Properties

Label 4598.2.a.cd.1.10
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $10$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, 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("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 118x^{6} - 220x^{5} - 270x^{4} + 512x^{3} + 176x^{2} - 392x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(3.30795\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.30795 q^{3} +1.00000 q^{4} -0.828943 q^{5} +3.30795 q^{6} +2.22864 q^{7} +1.00000 q^{8} +7.94251 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.30795 q^{3} +1.00000 q^{4} -0.828943 q^{5} +3.30795 q^{6} +2.22864 q^{7} +1.00000 q^{8} +7.94251 q^{9} -0.828943 q^{10} +3.30795 q^{12} +1.20735 q^{13} +2.22864 q^{14} -2.74210 q^{15} +1.00000 q^{16} +4.58220 q^{17} +7.94251 q^{18} +1.00000 q^{19} -0.828943 q^{20} +7.37221 q^{21} -1.76409 q^{23} +3.30795 q^{24} -4.31285 q^{25} +1.20735 q^{26} +16.3495 q^{27} +2.22864 q^{28} +3.91436 q^{29} -2.74210 q^{30} -8.68660 q^{31} +1.00000 q^{32} +4.58220 q^{34} -1.84741 q^{35} +7.94251 q^{36} -1.36981 q^{37} +1.00000 q^{38} +3.99384 q^{39} -0.828943 q^{40} -5.91652 q^{41} +7.37221 q^{42} -3.78596 q^{43} -6.58389 q^{45} -1.76409 q^{46} -7.92677 q^{47} +3.30795 q^{48} -2.03317 q^{49} -4.31285 q^{50} +15.1577 q^{51} +1.20735 q^{52} -8.13933 q^{53} +16.3495 q^{54} +2.22864 q^{56} +3.30795 q^{57} +3.91436 q^{58} -0.104572 q^{59} -2.74210 q^{60} +12.4843 q^{61} -8.68660 q^{62} +17.7010 q^{63} +1.00000 q^{64} -1.00082 q^{65} -11.6774 q^{67} +4.58220 q^{68} -5.83550 q^{69} -1.84741 q^{70} +5.40088 q^{71} +7.94251 q^{72} -13.2594 q^{73} -1.36981 q^{74} -14.2667 q^{75} +1.00000 q^{76} +3.99384 q^{78} -2.32315 q^{79} -0.828943 q^{80} +30.2559 q^{81} -5.91652 q^{82} -5.98648 q^{83} +7.37221 q^{84} -3.79838 q^{85} -3.78596 q^{86} +12.9485 q^{87} -6.07639 q^{89} -6.58389 q^{90} +2.69074 q^{91} -1.76409 q^{92} -28.7348 q^{93} -7.92677 q^{94} -0.828943 q^{95} +3.30795 q^{96} +17.3375 q^{97} -2.03317 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 3 q^{5} + 2 q^{6} + 11 q^{7} + 10 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 3 q^{5} + 2 q^{6} + 11 q^{7} + 10 q^{8} + 12 q^{9} - 3 q^{10} + 2 q^{12} + 11 q^{13} + 11 q^{14} + q^{15} + 10 q^{16} + 12 q^{17} + 12 q^{18} + 10 q^{19} - 3 q^{20} - q^{21} + 14 q^{23} + 2 q^{24} + 5 q^{25} + 11 q^{26} + 2 q^{27} + 11 q^{28} + 16 q^{29} + q^{30} + 12 q^{31} + 10 q^{32} + 12 q^{34} - 12 q^{35} + 12 q^{36} - q^{37} + 10 q^{38} + 11 q^{39} - 3 q^{40} - 5 q^{41} - q^{42} + 22 q^{43} - 2 q^{45} + 14 q^{46} + 8 q^{47} + 2 q^{48} - 3 q^{49} + 5 q^{50} + 8 q^{51} + 11 q^{52} + 2 q^{53} + 2 q^{54} + 11 q^{56} + 2 q^{57} + 16 q^{58} - 7 q^{59} + q^{60} + 35 q^{61} + 12 q^{62} + 38 q^{63} + 10 q^{64} + 4 q^{65} + 9 q^{67} + 12 q^{68} + 6 q^{69} - 12 q^{70} - 4 q^{71} + 12 q^{72} + 5 q^{73} - q^{74} - 15 q^{75} + 10 q^{76} + 11 q^{78} + 18 q^{79} - 3 q^{80} - 6 q^{81} - 5 q^{82} + 7 q^{83} - q^{84} + 35 q^{85} + 22 q^{86} + 8 q^{87} + 22 q^{89} - 2 q^{90} + 11 q^{91} + 14 q^{92} - 64 q^{93} + 8 q^{94} - 3 q^{95} + 2 q^{96} + 32 q^{97} - 3 q^{98}+O(q^{100}) \) 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
\(3\) 3.30795 1.90984 0.954922 0.296857i \(-0.0959386\pi\)
0.954922 + 0.296857i \(0.0959386\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.828943 −0.370715 −0.185357 0.982671i \(-0.559344\pi\)
−0.185357 + 0.982671i \(0.559344\pi\)
\(6\) 3.30795 1.35046
\(7\) 2.22864 0.842346 0.421173 0.906980i \(-0.361619\pi\)
0.421173 + 0.906980i \(0.361619\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.94251 2.64750
\(10\) −0.828943 −0.262135
\(11\) 0 0
\(12\) 3.30795 0.954922
\(13\) 1.20735 0.334858 0.167429 0.985884i \(-0.446454\pi\)
0.167429 + 0.985884i \(0.446454\pi\)
\(14\) 2.22864 0.595629
\(15\) −2.74210 −0.708007
\(16\) 1.00000 0.250000
\(17\) 4.58220 1.11135 0.555673 0.831401i \(-0.312461\pi\)
0.555673 + 0.831401i \(0.312461\pi\)
\(18\) 7.94251 1.87207
\(19\) 1.00000 0.229416
\(20\) −0.828943 −0.185357
\(21\) 7.37221 1.60875
\(22\) 0 0
\(23\) −1.76409 −0.367837 −0.183919 0.982941i \(-0.558878\pi\)
−0.183919 + 0.982941i \(0.558878\pi\)
\(24\) 3.30795 0.675232
\(25\) −4.31285 −0.862571
\(26\) 1.20735 0.236780
\(27\) 16.3495 3.14647
\(28\) 2.22864 0.421173
\(29\) 3.91436 0.726878 0.363439 0.931618i \(-0.381603\pi\)
0.363439 + 0.931618i \(0.381603\pi\)
\(30\) −2.74210 −0.500637
\(31\) −8.68660 −1.56016 −0.780080 0.625680i \(-0.784822\pi\)
−0.780080 + 0.625680i \(0.784822\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.58220 0.785841
\(35\) −1.84741 −0.312270
\(36\) 7.94251 1.32375
\(37\) −1.36981 −0.225195 −0.112597 0.993641i \(-0.535917\pi\)
−0.112597 + 0.993641i \(0.535917\pi\)
\(38\) 1.00000 0.162221
\(39\) 3.99384 0.639526
\(40\) −0.828943 −0.131067
\(41\) −5.91652 −0.924005 −0.462003 0.886879i \(-0.652869\pi\)
−0.462003 + 0.886879i \(0.652869\pi\)
\(42\) 7.37221 1.13756
\(43\) −3.78596 −0.577353 −0.288676 0.957427i \(-0.593215\pi\)
−0.288676 + 0.957427i \(0.593215\pi\)
\(44\) 0 0
\(45\) −6.58389 −0.981468
\(46\) −1.76409 −0.260100
\(47\) −7.92677 −1.15624 −0.578119 0.815953i \(-0.696213\pi\)
−0.578119 + 0.815953i \(0.696213\pi\)
\(48\) 3.30795 0.477461
\(49\) −2.03317 −0.290453
\(50\) −4.31285 −0.609930
\(51\) 15.1577 2.12250
\(52\) 1.20735 0.167429
\(53\) −8.13933 −1.11802 −0.559012 0.829160i \(-0.688819\pi\)
−0.559012 + 0.829160i \(0.688819\pi\)
\(54\) 16.3495 2.22489
\(55\) 0 0
\(56\) 2.22864 0.297814
\(57\) 3.30795 0.438148
\(58\) 3.91436 0.513980
\(59\) −0.104572 −0.0136142 −0.00680709 0.999977i \(-0.502167\pi\)
−0.00680709 + 0.999977i \(0.502167\pi\)
\(60\) −2.74210 −0.354004
\(61\) 12.4843 1.59846 0.799228 0.601028i \(-0.205242\pi\)
0.799228 + 0.601028i \(0.205242\pi\)
\(62\) −8.68660 −1.10320
\(63\) 17.7010 2.23011
\(64\) 1.00000 0.125000
\(65\) −1.00082 −0.124137
\(66\) 0 0
\(67\) −11.6774 −1.42662 −0.713308 0.700851i \(-0.752804\pi\)
−0.713308 + 0.700851i \(0.752804\pi\)
\(68\) 4.58220 0.555673
\(69\) −5.83550 −0.702512
\(70\) −1.84741 −0.220808
\(71\) 5.40088 0.640967 0.320483 0.947254i \(-0.396155\pi\)
0.320483 + 0.947254i \(0.396155\pi\)
\(72\) 7.94251 0.936034
\(73\) −13.2594 −1.55189 −0.775946 0.630799i \(-0.782727\pi\)
−0.775946 + 0.630799i \(0.782727\pi\)
\(74\) −1.36981 −0.159237
\(75\) −14.2667 −1.64737
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 3.99384 0.452213
\(79\) −2.32315 −0.261375 −0.130688 0.991424i \(-0.541719\pi\)
−0.130688 + 0.991424i \(0.541719\pi\)
\(80\) −0.828943 −0.0926787
\(81\) 30.2559 3.36177
\(82\) −5.91652 −0.653370
\(83\) −5.98648 −0.657101 −0.328551 0.944486i \(-0.606560\pi\)
−0.328551 + 0.944486i \(0.606560\pi\)
\(84\) 7.37221 0.804375
\(85\) −3.79838 −0.411992
\(86\) −3.78596 −0.408250
\(87\) 12.9485 1.38822
\(88\) 0 0
\(89\) −6.07639 −0.644096 −0.322048 0.946723i \(-0.604371\pi\)
−0.322048 + 0.946723i \(0.604371\pi\)
\(90\) −6.58389 −0.694003
\(91\) 2.69074 0.282066
\(92\) −1.76409 −0.183919
\(93\) −28.7348 −2.97966
\(94\) −7.92677 −0.817583
\(95\) −0.828943 −0.0850478
\(96\) 3.30795 0.337616
\(97\) 17.3375 1.76036 0.880178 0.474644i \(-0.157423\pi\)
0.880178 + 0.474644i \(0.157423\pi\)
\(98\) −2.03317 −0.205381
\(99\) 0 0
\(100\) −4.31285 −0.431285
\(101\) −0.0401979 −0.00399984 −0.00199992 0.999998i \(-0.500637\pi\)
−0.00199992 + 0.999998i \(0.500637\pi\)
\(102\) 15.1577 1.50083
\(103\) 17.2964 1.70426 0.852132 0.523327i \(-0.175309\pi\)
0.852132 + 0.523327i \(0.175309\pi\)
\(104\) 1.20735 0.118390
\(105\) −6.11115 −0.596387
\(106\) −8.13933 −0.790562
\(107\) 14.8051 1.43126 0.715631 0.698479i \(-0.246139\pi\)
0.715631 + 0.698479i \(0.246139\pi\)
\(108\) 16.3495 1.57324
\(109\) 0.176526 0.0169081 0.00845407 0.999964i \(-0.497309\pi\)
0.00845407 + 0.999964i \(0.497309\pi\)
\(110\) 0 0
\(111\) −4.53125 −0.430087
\(112\) 2.22864 0.210587
\(113\) 5.63764 0.530345 0.265172 0.964201i \(-0.414571\pi\)
0.265172 + 0.964201i \(0.414571\pi\)
\(114\) 3.30795 0.309818
\(115\) 1.46233 0.136363
\(116\) 3.91436 0.363439
\(117\) 9.58937 0.886537
\(118\) −0.104572 −0.00962667
\(119\) 10.2121 0.936138
\(120\) −2.74210 −0.250318
\(121\) 0 0
\(122\) 12.4843 1.13028
\(123\) −19.5715 −1.76471
\(124\) −8.68660 −0.780080
\(125\) 7.71983 0.690482
\(126\) 17.7010 1.57693
\(127\) 15.5597 1.38070 0.690351 0.723474i \(-0.257456\pi\)
0.690351 + 0.723474i \(0.257456\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.5237 −1.10265
\(130\) −1.00082 −0.0877780
\(131\) 14.8982 1.30166 0.650829 0.759224i \(-0.274421\pi\)
0.650829 + 0.759224i \(0.274421\pi\)
\(132\) 0 0
\(133\) 2.22864 0.193247
\(134\) −11.6774 −1.00877
\(135\) −13.5528 −1.16644
\(136\) 4.58220 0.392920
\(137\) 10.1562 0.867702 0.433851 0.900985i \(-0.357154\pi\)
0.433851 + 0.900985i \(0.357154\pi\)
\(138\) −5.83550 −0.496751
\(139\) 11.3949 0.966499 0.483250 0.875483i \(-0.339456\pi\)
0.483250 + 0.875483i \(0.339456\pi\)
\(140\) −1.84741 −0.156135
\(141\) −26.2213 −2.20823
\(142\) 5.40088 0.453232
\(143\) 0 0
\(144\) 7.94251 0.661876
\(145\) −3.24478 −0.269464
\(146\) −13.2594 −1.09735
\(147\) −6.72562 −0.554720
\(148\) −1.36981 −0.112597
\(149\) −16.6310 −1.36246 −0.681231 0.732068i \(-0.738555\pi\)
−0.681231 + 0.732068i \(0.738555\pi\)
\(150\) −14.2667 −1.16487
\(151\) −18.0434 −1.46835 −0.734174 0.678962i \(-0.762430\pi\)
−0.734174 + 0.678962i \(0.762430\pi\)
\(152\) 1.00000 0.0811107
\(153\) 36.3941 2.94229
\(154\) 0 0
\(155\) 7.20070 0.578374
\(156\) 3.99384 0.319763
\(157\) −17.1908 −1.37198 −0.685989 0.727612i \(-0.740630\pi\)
−0.685989 + 0.727612i \(0.740630\pi\)
\(158\) −2.32315 −0.184820
\(159\) −26.9245 −2.13525
\(160\) −0.828943 −0.0655337
\(161\) −3.93151 −0.309846
\(162\) 30.2559 2.37713
\(163\) 9.91065 0.776262 0.388131 0.921604i \(-0.373121\pi\)
0.388131 + 0.921604i \(0.373121\pi\)
\(164\) −5.91652 −0.462003
\(165\) 0 0
\(166\) −5.98648 −0.464641
\(167\) −13.6271 −1.05450 −0.527249 0.849711i \(-0.676776\pi\)
−0.527249 + 0.849711i \(0.676776\pi\)
\(168\) 7.37221 0.568779
\(169\) −11.5423 −0.887870
\(170\) −3.79838 −0.291323
\(171\) 7.94251 0.607379
\(172\) −3.78596 −0.288676
\(173\) −11.4678 −0.871881 −0.435941 0.899975i \(-0.643584\pi\)
−0.435941 + 0.899975i \(0.643584\pi\)
\(174\) 12.9485 0.981622
\(175\) −9.61179 −0.726583
\(176\) 0 0
\(177\) −0.345920 −0.0260009
\(178\) −6.07639 −0.455445
\(179\) 0.219190 0.0163830 0.00819152 0.999966i \(-0.497393\pi\)
0.00819152 + 0.999966i \(0.497393\pi\)
\(180\) −6.58389 −0.490734
\(181\) −24.9347 −1.85338 −0.926689 0.375828i \(-0.877358\pi\)
−0.926689 + 0.375828i \(0.877358\pi\)
\(182\) 2.69074 0.199451
\(183\) 41.2975 3.05280
\(184\) −1.76409 −0.130050
\(185\) 1.13549 0.0834831
\(186\) −28.7348 −2.10694
\(187\) 0 0
\(188\) −7.92677 −0.578119
\(189\) 36.4372 2.65042
\(190\) −0.828943 −0.0601379
\(191\) −9.25309 −0.669530 −0.334765 0.942302i \(-0.608657\pi\)
−0.334765 + 0.942302i \(0.608657\pi\)
\(192\) 3.30795 0.238730
\(193\) 16.1815 1.16477 0.582384 0.812914i \(-0.302120\pi\)
0.582384 + 0.812914i \(0.302120\pi\)
\(194\) 17.3375 1.24476
\(195\) −3.31067 −0.237082
\(196\) −2.03317 −0.145227
\(197\) 14.4283 1.02797 0.513985 0.857799i \(-0.328169\pi\)
0.513985 + 0.857799i \(0.328169\pi\)
\(198\) 0 0
\(199\) 12.4819 0.884821 0.442411 0.896813i \(-0.354123\pi\)
0.442411 + 0.896813i \(0.354123\pi\)
\(200\) −4.31285 −0.304965
\(201\) −38.6281 −2.72461
\(202\) −0.0401979 −0.00282831
\(203\) 8.72369 0.612283
\(204\) 15.1577 1.06125
\(205\) 4.90446 0.342542
\(206\) 17.2964 1.20510
\(207\) −14.0113 −0.973850
\(208\) 1.20735 0.0837145
\(209\) 0 0
\(210\) −6.11115 −0.421709
\(211\) 24.4896 1.68593 0.842967 0.537965i \(-0.180807\pi\)
0.842967 + 0.537965i \(0.180807\pi\)
\(212\) −8.13933 −0.559012
\(213\) 17.8658 1.22415
\(214\) 14.8051 1.01205
\(215\) 3.13834 0.214033
\(216\) 16.3495 1.11245
\(217\) −19.3593 −1.31419
\(218\) 0.176526 0.0119559
\(219\) −43.8613 −2.96387
\(220\) 0 0
\(221\) 5.53231 0.372143
\(222\) −4.53125 −0.304118
\(223\) −23.0348 −1.54252 −0.771262 0.636518i \(-0.780374\pi\)
−0.771262 + 0.636518i \(0.780374\pi\)
\(224\) 2.22864 0.148907
\(225\) −34.2549 −2.28366
\(226\) 5.63764 0.375010
\(227\) −11.6050 −0.770248 −0.385124 0.922865i \(-0.625841\pi\)
−0.385124 + 0.922865i \(0.625841\pi\)
\(228\) 3.30795 0.219074
\(229\) 10.8274 0.715493 0.357746 0.933819i \(-0.383545\pi\)
0.357746 + 0.933819i \(0.383545\pi\)
\(230\) 1.46233 0.0964230
\(231\) 0 0
\(232\) 3.91436 0.256990
\(233\) 0.972932 0.0637389 0.0318694 0.999492i \(-0.489854\pi\)
0.0318694 + 0.999492i \(0.489854\pi\)
\(234\) 9.58937 0.626877
\(235\) 6.57084 0.428634
\(236\) −0.104572 −0.00680709
\(237\) −7.68487 −0.499186
\(238\) 10.2121 0.661950
\(239\) −16.0095 −1.03557 −0.517784 0.855511i \(-0.673243\pi\)
−0.517784 + 0.855511i \(0.673243\pi\)
\(240\) −2.74210 −0.177002
\(241\) 11.2997 0.727878 0.363939 0.931423i \(-0.381432\pi\)
0.363939 + 0.931423i \(0.381432\pi\)
\(242\) 0 0
\(243\) 51.0363 3.27398
\(244\) 12.4843 0.799228
\(245\) 1.68538 0.107675
\(246\) −19.5715 −1.24784
\(247\) 1.20735 0.0768217
\(248\) −8.68660 −0.551600
\(249\) −19.8030 −1.25496
\(250\) 7.71983 0.488245
\(251\) 24.7199 1.56031 0.780153 0.625588i \(-0.215141\pi\)
0.780153 + 0.625588i \(0.215141\pi\)
\(252\) 17.7010 1.11506
\(253\) 0 0
\(254\) 15.5597 0.976304
\(255\) −12.5648 −0.786841
\(256\) 1.00000 0.0625000
\(257\) 0.535384 0.0333964 0.0166982 0.999861i \(-0.494685\pi\)
0.0166982 + 0.999861i \(0.494685\pi\)
\(258\) −12.5237 −0.779694
\(259\) −3.05281 −0.189692
\(260\) −1.00082 −0.0620684
\(261\) 31.0898 1.92441
\(262\) 14.8982 0.920412
\(263\) 9.20025 0.567312 0.283656 0.958926i \(-0.408453\pi\)
0.283656 + 0.958926i \(0.408453\pi\)
\(264\) 0 0
\(265\) 6.74704 0.414468
\(266\) 2.22864 0.136647
\(267\) −20.1004 −1.23012
\(268\) −11.6774 −0.713308
\(269\) −11.3073 −0.689420 −0.344710 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(270\) −13.5528 −0.824800
\(271\) 19.1989 1.16625 0.583124 0.812383i \(-0.301830\pi\)
0.583124 + 0.812383i \(0.301830\pi\)
\(272\) 4.58220 0.277837
\(273\) 8.90083 0.538703
\(274\) 10.1562 0.613558
\(275\) 0 0
\(276\) −5.83550 −0.351256
\(277\) 5.91887 0.355630 0.177815 0.984064i \(-0.443097\pi\)
0.177815 + 0.984064i \(0.443097\pi\)
\(278\) 11.3949 0.683418
\(279\) −68.9934 −4.13053
\(280\) −1.84741 −0.110404
\(281\) −17.5195 −1.04513 −0.522563 0.852600i \(-0.675024\pi\)
−0.522563 + 0.852600i \(0.675024\pi\)
\(282\) −26.2213 −1.56146
\(283\) 5.40443 0.321260 0.160630 0.987015i \(-0.448647\pi\)
0.160630 + 0.987015i \(0.448647\pi\)
\(284\) 5.40088 0.320483
\(285\) −2.74210 −0.162428
\(286\) 0 0
\(287\) −13.1858 −0.778332
\(288\) 7.94251 0.468017
\(289\) 3.99655 0.235091
\(290\) −3.24478 −0.190540
\(291\) 57.3515 3.36200
\(292\) −13.2594 −0.775946
\(293\) 6.26565 0.366043 0.183022 0.983109i \(-0.441412\pi\)
0.183022 + 0.983109i \(0.441412\pi\)
\(294\) −6.72562 −0.392246
\(295\) 0.0866846 0.00504697
\(296\) −1.36981 −0.0796185
\(297\) 0 0
\(298\) −16.6310 −0.963406
\(299\) −2.12986 −0.123173
\(300\) −14.2667 −0.823687
\(301\) −8.43752 −0.486331
\(302\) −18.0434 −1.03828
\(303\) −0.132973 −0.00763907
\(304\) 1.00000 0.0573539
\(305\) −10.3488 −0.592571
\(306\) 36.3941 2.08052
\(307\) −0.0433912 −0.00247647 −0.00123823 0.999999i \(-0.500394\pi\)
−0.00123823 + 0.999999i \(0.500394\pi\)
\(308\) 0 0
\(309\) 57.2155 3.25488
\(310\) 7.20070 0.408972
\(311\) 2.42902 0.137737 0.0688687 0.997626i \(-0.478061\pi\)
0.0688687 + 0.997626i \(0.478061\pi\)
\(312\) 3.99384 0.226107
\(313\) 0.817587 0.0462127 0.0231064 0.999733i \(-0.492644\pi\)
0.0231064 + 0.999733i \(0.492644\pi\)
\(314\) −17.1908 −0.970134
\(315\) −14.6731 −0.826736
\(316\) −2.32315 −0.130688
\(317\) −18.5495 −1.04184 −0.520922 0.853604i \(-0.674412\pi\)
−0.520922 + 0.853604i \(0.674412\pi\)
\(318\) −26.9245 −1.50985
\(319\) 0 0
\(320\) −0.828943 −0.0463393
\(321\) 48.9744 2.73349
\(322\) −3.93151 −0.219094
\(323\) 4.58220 0.254960
\(324\) 30.2559 1.68088
\(325\) −5.20711 −0.288839
\(326\) 9.91065 0.548900
\(327\) 0.583939 0.0322919
\(328\) −5.91652 −0.326685
\(329\) −17.6659 −0.973952
\(330\) 0 0
\(331\) −4.31311 −0.237070 −0.118535 0.992950i \(-0.537820\pi\)
−0.118535 + 0.992950i \(0.537820\pi\)
\(332\) −5.98648 −0.328551
\(333\) −10.8797 −0.596204
\(334\) −13.6271 −0.745642
\(335\) 9.67987 0.528868
\(336\) 7.37221 0.402187
\(337\) 1.35344 0.0737266 0.0368633 0.999320i \(-0.488263\pi\)
0.0368633 + 0.999320i \(0.488263\pi\)
\(338\) −11.5423 −0.627819
\(339\) 18.6490 1.01288
\(340\) −3.79838 −0.205996
\(341\) 0 0
\(342\) 7.94251 0.429482
\(343\) −20.1317 −1.08701
\(344\) −3.78596 −0.204125
\(345\) 4.83730 0.260431
\(346\) −11.4678 −0.616513
\(347\) −14.9394 −0.801988 −0.400994 0.916081i \(-0.631335\pi\)
−0.400994 + 0.916081i \(0.631335\pi\)
\(348\) 12.9485 0.694112
\(349\) 8.42846 0.451165 0.225583 0.974224i \(-0.427571\pi\)
0.225583 + 0.974224i \(0.427571\pi\)
\(350\) −9.61179 −0.513772
\(351\) 19.7396 1.05362
\(352\) 0 0
\(353\) −29.1828 −1.55324 −0.776621 0.629968i \(-0.783068\pi\)
−0.776621 + 0.629968i \(0.783068\pi\)
\(354\) −0.345920 −0.0183854
\(355\) −4.47702 −0.237616
\(356\) −6.07639 −0.322048
\(357\) 33.7810 1.78788
\(358\) 0.219190 0.0115846
\(359\) −26.4680 −1.39693 −0.698465 0.715644i \(-0.746133\pi\)
−0.698465 + 0.715644i \(0.746133\pi\)
\(360\) −6.58389 −0.347001
\(361\) 1.00000 0.0526316
\(362\) −24.9347 −1.31054
\(363\) 0 0
\(364\) 2.69074 0.141033
\(365\) 10.9913 0.575309
\(366\) 41.2975 2.15866
\(367\) −20.9635 −1.09429 −0.547143 0.837039i \(-0.684285\pi\)
−0.547143 + 0.837039i \(0.684285\pi\)
\(368\) −1.76409 −0.0919593
\(369\) −46.9920 −2.44631
\(370\) 1.13549 0.0590315
\(371\) −18.1396 −0.941762
\(372\) −28.7348 −1.48983
\(373\) −34.2399 −1.77288 −0.886438 0.462847i \(-0.846828\pi\)
−0.886438 + 0.462847i \(0.846828\pi\)
\(374\) 0 0
\(375\) 25.5368 1.31871
\(376\) −7.92677 −0.408792
\(377\) 4.72599 0.243401
\(378\) 36.4372 1.87413
\(379\) 8.74108 0.448999 0.224500 0.974474i \(-0.427925\pi\)
0.224500 + 0.974474i \(0.427925\pi\)
\(380\) −0.828943 −0.0425239
\(381\) 51.4707 2.63693
\(382\) −9.25309 −0.473429
\(383\) 4.17010 0.213082 0.106541 0.994308i \(-0.466022\pi\)
0.106541 + 0.994308i \(0.466022\pi\)
\(384\) 3.30795 0.168808
\(385\) 0 0
\(386\) 16.1815 0.823615
\(387\) −30.0700 −1.52854
\(388\) 17.3375 0.880178
\(389\) −7.70695 −0.390758 −0.195379 0.980728i \(-0.562594\pi\)
−0.195379 + 0.980728i \(0.562594\pi\)
\(390\) −3.31067 −0.167642
\(391\) −8.08339 −0.408795
\(392\) −2.03317 −0.102691
\(393\) 49.2823 2.48596
\(394\) 14.4283 0.726885
\(395\) 1.92576 0.0968957
\(396\) 0 0
\(397\) 27.6939 1.38991 0.694957 0.719051i \(-0.255423\pi\)
0.694957 + 0.719051i \(0.255423\pi\)
\(398\) 12.4819 0.625663
\(399\) 7.37221 0.369072
\(400\) −4.31285 −0.215643
\(401\) −28.3005 −1.41326 −0.706630 0.707583i \(-0.749786\pi\)
−0.706630 + 0.707583i \(0.749786\pi\)
\(402\) −38.6281 −1.92659
\(403\) −10.4878 −0.522432
\(404\) −0.0401979 −0.00199992
\(405\) −25.0804 −1.24626
\(406\) 8.72369 0.432949
\(407\) 0 0
\(408\) 15.1577 0.750416
\(409\) −7.15765 −0.353923 −0.176961 0.984218i \(-0.556627\pi\)
−0.176961 + 0.984218i \(0.556627\pi\)
\(410\) 4.90446 0.242214
\(411\) 33.5961 1.65718
\(412\) 17.2964 0.852132
\(413\) −0.233054 −0.0114678
\(414\) −14.0113 −0.688616
\(415\) 4.96245 0.243597
\(416\) 1.20735 0.0591951
\(417\) 37.6936 1.84586
\(418\) 0 0
\(419\) 9.45181 0.461751 0.230876 0.972983i \(-0.425841\pi\)
0.230876 + 0.972983i \(0.425841\pi\)
\(420\) −6.11115 −0.298193
\(421\) 27.2670 1.32891 0.664455 0.747328i \(-0.268664\pi\)
0.664455 + 0.747328i \(0.268664\pi\)
\(422\) 24.4896 1.19214
\(423\) −62.9584 −3.06114
\(424\) −8.13933 −0.395281
\(425\) −19.7624 −0.958615
\(426\) 17.8658 0.865602
\(427\) 27.8231 1.34645
\(428\) 14.8051 0.715631
\(429\) 0 0
\(430\) 3.13834 0.151344
\(431\) −1.36032 −0.0655245 −0.0327622 0.999463i \(-0.510430\pi\)
−0.0327622 + 0.999463i \(0.510430\pi\)
\(432\) 16.3495 0.786618
\(433\) 26.1728 1.25779 0.628893 0.777492i \(-0.283508\pi\)
0.628893 + 0.777492i \(0.283508\pi\)
\(434\) −19.3593 −0.929276
\(435\) −10.7336 −0.514635
\(436\) 0.176526 0.00845407
\(437\) −1.76409 −0.0843877
\(438\) −43.8613 −2.09577
\(439\) −9.79442 −0.467462 −0.233731 0.972301i \(-0.575094\pi\)
−0.233731 + 0.972301i \(0.575094\pi\)
\(440\) 0 0
\(441\) −16.1485 −0.768975
\(442\) 5.53231 0.263145
\(443\) −20.4676 −0.972447 −0.486224 0.873834i \(-0.661626\pi\)
−0.486224 + 0.873834i \(0.661626\pi\)
\(444\) −4.53125 −0.215044
\(445\) 5.03699 0.238776
\(446\) −23.0348 −1.09073
\(447\) −55.0144 −2.60209
\(448\) 2.22864 0.105293
\(449\) 40.4545 1.90917 0.954584 0.297943i \(-0.0963004\pi\)
0.954584 + 0.297943i \(0.0963004\pi\)
\(450\) −34.2549 −1.61479
\(451\) 0 0
\(452\) 5.63764 0.265172
\(453\) −59.6865 −2.80431
\(454\) −11.6050 −0.544648
\(455\) −2.23047 −0.104566
\(456\) 3.30795 0.154909
\(457\) 31.1469 1.45699 0.728495 0.685051i \(-0.240220\pi\)
0.728495 + 0.685051i \(0.240220\pi\)
\(458\) 10.8274 0.505930
\(459\) 74.9169 3.49682
\(460\) 1.46233 0.0681814
\(461\) −28.2437 −1.31544 −0.657719 0.753263i \(-0.728479\pi\)
−0.657719 + 0.753263i \(0.728479\pi\)
\(462\) 0 0
\(463\) −28.0244 −1.30240 −0.651201 0.758905i \(-0.725735\pi\)
−0.651201 + 0.758905i \(0.725735\pi\)
\(464\) 3.91436 0.181719
\(465\) 23.8195 1.10460
\(466\) 0.972932 0.0450702
\(467\) −36.0872 −1.66992 −0.834959 0.550312i \(-0.814509\pi\)
−0.834959 + 0.550312i \(0.814509\pi\)
\(468\) 9.58937 0.443269
\(469\) −26.0246 −1.20170
\(470\) 6.57084 0.303090
\(471\) −56.8663 −2.62026
\(472\) −0.104572 −0.00481334
\(473\) 0 0
\(474\) −7.68487 −0.352978
\(475\) −4.31285 −0.197887
\(476\) 10.2121 0.468069
\(477\) −64.6467 −2.95997
\(478\) −16.0095 −0.732258
\(479\) −26.3357 −1.20331 −0.601656 0.798756i \(-0.705492\pi\)
−0.601656 + 0.798756i \(0.705492\pi\)
\(480\) −2.74210 −0.125159
\(481\) −1.65383 −0.0754083
\(482\) 11.2997 0.514688
\(483\) −13.0052 −0.591758
\(484\) 0 0
\(485\) −14.3718 −0.652590
\(486\) 51.0363 2.31505
\(487\) −7.80731 −0.353783 −0.176891 0.984230i \(-0.556604\pi\)
−0.176891 + 0.984230i \(0.556604\pi\)
\(488\) 12.4843 0.565139
\(489\) 32.7839 1.48254
\(490\) 1.68538 0.0761379
\(491\) −8.97408 −0.404994 −0.202497 0.979283i \(-0.564906\pi\)
−0.202497 + 0.979283i \(0.564906\pi\)
\(492\) −19.5715 −0.882353
\(493\) 17.9364 0.807813
\(494\) 1.20735 0.0543211
\(495\) 0 0
\(496\) −8.68660 −0.390040
\(497\) 12.0366 0.539916
\(498\) −19.8030 −0.887391
\(499\) 29.6042 1.32527 0.662633 0.748944i \(-0.269439\pi\)
0.662633 + 0.748944i \(0.269439\pi\)
\(500\) 7.71983 0.345241
\(501\) −45.0777 −2.01393
\(502\) 24.7199 1.10330
\(503\) 5.34001 0.238099 0.119050 0.992888i \(-0.462015\pi\)
0.119050 + 0.992888i \(0.462015\pi\)
\(504\) 17.7010 0.788464
\(505\) 0.0333218 0.00148280
\(506\) 0 0
\(507\) −38.1813 −1.69569
\(508\) 15.5597 0.690351
\(509\) −21.7883 −0.965749 −0.482874 0.875690i \(-0.660407\pi\)
−0.482874 + 0.875690i \(0.660407\pi\)
\(510\) −12.5648 −0.556381
\(511\) −29.5504 −1.30723
\(512\) 1.00000 0.0441942
\(513\) 16.3495 0.721850
\(514\) 0.535384 0.0236148
\(515\) −14.3377 −0.631796
\(516\) −12.5237 −0.551327
\(517\) 0 0
\(518\) −3.05281 −0.134133
\(519\) −37.9349 −1.66516
\(520\) −1.00082 −0.0438890
\(521\) 15.5138 0.679674 0.339837 0.940484i \(-0.389628\pi\)
0.339837 + 0.940484i \(0.389628\pi\)
\(522\) 31.0898 1.36076
\(523\) 21.5143 0.940757 0.470378 0.882465i \(-0.344117\pi\)
0.470378 + 0.882465i \(0.344117\pi\)
\(524\) 14.8982 0.650829
\(525\) −31.7953 −1.38766
\(526\) 9.20025 0.401150
\(527\) −39.8037 −1.73388
\(528\) 0 0
\(529\) −19.8880 −0.864696
\(530\) 6.74704 0.293073
\(531\) −0.830567 −0.0360436
\(532\) 2.22864 0.0966237
\(533\) −7.14330 −0.309411
\(534\) −20.1004 −0.869829
\(535\) −12.2726 −0.530590
\(536\) −11.6774 −0.504385
\(537\) 0.725069 0.0312890
\(538\) −11.3073 −0.487494
\(539\) 0 0
\(540\) −13.5528 −0.583222
\(541\) 13.5322 0.581797 0.290898 0.956754i \(-0.406046\pi\)
0.290898 + 0.956754i \(0.406046\pi\)
\(542\) 19.1989 0.824662
\(543\) −82.4825 −3.53966
\(544\) 4.58220 0.196460
\(545\) −0.146330 −0.00626810
\(546\) 8.90083 0.380920
\(547\) 29.3177 1.25353 0.626767 0.779207i \(-0.284378\pi\)
0.626767 + 0.779207i \(0.284378\pi\)
\(548\) 10.1562 0.433851
\(549\) 99.1569 4.23191
\(550\) 0 0
\(551\) 3.91436 0.166757
\(552\) −5.83550 −0.248375
\(553\) −5.17747 −0.220168
\(554\) 5.91887 0.251469
\(555\) 3.75615 0.159440
\(556\) 11.3949 0.483250
\(557\) −3.84810 −0.163049 −0.0815246 0.996671i \(-0.525979\pi\)
−0.0815246 + 0.996671i \(0.525979\pi\)
\(558\) −68.9934 −2.92072
\(559\) −4.57096 −0.193331
\(560\) −1.84741 −0.0780675
\(561\) 0 0
\(562\) −17.5195 −0.739016
\(563\) −21.8709 −0.921750 −0.460875 0.887465i \(-0.652464\pi\)
−0.460875 + 0.887465i \(0.652464\pi\)
\(564\) −26.2213 −1.10412
\(565\) −4.67329 −0.196607
\(566\) 5.40443 0.227165
\(567\) 67.4295 2.83177
\(568\) 5.40088 0.226616
\(569\) −24.1303 −1.01159 −0.505797 0.862652i \(-0.668802\pi\)
−0.505797 + 0.862652i \(0.668802\pi\)
\(570\) −2.74210 −0.114854
\(571\) −24.3325 −1.01828 −0.509141 0.860683i \(-0.670037\pi\)
−0.509141 + 0.860683i \(0.670037\pi\)
\(572\) 0 0
\(573\) −30.6087 −1.27870
\(574\) −13.1858 −0.550364
\(575\) 7.60824 0.317286
\(576\) 7.94251 0.330938
\(577\) 0.513047 0.0213584 0.0106792 0.999943i \(-0.496601\pi\)
0.0106792 + 0.999943i \(0.496601\pi\)
\(578\) 3.99655 0.166234
\(579\) 53.5274 2.22452
\(580\) −3.24478 −0.134732
\(581\) −13.3417 −0.553507
\(582\) 57.3515 2.37730
\(583\) 0 0
\(584\) −13.2594 −0.548677
\(585\) −7.94904 −0.328652
\(586\) 6.26565 0.258832
\(587\) 9.67182 0.399199 0.199599 0.979878i \(-0.436036\pi\)
0.199599 + 0.979878i \(0.436036\pi\)
\(588\) −6.72562 −0.277360
\(589\) −8.68660 −0.357925
\(590\) 0.0866846 0.00356875
\(591\) 47.7279 1.96326
\(592\) −1.36981 −0.0562987
\(593\) 41.5780 1.70740 0.853701 0.520763i \(-0.174352\pi\)
0.853701 + 0.520763i \(0.174352\pi\)
\(594\) 0 0
\(595\) −8.46522 −0.347040
\(596\) −16.6310 −0.681231
\(597\) 41.2896 1.68987
\(598\) −2.12986 −0.0870967
\(599\) 9.65115 0.394335 0.197168 0.980370i \(-0.436826\pi\)
0.197168 + 0.980370i \(0.436826\pi\)
\(600\) −14.2667 −0.582435
\(601\) 10.0551 0.410157 0.205078 0.978746i \(-0.434255\pi\)
0.205078 + 0.978746i \(0.434255\pi\)
\(602\) −8.43752 −0.343888
\(603\) −92.7475 −3.77697
\(604\) −18.0434 −0.734174
\(605\) 0 0
\(606\) −0.132973 −0.00540164
\(607\) 43.7009 1.77377 0.886883 0.461995i \(-0.152866\pi\)
0.886883 + 0.461995i \(0.152866\pi\)
\(608\) 1.00000 0.0405554
\(609\) 28.8575 1.16936
\(610\) −10.3488 −0.419011
\(611\) −9.57036 −0.387175
\(612\) 36.3941 1.47115
\(613\) −4.41727 −0.178412 −0.0892060 0.996013i \(-0.528433\pi\)
−0.0892060 + 0.996013i \(0.528433\pi\)
\(614\) −0.0433912 −0.00175113
\(615\) 16.2237 0.654202
\(616\) 0 0
\(617\) 28.7853 1.15885 0.579427 0.815024i \(-0.303277\pi\)
0.579427 + 0.815024i \(0.303277\pi\)
\(618\) 57.2155 2.30155
\(619\) −12.6782 −0.509578 −0.254789 0.966997i \(-0.582006\pi\)
−0.254789 + 0.966997i \(0.582006\pi\)
\(620\) 7.20070 0.289187
\(621\) −28.8420 −1.15739
\(622\) 2.42902 0.0973950
\(623\) −13.5421 −0.542552
\(624\) 3.99384 0.159882
\(625\) 15.1650 0.606599
\(626\) 0.817587 0.0326773
\(627\) 0 0
\(628\) −17.1908 −0.685989
\(629\) −6.27673 −0.250270
\(630\) −14.6731 −0.584590
\(631\) 13.7759 0.548411 0.274205 0.961671i \(-0.411585\pi\)
0.274205 + 0.961671i \(0.411585\pi\)
\(632\) −2.32315 −0.0924101
\(633\) 81.0103 3.21987
\(634\) −18.5495 −0.736695
\(635\) −12.8981 −0.511847
\(636\) −26.9245 −1.06762
\(637\) −2.45475 −0.0972606
\(638\) 0 0
\(639\) 42.8965 1.69696
\(640\) −0.828943 −0.0327669
\(641\) −9.65449 −0.381329 −0.190665 0.981655i \(-0.561064\pi\)
−0.190665 + 0.981655i \(0.561064\pi\)
\(642\) 48.9744 1.93287
\(643\) −18.9601 −0.747713 −0.373857 0.927487i \(-0.621965\pi\)
−0.373857 + 0.927487i \(0.621965\pi\)
\(644\) −3.93151 −0.154923
\(645\) 10.3815 0.408770
\(646\) 4.58220 0.180284
\(647\) −1.35149 −0.0531325 −0.0265662 0.999647i \(-0.508457\pi\)
−0.0265662 + 0.999647i \(0.508457\pi\)
\(648\) 30.2559 1.18856
\(649\) 0 0
\(650\) −5.20711 −0.204240
\(651\) −64.0395 −2.50991
\(652\) 9.91065 0.388131
\(653\) 26.2017 1.02535 0.512676 0.858582i \(-0.328654\pi\)
0.512676 + 0.858582i \(0.328654\pi\)
\(654\) 0.583939 0.0228338
\(655\) −12.3497 −0.482544
\(656\) −5.91652 −0.231001
\(657\) −105.313 −4.10864
\(658\) −17.6659 −0.688688
\(659\) 10.9786 0.427666 0.213833 0.976870i \(-0.431405\pi\)
0.213833 + 0.976870i \(0.431405\pi\)
\(660\) 0 0
\(661\) 20.0499 0.779851 0.389926 0.920846i \(-0.372501\pi\)
0.389926 + 0.920846i \(0.372501\pi\)
\(662\) −4.31311 −0.167634
\(663\) 18.3006 0.710735
\(664\) −5.98648 −0.232320
\(665\) −1.84741 −0.0716397
\(666\) −10.8797 −0.421580
\(667\) −6.90526 −0.267373
\(668\) −13.6271 −0.527249
\(669\) −76.1978 −2.94598
\(670\) 9.67987 0.373966
\(671\) 0 0
\(672\) 7.37221 0.284389
\(673\) 1.57266 0.0606218 0.0303109 0.999541i \(-0.490350\pi\)
0.0303109 + 0.999541i \(0.490350\pi\)
\(674\) 1.35344 0.0521326
\(675\) −70.5132 −2.71405
\(676\) −11.5423 −0.443935
\(677\) 47.4839 1.82495 0.912476 0.409130i \(-0.134168\pi\)
0.912476 + 0.409130i \(0.134168\pi\)
\(678\) 18.6490 0.716211
\(679\) 38.6390 1.48283
\(680\) −3.79838 −0.145661
\(681\) −38.3886 −1.47105
\(682\) 0 0
\(683\) 30.3311 1.16059 0.580293 0.814408i \(-0.302938\pi\)
0.580293 + 0.814408i \(0.302938\pi\)
\(684\) 7.94251 0.303689
\(685\) −8.41891 −0.321670
\(686\) −20.1317 −0.768631
\(687\) 35.8163 1.36648
\(688\) −3.78596 −0.144338
\(689\) −9.82700 −0.374379
\(690\) 4.83730 0.184153
\(691\) 11.4962 0.437337 0.218669 0.975799i \(-0.429829\pi\)
0.218669 + 0.975799i \(0.429829\pi\)
\(692\) −11.4678 −0.435941
\(693\) 0 0
\(694\) −14.9394 −0.567091
\(695\) −9.44569 −0.358295
\(696\) 12.9485 0.490811
\(697\) −27.1107 −1.02689
\(698\) 8.42846 0.319022
\(699\) 3.21841 0.121731
\(700\) −9.61179 −0.363291
\(701\) −42.5115 −1.60564 −0.802818 0.596224i \(-0.796667\pi\)
−0.802818 + 0.596224i \(0.796667\pi\)
\(702\) 19.7396 0.745023
\(703\) −1.36981 −0.0516633
\(704\) 0 0
\(705\) 21.7360 0.818624
\(706\) −29.1828 −1.09831
\(707\) −0.0895866 −0.00336925
\(708\) −0.345920 −0.0130005
\(709\) 0.138545 0.00520315 0.00260158 0.999997i \(-0.499172\pi\)
0.00260158 + 0.999997i \(0.499172\pi\)
\(710\) −4.47702 −0.168020
\(711\) −18.4517 −0.691992
\(712\) −6.07639 −0.227722
\(713\) 15.3239 0.573885
\(714\) 33.7810 1.26422
\(715\) 0 0
\(716\) 0.219190 0.00819152
\(717\) −52.9586 −1.97777
\(718\) −26.4680 −0.987778
\(719\) 5.55518 0.207173 0.103587 0.994620i \(-0.466968\pi\)
0.103587 + 0.994620i \(0.466968\pi\)
\(720\) −6.58389 −0.245367
\(721\) 38.5474 1.43558
\(722\) 1.00000 0.0372161
\(723\) 37.3788 1.39013
\(724\) −24.9347 −0.926689
\(725\) −16.8820 −0.626984
\(726\) 0 0
\(727\) −42.2810 −1.56811 −0.784057 0.620688i \(-0.786853\pi\)
−0.784057 + 0.620688i \(0.786853\pi\)
\(728\) 2.69074 0.0997255
\(729\) 78.0575 2.89102
\(730\) 10.9913 0.406805
\(731\) −17.3480 −0.641639
\(732\) 41.2975 1.52640
\(733\) 30.8346 1.13890 0.569451 0.822025i \(-0.307156\pi\)
0.569451 + 0.822025i \(0.307156\pi\)
\(734\) −20.9635 −0.773778
\(735\) 5.57516 0.205643
\(736\) −1.76409 −0.0650251
\(737\) 0 0
\(738\) −46.9920 −1.72980
\(739\) 3.47185 0.127714 0.0638569 0.997959i \(-0.479660\pi\)
0.0638569 + 0.997959i \(0.479660\pi\)
\(740\) 1.13549 0.0417415
\(741\) 3.99384 0.146717
\(742\) −18.1396 −0.665926
\(743\) 41.6207 1.52692 0.763458 0.645857i \(-0.223500\pi\)
0.763458 + 0.645857i \(0.223500\pi\)
\(744\) −28.7348 −1.05347
\(745\) 13.7861 0.505085
\(746\) −34.2399 −1.25361
\(747\) −47.5477 −1.73968
\(748\) 0 0
\(749\) 32.9952 1.20562
\(750\) 25.5368 0.932471
\(751\) −21.1375 −0.771317 −0.385658 0.922642i \(-0.626026\pi\)
−0.385658 + 0.922642i \(0.626026\pi\)
\(752\) −7.92677 −0.289059
\(753\) 81.7721 2.97994
\(754\) 4.72599 0.172110
\(755\) 14.9569 0.544338
\(756\) 36.4372 1.32521
\(757\) −0.324892 −0.0118084 −0.00590419 0.999983i \(-0.501879\pi\)
−0.00590419 + 0.999983i \(0.501879\pi\)
\(758\) 8.74108 0.317490
\(759\) 0 0
\(760\) −0.828943 −0.0300689
\(761\) −7.39942 −0.268229 −0.134114 0.990966i \(-0.542819\pi\)
−0.134114 + 0.990966i \(0.542819\pi\)
\(762\) 51.4707 1.86459
\(763\) 0.393413 0.0142425
\(764\) −9.25309 −0.334765
\(765\) −30.1687 −1.09075
\(766\) 4.17010 0.150672
\(767\) −0.126255 −0.00455881
\(768\) 3.30795 0.119365
\(769\) −26.6836 −0.962235 −0.481118 0.876656i \(-0.659769\pi\)
−0.481118 + 0.876656i \(0.659769\pi\)
\(770\) 0 0
\(771\) 1.77102 0.0637818
\(772\) 16.1815 0.582384
\(773\) −27.4993 −0.989081 −0.494541 0.869154i \(-0.664664\pi\)
−0.494541 + 0.869154i \(0.664664\pi\)
\(774\) −30.0700 −1.08084
\(775\) 37.4640 1.34575
\(776\) 17.3375 0.622380
\(777\) −10.0985 −0.362282
\(778\) −7.70695 −0.276308
\(779\) −5.91652 −0.211981
\(780\) −3.31067 −0.118541
\(781\) 0 0
\(782\) −8.08339 −0.289062
\(783\) 63.9980 2.28710
\(784\) −2.03317 −0.0726133
\(785\) 14.2502 0.508612
\(786\) 49.2823 1.75784
\(787\) 51.3354 1.82991 0.914956 0.403555i \(-0.132225\pi\)
0.914956 + 0.403555i \(0.132225\pi\)
\(788\) 14.4283 0.513985
\(789\) 30.4339 1.08348
\(790\) 1.92576 0.0685156
\(791\) 12.5643 0.446734
\(792\) 0 0
\(793\) 15.0729 0.535256
\(794\) 27.6939 0.982818
\(795\) 22.3189 0.791568
\(796\) 12.4819 0.442411
\(797\) 9.56050 0.338650 0.169325 0.985560i \(-0.445841\pi\)
0.169325 + 0.985560i \(0.445841\pi\)
\(798\) 7.37221 0.260974
\(799\) −36.3220 −1.28498
\(800\) −4.31285 −0.152482
\(801\) −48.2618 −1.70525
\(802\) −28.3005 −0.999326
\(803\) 0 0
\(804\) −38.6281 −1.36231
\(805\) 3.25900 0.114865
\(806\) −10.4878 −0.369415
\(807\) −37.4041 −1.31669
\(808\) −0.0401979 −0.00141416
\(809\) −4.69556 −0.165087 −0.0825435 0.996587i \(-0.526304\pi\)
−0.0825435 + 0.996587i \(0.526304\pi\)
\(810\) −25.0804 −0.881236
\(811\) 51.9776 1.82518 0.912590 0.408876i \(-0.134079\pi\)
0.912590 + 0.408876i \(0.134079\pi\)
\(812\) 8.72369 0.306141
\(813\) 63.5088 2.22735
\(814\) 0 0
\(815\) −8.21536 −0.287772
\(816\) 15.1577 0.530624
\(817\) −3.78596 −0.132454
\(818\) −7.15765 −0.250261
\(819\) 21.3712 0.746771
\(820\) 4.90446 0.171271
\(821\) 44.9698 1.56946 0.784728 0.619840i \(-0.212802\pi\)
0.784728 + 0.619840i \(0.212802\pi\)
\(822\) 33.5961 1.17180
\(823\) 32.8043 1.14349 0.571744 0.820432i \(-0.306267\pi\)
0.571744 + 0.820432i \(0.306267\pi\)
\(824\) 17.2964 0.602548
\(825\) 0 0
\(826\) −0.233054 −0.00810899
\(827\) 0.752079 0.0261523 0.0130762 0.999915i \(-0.495838\pi\)
0.0130762 + 0.999915i \(0.495838\pi\)
\(828\) −14.0113 −0.486925
\(829\) 33.2350 1.15430 0.577150 0.816638i \(-0.304165\pi\)
0.577150 + 0.816638i \(0.304165\pi\)
\(830\) 4.96245 0.172249
\(831\) 19.5793 0.679199
\(832\) 1.20735 0.0418572
\(833\) −9.31640 −0.322794
\(834\) 37.6936 1.30522
\(835\) 11.2961 0.390918
\(836\) 0 0
\(837\) −142.022 −4.90900
\(838\) 9.45181 0.326507
\(839\) 6.91909 0.238873 0.119437 0.992842i \(-0.461891\pi\)
0.119437 + 0.992842i \(0.461891\pi\)
\(840\) −6.11115 −0.210855
\(841\) −13.6778 −0.471648
\(842\) 27.2670 0.939682
\(843\) −57.9536 −1.99603
\(844\) 24.4896 0.842967
\(845\) 9.56792 0.329147
\(846\) −62.9584 −2.16455
\(847\) 0 0
\(848\) −8.13933 −0.279506
\(849\) 17.8776 0.613556
\(850\) −19.7624 −0.677843
\(851\) 2.41646 0.0828351
\(852\) 17.8658 0.612073
\(853\) −15.7760 −0.540159 −0.270079 0.962838i \(-0.587050\pi\)
−0.270079 + 0.962838i \(0.587050\pi\)
\(854\) 27.8231 0.952086
\(855\) −6.58389 −0.225164
\(856\) 14.8051 0.506027
\(857\) 16.5727 0.566112 0.283056 0.959103i \(-0.408652\pi\)
0.283056 + 0.959103i \(0.408652\pi\)
\(858\) 0 0
\(859\) 6.87642 0.234620 0.117310 0.993095i \(-0.462573\pi\)
0.117310 + 0.993095i \(0.462573\pi\)
\(860\) 3.13834 0.107017
\(861\) −43.6179 −1.48649
\(862\) −1.36032 −0.0463328
\(863\) −40.5236 −1.37944 −0.689720 0.724076i \(-0.742266\pi\)
−0.689720 + 0.724076i \(0.742266\pi\)
\(864\) 16.3495 0.556223
\(865\) 9.50616 0.323219
\(866\) 26.1728 0.889389
\(867\) 13.2204 0.448987
\(868\) −19.3593 −0.657097
\(869\) 0 0
\(870\) −10.7336 −0.363902
\(871\) −14.0986 −0.477714
\(872\) 0.176526 0.00597793
\(873\) 137.703 4.66055
\(874\) −1.76409 −0.0596711
\(875\) 17.2047 0.581625
\(876\) −43.8613 −1.48194
\(877\) −27.9198 −0.942784 −0.471392 0.881924i \(-0.656248\pi\)
−0.471392 + 0.881924i \(0.656248\pi\)
\(878\) −9.79442 −0.330546
\(879\) 20.7264 0.699086
\(880\) 0 0
\(881\) 26.7069 0.899780 0.449890 0.893084i \(-0.351463\pi\)
0.449890 + 0.893084i \(0.351463\pi\)
\(882\) −16.1485 −0.543748
\(883\) 10.5785 0.355997 0.177998 0.984031i \(-0.443038\pi\)
0.177998 + 0.984031i \(0.443038\pi\)
\(884\) 5.53231 0.186072
\(885\) 0.286748 0.00963893
\(886\) −20.4676 −0.687624
\(887\) −21.3943 −0.718351 −0.359176 0.933270i \(-0.616942\pi\)
−0.359176 + 0.933270i \(0.616942\pi\)
\(888\) −4.53125 −0.152059
\(889\) 34.6770 1.16303
\(890\) 5.03699 0.168840
\(891\) 0 0
\(892\) −23.0348 −0.771262
\(893\) −7.92677 −0.265259
\(894\) −55.0144 −1.83996
\(895\) −0.181696 −0.00607343
\(896\) 2.22864 0.0744536
\(897\) −7.04548 −0.235242
\(898\) 40.4545 1.34999
\(899\) −34.0025 −1.13405
\(900\) −34.2549 −1.14183
\(901\) −37.2960 −1.24251
\(902\) 0 0
\(903\) −27.9109 −0.928816
\(904\) 5.63764 0.187505
\(905\) 20.6694 0.687075
\(906\) −59.6865 −1.98295
\(907\) 14.5370 0.482694 0.241347 0.970439i \(-0.422411\pi\)
0.241347 + 0.970439i \(0.422411\pi\)
\(908\) −11.6050 −0.385124
\(909\) −0.319272 −0.0105896
\(910\) −2.23047 −0.0739394
\(911\) 45.9200 1.52140 0.760698 0.649106i \(-0.224857\pi\)
0.760698 + 0.649106i \(0.224857\pi\)
\(912\) 3.30795 0.109537
\(913\) 0 0
\(914\) 31.1469 1.03025
\(915\) −34.2333 −1.13172
\(916\) 10.8274 0.357746
\(917\) 33.2026 1.09645
\(918\) 74.9169 2.47263
\(919\) 10.9057 0.359745 0.179872 0.983690i \(-0.442432\pi\)
0.179872 + 0.983690i \(0.442432\pi\)
\(920\) 1.46233 0.0482115
\(921\) −0.143536 −0.00472966
\(922\) −28.2437 −0.930155
\(923\) 6.52074 0.214633
\(924\) 0 0
\(925\) 5.90778 0.194247
\(926\) −28.0244 −0.920937
\(927\) 137.377 4.51204
\(928\) 3.91436 0.128495
\(929\) 8.41320 0.276028 0.138014 0.990430i \(-0.455928\pi\)
0.138014 + 0.990430i \(0.455928\pi\)
\(930\) 23.8195 0.781073
\(931\) −2.03317 −0.0666345
\(932\) 0.972932 0.0318694
\(933\) 8.03508 0.263057
\(934\) −36.0872 −1.18081
\(935\) 0 0
\(936\) 9.58937 0.313438
\(937\) −14.6747 −0.479401 −0.239701 0.970847i \(-0.577049\pi\)
−0.239701 + 0.970847i \(0.577049\pi\)
\(938\) −26.0246 −0.849733
\(939\) 2.70453 0.0882591
\(940\) 6.57084 0.214317
\(941\) 5.44917 0.177638 0.0888189 0.996048i \(-0.471691\pi\)
0.0888189 + 0.996048i \(0.471691\pi\)
\(942\) −56.8663 −1.85280
\(943\) 10.4372 0.339884
\(944\) −0.104572 −0.00340354
\(945\) −30.2044 −0.982549
\(946\) 0 0
\(947\) −29.9722 −0.973964 −0.486982 0.873412i \(-0.661902\pi\)
−0.486982 + 0.873412i \(0.661902\pi\)
\(948\) −7.68487 −0.249593
\(949\) −16.0087 −0.519664
\(950\) −4.31285 −0.139927
\(951\) −61.3607 −1.98976
\(952\) 10.2121 0.330975
\(953\) −18.2596 −0.591487 −0.295743 0.955267i \(-0.595567\pi\)
−0.295743 + 0.955267i \(0.595567\pi\)
\(954\) −64.6467 −2.09301
\(955\) 7.67029 0.248205
\(956\) −16.0095 −0.517784
\(957\) 0 0
\(958\) −26.3357 −0.850869
\(959\) 22.6345 0.730905
\(960\) −2.74210 −0.0885009
\(961\) 44.4571 1.43410
\(962\) −1.65383 −0.0533217
\(963\) 117.590 3.78927
\(964\) 11.2997 0.363939
\(965\) −13.4135 −0.431796
\(966\) −13.0052 −0.418436
\(967\) 29.4143 0.945899 0.472949 0.881090i \(-0.343189\pi\)
0.472949 + 0.881090i \(0.343189\pi\)
\(968\) 0 0
\(969\) 15.1577 0.486934
\(970\) −14.3718 −0.461451
\(971\) 14.7085 0.472019 0.236010 0.971751i \(-0.424160\pi\)
0.236010 + 0.971751i \(0.424160\pi\)
\(972\) 51.0363 1.63699
\(973\) 25.3950 0.814127
\(974\) −7.80731 −0.250162
\(975\) −17.2248 −0.551637
\(976\) 12.4843 0.399614
\(977\) 1.43626 0.0459501 0.0229751 0.999736i \(-0.492686\pi\)
0.0229751 + 0.999736i \(0.492686\pi\)
\(978\) 32.7839 1.04831
\(979\) 0 0
\(980\) 1.68538 0.0538376
\(981\) 1.40206 0.0447644
\(982\) −8.97408 −0.286374
\(983\) −51.6930 −1.64875 −0.824375 0.566044i \(-0.808473\pi\)
−0.824375 + 0.566044i \(0.808473\pi\)
\(984\) −19.5715 −0.623918
\(985\) −11.9602 −0.381084
\(986\) 17.9364 0.571210
\(987\) −58.4378 −1.86010
\(988\) 1.20735 0.0384108
\(989\) 6.67875 0.212372
\(990\) 0 0
\(991\) 35.4491 1.12608 0.563040 0.826430i \(-0.309632\pi\)
0.563040 + 0.826430i \(0.309632\pi\)
\(992\) −8.68660 −0.275800
\(993\) −14.2675 −0.452766
\(994\) 12.0366 0.381778
\(995\) −10.3468 −0.328016
\(996\) −19.8030 −0.627481
\(997\) −14.5269 −0.460071 −0.230035 0.973182i \(-0.573884\pi\)
−0.230035 + 0.973182i \(0.573884\pi\)
\(998\) 29.6042 0.937104
\(999\) −22.3957 −0.708570
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 4598.2.a.cd.1.10 10
11.2 odd 10 418.2.f.h.191.5 20
11.6 odd 10 418.2.f.h.267.5 yes 20
11.10 odd 2 4598.2.a.cc.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.h.191.5 20 11.2 odd 10
418.2.f.h.267.5 yes 20 11.6 odd 10
4598.2.a.cc.1.10 10 11.10 odd 2
4598.2.a.cd.1.10 10 1.1 even 1 trivial