Properties

Label 3549.2.a.bb.1.1
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $8$
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] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, 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("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 25x^{4} - 24x^{3} - 16x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.60802\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.60802 q^{2} +1.00000 q^{3} +4.80176 q^{4} +1.50528 q^{5} -2.60802 q^{6} -1.00000 q^{7} -7.30704 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.60802 q^{2} +1.00000 q^{3} +4.80176 q^{4} +1.50528 q^{5} -2.60802 q^{6} -1.00000 q^{7} -7.30704 q^{8} +1.00000 q^{9} -3.92579 q^{10} -0.0869524 q^{11} +4.80176 q^{12} +2.60802 q^{14} +1.50528 q^{15} +9.45336 q^{16} +6.49464 q^{17} -2.60802 q^{18} -5.29220 q^{19} +7.22798 q^{20} -1.00000 q^{21} +0.226773 q^{22} -5.20221 q^{23} -7.30704 q^{24} -2.73414 q^{25} +1.00000 q^{27} -4.80176 q^{28} -5.97602 q^{29} -3.92579 q^{30} -2.00289 q^{31} -10.0405 q^{32} -0.0869524 q^{33} -16.9381 q^{34} -1.50528 q^{35} +4.80176 q^{36} -9.52127 q^{37} +13.8022 q^{38} -10.9991 q^{40} +6.43896 q^{41} +2.60802 q^{42} +6.81071 q^{43} -0.417524 q^{44} +1.50528 q^{45} +13.5674 q^{46} +1.83051 q^{47} +9.45336 q^{48} +1.00000 q^{49} +7.13069 q^{50} +6.49464 q^{51} -3.28476 q^{53} -2.60802 q^{54} -0.130887 q^{55} +7.30704 q^{56} -5.29220 q^{57} +15.5856 q^{58} -2.67509 q^{59} +7.22798 q^{60} -6.54557 q^{61} +5.22357 q^{62} -1.00000 q^{63} +7.27901 q^{64} +0.226773 q^{66} -8.29459 q^{67} +31.1857 q^{68} -5.20221 q^{69} +3.92579 q^{70} +12.9178 q^{71} -7.30704 q^{72} -14.5507 q^{73} +24.8316 q^{74} -2.73414 q^{75} -25.4119 q^{76} +0.0869524 q^{77} -11.6780 q^{79} +14.2299 q^{80} +1.00000 q^{81} -16.7929 q^{82} -1.29310 q^{83} -4.80176 q^{84} +9.77623 q^{85} -17.7625 q^{86} -5.97602 q^{87} +0.635364 q^{88} -15.8454 q^{89} -3.92579 q^{90} -24.9797 q^{92} -2.00289 q^{93} -4.77401 q^{94} -7.96623 q^{95} -10.0405 q^{96} +5.86980 q^{97} -2.60802 q^{98} -0.0869524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} - 4 q^{10} - 8 q^{11} + 6 q^{12} + 2 q^{14} - 2 q^{15} + 10 q^{16} + 10 q^{17} - 2 q^{18} - 18 q^{19} - 2 q^{20} - 8 q^{21} + 2 q^{22} - 2 q^{23} - 12 q^{24} - 6 q^{25} + 8 q^{27} - 6 q^{28} - 12 q^{29} - 4 q^{30} - 16 q^{31} - 26 q^{32} - 8 q^{33} - 24 q^{34} + 2 q^{35} + 6 q^{36} - 24 q^{37} + 16 q^{38} - 30 q^{40} + 4 q^{41} + 2 q^{42} - 10 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 10 q^{47} + 10 q^{48} + 8 q^{49} + 16 q^{50} + 10 q^{51} + 6 q^{53} - 2 q^{54} - 10 q^{55} + 12 q^{56} - 18 q^{57} - 16 q^{58} - 6 q^{59} - 2 q^{60} + 6 q^{61} + 16 q^{62} - 8 q^{63} - 8 q^{64} + 2 q^{66} - 24 q^{67} + 20 q^{68} - 2 q^{69} + 4 q^{70} - 42 q^{71} - 12 q^{72} - 32 q^{73} - 18 q^{74} - 6 q^{75} - 28 q^{76} + 8 q^{77} - 2 q^{79} + 40 q^{80} + 8 q^{81} - 18 q^{82} + 2 q^{83} - 6 q^{84} - 4 q^{85} - 26 q^{86} - 12 q^{87} - 2 q^{88} - 12 q^{89} - 4 q^{90} - 10 q^{92} - 16 q^{93} - 16 q^{94} - 4 q^{95} - 26 q^{96} - 64 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.60802 −1.84415 −0.922074 0.387015i \(-0.873506\pi\)
−0.922074 + 0.387015i \(0.873506\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.80176 2.40088
\(5\) 1.50528 0.673181 0.336590 0.941651i \(-0.390726\pi\)
0.336590 + 0.941651i \(0.390726\pi\)
\(6\) −2.60802 −1.06472
\(7\) −1.00000 −0.377964
\(8\) −7.30704 −2.58343
\(9\) 1.00000 0.333333
\(10\) −3.92579 −1.24144
\(11\) −0.0869524 −0.0262171 −0.0131086 0.999914i \(-0.504173\pi\)
−0.0131086 + 0.999914i \(0.504173\pi\)
\(12\) 4.80176 1.38615
\(13\) 0 0
\(14\) 2.60802 0.697022
\(15\) 1.50528 0.388661
\(16\) 9.45336 2.36334
\(17\) 6.49464 1.57518 0.787591 0.616199i \(-0.211328\pi\)
0.787591 + 0.616199i \(0.211328\pi\)
\(18\) −2.60802 −0.614716
\(19\) −5.29220 −1.21411 −0.607057 0.794658i \(-0.707650\pi\)
−0.607057 + 0.794658i \(0.707650\pi\)
\(20\) 7.22798 1.61623
\(21\) −1.00000 −0.218218
\(22\) 0.226773 0.0483482
\(23\) −5.20221 −1.08473 −0.542367 0.840141i \(-0.682472\pi\)
−0.542367 + 0.840141i \(0.682472\pi\)
\(24\) −7.30704 −1.49154
\(25\) −2.73414 −0.546828
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.80176 −0.907447
\(29\) −5.97602 −1.10972 −0.554859 0.831944i \(-0.687228\pi\)
−0.554859 + 0.831944i \(0.687228\pi\)
\(30\) −3.92579 −0.716748
\(31\) −2.00289 −0.359730 −0.179865 0.983691i \(-0.557566\pi\)
−0.179865 + 0.983691i \(0.557566\pi\)
\(32\) −10.0405 −1.77492
\(33\) −0.0869524 −0.0151365
\(34\) −16.9381 −2.90487
\(35\) −1.50528 −0.254438
\(36\) 4.80176 0.800293
\(37\) −9.52127 −1.56529 −0.782643 0.622470i \(-0.786129\pi\)
−0.782643 + 0.622470i \(0.786129\pi\)
\(38\) 13.8022 2.23901
\(39\) 0 0
\(40\) −10.9991 −1.73911
\(41\) 6.43896 1.00560 0.502799 0.864404i \(-0.332304\pi\)
0.502799 + 0.864404i \(0.332304\pi\)
\(42\) 2.60802 0.402426
\(43\) 6.81071 1.03862 0.519312 0.854585i \(-0.326188\pi\)
0.519312 + 0.854585i \(0.326188\pi\)
\(44\) −0.417524 −0.0629441
\(45\) 1.50528 0.224394
\(46\) 13.5674 2.00041
\(47\) 1.83051 0.267008 0.133504 0.991048i \(-0.457377\pi\)
0.133504 + 0.991048i \(0.457377\pi\)
\(48\) 9.45336 1.36448
\(49\) 1.00000 0.142857
\(50\) 7.13069 1.00843
\(51\) 6.49464 0.909431
\(52\) 0 0
\(53\) −3.28476 −0.451197 −0.225598 0.974220i \(-0.572434\pi\)
−0.225598 + 0.974220i \(0.572434\pi\)
\(54\) −2.60802 −0.354906
\(55\) −0.130887 −0.0176489
\(56\) 7.30704 0.976444
\(57\) −5.29220 −0.700969
\(58\) 15.5856 2.04649
\(59\) −2.67509 −0.348267 −0.174133 0.984722i \(-0.555712\pi\)
−0.174133 + 0.984722i \(0.555712\pi\)
\(60\) 7.22798 0.933128
\(61\) −6.54557 −0.838074 −0.419037 0.907969i \(-0.637632\pi\)
−0.419037 + 0.907969i \(0.637632\pi\)
\(62\) 5.22357 0.663394
\(63\) −1.00000 −0.125988
\(64\) 7.27901 0.909876
\(65\) 0 0
\(66\) 0.226773 0.0279139
\(67\) −8.29459 −1.01335 −0.506673 0.862138i \(-0.669125\pi\)
−0.506673 + 0.862138i \(0.669125\pi\)
\(68\) 31.1857 3.78182
\(69\) −5.20221 −0.626272
\(70\) 3.92579 0.469222
\(71\) 12.9178 1.53307 0.766533 0.642205i \(-0.221980\pi\)
0.766533 + 0.642205i \(0.221980\pi\)
\(72\) −7.30704 −0.861142
\(73\) −14.5507 −1.70303 −0.851513 0.524334i \(-0.824314\pi\)
−0.851513 + 0.524334i \(0.824314\pi\)
\(74\) 24.8316 2.88662
\(75\) −2.73414 −0.315711
\(76\) −25.4119 −2.91494
\(77\) 0.0869524 0.00990914
\(78\) 0 0
\(79\) −11.6780 −1.31387 −0.656936 0.753946i \(-0.728148\pi\)
−0.656936 + 0.753946i \(0.728148\pi\)
\(80\) 14.2299 1.59096
\(81\) 1.00000 0.111111
\(82\) −16.7929 −1.85447
\(83\) −1.29310 −0.141937 −0.0709683 0.997479i \(-0.522609\pi\)
−0.0709683 + 0.997479i \(0.522609\pi\)
\(84\) −4.80176 −0.523915
\(85\) 9.77623 1.06038
\(86\) −17.7625 −1.91538
\(87\) −5.97602 −0.640697
\(88\) 0.635364 0.0677300
\(89\) −15.8454 −1.67961 −0.839805 0.542889i \(-0.817331\pi\)
−0.839805 + 0.542889i \(0.817331\pi\)
\(90\) −3.92579 −0.413815
\(91\) 0 0
\(92\) −24.9797 −2.60432
\(93\) −2.00289 −0.207690
\(94\) −4.77401 −0.492402
\(95\) −7.96623 −0.817318
\(96\) −10.0405 −1.02475
\(97\) 5.86980 0.595988 0.297994 0.954568i \(-0.403682\pi\)
0.297994 + 0.954568i \(0.403682\pi\)
\(98\) −2.60802 −0.263450
\(99\) −0.0869524 −0.00873904
\(100\) −13.1287 −1.31287
\(101\) −6.68172 −0.664856 −0.332428 0.943129i \(-0.607868\pi\)
−0.332428 + 0.943129i \(0.607868\pi\)
\(102\) −16.9381 −1.67713
\(103\) −4.52273 −0.445637 −0.222819 0.974860i \(-0.571526\pi\)
−0.222819 + 0.974860i \(0.571526\pi\)
\(104\) 0 0
\(105\) −1.50528 −0.146900
\(106\) 8.56672 0.832073
\(107\) −10.2448 −0.990404 −0.495202 0.868778i \(-0.664906\pi\)
−0.495202 + 0.868778i \(0.664906\pi\)
\(108\) 4.80176 0.462049
\(109\) 13.5552 1.29835 0.649177 0.760637i \(-0.275113\pi\)
0.649177 + 0.760637i \(0.275113\pi\)
\(110\) 0.341357 0.0325471
\(111\) −9.52127 −0.903719
\(112\) −9.45336 −0.893259
\(113\) 13.3874 1.25938 0.629691 0.776846i \(-0.283182\pi\)
0.629691 + 0.776846i \(0.283182\pi\)
\(114\) 13.8022 1.29269
\(115\) −7.83076 −0.730222
\(116\) −28.6954 −2.66430
\(117\) 0 0
\(118\) 6.97667 0.642255
\(119\) −6.49464 −0.595363
\(120\) −10.9991 −1.00408
\(121\) −10.9924 −0.999313
\(122\) 17.0710 1.54553
\(123\) 6.43896 0.580582
\(124\) −9.61739 −0.863667
\(125\) −11.6420 −1.04129
\(126\) 2.60802 0.232341
\(127\) 12.9447 1.14866 0.574328 0.818626i \(-0.305264\pi\)
0.574328 + 0.818626i \(0.305264\pi\)
\(128\) 1.09716 0.0969765
\(129\) 6.81071 0.599650
\(130\) 0 0
\(131\) 5.84887 0.511018 0.255509 0.966807i \(-0.417757\pi\)
0.255509 + 0.966807i \(0.417757\pi\)
\(132\) −0.417524 −0.0363408
\(133\) 5.29220 0.458892
\(134\) 21.6324 1.86876
\(135\) 1.50528 0.129554
\(136\) −47.4566 −4.06937
\(137\) −5.11089 −0.436653 −0.218327 0.975876i \(-0.570060\pi\)
−0.218327 + 0.975876i \(0.570060\pi\)
\(138\) 13.5674 1.15494
\(139\) −19.3275 −1.63934 −0.819669 0.572837i \(-0.805843\pi\)
−0.819669 + 0.572837i \(0.805843\pi\)
\(140\) −7.22798 −0.610876
\(141\) 1.83051 0.154157
\(142\) −33.6900 −2.82720
\(143\) 0 0
\(144\) 9.45336 0.787780
\(145\) −8.99557 −0.747041
\(146\) 37.9484 3.14063
\(147\) 1.00000 0.0824786
\(148\) −45.7188 −3.75806
\(149\) 9.51027 0.779112 0.389556 0.921003i \(-0.372628\pi\)
0.389556 + 0.921003i \(0.372628\pi\)
\(150\) 7.13069 0.582218
\(151\) 12.7770 1.03977 0.519887 0.854235i \(-0.325974\pi\)
0.519887 + 0.854235i \(0.325974\pi\)
\(152\) 38.6703 3.13658
\(153\) 6.49464 0.525060
\(154\) −0.226773 −0.0182739
\(155\) −3.01490 −0.242163
\(156\) 0 0
\(157\) −2.57466 −0.205480 −0.102740 0.994708i \(-0.532761\pi\)
−0.102740 + 0.994708i \(0.532761\pi\)
\(158\) 30.4563 2.42297
\(159\) −3.28476 −0.260499
\(160\) −15.1137 −1.19484
\(161\) 5.20221 0.409991
\(162\) −2.60802 −0.204905
\(163\) −19.5300 −1.52971 −0.764853 0.644205i \(-0.777188\pi\)
−0.764853 + 0.644205i \(0.777188\pi\)
\(164\) 30.9183 2.41432
\(165\) −0.130887 −0.0101896
\(166\) 3.37244 0.261752
\(167\) 1.39652 0.108066 0.0540331 0.998539i \(-0.482792\pi\)
0.0540331 + 0.998539i \(0.482792\pi\)
\(168\) 7.30704 0.563750
\(169\) 0 0
\(170\) −25.4966 −1.95550
\(171\) −5.29220 −0.404705
\(172\) 32.7034 2.49361
\(173\) 17.1814 1.30628 0.653138 0.757238i \(-0.273452\pi\)
0.653138 + 0.757238i \(0.273452\pi\)
\(174\) 15.5856 1.18154
\(175\) 2.73414 0.206682
\(176\) −0.821992 −0.0619600
\(177\) −2.67509 −0.201072
\(178\) 41.3251 3.09745
\(179\) −1.43708 −0.107412 −0.0537061 0.998557i \(-0.517103\pi\)
−0.0537061 + 0.998557i \(0.517103\pi\)
\(180\) 7.22798 0.538742
\(181\) −4.31882 −0.321015 −0.160508 0.987035i \(-0.551313\pi\)
−0.160508 + 0.987035i \(0.551313\pi\)
\(182\) 0 0
\(183\) −6.54557 −0.483862
\(184\) 38.0127 2.80233
\(185\) −14.3321 −1.05372
\(186\) 5.22357 0.383011
\(187\) −0.564724 −0.0412967
\(188\) 8.78969 0.641054
\(189\) −1.00000 −0.0727393
\(190\) 20.7761 1.50726
\(191\) −6.78719 −0.491104 −0.245552 0.969383i \(-0.578969\pi\)
−0.245552 + 0.969383i \(0.578969\pi\)
\(192\) 7.27901 0.525317
\(193\) 21.3749 1.53860 0.769300 0.638887i \(-0.220605\pi\)
0.769300 + 0.638887i \(0.220605\pi\)
\(194\) −15.3085 −1.09909
\(195\) 0 0
\(196\) 4.80176 0.342983
\(197\) 22.7982 1.62431 0.812153 0.583444i \(-0.198295\pi\)
0.812153 + 0.583444i \(0.198295\pi\)
\(198\) 0.226773 0.0161161
\(199\) 2.66365 0.188821 0.0944107 0.995533i \(-0.469903\pi\)
0.0944107 + 0.995533i \(0.469903\pi\)
\(200\) 19.9785 1.41269
\(201\) −8.29459 −0.585055
\(202\) 17.4260 1.22609
\(203\) 5.97602 0.419434
\(204\) 31.1857 2.18343
\(205\) 9.69243 0.676948
\(206\) 11.7953 0.821821
\(207\) −5.20221 −0.361578
\(208\) 0 0
\(209\) 0.460170 0.0318306
\(210\) 3.92579 0.270905
\(211\) −24.1832 −1.66484 −0.832419 0.554147i \(-0.813045\pi\)
−0.832419 + 0.554147i \(0.813045\pi\)
\(212\) −15.7726 −1.08327
\(213\) 12.9178 0.885116
\(214\) 26.7187 1.82645
\(215\) 10.2520 0.699181
\(216\) −7.30704 −0.497181
\(217\) 2.00289 0.135965
\(218\) −35.3523 −2.39436
\(219\) −14.5507 −0.983242
\(220\) −0.628490 −0.0423728
\(221\) 0 0
\(222\) 24.8316 1.66659
\(223\) −16.9080 −1.13224 −0.566122 0.824322i \(-0.691557\pi\)
−0.566122 + 0.824322i \(0.691557\pi\)
\(224\) 10.0405 0.670857
\(225\) −2.73414 −0.182276
\(226\) −34.9146 −2.32248
\(227\) −11.6431 −0.772782 −0.386391 0.922335i \(-0.626278\pi\)
−0.386391 + 0.922335i \(0.626278\pi\)
\(228\) −25.4119 −1.68294
\(229\) 7.90801 0.522576 0.261288 0.965261i \(-0.415853\pi\)
0.261288 + 0.965261i \(0.415853\pi\)
\(230\) 20.4228 1.34664
\(231\) 0.0869524 0.00572105
\(232\) 43.6670 2.86688
\(233\) −24.0854 −1.57789 −0.788945 0.614464i \(-0.789372\pi\)
−0.788945 + 0.614464i \(0.789372\pi\)
\(234\) 0 0
\(235\) 2.75543 0.179745
\(236\) −12.8451 −0.836146
\(237\) −11.6780 −0.758565
\(238\) 16.9381 1.09794
\(239\) 10.9354 0.707352 0.353676 0.935368i \(-0.384932\pi\)
0.353676 + 0.935368i \(0.384932\pi\)
\(240\) 14.2299 0.918538
\(241\) 19.9054 1.28222 0.641108 0.767450i \(-0.278475\pi\)
0.641108 + 0.767450i \(0.278475\pi\)
\(242\) 28.6685 1.84288
\(243\) 1.00000 0.0641500
\(244\) −31.4302 −2.01212
\(245\) 1.50528 0.0961687
\(246\) −16.7929 −1.07068
\(247\) 0 0
\(248\) 14.6352 0.929335
\(249\) −1.29310 −0.0819471
\(250\) 30.3626 1.92030
\(251\) −23.3883 −1.47626 −0.738129 0.674659i \(-0.764290\pi\)
−0.738129 + 0.674659i \(0.764290\pi\)
\(252\) −4.80176 −0.302482
\(253\) 0.452344 0.0284386
\(254\) −33.7600 −2.11829
\(255\) 9.77623 0.612212
\(256\) −17.4194 −1.08871
\(257\) 12.0595 0.752251 0.376125 0.926569i \(-0.377256\pi\)
0.376125 + 0.926569i \(0.377256\pi\)
\(258\) −17.7625 −1.10584
\(259\) 9.52127 0.591623
\(260\) 0 0
\(261\) −5.97602 −0.369906
\(262\) −15.2540 −0.942393
\(263\) 4.53226 0.279471 0.139736 0.990189i \(-0.455375\pi\)
0.139736 + 0.990189i \(0.455375\pi\)
\(264\) 0.635364 0.0391039
\(265\) −4.94448 −0.303737
\(266\) −13.8022 −0.846265
\(267\) −15.8454 −0.969723
\(268\) −39.8286 −2.43292
\(269\) −27.9600 −1.70475 −0.852375 0.522931i \(-0.824838\pi\)
−0.852375 + 0.522931i \(0.824838\pi\)
\(270\) −3.92579 −0.238916
\(271\) −4.18954 −0.254496 −0.127248 0.991871i \(-0.540614\pi\)
−0.127248 + 0.991871i \(0.540614\pi\)
\(272\) 61.3962 3.72269
\(273\) 0 0
\(274\) 13.3293 0.805252
\(275\) 0.237740 0.0143363
\(276\) −24.9797 −1.50360
\(277\) 9.86805 0.592913 0.296457 0.955046i \(-0.404195\pi\)
0.296457 + 0.955046i \(0.404195\pi\)
\(278\) 50.4065 3.02318
\(279\) −2.00289 −0.119910
\(280\) 10.9991 0.657323
\(281\) 2.76496 0.164944 0.0824719 0.996593i \(-0.473719\pi\)
0.0824719 + 0.996593i \(0.473719\pi\)
\(282\) −4.77401 −0.284288
\(283\) −5.74079 −0.341255 −0.170627 0.985336i \(-0.554579\pi\)
−0.170627 + 0.985336i \(0.554579\pi\)
\(284\) 62.0284 3.68071
\(285\) −7.96623 −0.471879
\(286\) 0 0
\(287\) −6.43896 −0.380080
\(288\) −10.0405 −0.591641
\(289\) 25.1803 1.48120
\(290\) 23.4606 1.37765
\(291\) 5.86980 0.344094
\(292\) −69.8687 −4.08876
\(293\) 15.8014 0.923131 0.461565 0.887106i \(-0.347288\pi\)
0.461565 + 0.887106i \(0.347288\pi\)
\(294\) −2.60802 −0.152103
\(295\) −4.02675 −0.234446
\(296\) 69.5722 4.04380
\(297\) −0.0869524 −0.00504549
\(298\) −24.8030 −1.43680
\(299\) 0 0
\(300\) −13.1287 −0.757985
\(301\) −6.81071 −0.392563
\(302\) −33.3226 −1.91750
\(303\) −6.68172 −0.383855
\(304\) −50.0291 −2.86937
\(305\) −9.85290 −0.564175
\(306\) −16.9381 −0.968289
\(307\) 10.1195 0.577549 0.288774 0.957397i \(-0.406752\pi\)
0.288774 + 0.957397i \(0.406752\pi\)
\(308\) 0.417524 0.0237906
\(309\) −4.52273 −0.257289
\(310\) 7.86293 0.446584
\(311\) 14.4989 0.822155 0.411077 0.911600i \(-0.365153\pi\)
0.411077 + 0.911600i \(0.365153\pi\)
\(312\) 0 0
\(313\) 13.2670 0.749897 0.374948 0.927046i \(-0.377660\pi\)
0.374948 + 0.927046i \(0.377660\pi\)
\(314\) 6.71477 0.378936
\(315\) −1.50528 −0.0848128
\(316\) −56.0747 −3.15445
\(317\) 30.4190 1.70850 0.854252 0.519860i \(-0.174016\pi\)
0.854252 + 0.519860i \(0.174016\pi\)
\(318\) 8.56672 0.480398
\(319\) 0.519629 0.0290936
\(320\) 10.9569 0.612511
\(321\) −10.2448 −0.571810
\(322\) −13.5674 −0.756084
\(323\) −34.3710 −1.91245
\(324\) 4.80176 0.266764
\(325\) 0 0
\(326\) 50.9345 2.82100
\(327\) 13.5552 0.749605
\(328\) −47.0497 −2.59789
\(329\) −1.83051 −0.100920
\(330\) 0.341357 0.0187911
\(331\) −9.30294 −0.511336 −0.255668 0.966765i \(-0.582295\pi\)
−0.255668 + 0.966765i \(0.582295\pi\)
\(332\) −6.20917 −0.340773
\(333\) −9.52127 −0.521762
\(334\) −3.64216 −0.199290
\(335\) −12.4857 −0.682165
\(336\) −9.45336 −0.515723
\(337\) 13.3054 0.724789 0.362395 0.932025i \(-0.381959\pi\)
0.362395 + 0.932025i \(0.381959\pi\)
\(338\) 0 0
\(339\) 13.3874 0.727104
\(340\) 46.9431 2.54585
\(341\) 0.174156 0.00943108
\(342\) 13.8022 0.746335
\(343\) −1.00000 −0.0539949
\(344\) −49.7661 −2.68321
\(345\) −7.83076 −0.421594
\(346\) −44.8094 −2.40897
\(347\) −16.8760 −0.905953 −0.452976 0.891523i \(-0.649638\pi\)
−0.452976 + 0.891523i \(0.649638\pi\)
\(348\) −28.6954 −1.53823
\(349\) 2.40720 0.128854 0.0644271 0.997922i \(-0.479478\pi\)
0.0644271 + 0.997922i \(0.479478\pi\)
\(350\) −7.13069 −0.381151
\(351\) 0 0
\(352\) 0.873043 0.0465333
\(353\) −30.7137 −1.63472 −0.817361 0.576125i \(-0.804564\pi\)
−0.817361 + 0.576125i \(0.804564\pi\)
\(354\) 6.97667 0.370806
\(355\) 19.4449 1.03203
\(356\) −76.0858 −4.03254
\(357\) −6.49464 −0.343733
\(358\) 3.74792 0.198084
\(359\) −10.6322 −0.561145 −0.280572 0.959833i \(-0.590524\pi\)
−0.280572 + 0.959833i \(0.590524\pi\)
\(360\) −10.9991 −0.579704
\(361\) 9.00742 0.474075
\(362\) 11.2636 0.592000
\(363\) −10.9924 −0.576953
\(364\) 0 0
\(365\) −21.9028 −1.14644
\(366\) 17.0710 0.892314
\(367\) 26.2062 1.36795 0.683977 0.729503i \(-0.260249\pi\)
0.683977 + 0.729503i \(0.260249\pi\)
\(368\) −49.1784 −2.56360
\(369\) 6.43896 0.335199
\(370\) 37.3785 1.94322
\(371\) 3.28476 0.170536
\(372\) −9.61739 −0.498639
\(373\) 18.2102 0.942888 0.471444 0.881896i \(-0.343733\pi\)
0.471444 + 0.881896i \(0.343733\pi\)
\(374\) 1.47281 0.0761572
\(375\) −11.6420 −0.601192
\(376\) −13.3756 −0.689796
\(377\) 0 0
\(378\) 2.60802 0.134142
\(379\) −28.8947 −1.48422 −0.742110 0.670278i \(-0.766175\pi\)
−0.742110 + 0.670278i \(0.766175\pi\)
\(380\) −38.2519 −1.96228
\(381\) 12.9447 0.663176
\(382\) 17.7011 0.905668
\(383\) 17.3262 0.885329 0.442665 0.896687i \(-0.354033\pi\)
0.442665 + 0.896687i \(0.354033\pi\)
\(384\) 1.09716 0.0559894
\(385\) 0.130887 0.00667064
\(386\) −55.7462 −2.83741
\(387\) 6.81071 0.346208
\(388\) 28.1854 1.43090
\(389\) 13.6737 0.693282 0.346641 0.937998i \(-0.387322\pi\)
0.346641 + 0.937998i \(0.387322\pi\)
\(390\) 0 0
\(391\) −33.7865 −1.70865
\(392\) −7.30704 −0.369061
\(393\) 5.84887 0.295037
\(394\) −59.4582 −2.99546
\(395\) −17.5786 −0.884473
\(396\) −0.417524 −0.0209814
\(397\) −15.4433 −0.775079 −0.387540 0.921853i \(-0.626675\pi\)
−0.387540 + 0.921853i \(0.626675\pi\)
\(398\) −6.94686 −0.348215
\(399\) 5.29220 0.264942
\(400\) −25.8468 −1.29234
\(401\) −14.2334 −0.710783 −0.355391 0.934718i \(-0.615652\pi\)
−0.355391 + 0.934718i \(0.615652\pi\)
\(402\) 21.6324 1.07893
\(403\) 0 0
\(404\) −32.0840 −1.59624
\(405\) 1.50528 0.0747978
\(406\) −15.5856 −0.773499
\(407\) 0.827897 0.0410373
\(408\) −47.4566 −2.34945
\(409\) −14.0858 −0.696499 −0.348249 0.937402i \(-0.613224\pi\)
−0.348249 + 0.937402i \(0.613224\pi\)
\(410\) −25.2780 −1.24839
\(411\) −5.11089 −0.252102
\(412\) −21.7170 −1.06992
\(413\) 2.67509 0.131632
\(414\) 13.5674 0.666804
\(415\) −1.94648 −0.0955489
\(416\) 0 0
\(417\) −19.3275 −0.946473
\(418\) −1.20013 −0.0587003
\(419\) −37.6723 −1.84041 −0.920207 0.391432i \(-0.871980\pi\)
−0.920207 + 0.391432i \(0.871980\pi\)
\(420\) −7.22798 −0.352689
\(421\) −2.28615 −0.111420 −0.0557101 0.998447i \(-0.517742\pi\)
−0.0557101 + 0.998447i \(0.517742\pi\)
\(422\) 63.0701 3.07021
\(423\) 1.83051 0.0890026
\(424\) 24.0019 1.16563
\(425\) −17.7573 −0.861353
\(426\) −33.6900 −1.63228
\(427\) 6.54557 0.316762
\(428\) −49.1931 −2.37784
\(429\) 0 0
\(430\) −26.7374 −1.28939
\(431\) 1.98118 0.0954301 0.0477150 0.998861i \(-0.484806\pi\)
0.0477150 + 0.998861i \(0.484806\pi\)
\(432\) 9.45336 0.454825
\(433\) −5.89088 −0.283098 −0.141549 0.989931i \(-0.545208\pi\)
−0.141549 + 0.989931i \(0.545208\pi\)
\(434\) −5.22357 −0.250740
\(435\) −8.99557 −0.431304
\(436\) 65.0889 3.11719
\(437\) 27.5311 1.31699
\(438\) 37.9484 1.81324
\(439\) −8.06563 −0.384951 −0.192476 0.981302i \(-0.561652\pi\)
−0.192476 + 0.981302i \(0.561652\pi\)
\(440\) 0.956399 0.0455945
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 6.65984 0.316419 0.158209 0.987406i \(-0.449428\pi\)
0.158209 + 0.987406i \(0.449428\pi\)
\(444\) −45.7188 −2.16972
\(445\) −23.8517 −1.13068
\(446\) 44.0964 2.08802
\(447\) 9.51027 0.449821
\(448\) −7.27901 −0.343901
\(449\) −19.8491 −0.936738 −0.468369 0.883533i \(-0.655158\pi\)
−0.468369 + 0.883533i \(0.655158\pi\)
\(450\) 7.13069 0.336144
\(451\) −0.559883 −0.0263639
\(452\) 64.2831 3.02362
\(453\) 12.7770 0.600314
\(454\) 30.3655 1.42512
\(455\) 0 0
\(456\) 38.6703 1.81090
\(457\) 2.58884 0.121101 0.0605505 0.998165i \(-0.480714\pi\)
0.0605505 + 0.998165i \(0.480714\pi\)
\(458\) −20.6242 −0.963708
\(459\) 6.49464 0.303144
\(460\) −37.6014 −1.75318
\(461\) −26.3667 −1.22802 −0.614010 0.789298i \(-0.710445\pi\)
−0.614010 + 0.789298i \(0.710445\pi\)
\(462\) −0.226773 −0.0105504
\(463\) −29.3082 −1.36207 −0.681034 0.732252i \(-0.738469\pi\)
−0.681034 + 0.732252i \(0.738469\pi\)
\(464\) −56.4935 −2.62264
\(465\) −3.01490 −0.139813
\(466\) 62.8153 2.90986
\(467\) 3.23005 0.149469 0.0747345 0.997203i \(-0.476189\pi\)
0.0747345 + 0.997203i \(0.476189\pi\)
\(468\) 0 0
\(469\) 8.29459 0.383009
\(470\) −7.18622 −0.331475
\(471\) −2.57466 −0.118634
\(472\) 19.5470 0.899721
\(473\) −0.592207 −0.0272297
\(474\) 30.4563 1.39890
\(475\) 14.4696 0.663912
\(476\) −31.1857 −1.42939
\(477\) −3.28476 −0.150399
\(478\) −28.5197 −1.30446
\(479\) −7.91296 −0.361552 −0.180776 0.983524i \(-0.557861\pi\)
−0.180776 + 0.983524i \(0.557861\pi\)
\(480\) −15.1137 −0.689843
\(481\) 0 0
\(482\) −51.9135 −2.36460
\(483\) 5.20221 0.236709
\(484\) −52.7830 −2.39923
\(485\) 8.83568 0.401208
\(486\) −2.60802 −0.118302
\(487\) 15.4451 0.699883 0.349941 0.936772i \(-0.386201\pi\)
0.349941 + 0.936772i \(0.386201\pi\)
\(488\) 47.8287 2.16510
\(489\) −19.5300 −0.883176
\(490\) −3.92579 −0.177349
\(491\) −14.9062 −0.672708 −0.336354 0.941736i \(-0.609194\pi\)
−0.336354 + 0.941736i \(0.609194\pi\)
\(492\) 30.9183 1.39391
\(493\) −38.8121 −1.74801
\(494\) 0 0
\(495\) −0.130887 −0.00588295
\(496\) −18.9340 −0.850164
\(497\) −12.9178 −0.579445
\(498\) 3.37244 0.151123
\(499\) −25.4306 −1.13843 −0.569214 0.822189i \(-0.692752\pi\)
−0.569214 + 0.822189i \(0.692752\pi\)
\(500\) −55.9022 −2.50002
\(501\) 1.39652 0.0623920
\(502\) 60.9972 2.72244
\(503\) 28.7912 1.28373 0.641867 0.766816i \(-0.278160\pi\)
0.641867 + 0.766816i \(0.278160\pi\)
\(504\) 7.30704 0.325481
\(505\) −10.0578 −0.447568
\(506\) −1.17972 −0.0524450
\(507\) 0 0
\(508\) 62.1573 2.75778
\(509\) −15.6649 −0.694334 −0.347167 0.937803i \(-0.612856\pi\)
−0.347167 + 0.937803i \(0.612856\pi\)
\(510\) −25.4966 −1.12901
\(511\) 14.5507 0.643683
\(512\) 43.2359 1.91077
\(513\) −5.29220 −0.233656
\(514\) −31.4514 −1.38726
\(515\) −6.80796 −0.299994
\(516\) 32.7034 1.43969
\(517\) −0.159168 −0.00700018
\(518\) −24.8316 −1.09104
\(519\) 17.1814 0.754179
\(520\) 0 0
\(521\) 4.38082 0.191927 0.0959636 0.995385i \(-0.469407\pi\)
0.0959636 + 0.995385i \(0.469407\pi\)
\(522\) 15.5856 0.682162
\(523\) −21.3207 −0.932289 −0.466144 0.884709i \(-0.654357\pi\)
−0.466144 + 0.884709i \(0.654357\pi\)
\(524\) 28.0849 1.22689
\(525\) 2.73414 0.119328
\(526\) −11.8202 −0.515386
\(527\) −13.0080 −0.566639
\(528\) −0.821992 −0.0357726
\(529\) 4.06295 0.176650
\(530\) 12.8953 0.560136
\(531\) −2.67509 −0.116089
\(532\) 25.4119 1.10174
\(533\) 0 0
\(534\) 41.3251 1.78831
\(535\) −15.4213 −0.666720
\(536\) 60.6089 2.61791
\(537\) −1.43708 −0.0620144
\(538\) 72.9201 3.14381
\(539\) −0.0869524 −0.00374530
\(540\) 7.22798 0.311043
\(541\) −18.7093 −0.804374 −0.402187 0.915557i \(-0.631750\pi\)
−0.402187 + 0.915557i \(0.631750\pi\)
\(542\) 10.9264 0.469329
\(543\) −4.31882 −0.185338
\(544\) −65.2093 −2.79582
\(545\) 20.4044 0.874027
\(546\) 0 0
\(547\) 26.4218 1.12972 0.564858 0.825188i \(-0.308931\pi\)
0.564858 + 0.825188i \(0.308931\pi\)
\(548\) −24.5413 −1.04835
\(549\) −6.54557 −0.279358
\(550\) −0.620030 −0.0264382
\(551\) 31.6263 1.34733
\(552\) 38.0127 1.61793
\(553\) 11.6780 0.496597
\(554\) −25.7360 −1.09342
\(555\) −14.3321 −0.608366
\(556\) −92.8061 −3.93585
\(557\) −18.0656 −0.765462 −0.382731 0.923860i \(-0.625016\pi\)
−0.382731 + 0.923860i \(0.625016\pi\)
\(558\) 5.22357 0.221131
\(559\) 0 0
\(560\) −14.2299 −0.601325
\(561\) −0.564724 −0.0238427
\(562\) −7.21107 −0.304181
\(563\) −32.9515 −1.38874 −0.694371 0.719617i \(-0.744317\pi\)
−0.694371 + 0.719617i \(0.744317\pi\)
\(564\) 8.78969 0.370113
\(565\) 20.1518 0.847791
\(566\) 14.9721 0.629324
\(567\) −1.00000 −0.0419961
\(568\) −94.3912 −3.96057
\(569\) 6.66475 0.279401 0.139700 0.990194i \(-0.455386\pi\)
0.139700 + 0.990194i \(0.455386\pi\)
\(570\) 20.7761 0.870214
\(571\) −40.5063 −1.69513 −0.847567 0.530688i \(-0.821934\pi\)
−0.847567 + 0.530688i \(0.821934\pi\)
\(572\) 0 0
\(573\) −6.78719 −0.283539
\(574\) 16.7929 0.700923
\(575\) 14.2236 0.593163
\(576\) 7.27901 0.303292
\(577\) 16.2052 0.674632 0.337316 0.941391i \(-0.390481\pi\)
0.337316 + 0.941391i \(0.390481\pi\)
\(578\) −65.6708 −2.73154
\(579\) 21.3749 0.888312
\(580\) −43.1945 −1.79356
\(581\) 1.29310 0.0536470
\(582\) −15.3085 −0.634560
\(583\) 0.285618 0.0118291
\(584\) 106.322 4.39964
\(585\) 0 0
\(586\) −41.2105 −1.70239
\(587\) 17.9423 0.740557 0.370278 0.928921i \(-0.379262\pi\)
0.370278 + 0.928921i \(0.379262\pi\)
\(588\) 4.80176 0.198021
\(589\) 10.5997 0.436753
\(590\) 10.5018 0.432354
\(591\) 22.7982 0.937794
\(592\) −90.0080 −3.69931
\(593\) 34.1889 1.40397 0.701984 0.712193i \(-0.252298\pi\)
0.701984 + 0.712193i \(0.252298\pi\)
\(594\) 0.226773 0.00930462
\(595\) −9.77623 −0.400787
\(596\) 45.6660 1.87055
\(597\) 2.66365 0.109016
\(598\) 0 0
\(599\) −9.68405 −0.395680 −0.197840 0.980234i \(-0.563393\pi\)
−0.197840 + 0.980234i \(0.563393\pi\)
\(600\) 19.9785 0.815617
\(601\) 12.8886 0.525738 0.262869 0.964832i \(-0.415331\pi\)
0.262869 + 0.964832i \(0.415331\pi\)
\(602\) 17.7625 0.723944
\(603\) −8.29459 −0.337782
\(604\) 61.3519 2.49637
\(605\) −16.5467 −0.672718
\(606\) 17.4260 0.707884
\(607\) 15.8179 0.642029 0.321014 0.947074i \(-0.395976\pi\)
0.321014 + 0.947074i \(0.395976\pi\)
\(608\) 53.1362 2.15496
\(609\) 5.97602 0.242161
\(610\) 25.6965 1.04042
\(611\) 0 0
\(612\) 31.1857 1.26061
\(613\) −26.1313 −1.05543 −0.527716 0.849421i \(-0.676951\pi\)
−0.527716 + 0.849421i \(0.676951\pi\)
\(614\) −26.3918 −1.06509
\(615\) 9.69243 0.390836
\(616\) −0.635364 −0.0255995
\(617\) −22.8276 −0.919003 −0.459502 0.888177i \(-0.651972\pi\)
−0.459502 + 0.888177i \(0.651972\pi\)
\(618\) 11.7953 0.474479
\(619\) 25.4700 1.02373 0.511863 0.859067i \(-0.328956\pi\)
0.511863 + 0.859067i \(0.328956\pi\)
\(620\) −14.4768 −0.581404
\(621\) −5.20221 −0.208757
\(622\) −37.8133 −1.51617
\(623\) 15.8454 0.634833
\(624\) 0 0
\(625\) −3.85378 −0.154151
\(626\) −34.6006 −1.38292
\(627\) 0.460170 0.0183774
\(628\) −12.3629 −0.493334
\(629\) −61.8372 −2.46561
\(630\) 3.92579 0.156407
\(631\) 15.4599 0.615448 0.307724 0.951476i \(-0.400433\pi\)
0.307724 + 0.951476i \(0.400433\pi\)
\(632\) 85.3312 3.39429
\(633\) −24.1832 −0.961195
\(634\) −79.3334 −3.15073
\(635\) 19.4853 0.773252
\(636\) −15.7726 −0.625426
\(637\) 0 0
\(638\) −1.35520 −0.0536529
\(639\) 12.9178 0.511022
\(640\) 1.65154 0.0652827
\(641\) −23.3837 −0.923600 −0.461800 0.886984i \(-0.652796\pi\)
−0.461800 + 0.886984i \(0.652796\pi\)
\(642\) 26.7187 1.05450
\(643\) 1.22303 0.0482316 0.0241158 0.999709i \(-0.492323\pi\)
0.0241158 + 0.999709i \(0.492323\pi\)
\(644\) 24.9797 0.984339
\(645\) 10.2520 0.403673
\(646\) 89.6401 3.52684
\(647\) 12.3618 0.485993 0.242996 0.970027i \(-0.421870\pi\)
0.242996 + 0.970027i \(0.421870\pi\)
\(648\) −7.30704 −0.287047
\(649\) 0.232605 0.00913055
\(650\) 0 0
\(651\) 2.00289 0.0784994
\(652\) −93.7782 −3.67264
\(653\) 9.86309 0.385972 0.192986 0.981201i \(-0.438183\pi\)
0.192986 + 0.981201i \(0.438183\pi\)
\(654\) −35.3523 −1.38238
\(655\) 8.80417 0.344008
\(656\) 60.8699 2.37657
\(657\) −14.5507 −0.567675
\(658\) 4.77401 0.186110
\(659\) 14.5491 0.566753 0.283376 0.959009i \(-0.408545\pi\)
0.283376 + 0.959009i \(0.408545\pi\)
\(660\) −0.628490 −0.0244639
\(661\) −19.4099 −0.754958 −0.377479 0.926018i \(-0.623209\pi\)
−0.377479 + 0.926018i \(0.623209\pi\)
\(662\) 24.2622 0.942979
\(663\) 0 0
\(664\) 9.44876 0.366683
\(665\) 7.96623 0.308917
\(666\) 24.8316 0.962206
\(667\) 31.0885 1.20375
\(668\) 6.70577 0.259454
\(669\) −16.9080 −0.653701
\(670\) 32.5628 1.25801
\(671\) 0.569153 0.0219719
\(672\) 10.0405 0.387320
\(673\) 35.3779 1.36372 0.681859 0.731484i \(-0.261172\pi\)
0.681859 + 0.731484i \(0.261172\pi\)
\(674\) −34.7006 −1.33662
\(675\) −2.73414 −0.105237
\(676\) 0 0
\(677\) 10.9841 0.422154 0.211077 0.977469i \(-0.432303\pi\)
0.211077 + 0.977469i \(0.432303\pi\)
\(678\) −34.9146 −1.34089
\(679\) −5.86980 −0.225262
\(680\) −71.4353 −2.73942
\(681\) −11.6431 −0.446166
\(682\) −0.454202 −0.0173923
\(683\) 6.41924 0.245625 0.122813 0.992430i \(-0.460809\pi\)
0.122813 + 0.992430i \(0.460809\pi\)
\(684\) −25.4119 −0.971648
\(685\) −7.69331 −0.293946
\(686\) 2.60802 0.0995746
\(687\) 7.90801 0.301710
\(688\) 64.3841 2.45462
\(689\) 0 0
\(690\) 20.4228 0.777482
\(691\) −27.6062 −1.05019 −0.525094 0.851044i \(-0.675970\pi\)
−0.525094 + 0.851044i \(0.675970\pi\)
\(692\) 82.5009 3.13621
\(693\) 0.0869524 0.00330305
\(694\) 44.0130 1.67071
\(695\) −29.0933 −1.10357
\(696\) 43.6670 1.65519
\(697\) 41.8187 1.58400
\(698\) −6.27801 −0.237626
\(699\) −24.0854 −0.910995
\(700\) 13.1287 0.496217
\(701\) 15.9426 0.602144 0.301072 0.953601i \(-0.402656\pi\)
0.301072 + 0.953601i \(0.402656\pi\)
\(702\) 0 0
\(703\) 50.3885 1.90044
\(704\) −0.632927 −0.0238543
\(705\) 2.75543 0.103776
\(706\) 80.1018 3.01467
\(707\) 6.68172 0.251292
\(708\) −12.8451 −0.482749
\(709\) −12.5273 −0.470472 −0.235236 0.971938i \(-0.575586\pi\)
−0.235236 + 0.971938i \(0.575586\pi\)
\(710\) −50.7128 −1.90322
\(711\) −11.6780 −0.437958
\(712\) 115.783 4.33915
\(713\) 10.4194 0.390211
\(714\) 16.9381 0.633894
\(715\) 0 0
\(716\) −6.90049 −0.257883
\(717\) 10.9354 0.408390
\(718\) 27.7289 1.03483
\(719\) −42.5632 −1.58734 −0.793670 0.608348i \(-0.791832\pi\)
−0.793670 + 0.608348i \(0.791832\pi\)
\(720\) 14.2299 0.530318
\(721\) 4.52273 0.168435
\(722\) −23.4915 −0.874263
\(723\) 19.9054 0.740288
\(724\) −20.7379 −0.770719
\(725\) 16.3393 0.606825
\(726\) 28.6685 1.06399
\(727\) −25.1186 −0.931596 −0.465798 0.884891i \(-0.654233\pi\)
−0.465798 + 0.884891i \(0.654233\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 57.1228 2.11421
\(731\) 44.2331 1.63602
\(732\) −31.4302 −1.16170
\(733\) −2.32992 −0.0860574 −0.0430287 0.999074i \(-0.513701\pi\)
−0.0430287 + 0.999074i \(0.513701\pi\)
\(734\) −68.3463 −2.52271
\(735\) 1.50528 0.0555230
\(736\) 52.2326 1.92532
\(737\) 0.721234 0.0265670
\(738\) −16.7929 −0.618156
\(739\) −26.8170 −0.986478 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(740\) −68.8195 −2.52986
\(741\) 0 0
\(742\) −8.56672 −0.314494
\(743\) −49.8687 −1.82951 −0.914753 0.404014i \(-0.867615\pi\)
−0.914753 + 0.404014i \(0.867615\pi\)
\(744\) 14.6352 0.536552
\(745\) 14.3156 0.524483
\(746\) −47.4925 −1.73882
\(747\) −1.29310 −0.0473122
\(748\) −2.71167 −0.0991484
\(749\) 10.2448 0.374337
\(750\) 30.3626 1.10869
\(751\) 2.78782 0.101729 0.0508645 0.998706i \(-0.483802\pi\)
0.0508645 + 0.998706i \(0.483802\pi\)
\(752\) 17.3045 0.631031
\(753\) −23.3883 −0.852318
\(754\) 0 0
\(755\) 19.2329 0.699956
\(756\) −4.80176 −0.174638
\(757\) 13.8497 0.503374 0.251687 0.967809i \(-0.419015\pi\)
0.251687 + 0.967809i \(0.419015\pi\)
\(758\) 75.3579 2.73712
\(759\) 0.452344 0.0164191
\(760\) 58.2096 2.11148
\(761\) −44.7203 −1.62111 −0.810555 0.585663i \(-0.800834\pi\)
−0.810555 + 0.585663i \(0.800834\pi\)
\(762\) −33.7600 −1.22299
\(763\) −13.5552 −0.490732
\(764\) −32.5904 −1.17908
\(765\) 9.77623 0.353461
\(766\) −45.1871 −1.63268
\(767\) 0 0
\(768\) −17.4194 −0.628570
\(769\) −12.8746 −0.464268 −0.232134 0.972684i \(-0.574571\pi\)
−0.232134 + 0.972684i \(0.574571\pi\)
\(770\) −0.341357 −0.0123016
\(771\) 12.0595 0.434312
\(772\) 102.637 3.69399
\(773\) 14.7887 0.531913 0.265957 0.963985i \(-0.414312\pi\)
0.265957 + 0.963985i \(0.414312\pi\)
\(774\) −17.7625 −0.638458
\(775\) 5.47618 0.196710
\(776\) −42.8909 −1.53969
\(777\) 9.52127 0.341574
\(778\) −35.6611 −1.27851
\(779\) −34.0763 −1.22091
\(780\) 0 0
\(781\) −1.12324 −0.0401926
\(782\) 88.1157 3.15101
\(783\) −5.97602 −0.213566
\(784\) 9.45336 0.337620
\(785\) −3.87558 −0.138325
\(786\) −15.2540 −0.544091
\(787\) 19.9581 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(788\) 109.472 3.89976
\(789\) 4.53226 0.161353
\(790\) 45.8452 1.63110
\(791\) −13.3874 −0.476001
\(792\) 0.635364 0.0225767
\(793\) 0 0
\(794\) 40.2765 1.42936
\(795\) −4.94448 −0.175363
\(796\) 12.7902 0.453337
\(797\) 11.4612 0.405977 0.202989 0.979181i \(-0.434935\pi\)
0.202989 + 0.979181i \(0.434935\pi\)
\(798\) −13.8022 −0.488591
\(799\) 11.8885 0.420586
\(800\) 27.4521 0.970577
\(801\) −15.8454 −0.559870
\(802\) 37.1210 1.31079
\(803\) 1.26521 0.0446484
\(804\) −39.8286 −1.40465
\(805\) 7.83076 0.275998
\(806\) 0 0
\(807\) −27.9600 −0.984238
\(808\) 48.8235 1.71761
\(809\) 21.4851 0.755375 0.377688 0.925933i \(-0.376719\pi\)
0.377688 + 0.925933i \(0.376719\pi\)
\(810\) −3.92579 −0.137938
\(811\) −25.8937 −0.909249 −0.454625 0.890683i \(-0.650227\pi\)
−0.454625 + 0.890683i \(0.650227\pi\)
\(812\) 28.6954 1.00701
\(813\) −4.18954 −0.146934
\(814\) −2.15917 −0.0756788
\(815\) −29.3980 −1.02977
\(816\) 61.3962 2.14930
\(817\) −36.0437 −1.26101
\(818\) 36.7361 1.28445
\(819\) 0 0
\(820\) 46.5407 1.62527
\(821\) 54.8269 1.91347 0.956736 0.290959i \(-0.0939743\pi\)
0.956736 + 0.290959i \(0.0939743\pi\)
\(822\) 13.3293 0.464913
\(823\) −5.98811 −0.208732 −0.104366 0.994539i \(-0.533281\pi\)
−0.104366 + 0.994539i \(0.533281\pi\)
\(824\) 33.0477 1.15127
\(825\) 0.237740 0.00827704
\(826\) −6.97667 −0.242750
\(827\) −13.1913 −0.458705 −0.229353 0.973343i \(-0.573661\pi\)
−0.229353 + 0.973343i \(0.573661\pi\)
\(828\) −24.9797 −0.868106
\(829\) 55.5439 1.92912 0.964559 0.263866i \(-0.0849975\pi\)
0.964559 + 0.263866i \(0.0849975\pi\)
\(830\) 5.07645 0.176206
\(831\) 9.86805 0.342319
\(832\) 0 0
\(833\) 6.49464 0.225026
\(834\) 50.4065 1.74544
\(835\) 2.10215 0.0727481
\(836\) 2.20962 0.0764214
\(837\) −2.00289 −0.0692300
\(838\) 98.2501 3.39399
\(839\) −1.28875 −0.0444928 −0.0222464 0.999753i \(-0.507082\pi\)
−0.0222464 + 0.999753i \(0.507082\pi\)
\(840\) 10.9991 0.379506
\(841\) 6.71280 0.231476
\(842\) 5.96233 0.205475
\(843\) 2.76496 0.0952304
\(844\) −116.122 −3.99707
\(845\) 0 0
\(846\) −4.77401 −0.164134
\(847\) 10.9924 0.377705
\(848\) −31.0521 −1.06633
\(849\) −5.74079 −0.197024
\(850\) 46.3112 1.58846
\(851\) 49.5316 1.69792
\(852\) 62.0284 2.12506
\(853\) −25.6774 −0.879178 −0.439589 0.898199i \(-0.644876\pi\)
−0.439589 + 0.898199i \(0.644876\pi\)
\(854\) −17.0710 −0.584156
\(855\) −7.96623 −0.272439
\(856\) 74.8592 2.55864
\(857\) 32.7226 1.11778 0.558891 0.829241i \(-0.311227\pi\)
0.558891 + 0.829241i \(0.311227\pi\)
\(858\) 0 0
\(859\) 20.4087 0.696335 0.348168 0.937432i \(-0.386804\pi\)
0.348168 + 0.937432i \(0.386804\pi\)
\(860\) 49.2277 1.67865
\(861\) −6.43896 −0.219439
\(862\) −5.16695 −0.175987
\(863\) 11.1079 0.378117 0.189059 0.981966i \(-0.439456\pi\)
0.189059 + 0.981966i \(0.439456\pi\)
\(864\) −10.0405 −0.341584
\(865\) 25.8628 0.879360
\(866\) 15.3635 0.522074
\(867\) 25.1803 0.855169
\(868\) 9.61739 0.326436
\(869\) 1.01543 0.0344460
\(870\) 23.4606 0.795389
\(871\) 0 0
\(872\) −99.0485 −3.35420
\(873\) 5.86980 0.198663
\(874\) −71.8017 −2.42873
\(875\) 11.6420 0.393572
\(876\) −69.8687 −2.36065
\(877\) −28.5938 −0.965545 −0.482772 0.875746i \(-0.660370\pi\)
−0.482772 + 0.875746i \(0.660370\pi\)
\(878\) 21.0353 0.709907
\(879\) 15.8014 0.532970
\(880\) −1.23733 −0.0417103
\(881\) 22.3516 0.753046 0.376523 0.926407i \(-0.377120\pi\)
0.376523 + 0.926407i \(0.377120\pi\)
\(882\) −2.60802 −0.0878165
\(883\) −2.58792 −0.0870905 −0.0435453 0.999051i \(-0.513865\pi\)
−0.0435453 + 0.999051i \(0.513865\pi\)
\(884\) 0 0
\(885\) −4.02675 −0.135358
\(886\) −17.3690 −0.583523
\(887\) −36.1778 −1.21473 −0.607365 0.794423i \(-0.707774\pi\)
−0.607365 + 0.794423i \(0.707774\pi\)
\(888\) 69.5722 2.33469
\(889\) −12.9447 −0.434151
\(890\) 62.2057 2.08514
\(891\) −0.0869524 −0.00291301
\(892\) −81.1881 −2.71838
\(893\) −9.68745 −0.324178
\(894\) −24.8030 −0.829535
\(895\) −2.16320 −0.0723077
\(896\) −1.09716 −0.0366537
\(897\) 0 0
\(898\) 51.7669 1.72748
\(899\) 11.9693 0.399199
\(900\) −13.1287 −0.437623
\(901\) −21.3333 −0.710717
\(902\) 1.46019 0.0486188
\(903\) −6.81071 −0.226646
\(904\) −97.8223 −3.25352
\(905\) −6.50102 −0.216101
\(906\) −33.3226 −1.10707
\(907\) 40.9607 1.36008 0.680039 0.733176i \(-0.261963\pi\)
0.680039 + 0.733176i \(0.261963\pi\)
\(908\) −55.9075 −1.85536
\(909\) −6.68172 −0.221619
\(910\) 0 0
\(911\) 2.19307 0.0726595 0.0363298 0.999340i \(-0.488433\pi\)
0.0363298 + 0.999340i \(0.488433\pi\)
\(912\) −50.0291 −1.65663
\(913\) 0.112438 0.00372117
\(914\) −6.75175 −0.223328
\(915\) −9.85290 −0.325727
\(916\) 37.9724 1.25464
\(917\) −5.84887 −0.193147
\(918\) −16.9381 −0.559042
\(919\) 51.0323 1.68340 0.841700 0.539945i \(-0.181555\pi\)
0.841700 + 0.539945i \(0.181555\pi\)
\(920\) 57.2197 1.88648
\(921\) 10.1195 0.333448
\(922\) 68.7649 2.26465
\(923\) 0 0
\(924\) 0.417524 0.0137355
\(925\) 26.0325 0.855942
\(926\) 76.4363 2.51185
\(927\) −4.52273 −0.148546
\(928\) 60.0021 1.96966
\(929\) 34.1988 1.12203 0.561014 0.827806i \(-0.310411\pi\)
0.561014 + 0.827806i \(0.310411\pi\)
\(930\) 7.86293 0.257836
\(931\) −5.29220 −0.173445
\(932\) −115.652 −3.78832
\(933\) 14.4989 0.474671
\(934\) −8.42404 −0.275643
\(935\) −0.850067 −0.0278002
\(936\) 0 0
\(937\) −56.0430 −1.83084 −0.915422 0.402496i \(-0.868143\pi\)
−0.915422 + 0.402496i \(0.868143\pi\)
\(938\) −21.6324 −0.706324
\(939\) 13.2670 0.432953
\(940\) 13.2309 0.431545
\(941\) 31.6322 1.03118 0.515591 0.856835i \(-0.327573\pi\)
0.515591 + 0.856835i \(0.327573\pi\)
\(942\) 6.71477 0.218779
\(943\) −33.4968 −1.09081
\(944\) −25.2886 −0.823073
\(945\) −1.50528 −0.0489667
\(946\) 1.54449 0.0502156
\(947\) −46.5004 −1.51106 −0.755529 0.655115i \(-0.772620\pi\)
−0.755529 + 0.655115i \(0.772620\pi\)
\(948\) −56.0747 −1.82122
\(949\) 0 0
\(950\) −37.7370 −1.22435
\(951\) 30.4190 0.986405
\(952\) 47.4566 1.53808
\(953\) −23.2511 −0.753178 −0.376589 0.926380i \(-0.622903\pi\)
−0.376589 + 0.926380i \(0.622903\pi\)
\(954\) 8.56672 0.277358
\(955\) −10.2166 −0.330601
\(956\) 52.5091 1.69827
\(957\) 0.519629 0.0167972
\(958\) 20.6371 0.666756
\(959\) 5.11089 0.165039
\(960\) 10.9569 0.353633
\(961\) −26.9884 −0.870595
\(962\) 0 0
\(963\) −10.2448 −0.330135
\(964\) 95.5807 3.07845
\(965\) 32.1752 1.03576
\(966\) −13.5674 −0.436525
\(967\) −13.6117 −0.437723 −0.218861 0.975756i \(-0.570234\pi\)
−0.218861 + 0.975756i \(0.570234\pi\)
\(968\) 80.3221 2.58165
\(969\) −34.3710 −1.10415
\(970\) −23.0436 −0.739886
\(971\) −9.93417 −0.318803 −0.159401 0.987214i \(-0.550956\pi\)
−0.159401 + 0.987214i \(0.550956\pi\)
\(972\) 4.80176 0.154016
\(973\) 19.3275 0.619612
\(974\) −40.2810 −1.29069
\(975\) 0 0
\(976\) −61.8777 −1.98066
\(977\) 39.1543 1.25266 0.626328 0.779559i \(-0.284557\pi\)
0.626328 + 0.779559i \(0.284557\pi\)
\(978\) 50.9345 1.62871
\(979\) 1.37780 0.0440345
\(980\) 7.22798 0.230889
\(981\) 13.5552 0.432785
\(982\) 38.8757 1.24057
\(983\) −4.01341 −0.128008 −0.0640039 0.997950i \(-0.520387\pi\)
−0.0640039 + 0.997950i \(0.520387\pi\)
\(984\) −47.0497 −1.49989
\(985\) 34.3177 1.09345
\(986\) 101.223 3.22359
\(987\) −1.83051 −0.0582659
\(988\) 0 0
\(989\) −35.4307 −1.12663
\(990\) 0.341357 0.0108490
\(991\) −34.3760 −1.09199 −0.545995 0.837789i \(-0.683848\pi\)
−0.545995 + 0.837789i \(0.683848\pi\)
\(992\) 20.1100 0.638492
\(993\) −9.30294 −0.295220
\(994\) 33.6900 1.06858
\(995\) 4.00954 0.127111
\(996\) −6.20917 −0.196745
\(997\) 4.46643 0.141453 0.0707266 0.997496i \(-0.477468\pi\)
0.0707266 + 0.997496i \(0.477468\pi\)
\(998\) 66.3233 2.09943
\(999\) −9.52127 −0.301240
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 3549.2.a.bb.1.1 8
13.2 odd 12 273.2.bd.a.43.1 16
13.7 odd 12 273.2.bd.a.127.1 yes 16
13.12 even 2 3549.2.a.bd.1.8 8
39.2 even 12 819.2.ct.b.316.8 16
39.20 even 12 819.2.ct.b.127.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.bd.a.43.1 16 13.2 odd 12
273.2.bd.a.127.1 yes 16 13.7 odd 12
819.2.ct.b.127.8 16 39.20 even 12
819.2.ct.b.316.8 16 39.2 even 12
3549.2.a.bb.1.1 8 1.1 even 1 trivial
3549.2.a.bd.1.8 8 13.12 even 2