Properties

Label 5225.2.a.k.1.4
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.577704\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.577704 q^{2} +0.746709 q^{3} -1.66626 q^{4} +0.431377 q^{6} +4.19297 q^{7} -2.11801 q^{8} -2.44243 q^{9} +O(q^{10})\) \(q+0.577704 q^{2} +0.746709 q^{3} -1.66626 q^{4} +0.431377 q^{6} +4.19297 q^{7} -2.11801 q^{8} -2.44243 q^{9} -1.00000 q^{11} -1.24421 q^{12} -0.324413 q^{13} +2.42230 q^{14} +2.10893 q^{16} +5.00810 q^{17} -1.41100 q^{18} -1.00000 q^{19} +3.13093 q^{21} -0.577704 q^{22} -8.11252 q^{23} -1.58154 q^{24} -0.187415 q^{26} -4.06391 q^{27} -6.98657 q^{28} -3.53579 q^{29} -3.45986 q^{31} +5.45436 q^{32} -0.746709 q^{33} +2.89320 q^{34} +4.06971 q^{36} -5.77902 q^{37} -0.577704 q^{38} -0.242242 q^{39} +0.484585 q^{41} +1.80875 q^{42} -5.86472 q^{43} +1.66626 q^{44} -4.68664 q^{46} -6.22580 q^{47} +1.57476 q^{48} +10.5810 q^{49} +3.73960 q^{51} +0.540556 q^{52} +10.2794 q^{53} -2.34774 q^{54} -8.88076 q^{56} -0.746709 q^{57} -2.04264 q^{58} +7.16900 q^{59} +5.33749 q^{61} -1.99877 q^{62} -10.2410 q^{63} -1.06685 q^{64} -0.431377 q^{66} -8.20951 q^{67} -8.34479 q^{68} -6.05769 q^{69} -10.0184 q^{71} +5.17309 q^{72} -2.88049 q^{73} -3.33857 q^{74} +1.66626 q^{76} -4.19297 q^{77} -0.139944 q^{78} +3.95809 q^{79} +4.29272 q^{81} +0.279947 q^{82} -10.2837 q^{83} -5.21693 q^{84} -3.38807 q^{86} -2.64021 q^{87} +2.11801 q^{88} +7.47277 q^{89} -1.36025 q^{91} +13.5175 q^{92} -2.58351 q^{93} -3.59667 q^{94} +4.07282 q^{96} +13.9617 q^{97} +6.11268 q^{98} +2.44243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.577704 0.408499 0.204249 0.978919i \(-0.434525\pi\)
0.204249 + 0.978919i \(0.434525\pi\)
\(3\) 0.746709 0.431113 0.215556 0.976491i \(-0.430844\pi\)
0.215556 + 0.976491i \(0.430844\pi\)
\(4\) −1.66626 −0.833129
\(5\) 0 0
\(6\) 0.431377 0.176109
\(7\) 4.19297 1.58479 0.792397 0.610006i \(-0.208833\pi\)
0.792397 + 0.610006i \(0.208833\pi\)
\(8\) −2.11801 −0.748830
\(9\) −2.44243 −0.814142
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.24421 −0.359172
\(13\) −0.324413 −0.0899760 −0.0449880 0.998988i \(-0.514325\pi\)
−0.0449880 + 0.998988i \(0.514325\pi\)
\(14\) 2.42230 0.647386
\(15\) 0 0
\(16\) 2.10893 0.527233
\(17\) 5.00810 1.21464 0.607322 0.794456i \(-0.292244\pi\)
0.607322 + 0.794456i \(0.292244\pi\)
\(18\) −1.41100 −0.332576
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.13093 0.683225
\(22\) −0.577704 −0.123167
\(23\) −8.11252 −1.69158 −0.845789 0.533518i \(-0.820870\pi\)
−0.845789 + 0.533518i \(0.820870\pi\)
\(24\) −1.58154 −0.322830
\(25\) 0 0
\(26\) −0.187415 −0.0367551
\(27\) −4.06391 −0.782100
\(28\) −6.98657 −1.32034
\(29\) −3.53579 −0.656580 −0.328290 0.944577i \(-0.606472\pi\)
−0.328290 + 0.944577i \(0.606472\pi\)
\(30\) 0 0
\(31\) −3.45986 −0.621409 −0.310704 0.950507i \(-0.600565\pi\)
−0.310704 + 0.950507i \(0.600565\pi\)
\(32\) 5.45436 0.964204
\(33\) −0.746709 −0.129985
\(34\) 2.89320 0.496180
\(35\) 0 0
\(36\) 4.06971 0.678285
\(37\) −5.77902 −0.950066 −0.475033 0.879968i \(-0.657564\pi\)
−0.475033 + 0.879968i \(0.657564\pi\)
\(38\) −0.577704 −0.0937160
\(39\) −0.242242 −0.0387898
\(40\) 0 0
\(41\) 0.484585 0.0756794 0.0378397 0.999284i \(-0.487952\pi\)
0.0378397 + 0.999284i \(0.487952\pi\)
\(42\) 1.80875 0.279096
\(43\) −5.86472 −0.894362 −0.447181 0.894444i \(-0.647572\pi\)
−0.447181 + 0.894444i \(0.647572\pi\)
\(44\) 1.66626 0.251198
\(45\) 0 0
\(46\) −4.68664 −0.691007
\(47\) −6.22580 −0.908126 −0.454063 0.890970i \(-0.650026\pi\)
−0.454063 + 0.890970i \(0.650026\pi\)
\(48\) 1.57476 0.227297
\(49\) 10.5810 1.51157
\(50\) 0 0
\(51\) 3.73960 0.523648
\(52\) 0.540556 0.0749616
\(53\) 10.2794 1.41198 0.705991 0.708221i \(-0.250502\pi\)
0.705991 + 0.708221i \(0.250502\pi\)
\(54\) −2.34774 −0.319487
\(55\) 0 0
\(56\) −8.88076 −1.18674
\(57\) −0.746709 −0.0989040
\(58\) −2.04264 −0.268212
\(59\) 7.16900 0.933325 0.466662 0.884435i \(-0.345456\pi\)
0.466662 + 0.884435i \(0.345456\pi\)
\(60\) 0 0
\(61\) 5.33749 0.683396 0.341698 0.939810i \(-0.388998\pi\)
0.341698 + 0.939810i \(0.388998\pi\)
\(62\) −1.99877 −0.253845
\(63\) −10.2410 −1.29025
\(64\) −1.06685 −0.133357
\(65\) 0 0
\(66\) −0.431377 −0.0530988
\(67\) −8.20951 −1.00295 −0.501476 0.865172i \(-0.667209\pi\)
−0.501476 + 0.865172i \(0.667209\pi\)
\(68\) −8.34479 −1.01195
\(69\) −6.05769 −0.729260
\(70\) 0 0
\(71\) −10.0184 −1.18897 −0.594484 0.804108i \(-0.702644\pi\)
−0.594484 + 0.804108i \(0.702644\pi\)
\(72\) 5.17309 0.609654
\(73\) −2.88049 −0.337136 −0.168568 0.985690i \(-0.553914\pi\)
−0.168568 + 0.985690i \(0.553914\pi\)
\(74\) −3.33857 −0.388100
\(75\) 0 0
\(76\) 1.66626 0.191133
\(77\) −4.19297 −0.477833
\(78\) −0.139944 −0.0158456
\(79\) 3.95809 0.445320 0.222660 0.974896i \(-0.428526\pi\)
0.222660 + 0.974896i \(0.428526\pi\)
\(80\) 0 0
\(81\) 4.29272 0.476969
\(82\) 0.279947 0.0309149
\(83\) −10.2837 −1.12878 −0.564389 0.825509i \(-0.690888\pi\)
−0.564389 + 0.825509i \(0.690888\pi\)
\(84\) −5.21693 −0.569214
\(85\) 0 0
\(86\) −3.38807 −0.365345
\(87\) −2.64021 −0.283060
\(88\) 2.11801 0.225781
\(89\) 7.47277 0.792112 0.396056 0.918226i \(-0.370379\pi\)
0.396056 + 0.918226i \(0.370379\pi\)
\(90\) 0 0
\(91\) −1.36025 −0.142593
\(92\) 13.5175 1.40930
\(93\) −2.58351 −0.267897
\(94\) −3.59667 −0.370968
\(95\) 0 0
\(96\) 4.07282 0.415681
\(97\) 13.9617 1.41759 0.708797 0.705412i \(-0.249238\pi\)
0.708797 + 0.705412i \(0.249238\pi\)
\(98\) 6.11268 0.617474
\(99\) 2.44243 0.245473
\(100\) 0 0
\(101\) 0.481481 0.0479091 0.0239546 0.999713i \(-0.492374\pi\)
0.0239546 + 0.999713i \(0.492374\pi\)
\(102\) 2.16038 0.213909
\(103\) −14.0759 −1.38694 −0.693468 0.720488i \(-0.743918\pi\)
−0.693468 + 0.720488i \(0.743918\pi\)
\(104\) 0.687111 0.0673768
\(105\) 0 0
\(106\) 5.93845 0.576793
\(107\) 0.995650 0.0962531 0.0481265 0.998841i \(-0.484675\pi\)
0.0481265 + 0.998841i \(0.484675\pi\)
\(108\) 6.77152 0.651590
\(109\) 5.36197 0.513584 0.256792 0.966467i \(-0.417334\pi\)
0.256792 + 0.966467i \(0.417334\pi\)
\(110\) 0 0
\(111\) −4.31525 −0.409585
\(112\) 8.84268 0.835555
\(113\) −2.57028 −0.241792 −0.120896 0.992665i \(-0.538577\pi\)
−0.120896 + 0.992665i \(0.538577\pi\)
\(114\) −0.431377 −0.0404021
\(115\) 0 0
\(116\) 5.89154 0.547016
\(117\) 0.792355 0.0732533
\(118\) 4.14156 0.381262
\(119\) 20.9988 1.92496
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.08349 0.279166
\(123\) 0.361844 0.0326264
\(124\) 5.76501 0.517714
\(125\) 0 0
\(126\) −5.91628 −0.527064
\(127\) −19.9231 −1.76789 −0.883947 0.467588i \(-0.845123\pi\)
−0.883947 + 0.467588i \(0.845123\pi\)
\(128\) −11.5251 −1.01868
\(129\) −4.37924 −0.385571
\(130\) 0 0
\(131\) −1.71963 −0.150245 −0.0751224 0.997174i \(-0.523935\pi\)
−0.0751224 + 0.997174i \(0.523935\pi\)
\(132\) 1.24421 0.108295
\(133\) −4.19297 −0.363577
\(134\) −4.74267 −0.409704
\(135\) 0 0
\(136\) −10.6072 −0.909562
\(137\) −12.6481 −1.08060 −0.540298 0.841473i \(-0.681689\pi\)
−0.540298 + 0.841473i \(0.681689\pi\)
\(138\) −3.49955 −0.297902
\(139\) −17.9111 −1.51920 −0.759601 0.650390i \(-0.774606\pi\)
−0.759601 + 0.650390i \(0.774606\pi\)
\(140\) 0 0
\(141\) −4.64886 −0.391505
\(142\) −5.78768 −0.485692
\(143\) 0.324413 0.0271288
\(144\) −5.15091 −0.429242
\(145\) 0 0
\(146\) −1.66407 −0.137720
\(147\) 7.90092 0.651657
\(148\) 9.62934 0.791527
\(149\) −17.1692 −1.40656 −0.703279 0.710914i \(-0.748282\pi\)
−0.703279 + 0.710914i \(0.748282\pi\)
\(150\) 0 0
\(151\) −15.6004 −1.26954 −0.634772 0.772699i \(-0.718906\pi\)
−0.634772 + 0.772699i \(0.718906\pi\)
\(152\) 2.11801 0.171793
\(153\) −12.2319 −0.988892
\(154\) −2.42230 −0.195194
\(155\) 0 0
\(156\) 0.403638 0.0323169
\(157\) −5.17981 −0.413394 −0.206697 0.978405i \(-0.566271\pi\)
−0.206697 + 0.978405i \(0.566271\pi\)
\(158\) 2.28660 0.181912
\(159\) 7.67571 0.608724
\(160\) 0 0
\(161\) −34.0155 −2.68080
\(162\) 2.47992 0.194841
\(163\) 2.66829 0.208997 0.104498 0.994525i \(-0.466676\pi\)
0.104498 + 0.994525i \(0.466676\pi\)
\(164\) −0.807443 −0.0630507
\(165\) 0 0
\(166\) −5.94091 −0.461104
\(167\) −12.7357 −0.985516 −0.492758 0.870166i \(-0.664011\pi\)
−0.492758 + 0.870166i \(0.664011\pi\)
\(168\) −6.63134 −0.511619
\(169\) −12.8948 −0.991904
\(170\) 0 0
\(171\) 2.44243 0.186777
\(172\) 9.77214 0.745119
\(173\) 3.51189 0.267004 0.133502 0.991049i \(-0.457378\pi\)
0.133502 + 0.991049i \(0.457378\pi\)
\(174\) −1.52526 −0.115630
\(175\) 0 0
\(176\) −2.10893 −0.158967
\(177\) 5.35316 0.402368
\(178\) 4.31705 0.323577
\(179\) −0.129487 −0.00967830 −0.00483915 0.999988i \(-0.501540\pi\)
−0.00483915 + 0.999988i \(0.501540\pi\)
\(180\) 0 0
\(181\) −1.43867 −0.106936 −0.0534679 0.998570i \(-0.517027\pi\)
−0.0534679 + 0.998570i \(0.517027\pi\)
\(182\) −0.785825 −0.0582492
\(183\) 3.98556 0.294621
\(184\) 17.1824 1.26670
\(185\) 0 0
\(186\) −1.49250 −0.109436
\(187\) −5.00810 −0.366229
\(188\) 10.3738 0.756586
\(189\) −17.0398 −1.23947
\(190\) 0 0
\(191\) 9.83771 0.711832 0.355916 0.934518i \(-0.384169\pi\)
0.355916 + 0.934518i \(0.384169\pi\)
\(192\) −0.796630 −0.0574918
\(193\) 21.0622 1.51609 0.758046 0.652201i \(-0.226154\pi\)
0.758046 + 0.652201i \(0.226154\pi\)
\(194\) 8.06572 0.579085
\(195\) 0 0
\(196\) −17.6307 −1.25933
\(197\) 15.7435 1.12168 0.560840 0.827925i \(-0.310478\pi\)
0.560840 + 0.827925i \(0.310478\pi\)
\(198\) 1.41100 0.100275
\(199\) −10.0628 −0.713330 −0.356665 0.934232i \(-0.616086\pi\)
−0.356665 + 0.934232i \(0.616086\pi\)
\(200\) 0 0
\(201\) −6.13012 −0.432385
\(202\) 0.278153 0.0195708
\(203\) −14.8255 −1.04054
\(204\) −6.23113 −0.436266
\(205\) 0 0
\(206\) −8.13168 −0.566561
\(207\) 19.8142 1.37718
\(208\) −0.684165 −0.0474383
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −11.0673 −0.761906 −0.380953 0.924594i \(-0.624404\pi\)
−0.380953 + 0.924594i \(0.624404\pi\)
\(212\) −17.1281 −1.17636
\(213\) −7.48085 −0.512579
\(214\) 0.575191 0.0393192
\(215\) 0 0
\(216\) 8.60741 0.585660
\(217\) −14.5071 −0.984804
\(218\) 3.09763 0.209798
\(219\) −2.15089 −0.145344
\(220\) 0 0
\(221\) −1.62469 −0.109289
\(222\) −2.49294 −0.167315
\(223\) −23.5560 −1.57743 −0.788713 0.614762i \(-0.789252\pi\)
−0.788713 + 0.614762i \(0.789252\pi\)
\(224\) 22.8700 1.52806
\(225\) 0 0
\(226\) −1.48486 −0.0987716
\(227\) −9.43858 −0.626461 −0.313230 0.949677i \(-0.601411\pi\)
−0.313230 + 0.949677i \(0.601411\pi\)
\(228\) 1.24421 0.0823998
\(229\) −17.0790 −1.12861 −0.564305 0.825566i \(-0.690856\pi\)
−0.564305 + 0.825566i \(0.690856\pi\)
\(230\) 0 0
\(231\) −3.13093 −0.206000
\(232\) 7.48885 0.491667
\(233\) 2.52469 0.165398 0.0826991 0.996575i \(-0.473646\pi\)
0.0826991 + 0.996575i \(0.473646\pi\)
\(234\) 0.457747 0.0299238
\(235\) 0 0
\(236\) −11.9454 −0.777580
\(237\) 2.95554 0.191983
\(238\) 12.1311 0.786343
\(239\) 16.4351 1.06310 0.531550 0.847027i \(-0.321610\pi\)
0.531550 + 0.847027i \(0.321610\pi\)
\(240\) 0 0
\(241\) 22.8885 1.47438 0.737189 0.675687i \(-0.236153\pi\)
0.737189 + 0.675687i \(0.236153\pi\)
\(242\) 0.577704 0.0371362
\(243\) 15.3971 0.987727
\(244\) −8.89364 −0.569357
\(245\) 0 0
\(246\) 0.209039 0.0133278
\(247\) 0.324413 0.0206419
\(248\) 7.32802 0.465330
\(249\) −7.67890 −0.486631
\(250\) 0 0
\(251\) 13.7873 0.870246 0.435123 0.900371i \(-0.356705\pi\)
0.435123 + 0.900371i \(0.356705\pi\)
\(252\) 17.0642 1.07494
\(253\) 8.11252 0.510030
\(254\) −11.5097 −0.722182
\(255\) 0 0
\(256\) −4.52436 −0.282773
\(257\) 22.2128 1.38560 0.692800 0.721130i \(-0.256377\pi\)
0.692800 + 0.721130i \(0.256377\pi\)
\(258\) −2.52991 −0.157505
\(259\) −24.2313 −1.50566
\(260\) 0 0
\(261\) 8.63591 0.534549
\(262\) −0.993438 −0.0613748
\(263\) −23.6220 −1.45660 −0.728298 0.685260i \(-0.759688\pi\)
−0.728298 + 0.685260i \(0.759688\pi\)
\(264\) 1.58154 0.0973370
\(265\) 0 0
\(266\) −2.42230 −0.148520
\(267\) 5.57999 0.341490
\(268\) 13.6792 0.835588
\(269\) 10.3081 0.628497 0.314249 0.949341i \(-0.398247\pi\)
0.314249 + 0.949341i \(0.398247\pi\)
\(270\) 0 0
\(271\) −11.0923 −0.673808 −0.336904 0.941539i \(-0.609380\pi\)
−0.336904 + 0.941539i \(0.609380\pi\)
\(272\) 10.5617 0.640400
\(273\) −1.01571 −0.0614738
\(274\) −7.30684 −0.441422
\(275\) 0 0
\(276\) 10.0937 0.607568
\(277\) −24.0828 −1.44700 −0.723498 0.690326i \(-0.757467\pi\)
−0.723498 + 0.690326i \(0.757467\pi\)
\(278\) −10.3473 −0.620591
\(279\) 8.45044 0.505915
\(280\) 0 0
\(281\) −22.2918 −1.32982 −0.664908 0.746926i \(-0.731529\pi\)
−0.664908 + 0.746926i \(0.731529\pi\)
\(282\) −2.68567 −0.159929
\(283\) 16.0712 0.955333 0.477666 0.878541i \(-0.341483\pi\)
0.477666 + 0.878541i \(0.341483\pi\)
\(284\) 16.6933 0.990564
\(285\) 0 0
\(286\) 0.187415 0.0110821
\(287\) 2.03185 0.119936
\(288\) −13.3219 −0.784999
\(289\) 8.08109 0.475358
\(290\) 0 0
\(291\) 10.4253 0.611143
\(292\) 4.79965 0.280878
\(293\) −22.1899 −1.29635 −0.648175 0.761491i \(-0.724468\pi\)
−0.648175 + 0.761491i \(0.724468\pi\)
\(294\) 4.56440 0.266201
\(295\) 0 0
\(296\) 12.2400 0.711438
\(297\) 4.06391 0.235812
\(298\) −9.91874 −0.574577
\(299\) 2.63181 0.152201
\(300\) 0 0
\(301\) −24.5906 −1.41738
\(302\) −9.01243 −0.518607
\(303\) 0.359526 0.0206542
\(304\) −2.10893 −0.120956
\(305\) 0 0
\(306\) −7.06643 −0.403961
\(307\) 13.5034 0.770680 0.385340 0.922775i \(-0.374084\pi\)
0.385340 + 0.922775i \(0.374084\pi\)
\(308\) 6.98657 0.398097
\(309\) −10.5106 −0.597925
\(310\) 0 0
\(311\) 26.8854 1.52453 0.762265 0.647265i \(-0.224087\pi\)
0.762265 + 0.647265i \(0.224087\pi\)
\(312\) 0.513072 0.0290470
\(313\) 24.0161 1.35747 0.678735 0.734383i \(-0.262529\pi\)
0.678735 + 0.734383i \(0.262529\pi\)
\(314\) −2.99240 −0.168871
\(315\) 0 0
\(316\) −6.59520 −0.371009
\(317\) 3.97085 0.223025 0.111513 0.993763i \(-0.464430\pi\)
0.111513 + 0.993763i \(0.464430\pi\)
\(318\) 4.43429 0.248663
\(319\) 3.53579 0.197966
\(320\) 0 0
\(321\) 0.743461 0.0414959
\(322\) −19.6509 −1.09510
\(323\) −5.00810 −0.278658
\(324\) −7.15278 −0.397377
\(325\) 0 0
\(326\) 1.54148 0.0853748
\(327\) 4.00383 0.221413
\(328\) −1.02636 −0.0566711
\(329\) −26.1046 −1.43919
\(330\) 0 0
\(331\) 10.2915 0.565672 0.282836 0.959168i \(-0.408725\pi\)
0.282836 + 0.959168i \(0.408725\pi\)
\(332\) 17.1352 0.940418
\(333\) 14.1148 0.773488
\(334\) −7.35745 −0.402582
\(335\) 0 0
\(336\) 6.60291 0.360218
\(337\) −14.8946 −0.811358 −0.405679 0.914016i \(-0.632965\pi\)
−0.405679 + 0.914016i \(0.632965\pi\)
\(338\) −7.44935 −0.405191
\(339\) −1.91925 −0.104240
\(340\) 0 0
\(341\) 3.45986 0.187362
\(342\) 1.41100 0.0762981
\(343\) 15.0150 0.810734
\(344\) 12.4216 0.669725
\(345\) 0 0
\(346\) 2.02883 0.109071
\(347\) 30.9460 1.66127 0.830635 0.556817i \(-0.187978\pi\)
0.830635 + 0.556817i \(0.187978\pi\)
\(348\) 4.39927 0.235826
\(349\) −4.85236 −0.259741 −0.129870 0.991531i \(-0.541456\pi\)
−0.129870 + 0.991531i \(0.541456\pi\)
\(350\) 0 0
\(351\) 1.31839 0.0703702
\(352\) −5.45436 −0.290719
\(353\) −35.5761 −1.89353 −0.946764 0.321930i \(-0.895669\pi\)
−0.946764 + 0.321930i \(0.895669\pi\)
\(354\) 3.09254 0.164367
\(355\) 0 0
\(356\) −12.4516 −0.659932
\(357\) 15.6800 0.829874
\(358\) −0.0748051 −0.00395357
\(359\) 18.1615 0.958525 0.479263 0.877672i \(-0.340904\pi\)
0.479263 + 0.877672i \(0.340904\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −0.831127 −0.0436831
\(363\) 0.746709 0.0391921
\(364\) 2.26654 0.118799
\(365\) 0 0
\(366\) 2.30247 0.120352
\(367\) 12.6666 0.661189 0.330595 0.943773i \(-0.392751\pi\)
0.330595 + 0.943773i \(0.392751\pi\)
\(368\) −17.1087 −0.891855
\(369\) −1.18356 −0.0616138
\(370\) 0 0
\(371\) 43.1012 2.23770
\(372\) 4.30479 0.223193
\(373\) −27.6947 −1.43398 −0.716988 0.697086i \(-0.754480\pi\)
−0.716988 + 0.697086i \(0.754480\pi\)
\(374\) −2.89320 −0.149604
\(375\) 0 0
\(376\) 13.1863 0.680033
\(377\) 1.14706 0.0590765
\(378\) −9.84399 −0.506320
\(379\) −5.84097 −0.300031 −0.150015 0.988684i \(-0.547932\pi\)
−0.150015 + 0.988684i \(0.547932\pi\)
\(380\) 0 0
\(381\) −14.8768 −0.762161
\(382\) 5.68329 0.290782
\(383\) −23.4576 −1.19863 −0.599313 0.800515i \(-0.704559\pi\)
−0.599313 + 0.800515i \(0.704559\pi\)
\(384\) −8.60586 −0.439166
\(385\) 0 0
\(386\) 12.1677 0.619321
\(387\) 14.3241 0.728137
\(388\) −23.2638 −1.18104
\(389\) −5.61130 −0.284504 −0.142252 0.989830i \(-0.545434\pi\)
−0.142252 + 0.989830i \(0.545434\pi\)
\(390\) 0 0
\(391\) −40.6283 −2.05466
\(392\) −22.4107 −1.13191
\(393\) −1.28406 −0.0647725
\(394\) 9.09509 0.458204
\(395\) 0 0
\(396\) −4.06971 −0.204511
\(397\) −2.81880 −0.141472 −0.0707358 0.997495i \(-0.522535\pi\)
−0.0707358 + 0.997495i \(0.522535\pi\)
\(398\) −5.81330 −0.291394
\(399\) −3.13093 −0.156742
\(400\) 0 0
\(401\) 32.8017 1.63804 0.819019 0.573767i \(-0.194518\pi\)
0.819019 + 0.573767i \(0.194518\pi\)
\(402\) −3.54139 −0.176629
\(403\) 1.12242 0.0559119
\(404\) −0.802271 −0.0399145
\(405\) 0 0
\(406\) −8.56474 −0.425061
\(407\) 5.77902 0.286456
\(408\) −7.92051 −0.392124
\(409\) −20.0663 −0.992214 −0.496107 0.868262i \(-0.665238\pi\)
−0.496107 + 0.868262i \(0.665238\pi\)
\(410\) 0 0
\(411\) −9.44442 −0.465859
\(412\) 23.4540 1.15550
\(413\) 30.0594 1.47913
\(414\) 11.4468 0.562577
\(415\) 0 0
\(416\) −1.76947 −0.0867553
\(417\) −13.3744 −0.654947
\(418\) 0.577704 0.0282564
\(419\) −7.51827 −0.367292 −0.183646 0.982992i \(-0.558790\pi\)
−0.183646 + 0.982992i \(0.558790\pi\)
\(420\) 0 0
\(421\) 6.89149 0.335870 0.167935 0.985798i \(-0.446290\pi\)
0.167935 + 0.985798i \(0.446290\pi\)
\(422\) −6.39364 −0.311238
\(423\) 15.2061 0.739344
\(424\) −21.7719 −1.05734
\(425\) 0 0
\(426\) −4.32172 −0.209388
\(427\) 22.3800 1.08304
\(428\) −1.65901 −0.0801912
\(429\) 0.242242 0.0116956
\(430\) 0 0
\(431\) 13.0438 0.628297 0.314149 0.949374i \(-0.398281\pi\)
0.314149 + 0.949374i \(0.398281\pi\)
\(432\) −8.57050 −0.412349
\(433\) −9.38251 −0.450895 −0.225447 0.974255i \(-0.572384\pi\)
−0.225447 + 0.974255i \(0.572384\pi\)
\(434\) −8.38080 −0.402291
\(435\) 0 0
\(436\) −8.93443 −0.427882
\(437\) 8.11252 0.388074
\(438\) −1.24258 −0.0593727
\(439\) −34.4559 −1.64449 −0.822244 0.569135i \(-0.807278\pi\)
−0.822244 + 0.569135i \(0.807278\pi\)
\(440\) 0 0
\(441\) −25.8433 −1.23063
\(442\) −0.938593 −0.0446443
\(443\) −15.6033 −0.741336 −0.370668 0.928765i \(-0.620871\pi\)
−0.370668 + 0.928765i \(0.620871\pi\)
\(444\) 7.19032 0.341237
\(445\) 0 0
\(446\) −13.6084 −0.644376
\(447\) −12.8204 −0.606385
\(448\) −4.47329 −0.211343
\(449\) 40.4269 1.90786 0.953932 0.300022i \(-0.0969940\pi\)
0.953932 + 0.300022i \(0.0969940\pi\)
\(450\) 0 0
\(451\) −0.484585 −0.0228182
\(452\) 4.28275 0.201444
\(453\) −11.6490 −0.547317
\(454\) −5.45271 −0.255908
\(455\) 0 0
\(456\) 1.58154 0.0740624
\(457\) −13.7522 −0.643301 −0.321650 0.946858i \(-0.604238\pi\)
−0.321650 + 0.946858i \(0.604238\pi\)
\(458\) −9.86660 −0.461036
\(459\) −20.3525 −0.949972
\(460\) 0 0
\(461\) 26.0534 1.21343 0.606715 0.794919i \(-0.292487\pi\)
0.606715 + 0.794919i \(0.292487\pi\)
\(462\) −1.80875 −0.0841507
\(463\) 5.34837 0.248560 0.124280 0.992247i \(-0.460338\pi\)
0.124280 + 0.992247i \(0.460338\pi\)
\(464\) −7.45674 −0.346171
\(465\) 0 0
\(466\) 1.45853 0.0675649
\(467\) −8.54901 −0.395601 −0.197800 0.980242i \(-0.563380\pi\)
−0.197800 + 0.980242i \(0.563380\pi\)
\(468\) −1.32027 −0.0610294
\(469\) −34.4222 −1.58947
\(470\) 0 0
\(471\) −3.86781 −0.178219
\(472\) −15.1840 −0.698902
\(473\) 5.86472 0.269660
\(474\) 1.70743 0.0784248
\(475\) 0 0
\(476\) −34.9895 −1.60374
\(477\) −25.1066 −1.14955
\(478\) 9.49464 0.434275
\(479\) −7.72446 −0.352939 −0.176470 0.984306i \(-0.556468\pi\)
−0.176470 + 0.984306i \(0.556468\pi\)
\(480\) 0 0
\(481\) 1.87479 0.0854831
\(482\) 13.2228 0.602281
\(483\) −25.3997 −1.15573
\(484\) −1.66626 −0.0757390
\(485\) 0 0
\(486\) 8.89499 0.403485
\(487\) −31.5919 −1.43157 −0.715784 0.698322i \(-0.753930\pi\)
−0.715784 + 0.698322i \(0.753930\pi\)
\(488\) −11.3049 −0.511748
\(489\) 1.99244 0.0901011
\(490\) 0 0
\(491\) −22.8113 −1.02946 −0.514729 0.857353i \(-0.672108\pi\)
−0.514729 + 0.857353i \(0.672108\pi\)
\(492\) −0.602925 −0.0271820
\(493\) −17.7076 −0.797511
\(494\) 0.187415 0.00843219
\(495\) 0 0
\(496\) −7.29660 −0.327627
\(497\) −42.0069 −1.88427
\(498\) −4.43613 −0.198788
\(499\) 41.7098 1.86719 0.933593 0.358336i \(-0.116656\pi\)
0.933593 + 0.358336i \(0.116656\pi\)
\(500\) 0 0
\(501\) −9.50984 −0.424868
\(502\) 7.96498 0.355494
\(503\) 5.43339 0.242263 0.121131 0.992636i \(-0.461348\pi\)
0.121131 + 0.992636i \(0.461348\pi\)
\(504\) 21.6906 0.966176
\(505\) 0 0
\(506\) 4.68664 0.208346
\(507\) −9.62863 −0.427623
\(508\) 33.1971 1.47288
\(509\) −20.8742 −0.925233 −0.462616 0.886559i \(-0.653089\pi\)
−0.462616 + 0.886559i \(0.653089\pi\)
\(510\) 0 0
\(511\) −12.0778 −0.534291
\(512\) 20.4364 0.903168
\(513\) 4.06391 0.179426
\(514\) 12.8325 0.566015
\(515\) 0 0
\(516\) 7.29694 0.321230
\(517\) 6.22580 0.273810
\(518\) −13.9985 −0.615059
\(519\) 2.62236 0.115109
\(520\) 0 0
\(521\) 40.4540 1.77232 0.886161 0.463377i \(-0.153362\pi\)
0.886161 + 0.463377i \(0.153362\pi\)
\(522\) 4.98900 0.218363
\(523\) −13.3864 −0.585345 −0.292672 0.956213i \(-0.594545\pi\)
−0.292672 + 0.956213i \(0.594545\pi\)
\(524\) 2.86535 0.125173
\(525\) 0 0
\(526\) −13.6465 −0.595017
\(527\) −17.3273 −0.754790
\(528\) −1.57476 −0.0685326
\(529\) 42.8130 1.86143
\(530\) 0 0
\(531\) −17.5098 −0.759859
\(532\) 6.98657 0.302906
\(533\) −0.157206 −0.00680934
\(534\) 3.22358 0.139498
\(535\) 0 0
\(536\) 17.3878 0.751041
\(537\) −0.0966890 −0.00417244
\(538\) 5.95505 0.256740
\(539\) −10.5810 −0.455756
\(540\) 0 0
\(541\) 33.5953 1.44437 0.722186 0.691699i \(-0.243137\pi\)
0.722186 + 0.691699i \(0.243137\pi\)
\(542\) −6.40805 −0.275249
\(543\) −1.07427 −0.0461014
\(544\) 27.3160 1.17116
\(545\) 0 0
\(546\) −0.586782 −0.0251120
\(547\) −16.5276 −0.706671 −0.353335 0.935497i \(-0.614953\pi\)
−0.353335 + 0.935497i \(0.614953\pi\)
\(548\) 21.0749 0.900277
\(549\) −13.0364 −0.556381
\(550\) 0 0
\(551\) 3.53579 0.150630
\(552\) 12.8303 0.546092
\(553\) 16.5961 0.705740
\(554\) −13.9127 −0.591096
\(555\) 0 0
\(556\) 29.8445 1.26569
\(557\) 39.5660 1.67647 0.838234 0.545311i \(-0.183589\pi\)
0.838234 + 0.545311i \(0.183589\pi\)
\(558\) 4.88186 0.206665
\(559\) 1.90259 0.0804711
\(560\) 0 0
\(561\) −3.73960 −0.157886
\(562\) −12.8780 −0.543228
\(563\) −27.8205 −1.17249 −0.586246 0.810133i \(-0.699395\pi\)
−0.586246 + 0.810133i \(0.699395\pi\)
\(564\) 7.74620 0.326174
\(565\) 0 0
\(566\) 9.28439 0.390252
\(567\) 17.9992 0.755897
\(568\) 21.2191 0.890335
\(569\) 25.3277 1.06179 0.530897 0.847436i \(-0.321855\pi\)
0.530897 + 0.847436i \(0.321855\pi\)
\(570\) 0 0
\(571\) 27.5290 1.15205 0.576027 0.817431i \(-0.304602\pi\)
0.576027 + 0.817431i \(0.304602\pi\)
\(572\) −0.540556 −0.0226018
\(573\) 7.34591 0.306880
\(574\) 1.17381 0.0489938
\(575\) 0 0
\(576\) 2.60571 0.108571
\(577\) −28.5982 −1.19056 −0.595280 0.803519i \(-0.702959\pi\)
−0.595280 + 0.803519i \(0.702959\pi\)
\(578\) 4.66848 0.194183
\(579\) 15.7273 0.653606
\(580\) 0 0
\(581\) −43.1191 −1.78888
\(582\) 6.02275 0.249651
\(583\) −10.2794 −0.425729
\(584\) 6.10092 0.252458
\(585\) 0 0
\(586\) −12.8192 −0.529557
\(587\) −27.7094 −1.14369 −0.571844 0.820362i \(-0.693772\pi\)
−0.571844 + 0.820362i \(0.693772\pi\)
\(588\) −13.1650 −0.542914
\(589\) 3.45986 0.142561
\(590\) 0 0
\(591\) 11.7558 0.483570
\(592\) −12.1876 −0.500906
\(593\) 15.3303 0.629539 0.314769 0.949168i \(-0.398073\pi\)
0.314769 + 0.949168i \(0.398073\pi\)
\(594\) 2.34774 0.0963288
\(595\) 0 0
\(596\) 28.6084 1.17184
\(597\) −7.51396 −0.307526
\(598\) 1.52041 0.0621740
\(599\) −6.66497 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(600\) 0 0
\(601\) 2.25703 0.0920661 0.0460331 0.998940i \(-0.485342\pi\)
0.0460331 + 0.998940i \(0.485342\pi\)
\(602\) −14.2061 −0.578997
\(603\) 20.0511 0.816545
\(604\) 25.9943 1.05769
\(605\) 0 0
\(606\) 0.207700 0.00843722
\(607\) −1.76278 −0.0715491 −0.0357745 0.999360i \(-0.511390\pi\)
−0.0357745 + 0.999360i \(0.511390\pi\)
\(608\) −5.45436 −0.221204
\(609\) −11.0703 −0.448592
\(610\) 0 0
\(611\) 2.01973 0.0817096
\(612\) 20.3815 0.823875
\(613\) 35.6831 1.44123 0.720614 0.693337i \(-0.243860\pi\)
0.720614 + 0.693337i \(0.243860\pi\)
\(614\) 7.80097 0.314822
\(615\) 0 0
\(616\) 8.88076 0.357816
\(617\) −3.86204 −0.155480 −0.0777400 0.996974i \(-0.524770\pi\)
−0.0777400 + 0.996974i \(0.524770\pi\)
\(618\) −6.07200 −0.244252
\(619\) 40.3816 1.62308 0.811538 0.584300i \(-0.198631\pi\)
0.811538 + 0.584300i \(0.198631\pi\)
\(620\) 0 0
\(621\) 32.9685 1.32298
\(622\) 15.5318 0.622768
\(623\) 31.3331 1.25533
\(624\) −0.510872 −0.0204513
\(625\) 0 0
\(626\) 13.8742 0.554524
\(627\) 0.746709 0.0298207
\(628\) 8.63089 0.344410
\(629\) −28.9419 −1.15399
\(630\) 0 0
\(631\) −31.9100 −1.27032 −0.635159 0.772382i \(-0.719065\pi\)
−0.635159 + 0.772382i \(0.719065\pi\)
\(632\) −8.38328 −0.333469
\(633\) −8.26407 −0.328467
\(634\) 2.29398 0.0911054
\(635\) 0 0
\(636\) −12.7897 −0.507145
\(637\) −3.43261 −0.136005
\(638\) 2.04264 0.0808690
\(639\) 24.4692 0.967988
\(640\) 0 0
\(641\) −42.9789 −1.69757 −0.848783 0.528741i \(-0.822664\pi\)
−0.848783 + 0.528741i \(0.822664\pi\)
\(642\) 0.429500 0.0169510
\(643\) −5.15134 −0.203149 −0.101574 0.994828i \(-0.532388\pi\)
−0.101574 + 0.994828i \(0.532388\pi\)
\(644\) 56.6787 2.23345
\(645\) 0 0
\(646\) −2.89320 −0.113831
\(647\) 11.7329 0.461268 0.230634 0.973041i \(-0.425920\pi\)
0.230634 + 0.973041i \(0.425920\pi\)
\(648\) −9.09203 −0.357169
\(649\) −7.16900 −0.281408
\(650\) 0 0
\(651\) −10.8326 −0.424562
\(652\) −4.44606 −0.174121
\(653\) −24.3265 −0.951971 −0.475986 0.879453i \(-0.657909\pi\)
−0.475986 + 0.879453i \(0.657909\pi\)
\(654\) 2.31303 0.0904467
\(655\) 0 0
\(656\) 1.02196 0.0399007
\(657\) 7.03539 0.274477
\(658\) −15.0807 −0.587908
\(659\) 24.4562 0.952680 0.476340 0.879261i \(-0.341963\pi\)
0.476340 + 0.879261i \(0.341963\pi\)
\(660\) 0 0
\(661\) 6.67388 0.259584 0.129792 0.991541i \(-0.458569\pi\)
0.129792 + 0.991541i \(0.458569\pi\)
\(662\) 5.94544 0.231076
\(663\) −1.21317 −0.0471158
\(664\) 21.7809 0.845263
\(665\) 0 0
\(666\) 8.15420 0.315969
\(667\) 28.6842 1.11066
\(668\) 21.2209 0.821062
\(669\) −17.5895 −0.680048
\(670\) 0 0
\(671\) −5.33749 −0.206052
\(672\) 17.0772 0.658768
\(673\) −2.22527 −0.0857778 −0.0428889 0.999080i \(-0.513656\pi\)
−0.0428889 + 0.999080i \(0.513656\pi\)
\(674\) −8.60465 −0.331439
\(675\) 0 0
\(676\) 21.4860 0.826384
\(677\) −22.6964 −0.872292 −0.436146 0.899876i \(-0.643657\pi\)
−0.436146 + 0.899876i \(0.643657\pi\)
\(678\) −1.10876 −0.0425817
\(679\) 58.5409 2.24659
\(680\) 0 0
\(681\) −7.04787 −0.270075
\(682\) 1.99877 0.0765370
\(683\) 39.2539 1.50201 0.751005 0.660296i \(-0.229569\pi\)
0.751005 + 0.660296i \(0.229569\pi\)
\(684\) −4.06971 −0.155609
\(685\) 0 0
\(686\) 8.67423 0.331183
\(687\) −12.7530 −0.486558
\(688\) −12.3683 −0.471537
\(689\) −3.33477 −0.127045
\(690\) 0 0
\(691\) 28.4393 1.08188 0.540941 0.841060i \(-0.318068\pi\)
0.540941 + 0.841060i \(0.318068\pi\)
\(692\) −5.85171 −0.222449
\(693\) 10.2410 0.389024
\(694\) 17.8777 0.678627
\(695\) 0 0
\(696\) 5.59199 0.211964
\(697\) 2.42685 0.0919235
\(698\) −2.80323 −0.106104
\(699\) 1.88521 0.0713053
\(700\) 0 0
\(701\) 13.1038 0.494925 0.247463 0.968897i \(-0.420403\pi\)
0.247463 + 0.968897i \(0.420403\pi\)
\(702\) 0.761637 0.0287461
\(703\) 5.77902 0.217960
\(704\) 1.06685 0.0402086
\(705\) 0 0
\(706\) −20.5525 −0.773503
\(707\) 2.01883 0.0759260
\(708\) −8.91975 −0.335225
\(709\) 18.3440 0.688924 0.344462 0.938800i \(-0.388061\pi\)
0.344462 + 0.938800i \(0.388061\pi\)
\(710\) 0 0
\(711\) −9.66734 −0.362553
\(712\) −15.8274 −0.593158
\(713\) 28.0682 1.05116
\(714\) 9.05841 0.339002
\(715\) 0 0
\(716\) 0.215759 0.00806327
\(717\) 12.2723 0.458316
\(718\) 10.4920 0.391556
\(719\) −1.88586 −0.0703309 −0.0351654 0.999382i \(-0.511196\pi\)
−0.0351654 + 0.999382i \(0.511196\pi\)
\(720\) 0 0
\(721\) −59.0196 −2.19801
\(722\) 0.577704 0.0214999
\(723\) 17.0910 0.635623
\(724\) 2.39720 0.0890913
\(725\) 0 0
\(726\) 0.431377 0.0160099
\(727\) 16.9917 0.630185 0.315093 0.949061i \(-0.397964\pi\)
0.315093 + 0.949061i \(0.397964\pi\)
\(728\) 2.88104 0.106778
\(729\) −1.38098 −0.0511473
\(730\) 0 0
\(731\) −29.3711 −1.08633
\(732\) −6.64096 −0.245457
\(733\) 17.8984 0.661091 0.330545 0.943790i \(-0.392767\pi\)
0.330545 + 0.943790i \(0.392767\pi\)
\(734\) 7.31753 0.270095
\(735\) 0 0
\(736\) −44.2486 −1.63103
\(737\) 8.20951 0.302401
\(738\) −0.683749 −0.0251691
\(739\) −51.7970 −1.90538 −0.952691 0.303940i \(-0.901698\pi\)
−0.952691 + 0.303940i \(0.901698\pi\)
\(740\) 0 0
\(741\) 0.242242 0.00889899
\(742\) 24.8997 0.914097
\(743\) 38.8105 1.42382 0.711909 0.702272i \(-0.247831\pi\)
0.711909 + 0.702272i \(0.247831\pi\)
\(744\) 5.47190 0.200610
\(745\) 0 0
\(746\) −15.9993 −0.585777
\(747\) 25.1171 0.918986
\(748\) 8.34479 0.305116
\(749\) 4.17473 0.152541
\(750\) 0 0
\(751\) 16.2035 0.591276 0.295638 0.955300i \(-0.404468\pi\)
0.295638 + 0.955300i \(0.404468\pi\)
\(752\) −13.1298 −0.478794
\(753\) 10.2951 0.375174
\(754\) 0.662660 0.0241327
\(755\) 0 0
\(756\) 28.3928 1.03264
\(757\) −2.83100 −0.102894 −0.0514471 0.998676i \(-0.516383\pi\)
−0.0514471 + 0.998676i \(0.516383\pi\)
\(758\) −3.37435 −0.122562
\(759\) 6.05769 0.219880
\(760\) 0 0
\(761\) −17.0317 −0.617399 −0.308700 0.951160i \(-0.599894\pi\)
−0.308700 + 0.951160i \(0.599894\pi\)
\(762\) −8.59439 −0.311342
\(763\) 22.4826 0.813925
\(764\) −16.3922 −0.593048
\(765\) 0 0
\(766\) −13.5515 −0.489637
\(767\) −2.32572 −0.0839769
\(768\) −3.37838 −0.121907
\(769\) −24.5256 −0.884415 −0.442208 0.896913i \(-0.645805\pi\)
−0.442208 + 0.896913i \(0.645805\pi\)
\(770\) 0 0
\(771\) 16.5865 0.597349
\(772\) −35.0951 −1.26310
\(773\) −2.29735 −0.0826299 −0.0413150 0.999146i \(-0.513155\pi\)
−0.0413150 + 0.999146i \(0.513155\pi\)
\(774\) 8.27512 0.297443
\(775\) 0 0
\(776\) −29.5710 −1.06154
\(777\) −18.0937 −0.649108
\(778\) −3.24167 −0.116220
\(779\) −0.484585 −0.0173621
\(780\) 0 0
\(781\) 10.0184 0.358487
\(782\) −23.4712 −0.839327
\(783\) 14.3691 0.513511
\(784\) 22.3146 0.796950
\(785\) 0 0
\(786\) −0.741809 −0.0264595
\(787\) 5.33806 0.190281 0.0951406 0.995464i \(-0.469670\pi\)
0.0951406 + 0.995464i \(0.469670\pi\)
\(788\) −26.2328 −0.934503
\(789\) −17.6388 −0.627957
\(790\) 0 0
\(791\) −10.7771 −0.383190
\(792\) −5.17309 −0.183818
\(793\) −1.73155 −0.0614893
\(794\) −1.62843 −0.0577909
\(795\) 0 0
\(796\) 16.7672 0.594296
\(797\) 51.5514 1.82604 0.913022 0.407911i \(-0.133743\pi\)
0.913022 + 0.407911i \(0.133743\pi\)
\(798\) −1.80875 −0.0640291
\(799\) −31.1794 −1.10305
\(800\) 0 0
\(801\) −18.2517 −0.644892
\(802\) 18.9497 0.669136
\(803\) 2.88049 0.101650
\(804\) 10.2144 0.360233
\(805\) 0 0
\(806\) 0.648429 0.0228399
\(807\) 7.69717 0.270953
\(808\) −1.01978 −0.0358758
\(809\) 20.4992 0.720712 0.360356 0.932815i \(-0.382655\pi\)
0.360356 + 0.932815i \(0.382655\pi\)
\(810\) 0 0
\(811\) 39.0473 1.37114 0.685568 0.728009i \(-0.259554\pi\)
0.685568 + 0.728009i \(0.259554\pi\)
\(812\) 24.7031 0.866907
\(813\) −8.28270 −0.290487
\(814\) 3.33857 0.117017
\(815\) 0 0
\(816\) 7.88655 0.276084
\(817\) 5.86472 0.205181
\(818\) −11.5924 −0.405318
\(819\) 3.32232 0.116091
\(820\) 0 0
\(821\) 6.52431 0.227700 0.113850 0.993498i \(-0.463682\pi\)
0.113850 + 0.993498i \(0.463682\pi\)
\(822\) −5.45608 −0.190303
\(823\) 13.0628 0.455339 0.227670 0.973738i \(-0.426889\pi\)
0.227670 + 0.973738i \(0.426889\pi\)
\(824\) 29.8128 1.03858
\(825\) 0 0
\(826\) 17.3655 0.604221
\(827\) −22.8198 −0.793522 −0.396761 0.917922i \(-0.629866\pi\)
−0.396761 + 0.917922i \(0.629866\pi\)
\(828\) −33.0156 −1.14737
\(829\) −43.7012 −1.51780 −0.758902 0.651205i \(-0.774264\pi\)
−0.758902 + 0.651205i \(0.774264\pi\)
\(830\) 0 0
\(831\) −17.9829 −0.623819
\(832\) 0.346102 0.0119989
\(833\) 52.9907 1.83602
\(834\) −7.72644 −0.267545
\(835\) 0 0
\(836\) −1.66626 −0.0576287
\(837\) 14.0605 0.486003
\(838\) −4.34334 −0.150038
\(839\) −12.2493 −0.422892 −0.211446 0.977390i \(-0.567817\pi\)
−0.211446 + 0.977390i \(0.567817\pi\)
\(840\) 0 0
\(841\) −16.4982 −0.568902
\(842\) 3.98124 0.137203
\(843\) −16.6455 −0.573300
\(844\) 18.4410 0.634766
\(845\) 0 0
\(846\) 8.78460 0.302021
\(847\) 4.19297 0.144072
\(848\) 21.6785 0.744444
\(849\) 12.0005 0.411856
\(850\) 0 0
\(851\) 46.8824 1.60711
\(852\) 12.4650 0.427044
\(853\) −2.70591 −0.0926487 −0.0463243 0.998926i \(-0.514751\pi\)
−0.0463243 + 0.998926i \(0.514751\pi\)
\(854\) 12.9290 0.442421
\(855\) 0 0
\(856\) −2.10880 −0.0720773
\(857\) −6.16547 −0.210608 −0.105304 0.994440i \(-0.533582\pi\)
−0.105304 + 0.994440i \(0.533582\pi\)
\(858\) 0.139944 0.00477762
\(859\) 48.5324 1.65590 0.827952 0.560800i \(-0.189506\pi\)
0.827952 + 0.560800i \(0.189506\pi\)
\(860\) 0 0
\(861\) 1.51720 0.0517061
\(862\) 7.53545 0.256659
\(863\) 41.0881 1.39866 0.699328 0.714801i \(-0.253483\pi\)
0.699328 + 0.714801i \(0.253483\pi\)
\(864\) −22.1660 −0.754104
\(865\) 0 0
\(866\) −5.42032 −0.184190
\(867\) 6.03422 0.204933
\(868\) 24.1725 0.820469
\(869\) −3.95809 −0.134269
\(870\) 0 0
\(871\) 2.66327 0.0902416
\(872\) −11.3567 −0.384587
\(873\) −34.1004 −1.15412
\(874\) 4.68664 0.158528
\(875\) 0 0
\(876\) 3.58394 0.121090
\(877\) 6.98501 0.235867 0.117934 0.993021i \(-0.462373\pi\)
0.117934 + 0.993021i \(0.462373\pi\)
\(878\) −19.9053 −0.671771
\(879\) −16.5694 −0.558873
\(880\) 0 0
\(881\) 34.5166 1.16289 0.581447 0.813584i \(-0.302487\pi\)
0.581447 + 0.813584i \(0.302487\pi\)
\(882\) −14.9298 −0.502712
\(883\) 42.9328 1.44480 0.722401 0.691474i \(-0.243038\pi\)
0.722401 + 0.691474i \(0.243038\pi\)
\(884\) 2.70716 0.0910517
\(885\) 0 0
\(886\) −9.01410 −0.302835
\(887\) 31.2234 1.04838 0.524189 0.851602i \(-0.324369\pi\)
0.524189 + 0.851602i \(0.324369\pi\)
\(888\) 9.13975 0.306710
\(889\) −83.5371 −2.80175
\(890\) 0 0
\(891\) −4.29272 −0.143812
\(892\) 39.2503 1.31420
\(893\) 6.22580 0.208338
\(894\) −7.40641 −0.247707
\(895\) 0 0
\(896\) −48.3242 −1.61440
\(897\) 1.96519 0.0656159
\(898\) 23.3548 0.779360
\(899\) 12.2333 0.408005
\(900\) 0 0
\(901\) 51.4802 1.71506
\(902\) −0.279947 −0.00932121
\(903\) −18.3620 −0.611050
\(904\) 5.44389 0.181061
\(905\) 0 0
\(906\) −6.72966 −0.223578
\(907\) −31.2576 −1.03789 −0.518946 0.854807i \(-0.673676\pi\)
−0.518946 + 0.854807i \(0.673676\pi\)
\(908\) 15.7271 0.521923
\(909\) −1.17598 −0.0390048
\(910\) 0 0
\(911\) 48.9833 1.62289 0.811445 0.584429i \(-0.198681\pi\)
0.811445 + 0.584429i \(0.198681\pi\)
\(912\) −1.57476 −0.0521455
\(913\) 10.2837 0.340339
\(914\) −7.94470 −0.262787
\(915\) 0 0
\(916\) 28.4580 0.940278
\(917\) −7.21036 −0.238107
\(918\) −11.7577 −0.388062
\(919\) −20.9524 −0.691157 −0.345578 0.938390i \(-0.612317\pi\)
−0.345578 + 0.938390i \(0.612317\pi\)
\(920\) 0 0
\(921\) 10.0831 0.332250
\(922\) 15.0512 0.495684
\(923\) 3.25011 0.106979
\(924\) 5.21693 0.171625
\(925\) 0 0
\(926\) 3.08977 0.101536
\(927\) 34.3792 1.12916
\(928\) −19.2855 −0.633077
\(929\) 56.0533 1.83905 0.919525 0.393031i \(-0.128574\pi\)
0.919525 + 0.393031i \(0.128574\pi\)
\(930\) 0 0
\(931\) −10.5810 −0.346778
\(932\) −4.20679 −0.137798
\(933\) 20.0756 0.657244
\(934\) −4.93880 −0.161602
\(935\) 0 0
\(936\) −1.67822 −0.0548543
\(937\) −7.53790 −0.246253 −0.123126 0.992391i \(-0.539292\pi\)
−0.123126 + 0.992391i \(0.539292\pi\)
\(938\) −19.8859 −0.649297
\(939\) 17.9330 0.585222
\(940\) 0 0
\(941\) 38.3962 1.25168 0.625840 0.779951i \(-0.284756\pi\)
0.625840 + 0.779951i \(0.284756\pi\)
\(942\) −2.23445 −0.0728023
\(943\) −3.93120 −0.128018
\(944\) 15.1189 0.492080
\(945\) 0 0
\(946\) 3.38807 0.110156
\(947\) 10.9350 0.355340 0.177670 0.984090i \(-0.443144\pi\)
0.177670 + 0.984090i \(0.443144\pi\)
\(948\) −4.92469 −0.159947
\(949\) 0.934470 0.0303342
\(950\) 0 0
\(951\) 2.96507 0.0961490
\(952\) −44.4758 −1.44147
\(953\) −45.9324 −1.48790 −0.743948 0.668238i \(-0.767049\pi\)
−0.743948 + 0.668238i \(0.767049\pi\)
\(954\) −14.5042 −0.469591
\(955\) 0 0
\(956\) −27.3852 −0.885700
\(957\) 2.64021 0.0853458
\(958\) −4.46245 −0.144175
\(959\) −53.0329 −1.71252
\(960\) 0 0
\(961\) −19.0294 −0.613851
\(962\) 1.08307 0.0349197
\(963\) −2.43180 −0.0783637
\(964\) −38.1381 −1.22835
\(965\) 0 0
\(966\) −14.6735 −0.472113
\(967\) 36.8982 1.18657 0.593283 0.804994i \(-0.297832\pi\)
0.593283 + 0.804994i \(0.297832\pi\)
\(968\) −2.11801 −0.0680755
\(969\) −3.73960 −0.120133
\(970\) 0 0
\(971\) −49.1483 −1.57724 −0.788622 0.614878i \(-0.789205\pi\)
−0.788622 + 0.614878i \(0.789205\pi\)
\(972\) −25.6556 −0.822904
\(973\) −75.1008 −2.40762
\(974\) −18.2508 −0.584793
\(975\) 0 0
\(976\) 11.2564 0.360309
\(977\) −27.8361 −0.890557 −0.445279 0.895392i \(-0.646895\pi\)
−0.445279 + 0.895392i \(0.646895\pi\)
\(978\) 1.15104 0.0368062
\(979\) −7.47277 −0.238831
\(980\) 0 0
\(981\) −13.0962 −0.418130
\(982\) −13.1782 −0.420532
\(983\) 12.7794 0.407598 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(984\) −0.766390 −0.0244316
\(985\) 0 0
\(986\) −10.2298 −0.325782
\(987\) −19.4925 −0.620454
\(988\) −0.540556 −0.0171974
\(989\) 47.5777 1.51288
\(990\) 0 0
\(991\) 62.7704 1.99397 0.996984 0.0776029i \(-0.0247266\pi\)
0.996984 + 0.0776029i \(0.0247266\pi\)
\(992\) −18.8713 −0.599165
\(993\) 7.68475 0.243868
\(994\) −24.2676 −0.769721
\(995\) 0 0
\(996\) 12.7950 0.405426
\(997\) −13.0365 −0.412871 −0.206435 0.978460i \(-0.566186\pi\)
−0.206435 + 0.978460i \(0.566186\pi\)
\(998\) 24.0959 0.762743
\(999\) 23.4854 0.743046
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 5225.2.a.k.1.4 6
5.4 even 2 1045.2.a.g.1.3 6
15.14 odd 2 9405.2.a.w.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.3 6 5.4 even 2
5225.2.a.k.1.4 6 1.1 even 1 trivial
9405.2.a.w.1.4 6 15.14 odd 2