 −0.403679
\(273\) −9.39490 −0.568605
\(274\) 12.9664 0.783326
\(275\) 3.93623 0.237364
\(276\) 7.69990 0.463480
\(277\) −4.92762 −0.296072 −0.148036 0.988982i \(-0.547295\pi\)
−0.148036 + 0.988982i \(0.547295\pi\)
\(278\) 0.692646 0.0415422
\(279\) −4.27129 −0.255715
\(280\) 3.65329 0.218326
\(281\) −21.3194 −1.27181 −0.635905 0.771767i \(-0.719373\pi\)
−0.635905 + 0.771767i \(0.719373\pi\)
\(282\) −9.13361 −0.543898
\(283\) −29.9997 −1.78330 −0.891649 0.452728i \(-0.850451\pi\)
−0.891649 + 0.452728i \(0.850451\pi\)
\(284\) 1.98957 0.118059
\(285\) −16.6018 −0.983408
\(286\) −25.9414 −1.53395
\(287\) 14.7399 0.870069
\(288\) 1.53742 0.0905931
\(289\) 27.3241 1.60730
\(290\) −14.4315 −0.847447
\(291\) −14.4354 −0.846218
\(292\) 12.7523 0.746269
\(293\) 16.9481 0.990120 0.495060 0.868859i \(-0.335146\pi\)
0.495060 + 0.868859i \(0.335146\pi\)
\(294\) −5.67084 −0.330730
\(295\) 15.1722 0.883360
\(296\) 1.59410 0.0926553
\(297\) 27.8563 1.61639
\(298\) −11.5986 −0.671891
\(299\) −32.5360 −1.88160
\(300\) −0.937750 −0.0541410
\(301\) −14.0786 −0.811479
\(302\) −1.14291 −0.0657672
\(303\) 17.5689 1.00931
\(304\) −5.71222 −0.327618
\(305\) 19.1756 1.09799
\(306\) −10.2356 −0.585128
\(307\) 18.4715 1.05422 0.527112 0.849796i \(-0.323275\pi\)
0.527112 + 0.849796i \(0.323275\pi\)
\(308\) 7.71699 0.439716
\(309\) 12.2205 0.695199
\(310\) 6.67665 0.379208
\(311\) 13.9141 0.788995 0.394497 0.918897i \(-0.370919\pi\)
0.394497 + 0.918897i \(0.370919\pi\)
\(312\) 6.18015 0.349882
\(313\) 21.9666 1.24162 0.620812 0.783959i \(-0.286803\pi\)
0.620812 + 0.783959i \(0.286803\pi\)
\(314\) 12.2350 0.690462
\(315\) 5.61663 0.316461
\(316\) 3.64283 0.204925
\(317\) −3.38070 −0.189879 −0.0949395 0.995483i \(-0.530266\pi\)
−0.0949395 + 0.995483i \(0.530266\pi\)
\(318\) −11.3930 −0.638887
\(319\) −30.4842 −1.70679
\(320\) −2.40321 −0.134343
\(321\) 14.8420 0.828402
\(322\) 9.67872 0.539374
\(323\) 38.0299 2.11604
\(324\) −2.02411 −0.112450
\(325\) 3.96246 0.219798
\(326\) 23.6815 1.31159
\(327\) −17.1928 −0.950765
\(328\) −9.69619 −0.535383
\(329\) −11.4809 −0.632961
\(330\) −14.7539 −0.812174
\(331\) 21.9585 1.20695 0.603476 0.797382i \(-0.293782\pi\)
0.603476 + 0.797382i \(0.293782\pi\)
\(332\) −0.689504 −0.0378415
\(333\) 2.45080 0.134303
\(334\) 3.54694 0.194080
\(335\) 6.12542 0.334667
\(336\) −1.83846 −0.100296
\(337\) 3.21883 0.175341 0.0876705 0.996150i \(-0.472058\pi\)
0.0876705 + 0.996150i \(0.472058\pi\)
\(338\) −13.1142 −0.713321
\(339\) −15.3619 −0.834346
\(340\) 15.9997 0.867704
\(341\) 14.1033 0.763738
\(342\) −8.78205 −0.474879
\(343\) −17.7694 −0.959459
\(344\) 9.26120 0.499330
\(345\) −18.5045 −0.996246
\(346\) −20.8927 −1.12320
\(347\) 15.8474 0.850734 0.425367 0.905021i \(-0.360145\pi\)
0.425367 + 0.905021i \(0.360145\pi\)
\(348\) 7.26242 0.389306
\(349\) −17.0834 −0.914451 −0.457226 0.889351i \(-0.651157\pi\)
−0.457226 + 0.889351i \(0.651157\pi\)
\(350\) −1.17874 −0.0630066
\(351\) 28.0419 1.49677
\(352\) −5.07638 −0.270572
\(353\) −4.73633 −0.252090 −0.126045 0.992025i \(-0.540228\pi\)
−0.126045 + 0.992025i \(0.540228\pi\)
\(354\) −7.63516 −0.405804
\(355\) −4.78134 −0.253767
\(356\) 2.78972 0.147855
\(357\) 12.2398 0.647798
\(358\) −2.13073 −0.112612
\(359\) 18.2662 0.964051 0.482025 0.876157i \(-0.339901\pi\)
0.482025 + 0.876157i \(0.339901\pi\)
\(360\) −3.69473 −0.194729
\(361\) 13.6294 0.717338
\(362\) −18.5589 −0.975437
\(363\) −17.8620 −0.937514
\(364\) 7.76841 0.407175
\(365\) −30.6463 −1.60410
\(366\) −9.64981 −0.504404
\(367\) 5.99172 0.312765 0.156383 0.987697i \(-0.450017\pi\)
0.156383 + 0.987697i \(0.450017\pi\)
\(368\) −6.36685 −0.331895
\(369\) −14.9071 −0.776032
\(370\) −3.83095 −0.199162
\(371\) −14.3209 −0.743505
\(372\) −3.35991 −0.174203
\(373\) −16.9416 −0.877203 −0.438601 0.898682i \(-0.644526\pi\)
−0.438601 + 0.898682i \(0.644526\pi\)
\(374\) 33.7967 1.74759
\(375\) −12.2783 −0.634047
\(376\) 7.55235 0.389483
\(377\) −30.6873 −1.58048
\(378\) −8.34185 −0.429058
\(379\) −16.7293 −0.859325 −0.429663 0.902989i \(-0.641368\pi\)
−0.429663 + 0.902989i \(0.641368\pi\)
\(380\) 13.7276 0.704213
\(381\) 20.7770 1.06444
\(382\) −15.6213 −0.799257
\(383\) −26.8421 −1.37157 −0.685785 0.727804i \(-0.740541\pi\)
−0.685785 + 0.727804i \(0.740541\pi\)
\(384\) 1.20937 0.0617156
\(385\) −18.5455 −0.945167
\(386\) 18.5417 0.943747
\(387\) 14.2383 0.723774
\(388\) 11.9363 0.605972
\(389\) 8.89100 0.450792 0.225396 0.974267i \(-0.427632\pi\)
0.225396 + 0.974267i \(0.427632\pi\)
\(390\) −14.8522 −0.752069
\(391\) 42.3882 2.14366
\(392\) 4.68907 0.236834
\(393\) −6.32405 −0.319006
\(394\) 9.98285 0.502929
\(395\) −8.75448 −0.440486
\(396\) −7.80451 −0.392191
\(397\) −15.3011 −0.767939 −0.383970 0.923346i \(-0.625443\pi\)
−0.383970 + 0.923346i \(0.625443\pi\)
\(398\) −5.33431 −0.267385
\(399\) 10.5017 0.525741
\(400\) 0.775401 0.0387701
\(401\) −9.91869 −0.495316 −0.247658 0.968848i \(-0.579661\pi\)
−0.247658 + 0.968848i \(0.579661\pi\)
\(402\) −3.08251 −0.153742
\(403\) 14.1973 0.707218
\(404\) −14.5273 −0.722759
\(405\) 4.86434 0.241711
\(406\) 9.12880 0.453055
\(407\) −8.09227 −0.401119
\(408\) −8.05157 −0.398612
\(409\) −13.4737 −0.666233 −0.333117 0.942886i \(-0.608100\pi\)
−0.333117 + 0.942886i \(0.608100\pi\)
\(410\) 23.3020 1.15080
\(411\) 15.6812 0.773495
\(412\) −10.1048 −0.497828
\(413\) −9.59734 −0.472254
\(414\) −9.78850 −0.481079
\(415\) 1.65702 0.0813400
\(416\) −5.11021 −0.250549
\(417\) 0.837668 0.0410208
\(418\) 28.9974 1.41831
\(419\) −24.5320 −1.19847 −0.599233 0.800574i \(-0.704528\pi\)
−0.599233 + 0.800574i \(0.704528\pi\)
\(420\) 4.41819 0.215586
\(421\) −27.8189 −1.35581 −0.677906 0.735149i \(-0.737112\pi\)
−0.677906 + 0.735149i \(0.737112\pi\)
\(422\) 19.7945 0.963581
\(423\) 11.6111 0.564551
\(424\) 9.42058 0.457504
\(425\) −5.16234 −0.250410
\(426\) 2.40613 0.116577
\(427\) −12.1297 −0.586999
\(428\) −12.2725 −0.593214
\(429\) −31.3728 −1.51469
\(430\) −22.2566 −1.07331
\(431\) 32.5743 1.56905 0.784525 0.620097i \(-0.212907\pi\)
0.784525 + 0.620097i \(0.212907\pi\)
\(432\) 5.48743 0.264014
\(433\) −5.11693 −0.245904 −0.122952 0.992413i \(-0.539236\pi\)
−0.122952 + 0.992413i \(0.539236\pi\)
\(434\) −4.22338 −0.202729
\(435\) −17.4531 −0.836811
\(436\) 14.2163 0.680838
\(437\) 36.3688 1.73976
\(438\) 15.4222 0.736903
\(439\) 18.8040 0.897466 0.448733 0.893666i \(-0.351875\pi\)
0.448733 + 0.893666i \(0.351875\pi\)
\(440\) 12.1996 0.581593
\(441\) 7.20905 0.343288
\(442\) 34.0219 1.61826
\(443\) 16.7817 0.797325 0.398662 0.917098i \(-0.369475\pi\)
0.398662 + 0.917098i \(0.369475\pi\)
\(444\) 1.92786 0.0914924
\(445\) −6.70426 −0.317812
\(446\) −29.2824 −1.38656
\(447\) −14.0271 −0.663458
\(448\) 1.52017 0.0718215
\(449\) 29.1983 1.37795 0.688977 0.724784i \(-0.258060\pi\)
0.688977 + 0.724784i \(0.258060\pi\)
\(450\) 1.19211 0.0561968
\(451\) 49.2216 2.31775
\(452\) 12.7024 0.597471
\(453\) −1.38221 −0.0649418
\(454\) 5.90479 0.277126
\(455\) −18.6691 −0.875220
\(456\) −6.90820 −0.323506
\(457\) 6.74219 0.315387 0.157693 0.987488i \(-0.449594\pi\)
0.157693 + 0.987488i \(0.449594\pi\)
\(458\) −14.2761 −0.667078
\(459\) −36.5333 −1.70523
\(460\) 15.3009 0.713406
\(461\) 17.8901 0.833226 0.416613 0.909084i \(-0.363217\pi\)
0.416613 + 0.909084i \(0.363217\pi\)
\(462\) 9.33272 0.434197
\(463\) 17.2936 0.803703 0.401851 0.915705i \(-0.368367\pi\)
0.401851 + 0.915705i \(0.368367\pi\)
\(464\) −6.00511 −0.278780
\(465\) 8.07456 0.374449
\(466\) 27.0850 1.25469
\(467\) −3.00187 −0.138910 −0.0694550 0.997585i \(-0.522126\pi\)
−0.0694550 + 0.997585i \(0.522126\pi\)
\(468\) −7.85652 −0.363168
\(469\) −3.87470 −0.178917
\(470\) −18.1498 −0.837190
\(471\) 14.7967 0.681796
\(472\) 6.31332 0.290594
\(473\) −47.0134 −2.16168
\(474\) 4.40555 0.202354
\(475\) −4.42926 −0.203228
\(476\) −10.1208 −0.463884
\(477\) 14.4833 0.663147
\(478\) −7.97200 −0.364631
\(479\) −25.7701 −1.17747 −0.588734 0.808327i \(-0.700373\pi\)
−0.588734 + 0.808327i \(0.700373\pi\)
\(480\) −2.90637 −0.132657
\(481\) −8.14619 −0.371434
\(482\) 2.36516 0.107730
\(483\) 11.7052 0.532605
\(484\) 14.7697 0.671349
\(485\) −28.6853 −1.30253
\(486\) 14.0144 0.635706
\(487\) 9.85904 0.446756 0.223378 0.974732i \(-0.428292\pi\)
0.223378 + 0.974732i \(0.428292\pi\)
\(488\) 7.97918 0.361200
\(489\) 28.6397 1.29513
\(490\) −11.2688 −0.509073
\(491\) 9.75565 0.440266 0.220133 0.975470i \(-0.429351\pi\)
0.220133 + 0.975470i \(0.429351\pi\)
\(492\) −11.7263 −0.528663
\(493\) 39.9798 1.80060
\(494\) 29.1906 1.31335
\(495\) 18.7559 0.843013
\(496\) 2.77822 0.124746
\(497\) 3.02448 0.135667
\(498\) −0.833868 −0.0373665
\(499\) −3.06163 −0.137058 −0.0685288 0.997649i \(-0.521830\pi\)
−0.0685288 + 0.997649i \(0.521830\pi\)
\(500\) 10.1526 0.454037
\(501\) 4.28957 0.191644
\(502\) 23.0435 1.02848
\(503\) −15.0940 −0.673010 −0.336505 0.941682i \(-0.609245\pi\)
−0.336505 + 0.941682i \(0.609245\pi\)
\(504\) 2.33714 0.104104
\(505\) 34.9121 1.55357
\(506\) 32.3206 1.43683
\(507\) −15.8600 −0.704368
\(508\) −17.1800 −0.762238
\(509\) 13.2575 0.587629 0.293814 0.955863i \(-0.405075\pi\)
0.293814 + 0.955863i \(0.405075\pi\)
\(510\) 19.3496 0.856814
\(511\) 19.3856 0.857570
\(512\) −1.00000 −0.0441942
\(513\) −31.3454 −1.38393
\(514\) −23.2436 −1.02523
\(515\) 24.2839 1.07008
\(516\) 11.2002 0.493063
\(517\) −38.3386 −1.68613
\(518\) 2.42331 0.106474
\(519\) −25.2670 −1.10910
\(520\) 12.2809 0.538553
\(521\) 20.2639 0.887776 0.443888 0.896082i \(-0.353599\pi\)
0.443888 + 0.896082i \(0.353599\pi\)
\(522\) −9.23234 −0.404089
\(523\) −2.18471 −0.0955307 −0.0477654 0.998859i \(-0.515210\pi\)
−0.0477654 + 0.998859i \(0.515210\pi\)
\(524\) 5.22919 0.228438
\(525\) −1.42554 −0.0622158
\(526\) −4.62308 −0.201576
\(527\) −18.4964 −0.805716
\(528\) −6.13924 −0.267176
\(529\) 17.5368 0.762470
\(530\) −22.6396 −0.983401
\(531\) 9.70620 0.421213
\(532\) −8.68356 −0.376480
\(533\) 49.5496 2.14623
\(534\) 3.37381 0.145999
\(535\) 29.4933 1.27511
\(536\) 2.54885 0.110094
\(537\) −2.57685 −0.111199
\(538\) 1.00000 0.0431131
\(539\) −23.8035 −1.02529
\(540\) −13.1874 −0.567497
\(541\) 21.9779 0.944903 0.472452 0.881357i \(-0.343369\pi\)
0.472452 + 0.881357i \(0.343369\pi\)
\(542\) −12.4311 −0.533961
\(543\) −22.4447 −0.963194
\(544\) 6.65764 0.285444
\(545\) −34.1647 −1.46346
\(546\) 9.39490 0.402065
\(547\) −2.59714 −0.111046 −0.0555229 0.998457i \(-0.517683\pi\)
−0.0555229 + 0.998457i \(0.517683\pi\)
\(548\) −12.9664 −0.553895
\(549\) 12.2673 0.523556
\(550\) −3.93623 −0.167842
\(551\) 34.3025 1.46133
\(552\) −7.69990 −0.327730
\(553\) 5.53774 0.235489
\(554\) 4.92762 0.209354
\(555\) −4.63305 −0.196662
\(556\) −0.692646 −0.0293747
\(557\) 32.9165 1.39471 0.697357 0.716724i \(-0.254359\pi\)
0.697357 + 0.716724i \(0.254359\pi\)
\(558\) 4.27129 0.180818
\(559\) −47.3266 −2.00170
\(560\) −3.65329 −0.154380
\(561\) 40.8729 1.72565
\(562\) 21.3194 0.899306
\(563\) −16.1700 −0.681485 −0.340742 0.940157i \(-0.610678\pi\)
−0.340742 + 0.940157i \(0.610678\pi\)
\(564\) 9.13361 0.384594
\(565\) −30.5265 −1.28426
\(566\) 29.9997 1.26098
\(567\) −3.07699 −0.129222
\(568\) −1.98957 −0.0834803
\(569\) −11.9495 −0.500951 −0.250476 0.968123i \(-0.580587\pi\)
−0.250476 + 0.968123i \(0.580587\pi\)
\(570\) 16.6018 0.695375
\(571\) −18.1326 −0.758824 −0.379412 0.925228i \(-0.623874\pi\)
−0.379412 + 0.925228i \(0.623874\pi\)
\(572\) 25.9414 1.08466
\(573\) −18.8920 −0.789225
\(574\) −14.7399 −0.615232
\(575\) −4.93687 −0.205882
\(576\) −1.53742 −0.0640590
\(577\) 22.5512 0.938817 0.469408 0.882981i \(-0.344467\pi\)
0.469408 + 0.882981i \(0.344467\pi\)
\(578\) −27.3241 −1.13653
\(579\) 22.4238 0.931903
\(580\) 14.4315 0.599236
\(581\) −1.04817 −0.0434853
\(582\) 14.4354 0.598367
\(583\) −47.8225 −1.98060
\(584\) −12.7523 −0.527692
\(585\) 18.8808 0.780626
\(586\) −16.9481 −0.700121
\(587\) 22.1880 0.915796 0.457898 0.889005i \(-0.348603\pi\)
0.457898 + 0.889005i \(0.348603\pi\)
\(588\) 5.67084 0.233861
\(589\) −15.8698 −0.653905
\(590\) −15.1722 −0.624630
\(591\) 12.0730 0.496617
\(592\) −1.59410 −0.0655172
\(593\) 30.4282 1.24954 0.624768 0.780811i \(-0.285194\pi\)
0.624768 + 0.780811i \(0.285194\pi\)
\(594\) −27.8563 −1.14296
\(595\) 24.3223 0.997117
\(596\) 11.5986 0.475099
\(597\) −6.45118 −0.264029
\(598\) 32.5360 1.33049
\(599\) 15.6413 0.639086 0.319543 0.947572i \(-0.396471\pi\)
0.319543 + 0.947572i \(0.396471\pi\)
\(600\) 0.937750 0.0382835
\(601\) −19.9435 −0.813512 −0.406756 0.913537i \(-0.633340\pi\)
−0.406756 + 0.913537i \(0.633340\pi\)
\(602\) 14.0786 0.573802
\(603\) 3.91864 0.159580
\(604\) 1.14291 0.0465045
\(605\) −35.4946 −1.44306
\(606\) −17.5689 −0.713688
\(607\) −11.4793 −0.465932 −0.232966 0.972485i \(-0.574843\pi\)
−0.232966 + 0.972485i \(0.574843\pi\)
\(608\) 5.71222 0.231661
\(609\) 11.0401 0.447369
\(610\) −19.1756 −0.776398
\(611\) −38.5941 −1.56135
\(612\) 10.2356 0.413748
\(613\) 11.2663 0.455044 0.227522 0.973773i \(-0.426938\pi\)
0.227522 + 0.973773i \(0.426938\pi\)
\(614\) −18.4715 −0.745449
\(615\) 28.1808 1.13636
\(616\) −7.71699 −0.310926
\(617\) 10.0317 0.403861 0.201930 0.979400i \(-0.435279\pi\)
0.201930 + 0.979400i \(0.435279\pi\)
\(618\) −12.2205 −0.491580
\(619\) 10.0658 0.404579 0.202289 0.979326i \(-0.435162\pi\)
0.202289 + 0.979326i \(0.435162\pi\)
\(620\) −6.67665 −0.268141
\(621\) −34.9377 −1.40200
\(622\) −13.9141 −0.557904
\(623\) 4.24085 0.169906
\(624\) −6.18015 −0.247404
\(625\) −28.2758 −1.13103
\(626\) −21.9666 −0.877961
\(627\) 35.0687 1.40051
\(628\) −12.2350 −0.488230
\(629\) 10.6129 0.423166
\(630\) −5.61663 −0.223772
\(631\) −35.3530 −1.40738 −0.703691 0.710506i \(-0.748466\pi\)
−0.703691 + 0.710506i \(0.748466\pi\)
\(632\) −3.64283 −0.144904
\(633\) 23.9389 0.951488
\(634\) 3.38070 0.134265
\(635\) 41.2870 1.63842
\(636\) 11.3930 0.451762
\(637\) −23.9621 −0.949414
\(638\) 30.4842 1.20688
\(639\) −3.05879 −0.121004
\(640\) 2.40321 0.0949951
\(641\) 0.166926 0.00659316 0.00329658 0.999995i \(-0.498951\pi\)
0.00329658 + 0.999995i \(0.498951\pi\)
\(642\) −14.8420 −0.585768
\(643\) −30.9571 −1.22083 −0.610415 0.792082i \(-0.708997\pi\)
−0.610415 + 0.792082i \(0.708997\pi\)
\(644\) −9.67872 −0.381395
\(645\) −26.9165 −1.05984
\(646\) −38.0299 −1.49626
\(647\) −0.959927 −0.0377386 −0.0188693 0.999822i \(-0.506007\pi\)
−0.0188693 + 0.999822i \(0.506007\pi\)
\(648\) 2.02411 0.0795144
\(649\) −32.0488 −1.25803
\(650\) −3.96246 −0.155421
\(651\) −5.10765 −0.200185
\(652\) −23.6815 −0.927438
\(653\) 27.5173 1.07684 0.538418 0.842678i \(-0.319022\pi\)
0.538418 + 0.842678i \(0.319022\pi\)
\(654\) 17.1928 0.672293
\(655\) −12.5668 −0.491027
\(656\) 9.69619 0.378573
\(657\) −19.6055 −0.764884
\(658\) 11.4809 0.447571
\(659\) 3.87281 0.150863 0.0754316 0.997151i \(-0.475967\pi\)
0.0754316 + 0.997151i \(0.475967\pi\)
\(660\) 14.7539 0.574294
\(661\) 2.60845 0.101457 0.0507285 0.998712i \(-0.483846\pi\)
0.0507285 + 0.998712i \(0.483846\pi\)
\(662\) −21.9585 −0.853443
\(663\) 41.1452 1.59795
\(664\) 0.689504 0.0267580
\(665\) 20.8684 0.809242
\(666\) −2.45080 −0.0949664
\(667\) 38.2336 1.48041
\(668\) −3.54694 −0.137235
\(669\) −35.4134 −1.36916
\(670\) −6.12542 −0.236645
\(671\) −40.5054 −1.56369
\(672\) 1.83846 0.0709201
\(673\) −12.9907 −0.500753 −0.250376 0.968149i \(-0.580554\pi\)
−0.250376 + 0.968149i \(0.580554\pi\)
\(674\) −3.21883 −0.123985
\(675\) 4.25496 0.163774
\(676\) 13.1142 0.504394
\(677\) 30.7406 1.18146 0.590728 0.806871i \(-0.298841\pi\)
0.590728 + 0.806871i \(0.298841\pi\)
\(678\) 15.3619 0.589972
\(679\) 18.1452 0.696349
\(680\) −15.9997 −0.613560
\(681\) 7.14110 0.273648
\(682\) −14.1033 −0.540045
\(683\) −6.20328 −0.237362 −0.118681 0.992932i \(-0.537867\pi\)
−0.118681 + 0.992932i \(0.537867\pi\)
\(684\) 8.78205 0.335790
\(685\) 31.1608 1.19059
\(686\) 17.7694 0.678440
\(687\) −17.2651 −0.658706
\(688\) −9.26120 −0.353080
\(689\) −48.1411 −1.83403
\(690\) 18.5045 0.704453
\(691\) 18.1544 0.690626 0.345313 0.938488i \(-0.387773\pi\)
0.345313 + 0.938488i \(0.387773\pi\)
\(692\) 20.8927 0.794220
\(693\) −11.8642 −0.450684
\(694\) −15.8474 −0.601560
\(695\) 1.66457 0.0631408
\(696\) −7.26242 −0.275281
\(697\) −64.5537 −2.44515
\(698\) 17.0834 0.646615
\(699\) 32.7559 1.23894
\(700\) 1.17874 0.0445524
\(701\) −46.2492 −1.74681 −0.873404 0.486996i \(-0.838093\pi\)
−0.873404 + 0.486996i \(0.838093\pi\)
\(702\) −28.0419 −1.05837
\(703\) 9.10585 0.343434
\(704\) 5.07638 0.191323
\(705\) −21.9499 −0.826683
\(706\) 4.73633 0.178254
\(707\) −22.0840 −0.830554
\(708\) 7.63516 0.286947
\(709\) −35.6219 −1.33781 −0.668904 0.743349i \(-0.733236\pi\)
−0.668904 + 0.743349i \(0.733236\pi\)
\(710\) 4.78134 0.179440
\(711\) −5.60055 −0.210037
\(712\) −2.78972 −0.104549
\(713\) −17.6885 −0.662441
\(714\) −12.2398 −0.458062
\(715\) −62.3425 −2.33148
\(716\) 2.13073 0.0796290
\(717\) −9.64112 −0.360054
\(718\) −18.2662 −0.681687
\(719\) −37.4220 −1.39561 −0.697803 0.716290i \(-0.745839\pi\)
−0.697803 + 0.716290i \(0.745839\pi\)
\(720\) 3.69473 0.137694
\(721\) −15.3611 −0.572076
\(722\) −13.6294 −0.507234
\(723\) 2.86036 0.106378
\(724\) 18.5589 0.689738
\(725\) −4.65637 −0.172933
\(726\) 17.8620 0.662923
\(727\) −24.4331 −0.906172 −0.453086 0.891467i \(-0.649677\pi\)
−0.453086 + 0.891467i \(0.649677\pi\)
\(728\) −7.76841 −0.287916
\(729\) 23.0209 0.852628
\(730\) 30.6463 1.13427
\(731\) 61.6577 2.28049
\(732\) 9.64981 0.356667
\(733\) 37.4726 1.38408 0.692042 0.721857i \(-0.256711\pi\)
0.692042 + 0.721857i \(0.256711\pi\)
\(734\) −5.99172 −0.221159
\(735\) −13.6282 −0.502684
\(736\) 6.36685 0.234685
\(737\) −12.9389 −0.476612
\(738\) 14.9071 0.548737
\(739\) −18.6134 −0.684707 −0.342353 0.939571i \(-0.611224\pi\)
−0.342353 + 0.939571i \(0.611224\pi\)
\(740\) 3.83095 0.140829
\(741\) 35.3024 1.29686
\(742\) 14.3209 0.525737
\(743\) 0.773612 0.0283811 0.0141905 0.999899i \(-0.495483\pi\)
0.0141905 + 0.999899i \(0.495483\pi\)
\(744\) 3.35991 0.123180
\(745\) −27.8739 −1.02122
\(746\) 16.9416 0.620276
\(747\) 1.06005 0.0387854
\(748\) −33.7967 −1.23573
\(749\) −18.6563 −0.681688
\(750\) 12.2783 0.448339
\(751\) −11.0543 −0.403377 −0.201689 0.979450i \(-0.564643\pi\)
−0.201689 + 0.979450i \(0.564643\pi\)
\(752\) −7.55235 −0.275406
\(753\) 27.8682 1.01557
\(754\) 30.6873 1.11757
\(755\) −2.74666 −0.0999610
\(756\) 8.34185 0.303390
\(757\) 31.9235 1.16028 0.580140 0.814517i \(-0.302998\pi\)
0.580140 + 0.814517i \(0.302998\pi\)
\(758\) 16.7293 0.607635
\(759\) 39.0877 1.41879
\(760\) −13.7276 −0.497954
\(761\) 14.7166 0.533477 0.266739 0.963769i \(-0.414054\pi\)
0.266739 + 0.963769i \(0.414054\pi\)
\(762\) −20.7770 −0.752671
\(763\) 21.6113 0.782380
\(764\) 15.6213 0.565160
\(765\) −24.5982 −0.889348
\(766\) 26.8421 0.969846
\(767\) −32.2624 −1.16493
\(768\) −1.20937 −0.0436395
\(769\) −18.0186 −0.649768 −0.324884 0.945754i \(-0.605325\pi\)
−0.324884 + 0.945754i \(0.605325\pi\)
\(770\) 18.5455 0.668334
\(771\) −28.1102 −1.01236
\(772\) −18.5417 −0.667330
\(773\) 48.8663 1.75760 0.878799 0.477193i \(-0.158346\pi\)
0.878799 + 0.477193i \(0.158346\pi\)
\(774\) −14.2383 −0.511785
\(775\) 2.15424 0.0773825
\(776\) −11.9363 −0.428487
\(777\) 2.93069 0.105138
\(778\) −8.89100 −0.318758
\(779\) −55.3868 −1.98444
\(780\) 14.8522 0.531793
\(781\) 10.0998 0.361399
\(782\) −42.3882 −1.51580
\(783\) −32.9526 −1.17763
\(784\) −4.68907 −0.167467
\(785\) 29.4033 1.04945
\(786\) 6.32405 0.225571
\(787\) −31.8942 −1.13691 −0.568453 0.822715i \(-0.692458\pi\)
−0.568453 + 0.822715i \(0.692458\pi\)
\(788\) −9.98285 −0.355624
\(789\) −5.59103 −0.199046
\(790\) 8.75448 0.311471
\(791\) 19.3099 0.686579
\(792\) 7.80451 0.277321
\(793\) −40.7753 −1.44797
\(794\) 15.3011 0.543015
\(795\) −27.3797 −0.971059
\(796\) 5.33431 0.189070
\(797\) −26.0846 −0.923964 −0.461982 0.886889i \(-0.652861\pi\)
−0.461982 + 0.886889i \(0.652861\pi\)
\(798\) −10.5017 −0.371755
\(799\) 50.2808 1.77881
\(800\) −0.775401 −0.0274146
\(801\) −4.28895 −0.151543
\(802\) 9.91869 0.350241
\(803\) 64.7353 2.28446
\(804\) 3.08251 0.108712
\(805\) 23.2600 0.819806
\(806\) −14.1973 −0.500079
\(807\) 1.20937 0.0425720
\(808\) 14.5273 0.511068
\(809\) −2.45878 −0.0864461 −0.0432231 0.999065i \(-0.513763\pi\)
−0.0432231 + 0.999065i \(0.513763\pi\)
\(810\) −4.86434 −0.170916
\(811\) −20.3841 −0.715783 −0.357891 0.933763i \(-0.616504\pi\)
−0.357891 + 0.933763i \(0.616504\pi\)
\(812\) −9.12880 −0.320358
\(813\) −15.0338 −0.527260
\(814\) 8.09227 0.283634
\(815\) 56.9114 1.99352
\(816\) 8.05157 0.281861
\(817\) 52.9020 1.85081
\(818\) 13.4737 0.471098
\(819\) −11.9433 −0.417332
\(820\) −23.3020 −0.813740
\(821\) −8.31824 −0.290309 −0.145154 0.989409i \(-0.546368\pi\)
−0.145154 + 0.989409i \(0.546368\pi\)
\(822\) −15.6812 −0.546943
\(823\) −25.5967 −0.892244 −0.446122 0.894972i \(-0.647195\pi\)
−0.446122 + 0.894972i \(0.647195\pi\)
\(824\) 10.1048 0.352017
\(825\) −4.76038 −0.165735
\(826\) 9.59734 0.333934
\(827\) 28.4900 0.990693 0.495347 0.868695i \(-0.335041\pi\)
0.495347 + 0.868695i \(0.335041\pi\)
\(828\) 9.78850 0.340174
\(829\) −15.2875 −0.530956 −0.265478 0.964117i \(-0.585530\pi\)
−0.265478 + 0.964117i \(0.585530\pi\)
\(830\) −1.65702 −0.0575161
\(831\) 5.95933 0.206727
\(832\) 5.11021 0.177165
\(833\) 31.2181 1.08164
\(834\) −0.837668 −0.0290061
\(835\) 8.52403 0.294986
\(836\) −28.9974 −1.00290
\(837\) 15.2453 0.526955
\(838\) 24.5320 0.847444
\(839\) 33.9640 1.17257 0.586283 0.810106i \(-0.300591\pi\)
0.586283 + 0.810106i \(0.300591\pi\)
\(840\) −4.41819 −0.152442
\(841\) 7.06129 0.243493
\(842\) 27.8189 0.958704
\(843\) 25.7832 0.888019
\(844\) −19.7945 −0.681355
\(845\) −31.5162 −1.08419
\(846\) −11.6111 −0.399198
\(847\) 22.4525 0.771476
\(848\) −9.42058 −0.323504
\(849\) 36.2808 1.24516
\(850\) 5.16234 0.177067
\(851\) 10.1494 0.347917
\(852\) −2.40613 −0.0824326
\(853\) −18.6395 −0.638203 −0.319102 0.947720i \(-0.603381\pi\)
−0.319102 + 0.947720i \(0.603381\pi\)
\(854\) 12.1297 0.415071
\(855\) −21.1051 −0.721779
\(856\) 12.2725 0.419465
\(857\) 38.5929 1.31831 0.659154 0.752008i \(-0.270915\pi\)
0.659154 + 0.752008i \(0.270915\pi\)
\(858\) 31.3728 1.07105
\(859\) 36.7533 1.25400 0.627002 0.779017i \(-0.284282\pi\)
0.627002 + 0.779017i \(0.284282\pi\)
\(860\) 22.2566 0.758943
\(861\) −17.8260 −0.607510
\(862\) −32.5743 −1.10949
\(863\) 39.5677 1.34690 0.673450 0.739233i \(-0.264812\pi\)
0.673450 + 0.739233i \(0.264812\pi\)
\(864\) −5.48743 −0.186686
\(865\) −50.2094 −1.70717
\(866\) 5.11693 0.173880
\(867\) −33.0451 −1.12227
\(868\) 4.22338 0.143351
\(869\) 18.4924 0.627313
\(870\) 17.4531 0.591715
\(871\) −13.0252 −0.441341
\(872\) −14.2163 −0.481425
\(873\) −18.3510 −0.621087
\(874\) −36.3688 −1.23019
\(875\) 15.4337 0.521754
\(876\) −15.4222 −0.521069
\(877\) 4.55367 0.153766 0.0768832 0.997040i \(-0.475503\pi\)
0.0768832 + 0.997040i \(0.475503\pi\)
\(878\) −18.8040 −0.634604
\(879\) −20.4966 −0.691334
\(880\) −12.1996 −0.411248
\(881\) −16.4979 −0.555828 −0.277914 0.960606i \(-0.589643\pi\)
−0.277914 + 0.960606i \(0.589643\pi\)
\(882\) −7.20905 −0.242741
\(883\) −43.4286 −1.46149 −0.730744 0.682652i \(-0.760827\pi\)
−0.730744 + 0.682652i \(0.760827\pi\)
\(884\) −34.0219 −1.14428
\(885\) −18.3489 −0.616790
\(886\) −16.7817 −0.563794
\(887\) −9.36355 −0.314397 −0.157199 0.987567i \(-0.550246\pi\)
−0.157199 + 0.987567i \(0.550246\pi\)
\(888\) −1.92786 −0.0646949
\(889\) −26.1165 −0.875920
\(890\) 6.70426 0.224727
\(891\) −10.2751 −0.344230
\(892\) 29.2824 0.980447
\(893\) 43.1406 1.44365
\(894\) 14.0271 0.469136
\(895\) −5.12058 −0.171162
\(896\) −1.52017 −0.0507854
\(897\) 39.3481 1.31380
\(898\) −29.1983 −0.974360
\(899\) −16.6835 −0.556427
\(900\) −1.19211 −0.0397371
\(901\) 62.7188 2.08947
\(902\) −49.2216 −1.63890
\(903\) 17.0263 0.566600
\(904\) −12.7024 −0.422476
\(905\) −44.6010 −1.48259
\(906\) 1.38221 0.0459208
\(907\) −43.0813 −1.43049 −0.715246 0.698873i \(-0.753685\pi\)
−0.715246 + 0.698873i \(0.753685\pi\)
\(908\) −5.90479 −0.195957
\(909\) 22.3345 0.740788
\(910\) 18.6691 0.618874
\(911\) −3.70489 −0.122749 −0.0613743 0.998115i \(-0.519548\pi\)
−0.0613743 + 0.998115i \(0.519548\pi\)
\(912\) 6.90820 0.228753
\(913\) −3.50019 −0.115839
\(914\) −6.74219 −0.223012
\(915\) −23.1905 −0.766654
\(916\) 14.2761 0.471695
\(917\) 7.94928 0.262508
\(918\) 36.5333 1.20578
\(919\) 42.2046 1.39220 0.696101 0.717944i \(-0.254917\pi\)
0.696101 + 0.717944i \(0.254917\pi\)
\(920\) −15.3009 −0.504454
\(921\) −22.3390 −0.736093
\(922\) −17.8901 −0.589180
\(923\) 10.1671 0.334654
\(924\) −9.33272 −0.307024
\(925\) −1.23607 −0.0406417
\(926\) −17.2936 −0.568304
\(927\) 15.5353 0.510246
\(928\) 6.00511 0.197127
\(929\) −40.9616 −1.34391 −0.671954 0.740593i \(-0.734545\pi\)
−0.671954 + 0.740593i \(0.734545\pi\)
\(930\) −8.07456 −0.264775
\(931\) 26.7850 0.877843
\(932\) −27.0850 −0.887198
\(933\) −16.8273 −0.550901
\(934\) 3.00187 0.0982243
\(935\) 81.2205 2.65619
\(936\) 7.85652 0.256798
\(937\) −11.2419 −0.367258 −0.183629 0.982996i \(-0.558785\pi\)
−0.183629 + 0.982996i \(0.558785\pi\)
\(938\) 3.87470 0.126513
\(939\) −26.5658 −0.866942
\(940\) 18.1498 0.591983
\(941\) 1.14007 0.0371652 0.0185826 0.999827i \(-0.494085\pi\)
0.0185826 + 0.999827i \(0.494085\pi\)
\(942\) −14.7967 −0.482102
\(943\) −61.7342 −2.01034
\(944\) −6.31332 −0.205481
\(945\) −20.0472 −0.652135
\(946\) 47.0134 1.52854
\(947\) −20.5389 −0.667424 −0.333712 0.942675i \(-0.608301\pi\)
−0.333712 + 0.942675i \(0.608301\pi\)
\(948\) −4.40555 −0.143086
\(949\) 65.1667 2.11540
\(950\) 4.42926 0.143704
\(951\) 4.08853 0.132580
\(952\) 10.1208 0.328016
\(953\) −18.6016 −0.602567 −0.301283 0.953535i \(-0.597415\pi\)
−0.301283 + 0.953535i \(0.597415\pi\)
\(954\) −14.4833 −0.468916
\(955\) −37.5413 −1.21481
\(956\) 7.97200 0.257833
\(957\) 36.8668 1.19173
\(958\) 25.7701 0.832595
\(959\) −19.7111 −0.636505
\(960\) 2.90637 0.0938028
\(961\) −23.2815 −0.751015
\(962\) 8.14619 0.262644
\(963\) 18.8679 0.608011
\(964\) −2.36516 −0.0761767
\(965\) 44.5595 1.43442
\(966\) −11.7052 −0.376608
\(967\) −21.3331 −0.686026 −0.343013 0.939331i \(-0.611447\pi\)
−0.343013 + 0.939331i \(0.611447\pi\)
\(968\) −14.7697 −0.474715
\(969\) −45.9923 −1.47749
\(970\) 28.6853 0.921030
\(971\) 13.0827 0.419844 0.209922 0.977718i \(-0.432679\pi\)
0.209922 + 0.977718i \(0.432679\pi\)
\(972\) −14.0144 −0.449512
\(973\) −1.05294 −0.0337558
\(974\) −9.85904 −0.315904
\(975\) −4.79210 −0.153470
\(976\) −7.97918 −0.255407
\(977\) −22.6902 −0.725924 −0.362962 0.931804i \(-0.618234\pi\)
−0.362962 + 0.931804i \(0.618234\pi\)
\(978\) −28.6397 −0.915798
\(979\) 14.1617 0.452609
\(980\) 11.2688 0.359969
\(981\) −21.8564 −0.697820
\(982\) −9.75565 −0.311315
\(983\) −51.3612 −1.63817 −0.819084 0.573674i \(-0.805518\pi\)
−0.819084 + 0.573674i \(0.805518\pi\)
\(984\) 11.7263 0.373821
\(985\) 23.9909 0.764412
\(986\) −39.9798 −1.27322
\(987\) 13.8847 0.441954
\(988\) −29.1906 −0.928677
\(989\) 58.9647 1.87497
\(990\) −18.7559 −0.596100
\(991\) −11.0015 −0.349475 −0.174737 0.984615i \(-0.555908\pi\)
−0.174737 + 0.984615i \(0.555908\pi\)
\(992\) −2.77822 −0.0882087
\(993\) −26.5561 −0.842732
\(994\) −3.02448 −0.0959308
\(995\) −12.8195 −0.406404
\(996\) 0.833868 0.0264221
\(997\) 12.2415 0.387693 0.193846 0.981032i \(-0.437904\pi\)
0.193846 + 0.981032i \(0.437904\pi\)
\(998\) 3.06163 0.0969143
\(999\) −8.74752 −0.276759
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 538.2.a.d.1.3 7
3.2 odd 2 4842.2.a.o.1.5 7
4.3 odd 2 4304.2.a.i.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.d.1.3 7 1.1 even 1 trivial
4304.2.a.i.1.5 7 4.3 odd 2
4842.2.a.o.1.5 7 3.2 odd 2