Properties

Label 6034.2.a.k.1.6
Level $6034$
Weight $2$
Character 6034.1
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $20$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + 2170 x^{12} - 42069 x^{11} + 19553 x^{10} + 84697 x^{9} - 82713 x^{8} + \cdots - 44 \) 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 1.6
Root \(-1.63154\) of defining polynomial
Character \(\chi\) \(=\) 6034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.63154 q^{3} +1.00000 q^{4} +1.13171 q^{5} +1.63154 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.338090 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.63154 q^{3} +1.00000 q^{4} +1.13171 q^{5} +1.63154 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.338090 q^{9} -1.13171 q^{10} +4.25957 q^{11} -1.63154 q^{12} +4.57626 q^{13} -1.00000 q^{14} -1.84643 q^{15} +1.00000 q^{16} -4.46210 q^{17} +0.338090 q^{18} +4.98269 q^{19} +1.13171 q^{20} -1.63154 q^{21} -4.25957 q^{22} -2.19484 q^{23} +1.63154 q^{24} -3.71923 q^{25} -4.57626 q^{26} +5.44621 q^{27} +1.00000 q^{28} -8.18975 q^{29} +1.84643 q^{30} -0.537753 q^{31} -1.00000 q^{32} -6.94963 q^{33} +4.46210 q^{34} +1.13171 q^{35} -0.338090 q^{36} -0.285823 q^{37} -4.98269 q^{38} -7.46634 q^{39} -1.13171 q^{40} -6.78124 q^{41} +1.63154 q^{42} +0.00276006 q^{43} +4.25957 q^{44} -0.382620 q^{45} +2.19484 q^{46} -7.84608 q^{47} -1.63154 q^{48} +1.00000 q^{49} +3.71923 q^{50} +7.28007 q^{51} +4.57626 q^{52} -0.524949 q^{53} -5.44621 q^{54} +4.82059 q^{55} -1.00000 q^{56} -8.12943 q^{57} +8.18975 q^{58} -11.0613 q^{59} -1.84643 q^{60} -11.2944 q^{61} +0.537753 q^{62} -0.338090 q^{63} +1.00000 q^{64} +5.17900 q^{65} +6.94963 q^{66} -5.88748 q^{67} -4.46210 q^{68} +3.58097 q^{69} -1.13171 q^{70} +7.34167 q^{71} +0.338090 q^{72} +2.76410 q^{73} +0.285823 q^{74} +6.06806 q^{75} +4.98269 q^{76} +4.25957 q^{77} +7.46634 q^{78} +7.07373 q^{79} +1.13171 q^{80} -7.87142 q^{81} +6.78124 q^{82} -10.8428 q^{83} -1.63154 q^{84} -5.04980 q^{85} -0.00276006 q^{86} +13.3619 q^{87} -4.25957 q^{88} -7.78810 q^{89} +0.382620 q^{90} +4.57626 q^{91} -2.19484 q^{92} +0.877364 q^{93} +7.84608 q^{94} +5.63896 q^{95} +1.63154 q^{96} +15.5210 q^{97} -1.00000 q^{98} -1.44012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 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
\(3\) −1.63154 −0.941968 −0.470984 0.882142i \(-0.656101\pi\)
−0.470984 + 0.882142i \(0.656101\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.13171 0.506116 0.253058 0.967451i \(-0.418564\pi\)
0.253058 + 0.967451i \(0.418564\pi\)
\(6\) 1.63154 0.666072
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −0.338090 −0.112697
\(10\) −1.13171 −0.357878
\(11\) 4.25957 1.28431 0.642154 0.766576i \(-0.278041\pi\)
0.642154 + 0.766576i \(0.278041\pi\)
\(12\) −1.63154 −0.470984
\(13\) 4.57626 1.26923 0.634614 0.772830i \(-0.281159\pi\)
0.634614 + 0.772830i \(0.281159\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.84643 −0.476745
\(16\) 1.00000 0.250000
\(17\) −4.46210 −1.08222 −0.541109 0.840953i \(-0.681995\pi\)
−0.541109 + 0.840953i \(0.681995\pi\)
\(18\) 0.338090 0.0796887
\(19\) 4.98269 1.14311 0.571553 0.820565i \(-0.306341\pi\)
0.571553 + 0.820565i \(0.306341\pi\)
\(20\) 1.13171 0.253058
\(21\) −1.63154 −0.356030
\(22\) −4.25957 −0.908142
\(23\) −2.19484 −0.457657 −0.228828 0.973467i \(-0.573489\pi\)
−0.228828 + 0.973467i \(0.573489\pi\)
\(24\) 1.63154 0.333036
\(25\) −3.71923 −0.743846
\(26\) −4.57626 −0.897479
\(27\) 5.44621 1.04812
\(28\) 1.00000 0.188982
\(29\) −8.18975 −1.52080 −0.760399 0.649456i \(-0.774997\pi\)
−0.760399 + 0.649456i \(0.774997\pi\)
\(30\) 1.84643 0.337110
\(31\) −0.537753 −0.0965833 −0.0482917 0.998833i \(-0.515378\pi\)
−0.0482917 + 0.998833i \(0.515378\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.94963 −1.20978
\(34\) 4.46210 0.765243
\(35\) 1.13171 0.191294
\(36\) −0.338090 −0.0563484
\(37\) −0.285823 −0.0469889 −0.0234945 0.999724i \(-0.507479\pi\)
−0.0234945 + 0.999724i \(0.507479\pi\)
\(38\) −4.98269 −0.808299
\(39\) −7.46634 −1.19557
\(40\) −1.13171 −0.178939
\(41\) −6.78124 −1.05905 −0.529526 0.848294i \(-0.677630\pi\)
−0.529526 + 0.848294i \(0.677630\pi\)
\(42\) 1.63154 0.251751
\(43\) 0.00276006 0.000420905 0 0.000210453 1.00000i \(-0.499933\pi\)
0.000210453 1.00000i \(0.499933\pi\)
\(44\) 4.25957 0.642154
\(45\) −0.382620 −0.0570377
\(46\) 2.19484 0.323612
\(47\) −7.84608 −1.14447 −0.572234 0.820090i \(-0.693923\pi\)
−0.572234 + 0.820090i \(0.693923\pi\)
\(48\) −1.63154 −0.235492
\(49\) 1.00000 0.142857
\(50\) 3.71923 0.525979
\(51\) 7.28007 1.01941
\(52\) 4.57626 0.634614
\(53\) −0.524949 −0.0721073 −0.0360537 0.999350i \(-0.511479\pi\)
−0.0360537 + 0.999350i \(0.511479\pi\)
\(54\) −5.44621 −0.741136
\(55\) 4.82059 0.650009
\(56\) −1.00000 −0.133631
\(57\) −8.12943 −1.07677
\(58\) 8.18975 1.07537
\(59\) −11.0613 −1.44006 −0.720028 0.693945i \(-0.755871\pi\)
−0.720028 + 0.693945i \(0.755871\pi\)
\(60\) −1.84643 −0.238373
\(61\) −11.2944 −1.44610 −0.723048 0.690798i \(-0.757260\pi\)
−0.723048 + 0.690798i \(0.757260\pi\)
\(62\) 0.537753 0.0682947
\(63\) −0.338090 −0.0425954
\(64\) 1.00000 0.125000
\(65\) 5.17900 0.642376
\(66\) 6.94963 0.855441
\(67\) −5.88748 −0.719270 −0.359635 0.933093i \(-0.617099\pi\)
−0.359635 + 0.933093i \(0.617099\pi\)
\(68\) −4.46210 −0.541109
\(69\) 3.58097 0.431098
\(70\) −1.13171 −0.135265
\(71\) 7.34167 0.871296 0.435648 0.900117i \(-0.356519\pi\)
0.435648 + 0.900117i \(0.356519\pi\)
\(72\) 0.338090 0.0398443
\(73\) 2.76410 0.323513 0.161756 0.986831i \(-0.448284\pi\)
0.161756 + 0.986831i \(0.448284\pi\)
\(74\) 0.285823 0.0332262
\(75\) 6.06806 0.700679
\(76\) 4.98269 0.571553
\(77\) 4.25957 0.485423
\(78\) 7.46634 0.845396
\(79\) 7.07373 0.795857 0.397928 0.917417i \(-0.369729\pi\)
0.397928 + 0.917417i \(0.369729\pi\)
\(80\) 1.13171 0.126529
\(81\) −7.87142 −0.874603
\(82\) 6.78124 0.748862
\(83\) −10.8428 −1.19015 −0.595074 0.803671i \(-0.702877\pi\)
−0.595074 + 0.803671i \(0.702877\pi\)
\(84\) −1.63154 −0.178015
\(85\) −5.04980 −0.547728
\(86\) −0.00276006 −0.000297625 0
\(87\) 13.3619 1.43254
\(88\) −4.25957 −0.454071
\(89\) −7.78810 −0.825537 −0.412768 0.910836i \(-0.635438\pi\)
−0.412768 + 0.910836i \(0.635438\pi\)
\(90\) 0.382620 0.0403317
\(91\) 4.57626 0.479723
\(92\) −2.19484 −0.228828
\(93\) 0.877364 0.0909784
\(94\) 7.84608 0.809261
\(95\) 5.63896 0.578545
\(96\) 1.63154 0.166518
\(97\) 15.5210 1.57592 0.787959 0.615728i \(-0.211138\pi\)
0.787959 + 0.615728i \(0.211138\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.44012 −0.144737
\(100\) −3.71923 −0.371923
\(101\) −11.4781 −1.14211 −0.571055 0.820912i \(-0.693466\pi\)
−0.571055 + 0.820912i \(0.693466\pi\)
\(102\) −7.28007 −0.720834
\(103\) −0.837340 −0.0825056 −0.0412528 0.999149i \(-0.513135\pi\)
−0.0412528 + 0.999149i \(0.513135\pi\)
\(104\) −4.57626 −0.448740
\(105\) −1.84643 −0.180193
\(106\) 0.524949 0.0509876
\(107\) −4.85453 −0.469305 −0.234652 0.972079i \(-0.575395\pi\)
−0.234652 + 0.972079i \(0.575395\pi\)
\(108\) 5.44621 0.524062
\(109\) −0.840181 −0.0804747 −0.0402373 0.999190i \(-0.512811\pi\)
−0.0402373 + 0.999190i \(0.512811\pi\)
\(110\) −4.82059 −0.459626
\(111\) 0.466330 0.0442621
\(112\) 1.00000 0.0944911
\(113\) 6.08513 0.572441 0.286220 0.958164i \(-0.407601\pi\)
0.286220 + 0.958164i \(0.407601\pi\)
\(114\) 8.12943 0.761391
\(115\) −2.48393 −0.231627
\(116\) −8.18975 −0.760399
\(117\) −1.54719 −0.143038
\(118\) 11.0613 1.01827
\(119\) −4.46210 −0.409040
\(120\) 1.84643 0.168555
\(121\) 7.14390 0.649445
\(122\) 11.2944 1.02254
\(123\) 11.0638 0.997592
\(124\) −0.537753 −0.0482917
\(125\) −9.86764 −0.882589
\(126\) 0.338090 0.0301195
\(127\) −7.54825 −0.669799 −0.334899 0.942254i \(-0.608702\pi\)
−0.334899 + 0.942254i \(0.608702\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.00450314 −0.000396479 0
\(130\) −5.17900 −0.454229
\(131\) 2.63117 0.229887 0.114943 0.993372i \(-0.463331\pi\)
0.114943 + 0.993372i \(0.463331\pi\)
\(132\) −6.94963 −0.604888
\(133\) 4.98269 0.432054
\(134\) 5.88748 0.508601
\(135\) 6.16354 0.530473
\(136\) 4.46210 0.382622
\(137\) 0.401572 0.0343086 0.0171543 0.999853i \(-0.494539\pi\)
0.0171543 + 0.999853i \(0.494539\pi\)
\(138\) −3.58097 −0.304832
\(139\) −6.88303 −0.583811 −0.291906 0.956447i \(-0.594289\pi\)
−0.291906 + 0.956447i \(0.594289\pi\)
\(140\) 1.13171 0.0956470
\(141\) 12.8012 1.07805
\(142\) −7.34167 −0.616099
\(143\) 19.4929 1.63008
\(144\) −0.338090 −0.0281742
\(145\) −9.26843 −0.769701
\(146\) −2.76410 −0.228758
\(147\) −1.63154 −0.134567
\(148\) −0.285823 −0.0234945
\(149\) 16.0699 1.31650 0.658248 0.752801i \(-0.271298\pi\)
0.658248 + 0.752801i \(0.271298\pi\)
\(150\) −6.06806 −0.495455
\(151\) −20.5194 −1.66984 −0.834921 0.550370i \(-0.814487\pi\)
−0.834921 + 0.550370i \(0.814487\pi\)
\(152\) −4.98269 −0.404149
\(153\) 1.50859 0.121962
\(154\) −4.25957 −0.343246
\(155\) −0.608581 −0.0488824
\(156\) −7.46634 −0.597785
\(157\) 9.01726 0.719655 0.359828 0.933019i \(-0.382835\pi\)
0.359828 + 0.933019i \(0.382835\pi\)
\(158\) −7.07373 −0.562756
\(159\) 0.856473 0.0679228
\(160\) −1.13171 −0.0894695
\(161\) −2.19484 −0.172978
\(162\) 7.87142 0.618437
\(163\) −4.76823 −0.373476 −0.186738 0.982410i \(-0.559792\pi\)
−0.186738 + 0.982410i \(0.559792\pi\)
\(164\) −6.78124 −0.529526
\(165\) −7.86497 −0.612287
\(166\) 10.8428 0.841562
\(167\) −2.66178 −0.205975 −0.102988 0.994683i \(-0.532840\pi\)
−0.102988 + 0.994683i \(0.532840\pi\)
\(168\) 1.63154 0.125876
\(169\) 7.94219 0.610937
\(170\) 5.04980 0.387302
\(171\) −1.68460 −0.128824
\(172\) 0.00276006 0.000210453 0
\(173\) 20.8332 1.58392 0.791961 0.610572i \(-0.209060\pi\)
0.791961 + 0.610572i \(0.209060\pi\)
\(174\) −13.3619 −1.01296
\(175\) −3.71923 −0.281148
\(176\) 4.25957 0.321077
\(177\) 18.0469 1.35649
\(178\) 7.78810 0.583743
\(179\) 9.64774 0.721106 0.360553 0.932739i \(-0.382588\pi\)
0.360553 + 0.932739i \(0.382588\pi\)
\(180\) −0.382620 −0.0285188
\(181\) −14.4368 −1.07308 −0.536540 0.843875i \(-0.680269\pi\)
−0.536540 + 0.843875i \(0.680269\pi\)
\(182\) −4.57626 −0.339215
\(183\) 18.4272 1.36218
\(184\) 2.19484 0.161806
\(185\) −0.323468 −0.0237819
\(186\) −0.877364 −0.0643314
\(187\) −19.0066 −1.38990
\(188\) −7.84608 −0.572234
\(189\) 5.44621 0.396154
\(190\) −5.63896 −0.409093
\(191\) 14.8679 1.07581 0.537903 0.843007i \(-0.319217\pi\)
0.537903 + 0.843007i \(0.319217\pi\)
\(192\) −1.63154 −0.117746
\(193\) −3.49277 −0.251415 −0.125707 0.992067i \(-0.540120\pi\)
−0.125707 + 0.992067i \(0.540120\pi\)
\(194\) −15.5210 −1.11434
\(195\) −8.44973 −0.605098
\(196\) 1.00000 0.0714286
\(197\) −10.3062 −0.734289 −0.367144 0.930164i \(-0.619664\pi\)
−0.367144 + 0.930164i \(0.619664\pi\)
\(198\) 1.44012 0.102345
\(199\) −8.74513 −0.619926 −0.309963 0.950749i \(-0.600317\pi\)
−0.309963 + 0.950749i \(0.600317\pi\)
\(200\) 3.71923 0.262989
\(201\) 9.60564 0.677529
\(202\) 11.4781 0.807594
\(203\) −8.18975 −0.574808
\(204\) 7.28007 0.509707
\(205\) −7.67440 −0.536003
\(206\) 0.837340 0.0583402
\(207\) 0.742055 0.0515764
\(208\) 4.57626 0.317307
\(209\) 21.2241 1.46810
\(210\) 1.84643 0.127415
\(211\) 7.63204 0.525411 0.262706 0.964876i \(-0.415385\pi\)
0.262706 + 0.964876i \(0.415385\pi\)
\(212\) −0.524949 −0.0360537
\(213\) −11.9782 −0.820733
\(214\) 4.85453 0.331849
\(215\) 0.00312359 0.000213027 0
\(216\) −5.44621 −0.370568
\(217\) −0.537753 −0.0365051
\(218\) 0.840181 0.0569042
\(219\) −4.50972 −0.304739
\(220\) 4.82059 0.325004
\(221\) −20.4197 −1.37358
\(222\) −0.466330 −0.0312980
\(223\) 0.0143017 0.000957713 0 0.000478857 1.00000i \(-0.499848\pi\)
0.000478857 1.00000i \(0.499848\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.25744 0.0838291
\(226\) −6.08513 −0.404777
\(227\) −11.4591 −0.760570 −0.380285 0.924869i \(-0.624174\pi\)
−0.380285 + 0.924869i \(0.624174\pi\)
\(228\) −8.12943 −0.538385
\(229\) −19.4007 −1.28203 −0.641016 0.767528i \(-0.721487\pi\)
−0.641016 + 0.767528i \(0.721487\pi\)
\(230\) 2.48393 0.163785
\(231\) −6.94963 −0.457252
\(232\) 8.18975 0.537684
\(233\) 2.29636 0.150440 0.0752199 0.997167i \(-0.476034\pi\)
0.0752199 + 0.997167i \(0.476034\pi\)
\(234\) 1.54719 0.101143
\(235\) −8.87949 −0.579234
\(236\) −11.0613 −0.720028
\(237\) −11.5410 −0.749671
\(238\) 4.46210 0.289235
\(239\) 2.01797 0.130532 0.0652659 0.997868i \(-0.479210\pi\)
0.0652659 + 0.997868i \(0.479210\pi\)
\(240\) −1.84643 −0.119186
\(241\) 18.9709 1.22202 0.611012 0.791621i \(-0.290763\pi\)
0.611012 + 0.791621i \(0.290763\pi\)
\(242\) −7.14390 −0.459227
\(243\) −3.49613 −0.224277
\(244\) −11.2944 −0.723048
\(245\) 1.13171 0.0723023
\(246\) −11.0638 −0.705404
\(247\) 22.8021 1.45086
\(248\) 0.537753 0.0341474
\(249\) 17.6904 1.12108
\(250\) 9.86764 0.624085
\(251\) −18.2486 −1.15184 −0.575920 0.817506i \(-0.695356\pi\)
−0.575920 + 0.817506i \(0.695356\pi\)
\(252\) −0.338090 −0.0212977
\(253\) −9.34908 −0.587772
\(254\) 7.54825 0.473619
\(255\) 8.23893 0.515942
\(256\) 1.00000 0.0625000
\(257\) 13.5693 0.846428 0.423214 0.906030i \(-0.360902\pi\)
0.423214 + 0.906030i \(0.360902\pi\)
\(258\) 0.00450314 0.000280353 0
\(259\) −0.285823 −0.0177602
\(260\) 5.17900 0.321188
\(261\) 2.76888 0.171389
\(262\) −2.63117 −0.162554
\(263\) −6.14923 −0.379178 −0.189589 0.981864i \(-0.560716\pi\)
−0.189589 + 0.981864i \(0.560716\pi\)
\(264\) 6.94963 0.427720
\(265\) −0.594090 −0.0364947
\(266\) −4.98269 −0.305508
\(267\) 12.7066 0.777629
\(268\) −5.88748 −0.359635
\(269\) −21.1560 −1.28991 −0.644953 0.764223i \(-0.723123\pi\)
−0.644953 + 0.764223i \(0.723123\pi\)
\(270\) −6.16354 −0.375101
\(271\) 3.35479 0.203789 0.101895 0.994795i \(-0.467510\pi\)
0.101895 + 0.994795i \(0.467510\pi\)
\(272\) −4.46210 −0.270554
\(273\) −7.46634 −0.451883
\(274\) −0.401572 −0.0242598
\(275\) −15.8423 −0.955327
\(276\) 3.58097 0.215549
\(277\) 13.2730 0.797498 0.398749 0.917060i \(-0.369444\pi\)
0.398749 + 0.917060i \(0.369444\pi\)
\(278\) 6.88303 0.412817
\(279\) 0.181809 0.0108846
\(280\) −1.13171 −0.0676326
\(281\) −3.31705 −0.197879 −0.0989394 0.995093i \(-0.531545\pi\)
−0.0989394 + 0.995093i \(0.531545\pi\)
\(282\) −12.8012 −0.762298
\(283\) 32.6018 1.93798 0.968988 0.247106i \(-0.0794796\pi\)
0.968988 + 0.247106i \(0.0794796\pi\)
\(284\) 7.34167 0.435648
\(285\) −9.20016 −0.544971
\(286\) −19.4929 −1.15264
\(287\) −6.78124 −0.400284
\(288\) 0.338090 0.0199222
\(289\) 2.91030 0.171194
\(290\) 9.26843 0.544261
\(291\) −25.3231 −1.48446
\(292\) 2.76410 0.161756
\(293\) 20.4707 1.19591 0.597955 0.801529i \(-0.295980\pi\)
0.597955 + 0.801529i \(0.295980\pi\)
\(294\) 1.63154 0.0951531
\(295\) −12.5182 −0.728836
\(296\) 0.285823 0.0166131
\(297\) 23.1985 1.34611
\(298\) −16.0699 −0.930903
\(299\) −10.0442 −0.580870
\(300\) 6.06806 0.350340
\(301\) 0.00276006 0.000159087 0
\(302\) 20.5194 1.18076
\(303\) 18.7269 1.07583
\(304\) 4.98269 0.285777
\(305\) −12.7820 −0.731893
\(306\) −1.50859 −0.0862404
\(307\) 32.9099 1.87826 0.939132 0.343556i \(-0.111632\pi\)
0.939132 + 0.343556i \(0.111632\pi\)
\(308\) 4.25957 0.242711
\(309\) 1.36615 0.0777176
\(310\) 0.608581 0.0345651
\(311\) 8.56387 0.485612 0.242806 0.970075i \(-0.421932\pi\)
0.242806 + 0.970075i \(0.421932\pi\)
\(312\) 7.46634 0.422698
\(313\) 19.8106 1.11976 0.559880 0.828574i \(-0.310847\pi\)
0.559880 + 0.828574i \(0.310847\pi\)
\(314\) −9.01726 −0.508873
\(315\) −0.382620 −0.0215582
\(316\) 7.07373 0.397928
\(317\) 3.65393 0.205225 0.102613 0.994721i \(-0.467280\pi\)
0.102613 + 0.994721i \(0.467280\pi\)
\(318\) −0.856473 −0.0480286
\(319\) −34.8848 −1.95317
\(320\) 1.13171 0.0632645
\(321\) 7.92034 0.442070
\(322\) 2.19484 0.122314
\(323\) −22.2332 −1.23709
\(324\) −7.87142 −0.437301
\(325\) −17.0202 −0.944110
\(326\) 4.76823 0.264088
\(327\) 1.37078 0.0758046
\(328\) 6.78124 0.374431
\(329\) −7.84608 −0.432568
\(330\) 7.86497 0.432952
\(331\) −20.1991 −1.11024 −0.555121 0.831769i \(-0.687328\pi\)
−0.555121 + 0.831769i \(0.687328\pi\)
\(332\) −10.8428 −0.595074
\(333\) 0.0966339 0.00529550
\(334\) 2.66178 0.145646
\(335\) −6.66292 −0.364034
\(336\) −1.63154 −0.0890076
\(337\) −15.9701 −0.869945 −0.434973 0.900444i \(-0.643242\pi\)
−0.434973 + 0.900444i \(0.643242\pi\)
\(338\) −7.94219 −0.431998
\(339\) −9.92811 −0.539221
\(340\) −5.04980 −0.273864
\(341\) −2.29059 −0.124043
\(342\) 1.68460 0.0910927
\(343\) 1.00000 0.0539949
\(344\) −0.00276006 −0.000148812 0
\(345\) 4.05262 0.218186
\(346\) −20.8332 −1.12000
\(347\) −28.5250 −1.53130 −0.765650 0.643257i \(-0.777583\pi\)
−0.765650 + 0.643257i \(0.777583\pi\)
\(348\) 13.3619 0.716272
\(349\) 29.0357 1.55424 0.777122 0.629350i \(-0.216679\pi\)
0.777122 + 0.629350i \(0.216679\pi\)
\(350\) 3.71923 0.198801
\(351\) 24.9233 1.33031
\(352\) −4.25957 −0.227036
\(353\) −21.1847 −1.12755 −0.563773 0.825930i \(-0.690651\pi\)
−0.563773 + 0.825930i \(0.690651\pi\)
\(354\) −18.0469 −0.959181
\(355\) 8.30864 0.440977
\(356\) −7.78810 −0.412768
\(357\) 7.28007 0.385302
\(358\) −9.64774 −0.509899
\(359\) −22.9523 −1.21137 −0.605687 0.795703i \(-0.707102\pi\)
−0.605687 + 0.795703i \(0.707102\pi\)
\(360\) 0.382620 0.0201659
\(361\) 5.82717 0.306693
\(362\) 14.4368 0.758782
\(363\) −11.6555 −0.611757
\(364\) 4.57626 0.239861
\(365\) 3.12816 0.163735
\(366\) −18.4272 −0.963204
\(367\) −17.8076 −0.929547 −0.464773 0.885430i \(-0.653864\pi\)
−0.464773 + 0.885430i \(0.653864\pi\)
\(368\) −2.19484 −0.114414
\(369\) 2.29267 0.119352
\(370\) 0.323468 0.0168163
\(371\) −0.524949 −0.0272540
\(372\) 0.877364 0.0454892
\(373\) −20.3958 −1.05606 −0.528028 0.849227i \(-0.677069\pi\)
−0.528028 + 0.849227i \(0.677069\pi\)
\(374\) 19.0066 0.982807
\(375\) 16.0994 0.831370
\(376\) 7.84608 0.404631
\(377\) −37.4785 −1.93024
\(378\) −5.44621 −0.280123
\(379\) −15.7131 −0.807127 −0.403563 0.914952i \(-0.632229\pi\)
−0.403563 + 0.914952i \(0.632229\pi\)
\(380\) 5.63896 0.289272
\(381\) 12.3152 0.630929
\(382\) −14.8679 −0.760710
\(383\) −13.6780 −0.698914 −0.349457 0.936952i \(-0.613634\pi\)
−0.349457 + 0.936952i \(0.613634\pi\)
\(384\) 1.63154 0.0832590
\(385\) 4.82059 0.245680
\(386\) 3.49277 0.177777
\(387\) −0.000933149 0 −4.74346e−5 0
\(388\) 15.5210 0.787959
\(389\) 3.11248 0.157809 0.0789044 0.996882i \(-0.474858\pi\)
0.0789044 + 0.996882i \(0.474858\pi\)
\(390\) 8.44973 0.427869
\(391\) 9.79360 0.495284
\(392\) −1.00000 −0.0505076
\(393\) −4.29285 −0.216546
\(394\) 10.3062 0.519220
\(395\) 8.00541 0.402796
\(396\) −1.44012 −0.0723687
\(397\) −12.8864 −0.646748 −0.323374 0.946271i \(-0.604817\pi\)
−0.323374 + 0.946271i \(0.604817\pi\)
\(398\) 8.74513 0.438354
\(399\) −8.12943 −0.406981
\(400\) −3.71923 −0.185962
\(401\) 9.90827 0.494796 0.247398 0.968914i \(-0.420425\pi\)
0.247398 + 0.968914i \(0.420425\pi\)
\(402\) −9.60564 −0.479086
\(403\) −2.46090 −0.122586
\(404\) −11.4781 −0.571055
\(405\) −8.90817 −0.442651
\(406\) 8.18975 0.406451
\(407\) −1.21748 −0.0603483
\(408\) −7.28007 −0.360417
\(409\) 15.4593 0.764414 0.382207 0.924077i \(-0.375164\pi\)
0.382207 + 0.924077i \(0.375164\pi\)
\(410\) 7.67440 0.379011
\(411\) −0.655179 −0.0323176
\(412\) −0.837340 −0.0412528
\(413\) −11.0613 −0.544290
\(414\) −0.742055 −0.0364700
\(415\) −12.2709 −0.602353
\(416\) −4.57626 −0.224370
\(417\) 11.2299 0.549931
\(418\) −21.2241 −1.03810
\(419\) −3.59983 −0.175863 −0.0879315 0.996127i \(-0.528026\pi\)
−0.0879315 + 0.996127i \(0.528026\pi\)
\(420\) −1.84643 −0.0900964
\(421\) −7.04424 −0.343315 −0.171658 0.985157i \(-0.554912\pi\)
−0.171658 + 0.985157i \(0.554912\pi\)
\(422\) −7.63204 −0.371522
\(423\) 2.65268 0.128978
\(424\) 0.524949 0.0254938
\(425\) 16.5956 0.805003
\(426\) 11.9782 0.580346
\(427\) −11.2944 −0.546573
\(428\) −4.85453 −0.234652
\(429\) −31.8034 −1.53548
\(430\) −0.00312359 −0.000150633 0
\(431\) 1.00000 0.0481683
\(432\) 5.44621 0.262031
\(433\) 26.0320 1.25102 0.625510 0.780216i \(-0.284891\pi\)
0.625510 + 0.780216i \(0.284891\pi\)
\(434\) 0.537753 0.0258130
\(435\) 15.1218 0.725033
\(436\) −0.840181 −0.0402373
\(437\) −10.9362 −0.523150
\(438\) 4.50972 0.215483
\(439\) −21.6498 −1.03329 −0.516645 0.856199i \(-0.672819\pi\)
−0.516645 + 0.856199i \(0.672819\pi\)
\(440\) −4.82059 −0.229813
\(441\) −0.338090 −0.0160995
\(442\) 20.4197 0.971267
\(443\) −4.41468 −0.209748 −0.104874 0.994486i \(-0.533444\pi\)
−0.104874 + 0.994486i \(0.533444\pi\)
\(444\) 0.466330 0.0221310
\(445\) −8.81387 −0.417818
\(446\) −0.0143017 −0.000677206 0
\(447\) −26.2186 −1.24010
\(448\) 1.00000 0.0472456
\(449\) 0.922409 0.0435312 0.0217656 0.999763i \(-0.493071\pi\)
0.0217656 + 0.999763i \(0.493071\pi\)
\(450\) −1.25744 −0.0592761
\(451\) −28.8851 −1.36015
\(452\) 6.08513 0.286220
\(453\) 33.4781 1.57294
\(454\) 11.4591 0.537804
\(455\) 5.17900 0.242795
\(456\) 8.12943 0.380696
\(457\) 14.9046 0.697208 0.348604 0.937270i \(-0.386656\pi\)
0.348604 + 0.937270i \(0.386656\pi\)
\(458\) 19.4007 0.906533
\(459\) −24.3015 −1.13430
\(460\) −2.48393 −0.115814
\(461\) 22.1832 1.03317 0.516587 0.856235i \(-0.327202\pi\)
0.516587 + 0.856235i \(0.327202\pi\)
\(462\) 6.94963 0.323326
\(463\) −33.8708 −1.57411 −0.787054 0.616884i \(-0.788395\pi\)
−0.787054 + 0.616884i \(0.788395\pi\)
\(464\) −8.18975 −0.380200
\(465\) 0.992921 0.0460456
\(466\) −2.29636 −0.106377
\(467\) 29.2811 1.35497 0.677485 0.735537i \(-0.263070\pi\)
0.677485 + 0.735537i \(0.263070\pi\)
\(468\) −1.54719 −0.0715189
\(469\) −5.88748 −0.271859
\(470\) 8.87949 0.409580
\(471\) −14.7120 −0.677892
\(472\) 11.0613 0.509137
\(473\) 0.0117567 0.000540571 0
\(474\) 11.5410 0.530098
\(475\) −18.5318 −0.850296
\(476\) −4.46210 −0.204520
\(477\) 0.177480 0.00812626
\(478\) −2.01797 −0.0922999
\(479\) −30.9658 −1.41486 −0.707431 0.706782i \(-0.750146\pi\)
−0.707431 + 0.706782i \(0.750146\pi\)
\(480\) 1.84643 0.0842774
\(481\) −1.30800 −0.0596396
\(482\) −18.9709 −0.864102
\(483\) 3.58097 0.162940
\(484\) 7.14390 0.324723
\(485\) 17.5653 0.797598
\(486\) 3.49613 0.158588
\(487\) −18.9577 −0.859057 −0.429529 0.903053i \(-0.641320\pi\)
−0.429529 + 0.903053i \(0.641320\pi\)
\(488\) 11.2944 0.511272
\(489\) 7.77953 0.351803
\(490\) −1.13171 −0.0511255
\(491\) −3.89068 −0.175584 −0.0877919 0.996139i \(-0.527981\pi\)
−0.0877919 + 0.996139i \(0.527981\pi\)
\(492\) 11.0638 0.498796
\(493\) 36.5435 1.64583
\(494\) −22.8021 −1.02591
\(495\) −1.62980 −0.0732539
\(496\) −0.537753 −0.0241458
\(497\) 7.34167 0.329319
\(498\) −17.6904 −0.792724
\(499\) −26.0709 −1.16709 −0.583546 0.812080i \(-0.698335\pi\)
−0.583546 + 0.812080i \(0.698335\pi\)
\(500\) −9.86764 −0.441294
\(501\) 4.34280 0.194022
\(502\) 18.2486 0.814475
\(503\) 34.7034 1.54735 0.773673 0.633585i \(-0.218417\pi\)
0.773673 + 0.633585i \(0.218417\pi\)
\(504\) 0.338090 0.0150597
\(505\) −12.9898 −0.578040
\(506\) 9.34908 0.415617
\(507\) −12.9580 −0.575483
\(508\) −7.54825 −0.334899
\(509\) 25.4382 1.12753 0.563764 0.825936i \(-0.309353\pi\)
0.563764 + 0.825936i \(0.309353\pi\)
\(510\) −8.23893 −0.364826
\(511\) 2.76410 0.122276
\(512\) −1.00000 −0.0441942
\(513\) 27.1368 1.19812
\(514\) −13.5693 −0.598515
\(515\) −0.947626 −0.0417574
\(516\) −0.00450314 −0.000198239 0
\(517\) −33.4209 −1.46985
\(518\) 0.285823 0.0125583
\(519\) −33.9902 −1.49200
\(520\) −5.17900 −0.227114
\(521\) −20.2813 −0.888542 −0.444271 0.895893i \(-0.646537\pi\)
−0.444271 + 0.895893i \(0.646537\pi\)
\(522\) −2.76888 −0.121190
\(523\) 13.9852 0.611529 0.305764 0.952107i \(-0.401088\pi\)
0.305764 + 0.952107i \(0.401088\pi\)
\(524\) 2.63117 0.114943
\(525\) 6.06806 0.264832
\(526\) 6.14923 0.268119
\(527\) 2.39951 0.104524
\(528\) −6.94963 −0.302444
\(529\) −18.1827 −0.790550
\(530\) 0.594090 0.0258056
\(531\) 3.73971 0.162290
\(532\) 4.98269 0.216027
\(533\) −31.0327 −1.34418
\(534\) −12.7066 −0.549867
\(535\) −5.49392 −0.237523
\(536\) 5.88748 0.254300
\(537\) −15.7406 −0.679258
\(538\) 21.1560 0.912101
\(539\) 4.25957 0.183472
\(540\) 6.16354 0.265236
\(541\) −29.5588 −1.27083 −0.635417 0.772169i \(-0.719172\pi\)
−0.635417 + 0.772169i \(0.719172\pi\)
\(542\) −3.35479 −0.144101
\(543\) 23.5542 1.01081
\(544\) 4.46210 0.191311
\(545\) −0.950841 −0.0407295
\(546\) 7.46634 0.319530
\(547\) −23.1185 −0.988476 −0.494238 0.869327i \(-0.664553\pi\)
−0.494238 + 0.869327i \(0.664553\pi\)
\(548\) 0.401572 0.0171543
\(549\) 3.81852 0.162970
\(550\) 15.8423 0.675519
\(551\) −40.8070 −1.73844
\(552\) −3.58097 −0.152416
\(553\) 7.07373 0.300805
\(554\) −13.2730 −0.563916
\(555\) 0.527750 0.0224018
\(556\) −6.88303 −0.291906
\(557\) −37.9454 −1.60780 −0.803900 0.594765i \(-0.797245\pi\)
−0.803900 + 0.594765i \(0.797245\pi\)
\(558\) −0.181809 −0.00769660
\(559\) 0.0126308 0.000534224 0
\(560\) 1.13171 0.0478235
\(561\) 31.0099 1.30924
\(562\) 3.31705 0.139921
\(563\) −1.50645 −0.0634894 −0.0317447 0.999496i \(-0.510106\pi\)
−0.0317447 + 0.999496i \(0.510106\pi\)
\(564\) 12.8012 0.539026
\(565\) 6.88660 0.289721
\(566\) −32.6018 −1.37036
\(567\) −7.87142 −0.330569
\(568\) −7.34167 −0.308050
\(569\) −0.668840 −0.0280392 −0.0140196 0.999902i \(-0.504463\pi\)
−0.0140196 + 0.999902i \(0.504463\pi\)
\(570\) 9.20016 0.385352
\(571\) −12.7813 −0.534882 −0.267441 0.963574i \(-0.586178\pi\)
−0.267441 + 0.963574i \(0.586178\pi\)
\(572\) 19.4929 0.815039
\(573\) −24.2576 −1.01337
\(574\) 6.78124 0.283043
\(575\) 8.16313 0.340426
\(576\) −0.338090 −0.0140871
\(577\) 21.5570 0.897428 0.448714 0.893675i \(-0.351882\pi\)
0.448714 + 0.893675i \(0.351882\pi\)
\(578\) −2.91030 −0.121052
\(579\) 5.69857 0.236825
\(580\) −9.26843 −0.384850
\(581\) −10.8428 −0.449834
\(582\) 25.3231 1.04967
\(583\) −2.23606 −0.0926079
\(584\) −2.76410 −0.114379
\(585\) −1.75097 −0.0723938
\(586\) −20.4707 −0.845637
\(587\) 8.35060 0.344666 0.172333 0.985039i \(-0.444869\pi\)
0.172333 + 0.985039i \(0.444869\pi\)
\(588\) −1.63154 −0.0672834
\(589\) −2.67946 −0.110405
\(590\) 12.5182 0.515365
\(591\) 16.8150 0.691676
\(592\) −0.285823 −0.0117472
\(593\) 25.9889 1.06724 0.533618 0.845726i \(-0.320832\pi\)
0.533618 + 0.845726i \(0.320832\pi\)
\(594\) −23.1985 −0.951846
\(595\) −5.04980 −0.207022
\(596\) 16.0699 0.658248
\(597\) 14.2680 0.583950
\(598\) 10.0442 0.410737
\(599\) −10.8328 −0.442617 −0.221308 0.975204i \(-0.571033\pi\)
−0.221308 + 0.975204i \(0.571033\pi\)
\(600\) −6.06806 −0.247728
\(601\) 6.90108 0.281501 0.140751 0.990045i \(-0.455048\pi\)
0.140751 + 0.990045i \(0.455048\pi\)
\(602\) −0.00276006 −0.000112492 0
\(603\) 1.99050 0.0810595
\(604\) −20.5194 −0.834921
\(605\) 8.08482 0.328695
\(606\) −18.7269 −0.760727
\(607\) −37.9678 −1.54106 −0.770532 0.637401i \(-0.780009\pi\)
−0.770532 + 0.637401i \(0.780009\pi\)
\(608\) −4.98269 −0.202075
\(609\) 13.3619 0.541451
\(610\) 12.7820 0.517526
\(611\) −35.9057 −1.45259
\(612\) 1.50859 0.0609812
\(613\) −40.6488 −1.64179 −0.820895 0.571079i \(-0.806525\pi\)
−0.820895 + 0.571079i \(0.806525\pi\)
\(614\) −32.9099 −1.32813
\(615\) 12.5211 0.504898
\(616\) −4.25957 −0.171623
\(617\) −2.27059 −0.0914103 −0.0457052 0.998955i \(-0.514553\pi\)
−0.0457052 + 0.998955i \(0.514553\pi\)
\(618\) −1.36615 −0.0549546
\(619\) −39.0898 −1.57115 −0.785575 0.618767i \(-0.787633\pi\)
−0.785575 + 0.618767i \(0.787633\pi\)
\(620\) −0.608581 −0.0244412
\(621\) −11.9536 −0.479681
\(622\) −8.56387 −0.343380
\(623\) −7.78810 −0.312024
\(624\) −7.46634 −0.298893
\(625\) 7.42885 0.297154
\(626\) −19.8106 −0.791790
\(627\) −34.6279 −1.38290
\(628\) 9.01726 0.359828
\(629\) 1.27537 0.0508522
\(630\) 0.382620 0.0152440
\(631\) 23.0675 0.918304 0.459152 0.888358i \(-0.348153\pi\)
0.459152 + 0.888358i \(0.348153\pi\)
\(632\) −7.07373 −0.281378
\(633\) −12.4520 −0.494921
\(634\) −3.65393 −0.145116
\(635\) −8.54243 −0.338996
\(636\) 0.856473 0.0339614
\(637\) 4.57626 0.181318
\(638\) 34.8848 1.38110
\(639\) −2.48215 −0.0981922
\(640\) −1.13171 −0.0447348
\(641\) −42.4129 −1.67521 −0.837604 0.546278i \(-0.816044\pi\)
−0.837604 + 0.546278i \(0.816044\pi\)
\(642\) −7.92034 −0.312591
\(643\) −5.50822 −0.217223 −0.108612 0.994084i \(-0.534640\pi\)
−0.108612 + 0.994084i \(0.534640\pi\)
\(644\) −2.19484 −0.0864890
\(645\) −0.00509624 −0.000200664 0
\(646\) 22.2332 0.874755
\(647\) 9.96669 0.391831 0.195916 0.980621i \(-0.437232\pi\)
0.195916 + 0.980621i \(0.437232\pi\)
\(648\) 7.87142 0.309219
\(649\) −47.1162 −1.84947
\(650\) 17.0202 0.667587
\(651\) 0.877364 0.0343866
\(652\) −4.76823 −0.186738
\(653\) −7.20354 −0.281896 −0.140948 0.990017i \(-0.545015\pi\)
−0.140948 + 0.990017i \(0.545015\pi\)
\(654\) −1.37078 −0.0536019
\(655\) 2.97772 0.116349
\(656\) −6.78124 −0.264763
\(657\) −0.934514 −0.0364589
\(658\) 7.84608 0.305872
\(659\) −25.8502 −1.00698 −0.503491 0.864000i \(-0.667951\pi\)
−0.503491 + 0.864000i \(0.667951\pi\)
\(660\) −7.86497 −0.306144
\(661\) −34.7135 −1.35020 −0.675099 0.737727i \(-0.735899\pi\)
−0.675099 + 0.737727i \(0.735899\pi\)
\(662\) 20.1991 0.785060
\(663\) 33.3155 1.29387
\(664\) 10.8428 0.420781
\(665\) 5.63896 0.218669
\(666\) −0.0966339 −0.00374449
\(667\) 17.9752 0.696004
\(668\) −2.66178 −0.102988
\(669\) −0.0233338 −0.000902135 0
\(670\) 6.66292 0.257411
\(671\) −48.1091 −1.85723
\(672\) 1.63154 0.0629379
\(673\) −22.2863 −0.859073 −0.429537 0.903049i \(-0.641323\pi\)
−0.429537 + 0.903049i \(0.641323\pi\)
\(674\) 15.9701 0.615144
\(675\) −20.2557 −0.779644
\(676\) 7.94219 0.305469
\(677\) −20.4389 −0.785531 −0.392765 0.919639i \(-0.628482\pi\)
−0.392765 + 0.919639i \(0.628482\pi\)
\(678\) 9.92811 0.381287
\(679\) 15.5210 0.595641
\(680\) 5.04980 0.193651
\(681\) 18.6960 0.716432
\(682\) 2.29059 0.0877114
\(683\) −20.6173 −0.788898 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(684\) −1.68460 −0.0644122
\(685\) 0.454463 0.0173641
\(686\) −1.00000 −0.0381802
\(687\) 31.6529 1.20763
\(688\) 0.00276006 0.000105226 0
\(689\) −2.40231 −0.0915206
\(690\) −4.05262 −0.154280
\(691\) 42.0521 1.59974 0.799869 0.600175i \(-0.204902\pi\)
0.799869 + 0.600175i \(0.204902\pi\)
\(692\) 20.8332 0.791961
\(693\) −1.44012 −0.0547056
\(694\) 28.5250 1.08279
\(695\) −7.78960 −0.295476
\(696\) −13.3619 −0.506481
\(697\) 30.2585 1.14612
\(698\) −29.0357 −1.09902
\(699\) −3.74660 −0.141709
\(700\) −3.71923 −0.140574
\(701\) 9.58941 0.362187 0.181093 0.983466i \(-0.442036\pi\)
0.181093 + 0.983466i \(0.442036\pi\)
\(702\) −24.9233 −0.940670
\(703\) −1.42416 −0.0537134
\(704\) 4.25957 0.160538
\(705\) 14.4872 0.545620
\(706\) 21.1847 0.797296
\(707\) −11.4781 −0.431677
\(708\) 18.0469 0.678243
\(709\) 3.96897 0.149058 0.0745289 0.997219i \(-0.476255\pi\)
0.0745289 + 0.997219i \(0.476255\pi\)
\(710\) −8.30864 −0.311818
\(711\) −2.39156 −0.0896905
\(712\) 7.78810 0.291871
\(713\) 1.18028 0.0442020
\(714\) −7.28007 −0.272450
\(715\) 22.0603 0.825009
\(716\) 9.64774 0.360553
\(717\) −3.29239 −0.122957
\(718\) 22.9523 0.856571
\(719\) 22.3423 0.833226 0.416613 0.909084i \(-0.363217\pi\)
0.416613 + 0.909084i \(0.363217\pi\)
\(720\) −0.382620 −0.0142594
\(721\) −0.837340 −0.0311842
\(722\) −5.82717 −0.216865
\(723\) −30.9518 −1.15111
\(724\) −14.4368 −0.536540
\(725\) 30.4596 1.13124
\(726\) 11.6555 0.432577
\(727\) 18.9447 0.702619 0.351310 0.936259i \(-0.385736\pi\)
0.351310 + 0.936259i \(0.385736\pi\)
\(728\) −4.57626 −0.169608
\(729\) 29.3183 1.08586
\(730\) −3.12816 −0.115778
\(731\) −0.0123156 −0.000455511 0
\(732\) 18.4272 0.681088
\(733\) 37.8644 1.39855 0.699277 0.714851i \(-0.253505\pi\)
0.699277 + 0.714851i \(0.253505\pi\)
\(734\) 17.8076 0.657289
\(735\) −1.84643 −0.0681064
\(736\) 2.19484 0.0809030
\(737\) −25.0781 −0.923764
\(738\) −2.29267 −0.0843944
\(739\) 3.08276 0.113401 0.0567006 0.998391i \(-0.481942\pi\)
0.0567006 + 0.998391i \(0.481942\pi\)
\(740\) −0.323468 −0.0118909
\(741\) −37.2024 −1.36667
\(742\) 0.524949 0.0192715
\(743\) −12.0168 −0.440854 −0.220427 0.975404i \(-0.570745\pi\)
−0.220427 + 0.975404i \(0.570745\pi\)
\(744\) −0.877364 −0.0321657
\(745\) 18.1864 0.666300
\(746\) 20.3958 0.746745
\(747\) 3.66584 0.134126
\(748\) −19.0066 −0.694950
\(749\) −4.85453 −0.177381
\(750\) −16.0994 −0.587868
\(751\) 29.2469 1.06724 0.533618 0.845726i \(-0.320832\pi\)
0.533618 + 0.845726i \(0.320832\pi\)
\(752\) −7.84608 −0.286117
\(753\) 29.7732 1.08500
\(754\) 37.4785 1.36489
\(755\) −23.2220 −0.845134
\(756\) 5.44621 0.198077
\(757\) −13.3418 −0.484915 −0.242458 0.970162i \(-0.577954\pi\)
−0.242458 + 0.970162i \(0.577954\pi\)
\(758\) 15.7131 0.570725
\(759\) 15.2534 0.553662
\(760\) −5.63896 −0.204546
\(761\) 9.74403 0.353221 0.176610 0.984281i \(-0.443487\pi\)
0.176610 + 0.984281i \(0.443487\pi\)
\(762\) −12.3152 −0.446134
\(763\) −0.840181 −0.0304166
\(764\) 14.8679 0.537903
\(765\) 1.70729 0.0617271
\(766\) 13.6780 0.494207
\(767\) −50.6193 −1.82776
\(768\) −1.63154 −0.0588730
\(769\) 3.62512 0.130725 0.0653626 0.997862i \(-0.479180\pi\)
0.0653626 + 0.997862i \(0.479180\pi\)
\(770\) −4.82059 −0.173722
\(771\) −22.1388 −0.797308
\(772\) −3.49277 −0.125707
\(773\) −14.9357 −0.537200 −0.268600 0.963252i \(-0.586561\pi\)
−0.268600 + 0.963252i \(0.586561\pi\)
\(774\) 0.000933149 0 3.35414e−5 0
\(775\) 2.00003 0.0718432
\(776\) −15.5210 −0.557171
\(777\) 0.466330 0.0167295
\(778\) −3.11248 −0.111588
\(779\) −33.7888 −1.21061
\(780\) −8.44973 −0.302549
\(781\) 31.2723 1.11901
\(782\) −9.79360 −0.350219
\(783\) −44.6032 −1.59399
\(784\) 1.00000 0.0357143
\(785\) 10.2049 0.364229
\(786\) 4.29285 0.153121
\(787\) −6.11167 −0.217858 −0.108929 0.994050i \(-0.534742\pi\)
−0.108929 + 0.994050i \(0.534742\pi\)
\(788\) −10.3062 −0.367144
\(789\) 10.0327 0.357173
\(790\) −8.00541 −0.284820
\(791\) 6.08513 0.216362
\(792\) 1.44012 0.0511724
\(793\) −51.6860 −1.83542
\(794\) 12.8864 0.457320
\(795\) 0.969280 0.0343768
\(796\) −8.74513 −0.309963
\(797\) −31.0721 −1.10063 −0.550314 0.834958i \(-0.685492\pi\)
−0.550314 + 0.834958i \(0.685492\pi\)
\(798\) 8.12943 0.287779
\(799\) 35.0100 1.23856
\(800\) 3.71923 0.131495
\(801\) 2.63308 0.0930354
\(802\) −9.90827 −0.349873
\(803\) 11.7738 0.415490
\(804\) 9.60564 0.338765
\(805\) −2.48393 −0.0875469
\(806\) 2.46090 0.0866815
\(807\) 34.5168 1.21505
\(808\) 11.4781 0.403797
\(809\) −24.2082 −0.851116 −0.425558 0.904931i \(-0.639922\pi\)
−0.425558 + 0.904931i \(0.639922\pi\)
\(810\) 8.90817 0.313001
\(811\) −31.3099 −1.09944 −0.549720 0.835349i \(-0.685266\pi\)
−0.549720 + 0.835349i \(0.685266\pi\)
\(812\) −8.18975 −0.287404
\(813\) −5.47347 −0.191963
\(814\) 1.21748 0.0426727
\(815\) −5.39625 −0.189022
\(816\) 7.28007 0.254853
\(817\) 0.0137525 0.000481139 0
\(818\) −15.4593 −0.540522
\(819\) −1.54719 −0.0540632
\(820\) −7.67440 −0.268002
\(821\) 35.0922 1.22473 0.612363 0.790577i \(-0.290219\pi\)
0.612363 + 0.790577i \(0.290219\pi\)
\(822\) 0.655179 0.0228520
\(823\) 16.2807 0.567510 0.283755 0.958897i \(-0.408420\pi\)
0.283755 + 0.958897i \(0.408420\pi\)
\(824\) 0.837340 0.0291701
\(825\) 25.8473 0.899888
\(826\) 11.0613 0.384871
\(827\) 17.9961 0.625787 0.312893 0.949788i \(-0.398702\pi\)
0.312893 + 0.949788i \(0.398702\pi\)
\(828\) 0.742055 0.0257882
\(829\) 28.5139 0.990328 0.495164 0.868800i \(-0.335108\pi\)
0.495164 + 0.868800i \(0.335108\pi\)
\(830\) 12.2709 0.425928
\(831\) −21.6554 −0.751218
\(832\) 4.57626 0.158653
\(833\) −4.46210 −0.154602
\(834\) −11.2299 −0.388860
\(835\) −3.01237 −0.104247
\(836\) 21.2241 0.734050
\(837\) −2.92872 −0.101231
\(838\) 3.59983 0.124354
\(839\) 51.2089 1.76793 0.883964 0.467556i \(-0.154865\pi\)
0.883964 + 0.467556i \(0.154865\pi\)
\(840\) 1.84643 0.0637077
\(841\) 38.0721 1.31283
\(842\) 7.04424 0.242761
\(843\) 5.41189 0.186395
\(844\) 7.63204 0.262706
\(845\) 8.98825 0.309205
\(846\) −2.65268 −0.0912011
\(847\) 7.14390 0.245467
\(848\) −0.524949 −0.0180268
\(849\) −53.1910 −1.82551
\(850\) −16.5956 −0.569223
\(851\) 0.627336 0.0215048
\(852\) −11.9782 −0.410366
\(853\) −23.5534 −0.806453 −0.403226 0.915100i \(-0.632111\pi\)
−0.403226 + 0.915100i \(0.632111\pi\)
\(854\) 11.2944 0.386486
\(855\) −1.90648 −0.0652001
\(856\) 4.85453 0.165924
\(857\) −42.7907 −1.46170 −0.730851 0.682537i \(-0.760877\pi\)
−0.730851 + 0.682537i \(0.760877\pi\)
\(858\) 31.8034 1.08575
\(859\) −3.12743 −0.106707 −0.0533533 0.998576i \(-0.516991\pi\)
−0.0533533 + 0.998576i \(0.516991\pi\)
\(860\) 0.00312359 0.000106513 0
\(861\) 11.0638 0.377054
\(862\) −1.00000 −0.0340601
\(863\) 11.4305 0.389098 0.194549 0.980893i \(-0.437676\pi\)
0.194549 + 0.980893i \(0.437676\pi\)
\(864\) −5.44621 −0.185284
\(865\) 23.5772 0.801648
\(866\) −26.0320 −0.884605
\(867\) −4.74826 −0.161259
\(868\) −0.537753 −0.0182525
\(869\) 30.1310 1.02212
\(870\) −15.1218 −0.512676
\(871\) −26.9427 −0.912917
\(872\) 0.840181 0.0284521
\(873\) −5.24750 −0.177601
\(874\) 10.9362 0.369923
\(875\) −9.86764 −0.333587
\(876\) −4.50972 −0.152369
\(877\) 27.2350 0.919662 0.459831 0.888006i \(-0.347910\pi\)
0.459831 + 0.888006i \(0.347910\pi\)
\(878\) 21.6498 0.730647
\(879\) −33.3987 −1.12651
\(880\) 4.82059 0.162502
\(881\) −1.56039 −0.0525709 −0.0262854 0.999654i \(-0.508368\pi\)
−0.0262854 + 0.999654i \(0.508368\pi\)
\(882\) 0.338090 0.0113841
\(883\) 21.6417 0.728303 0.364151 0.931340i \(-0.381359\pi\)
0.364151 + 0.931340i \(0.381359\pi\)
\(884\) −20.4197 −0.686790
\(885\) 20.4238 0.686540
\(886\) 4.41468 0.148314
\(887\) −36.2668 −1.21772 −0.608860 0.793278i \(-0.708373\pi\)
−0.608860 + 0.793278i \(0.708373\pi\)
\(888\) −0.466330 −0.0156490
\(889\) −7.54825 −0.253160
\(890\) 8.81387 0.295442
\(891\) −33.5288 −1.12326
\(892\) 0.0143017 0.000478857 0
\(893\) −39.0946 −1.30825
\(894\) 26.2186 0.876881
\(895\) 10.9184 0.364963
\(896\) −1.00000 −0.0334077
\(897\) 16.3874 0.547161
\(898\) −0.922409 −0.0307812
\(899\) 4.40407 0.146884
\(900\) 1.25744 0.0419146
\(901\) 2.34237 0.0780358
\(902\) 28.8851 0.961770
\(903\) −0.00450314 −0.000149855 0
\(904\) −6.08513 −0.202388
\(905\) −16.3383 −0.543103
\(906\) −33.4781 −1.11223
\(907\) 7.17032 0.238087 0.119043 0.992889i \(-0.462017\pi\)
0.119043 + 0.992889i \(0.462017\pi\)
\(908\) −11.4591 −0.380285
\(909\) 3.88062 0.128712
\(910\) −5.17900 −0.171682
\(911\) −25.8384 −0.856065 −0.428032 0.903763i \(-0.640793\pi\)
−0.428032 + 0.903763i \(0.640793\pi\)
\(912\) −8.12943 −0.269192
\(913\) −46.1855 −1.52852
\(914\) −14.9046 −0.493000
\(915\) 20.8542 0.689419
\(916\) −19.4007 −0.641016
\(917\) 2.63117 0.0868889
\(918\) 24.3015 0.802070
\(919\) 13.9390 0.459804 0.229902 0.973214i \(-0.426159\pi\)
0.229902 + 0.973214i \(0.426159\pi\)
\(920\) 2.48393 0.0818926
\(921\) −53.6936 −1.76926
\(922\) −22.1832 −0.730564
\(923\) 33.5974 1.10587
\(924\) −6.94963 −0.228626
\(925\) 1.06304 0.0349526
\(926\) 33.8708 1.11306
\(927\) 0.283097 0.00929811
\(928\) 8.18975 0.268842
\(929\) −41.0531 −1.34691 −0.673455 0.739229i \(-0.735190\pi\)
−0.673455 + 0.739229i \(0.735190\pi\)
\(930\) −0.992921 −0.0325592
\(931\) 4.98269 0.163301
\(932\) 2.29636 0.0752199
\(933\) −13.9723 −0.457431
\(934\) −29.2811 −0.958108
\(935\) −21.5100 −0.703451
\(936\) 1.54719 0.0505715
\(937\) 21.5908 0.705341 0.352671 0.935748i \(-0.385274\pi\)
0.352671 + 0.935748i \(0.385274\pi\)
\(938\) 5.88748 0.192233
\(939\) −32.3217 −1.05478
\(940\) −8.87949 −0.289617
\(941\) −37.7679 −1.23120 −0.615600 0.788059i \(-0.711086\pi\)
−0.615600 + 0.788059i \(0.711086\pi\)
\(942\) 14.7120 0.479342
\(943\) 14.8838 0.484682
\(944\) −11.0613 −0.360014
\(945\) 6.16354 0.200500
\(946\) −0.0117567 −0.000382242 0
\(947\) 12.6737 0.411841 0.205921 0.978569i \(-0.433981\pi\)
0.205921 + 0.978569i \(0.433981\pi\)
\(948\) −11.5410 −0.374836
\(949\) 12.6492 0.410611
\(950\) 18.5318 0.601250
\(951\) −5.96152 −0.193316
\(952\) 4.46210 0.144617
\(953\) 26.8236 0.868901 0.434450 0.900696i \(-0.356943\pi\)
0.434450 + 0.900696i \(0.356943\pi\)
\(954\) −0.177480 −0.00574614
\(955\) 16.8262 0.544483
\(956\) 2.01797 0.0652659
\(957\) 56.9158 1.83983
\(958\) 30.9658 1.00046
\(959\) 0.401572 0.0129674
\(960\) −1.84643 −0.0595931
\(961\) −30.7108 −0.990672
\(962\) 1.30800 0.0421716
\(963\) 1.64127 0.0528892
\(964\) 18.9709 0.611012
\(965\) −3.95280 −0.127245
\(966\) −3.58097 −0.115216
\(967\) 28.9365 0.930535 0.465267 0.885170i \(-0.345958\pi\)
0.465267 + 0.885170i \(0.345958\pi\)
\(968\) −7.14390 −0.229614
\(969\) 36.2743 1.16530
\(970\) −17.5653 −0.563987
\(971\) 24.0696 0.772430 0.386215 0.922409i \(-0.373782\pi\)
0.386215 + 0.922409i \(0.373782\pi\)
\(972\) −3.49613 −0.112139
\(973\) −6.88303 −0.220660
\(974\) 18.9577 0.607445
\(975\) 27.7690 0.889321
\(976\) −11.2944 −0.361524
\(977\) 17.9735 0.575023 0.287511 0.957777i \(-0.407172\pi\)
0.287511 + 0.957777i \(0.407172\pi\)
\(978\) −7.77953 −0.248762
\(979\) −33.1739 −1.06024
\(980\) 1.13171 0.0361512
\(981\) 0.284057 0.00906924
\(982\) 3.89068 0.124157
\(983\) 56.2876 1.79530 0.897648 0.440714i \(-0.145275\pi\)
0.897648 + 0.440714i \(0.145275\pi\)
\(984\) −11.0638 −0.352702
\(985\) −11.6637 −0.371635
\(986\) −36.5435 −1.16378
\(987\) 12.8012 0.407465
\(988\) 22.8021 0.725431
\(989\) −0.00605790 −0.000192630 0
\(990\) 1.62980 0.0517983
\(991\) 50.9832 1.61953 0.809767 0.586751i \(-0.199593\pi\)
0.809767 + 0.586751i \(0.199593\pi\)
\(992\) 0.537753 0.0170737
\(993\) 32.9556 1.04581
\(994\) −7.34167 −0.232864
\(995\) −9.89695 −0.313754
\(996\) 17.6904 0.560541
\(997\) −24.8437 −0.786807 −0.393404 0.919366i \(-0.628702\pi\)
−0.393404 + 0.919366i \(0.628702\pi\)
\(998\) 26.0709 0.825259
\(999\) −1.55665 −0.0492503
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 6034.2.a.k.1.6 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6034.2.a.k.1.6 20 1.1 even 1 trivial