Properties

Label 2004.2.a.c.1.4
Level $2004$
Weight $2$
Character 2004.1
Self dual yes
Analytic conductor $16.002$
Analytic rank $0$
Dimension $9$
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] = [2004,2,Mod(1,2004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2004, 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("2004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.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: \(16.0020205651\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.419033\) of defining polynomial
Character \(\chi\) \(=\) 2004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.419033 q^{5} -4.23221 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.419033 q^{5} -4.23221 q^{7} +1.00000 q^{9} -2.07280 q^{11} -5.80837 q^{13} +0.419033 q^{15} -0.0751145 q^{17} -3.35395 q^{19} +4.23221 q^{21} +4.32486 q^{23} -4.82441 q^{25} -1.00000 q^{27} +8.99097 q^{29} +6.91570 q^{31} +2.07280 q^{33} +1.77344 q^{35} +6.31045 q^{37} +5.80837 q^{39} +4.28666 q^{41} -1.06448 q^{43} -0.419033 q^{45} -9.61226 q^{47} +10.9116 q^{49} +0.0751145 q^{51} -14.0744 q^{53} +0.868572 q^{55} +3.35395 q^{57} +3.20748 q^{59} +11.7562 q^{61} -4.23221 q^{63} +2.43390 q^{65} -5.13807 q^{67} -4.32486 q^{69} -0.458914 q^{71} +11.5309 q^{73} +4.82441 q^{75} +8.77254 q^{77} -4.93027 q^{79} +1.00000 q^{81} -9.02732 q^{83} +0.0314754 q^{85} -8.99097 q^{87} +13.3953 q^{89} +24.5823 q^{91} -6.91570 q^{93} +1.40541 q^{95} +10.7908 q^{97} -2.07280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9} - 9 q^{11} + 10 q^{13} - q^{15} + 7 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 18 q^{25} - 9 q^{27} + 5 q^{29} + 12 q^{31} + 9 q^{33} - 6 q^{35} + 15 q^{37} - 10 q^{39} + 14 q^{41} + 6 q^{43} + q^{45} - 3 q^{47} + 27 q^{49} - 7 q^{51} + 9 q^{53} + 19 q^{55} + 2 q^{57} - 9 q^{59} + 30 q^{61} + 2 q^{63} + 28 q^{65} + 16 q^{67} + 3 q^{69} - 3 q^{71} + 32 q^{73} - 18 q^{75} + 18 q^{77} + 24 q^{79} + 9 q^{81} - 3 q^{83} + 37 q^{85} - 5 q^{87} + 46 q^{89} + 33 q^{91} - 12 q^{93} + 11 q^{95} + 43 q^{97} - 9 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.419033 −0.187397 −0.0936986 0.995601i \(-0.529869\pi\)
−0.0936986 + 0.995601i \(0.529869\pi\)
\(6\) 0 0
\(7\) −4.23221 −1.59963 −0.799813 0.600249i \(-0.795068\pi\)
−0.799813 + 0.600249i \(0.795068\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.07280 −0.624973 −0.312487 0.949922i \(-0.601162\pi\)
−0.312487 + 0.949922i \(0.601162\pi\)
\(12\) 0 0
\(13\) −5.80837 −1.61095 −0.805476 0.592628i \(-0.798090\pi\)
−0.805476 + 0.592628i \(0.798090\pi\)
\(14\) 0 0
\(15\) 0.419033 0.108194
\(16\) 0 0
\(17\) −0.0751145 −0.0182179 −0.00910897 0.999959i \(-0.502900\pi\)
−0.00910897 + 0.999959i \(0.502900\pi\)
\(18\) 0 0
\(19\) −3.35395 −0.769448 −0.384724 0.923032i \(-0.625703\pi\)
−0.384724 + 0.923032i \(0.625703\pi\)
\(20\) 0 0
\(21\) 4.23221 0.923545
\(22\) 0 0
\(23\) 4.32486 0.901796 0.450898 0.892576i \(-0.351104\pi\)
0.450898 + 0.892576i \(0.351104\pi\)
\(24\) 0 0
\(25\) −4.82441 −0.964882
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.99097 1.66958 0.834791 0.550567i \(-0.185589\pi\)
0.834791 + 0.550567i \(0.185589\pi\)
\(30\) 0 0
\(31\) 6.91570 1.24210 0.621048 0.783773i \(-0.286707\pi\)
0.621048 + 0.783773i \(0.286707\pi\)
\(32\) 0 0
\(33\) 2.07280 0.360829
\(34\) 0 0
\(35\) 1.77344 0.299765
\(36\) 0 0
\(37\) 6.31045 1.03743 0.518716 0.854947i \(-0.326410\pi\)
0.518716 + 0.854947i \(0.326410\pi\)
\(38\) 0 0
\(39\) 5.80837 0.930084
\(40\) 0 0
\(41\) 4.28666 0.669464 0.334732 0.942313i \(-0.391354\pi\)
0.334732 + 0.942313i \(0.391354\pi\)
\(42\) 0 0
\(43\) −1.06448 −0.162332 −0.0811659 0.996701i \(-0.525864\pi\)
−0.0811659 + 0.996701i \(0.525864\pi\)
\(44\) 0 0
\(45\) −0.419033 −0.0624657
\(46\) 0 0
\(47\) −9.61226 −1.40209 −0.701046 0.713116i \(-0.747283\pi\)
−0.701046 + 0.713116i \(0.747283\pi\)
\(48\) 0 0
\(49\) 10.9116 1.55880
\(50\) 0 0
\(51\) 0.0751145 0.0105181
\(52\) 0 0
\(53\) −14.0744 −1.93327 −0.966636 0.256155i \(-0.917544\pi\)
−0.966636 + 0.256155i \(0.917544\pi\)
\(54\) 0 0
\(55\) 0.868572 0.117118
\(56\) 0 0
\(57\) 3.35395 0.444241
\(58\) 0 0
\(59\) 3.20748 0.417578 0.208789 0.977961i \(-0.433048\pi\)
0.208789 + 0.977961i \(0.433048\pi\)
\(60\) 0 0
\(61\) 11.7562 1.50523 0.752616 0.658460i \(-0.228792\pi\)
0.752616 + 0.658460i \(0.228792\pi\)
\(62\) 0 0
\(63\) −4.23221 −0.533209
\(64\) 0 0
\(65\) 2.43390 0.301888
\(66\) 0 0
\(67\) −5.13807 −0.627715 −0.313858 0.949470i \(-0.601621\pi\)
−0.313858 + 0.949470i \(0.601621\pi\)
\(68\) 0 0
\(69\) −4.32486 −0.520652
\(70\) 0 0
\(71\) −0.458914 −0.0544630 −0.0272315 0.999629i \(-0.508669\pi\)
−0.0272315 + 0.999629i \(0.508669\pi\)
\(72\) 0 0
\(73\) 11.5309 1.34960 0.674798 0.738003i \(-0.264231\pi\)
0.674798 + 0.738003i \(0.264231\pi\)
\(74\) 0 0
\(75\) 4.82441 0.557075
\(76\) 0 0
\(77\) 8.77254 0.999724
\(78\) 0 0
\(79\) −4.93027 −0.554698 −0.277349 0.960769i \(-0.589456\pi\)
−0.277349 + 0.960769i \(0.589456\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.02732 −0.990877 −0.495439 0.868643i \(-0.664993\pi\)
−0.495439 + 0.868643i \(0.664993\pi\)
\(84\) 0 0
\(85\) 0.0314754 0.00341399
\(86\) 0 0
\(87\) −8.99097 −0.963933
\(88\) 0 0
\(89\) 13.3953 1.41990 0.709948 0.704254i \(-0.248718\pi\)
0.709948 + 0.704254i \(0.248718\pi\)
\(90\) 0 0
\(91\) 24.5823 2.57692
\(92\) 0 0
\(93\) −6.91570 −0.717124
\(94\) 0 0
\(95\) 1.40541 0.144192
\(96\) 0 0
\(97\) 10.7908 1.09564 0.547818 0.836598i \(-0.315459\pi\)
0.547818 + 0.836598i \(0.315459\pi\)
\(98\) 0 0
\(99\) −2.07280 −0.208324
\(100\) 0 0
\(101\) −1.08758 −0.108218 −0.0541092 0.998535i \(-0.517232\pi\)
−0.0541092 + 0.998535i \(0.517232\pi\)
\(102\) 0 0
\(103\) 8.55080 0.842535 0.421268 0.906936i \(-0.361585\pi\)
0.421268 + 0.906936i \(0.361585\pi\)
\(104\) 0 0
\(105\) −1.77344 −0.173070
\(106\) 0 0
\(107\) 1.28422 0.124150 0.0620752 0.998071i \(-0.480228\pi\)
0.0620752 + 0.998071i \(0.480228\pi\)
\(108\) 0 0
\(109\) −2.12627 −0.203660 −0.101830 0.994802i \(-0.532470\pi\)
−0.101830 + 0.994802i \(0.532470\pi\)
\(110\) 0 0
\(111\) −6.31045 −0.598961
\(112\) 0 0
\(113\) 9.70039 0.912536 0.456268 0.889842i \(-0.349186\pi\)
0.456268 + 0.889842i \(0.349186\pi\)
\(114\) 0 0
\(115\) −1.81226 −0.168994
\(116\) 0 0
\(117\) −5.80837 −0.536984
\(118\) 0 0
\(119\) 0.317900 0.0291419
\(120\) 0 0
\(121\) −6.70349 −0.609408
\(122\) 0 0
\(123\) −4.28666 −0.386515
\(124\) 0 0
\(125\) 4.11675 0.368213
\(126\) 0 0
\(127\) −7.62835 −0.676907 −0.338454 0.940983i \(-0.609904\pi\)
−0.338454 + 0.940983i \(0.609904\pi\)
\(128\) 0 0
\(129\) 1.06448 0.0937223
\(130\) 0 0
\(131\) 2.58643 0.225977 0.112989 0.993596i \(-0.463958\pi\)
0.112989 + 0.993596i \(0.463958\pi\)
\(132\) 0 0
\(133\) 14.1946 1.23083
\(134\) 0 0
\(135\) 0.419033 0.0360646
\(136\) 0 0
\(137\) −4.21345 −0.359979 −0.179990 0.983668i \(-0.557606\pi\)
−0.179990 + 0.983668i \(0.557606\pi\)
\(138\) 0 0
\(139\) −7.30842 −0.619892 −0.309946 0.950754i \(-0.600311\pi\)
−0.309946 + 0.950754i \(0.600311\pi\)
\(140\) 0 0
\(141\) 9.61226 0.809498
\(142\) 0 0
\(143\) 12.0396 1.00680
\(144\) 0 0
\(145\) −3.76751 −0.312875
\(146\) 0 0
\(147\) −10.9116 −0.899976
\(148\) 0 0
\(149\) 8.66285 0.709689 0.354844 0.934925i \(-0.384534\pi\)
0.354844 + 0.934925i \(0.384534\pi\)
\(150\) 0 0
\(151\) 7.01818 0.571132 0.285566 0.958359i \(-0.407818\pi\)
0.285566 + 0.958359i \(0.407818\pi\)
\(152\) 0 0
\(153\) −0.0751145 −0.00607265
\(154\) 0 0
\(155\) −2.89790 −0.232765
\(156\) 0 0
\(157\) 13.1769 1.05163 0.525814 0.850599i \(-0.323761\pi\)
0.525814 + 0.850599i \(0.323761\pi\)
\(158\) 0 0
\(159\) 14.0744 1.11617
\(160\) 0 0
\(161\) −18.3037 −1.44254
\(162\) 0 0
\(163\) 8.40709 0.658494 0.329247 0.944244i \(-0.393205\pi\)
0.329247 + 0.944244i \(0.393205\pi\)
\(164\) 0 0
\(165\) −0.868572 −0.0676182
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 20.7372 1.59517
\(170\) 0 0
\(171\) −3.35395 −0.256483
\(172\) 0 0
\(173\) 0.397914 0.0302529 0.0151264 0.999886i \(-0.495185\pi\)
0.0151264 + 0.999886i \(0.495185\pi\)
\(174\) 0 0
\(175\) 20.4179 1.54345
\(176\) 0 0
\(177\) −3.20748 −0.241089
\(178\) 0 0
\(179\) −22.3288 −1.66893 −0.834466 0.551059i \(-0.814224\pi\)
−0.834466 + 0.551059i \(0.814224\pi\)
\(180\) 0 0
\(181\) −6.82249 −0.507112 −0.253556 0.967321i \(-0.581600\pi\)
−0.253556 + 0.967321i \(0.581600\pi\)
\(182\) 0 0
\(183\) −11.7562 −0.869046
\(184\) 0 0
\(185\) −2.64428 −0.194412
\(186\) 0 0
\(187\) 0.155697 0.0113857
\(188\) 0 0
\(189\) 4.23221 0.307848
\(190\) 0 0
\(191\) −10.9092 −0.789361 −0.394681 0.918818i \(-0.629145\pi\)
−0.394681 + 0.918818i \(0.629145\pi\)
\(192\) 0 0
\(193\) −9.78214 −0.704134 −0.352067 0.935975i \(-0.614521\pi\)
−0.352067 + 0.935975i \(0.614521\pi\)
\(194\) 0 0
\(195\) −2.43390 −0.174295
\(196\) 0 0
\(197\) 15.2135 1.08391 0.541957 0.840406i \(-0.317684\pi\)
0.541957 + 0.840406i \(0.317684\pi\)
\(198\) 0 0
\(199\) 21.8623 1.54978 0.774888 0.632099i \(-0.217806\pi\)
0.774888 + 0.632099i \(0.217806\pi\)
\(200\) 0 0
\(201\) 5.13807 0.362411
\(202\) 0 0
\(203\) −38.0517 −2.67071
\(204\) 0 0
\(205\) −1.79625 −0.125456
\(206\) 0 0
\(207\) 4.32486 0.300599
\(208\) 0 0
\(209\) 6.95207 0.480885
\(210\) 0 0
\(211\) 5.76231 0.396693 0.198347 0.980132i \(-0.436443\pi\)
0.198347 + 0.980132i \(0.436443\pi\)
\(212\) 0 0
\(213\) 0.458914 0.0314443
\(214\) 0 0
\(215\) 0.446052 0.0304205
\(216\) 0 0
\(217\) −29.2687 −1.98689
\(218\) 0 0
\(219\) −11.5309 −0.779189
\(220\) 0 0
\(221\) 0.436293 0.0293482
\(222\) 0 0
\(223\) 13.1968 0.883722 0.441861 0.897083i \(-0.354318\pi\)
0.441861 + 0.897083i \(0.354318\pi\)
\(224\) 0 0
\(225\) −4.82441 −0.321627
\(226\) 0 0
\(227\) 22.5195 1.49467 0.747335 0.664447i \(-0.231333\pi\)
0.747335 + 0.664447i \(0.231333\pi\)
\(228\) 0 0
\(229\) 5.47130 0.361554 0.180777 0.983524i \(-0.442139\pi\)
0.180777 + 0.983524i \(0.442139\pi\)
\(230\) 0 0
\(231\) −8.77254 −0.577191
\(232\) 0 0
\(233\) −21.9451 −1.43767 −0.718837 0.695178i \(-0.755325\pi\)
−0.718837 + 0.695178i \(0.755325\pi\)
\(234\) 0 0
\(235\) 4.02785 0.262748
\(236\) 0 0
\(237\) 4.93027 0.320255
\(238\) 0 0
\(239\) −3.51693 −0.227491 −0.113745 0.993510i \(-0.536285\pi\)
−0.113745 + 0.993510i \(0.536285\pi\)
\(240\) 0 0
\(241\) −21.6462 −1.39435 −0.697177 0.716899i \(-0.745561\pi\)
−0.697177 + 0.716899i \(0.745561\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.57233 −0.292115
\(246\) 0 0
\(247\) 19.4810 1.23955
\(248\) 0 0
\(249\) 9.02732 0.572083
\(250\) 0 0
\(251\) −19.1241 −1.20710 −0.603551 0.797324i \(-0.706248\pi\)
−0.603551 + 0.797324i \(0.706248\pi\)
\(252\) 0 0
\(253\) −8.96458 −0.563598
\(254\) 0 0
\(255\) −0.0314754 −0.00197107
\(256\) 0 0
\(257\) 27.7536 1.73122 0.865612 0.500716i \(-0.166930\pi\)
0.865612 + 0.500716i \(0.166930\pi\)
\(258\) 0 0
\(259\) −26.7072 −1.65950
\(260\) 0 0
\(261\) 8.99097 0.556527
\(262\) 0 0
\(263\) −0.701597 −0.0432623 −0.0216312 0.999766i \(-0.506886\pi\)
−0.0216312 + 0.999766i \(0.506886\pi\)
\(264\) 0 0
\(265\) 5.89765 0.362290
\(266\) 0 0
\(267\) −13.3953 −0.819777
\(268\) 0 0
\(269\) −10.5331 −0.642213 −0.321107 0.947043i \(-0.604055\pi\)
−0.321107 + 0.947043i \(0.604055\pi\)
\(270\) 0 0
\(271\) 24.9096 1.51315 0.756576 0.653906i \(-0.226871\pi\)
0.756576 + 0.653906i \(0.226871\pi\)
\(272\) 0 0
\(273\) −24.5823 −1.48779
\(274\) 0 0
\(275\) 10.0001 0.603026
\(276\) 0 0
\(277\) 15.3100 0.919891 0.459945 0.887947i \(-0.347869\pi\)
0.459945 + 0.887947i \(0.347869\pi\)
\(278\) 0 0
\(279\) 6.91570 0.414032
\(280\) 0 0
\(281\) 20.2673 1.20904 0.604522 0.796589i \(-0.293364\pi\)
0.604522 + 0.796589i \(0.293364\pi\)
\(282\) 0 0
\(283\) 29.3380 1.74396 0.871982 0.489538i \(-0.162835\pi\)
0.871982 + 0.489538i \(0.162835\pi\)
\(284\) 0 0
\(285\) −1.40541 −0.0832495
\(286\) 0 0
\(287\) −18.1421 −1.07089
\(288\) 0 0
\(289\) −16.9944 −0.999668
\(290\) 0 0
\(291\) −10.7908 −0.632565
\(292\) 0 0
\(293\) 8.29085 0.484357 0.242178 0.970232i \(-0.422138\pi\)
0.242178 + 0.970232i \(0.422138\pi\)
\(294\) 0 0
\(295\) −1.34404 −0.0782529
\(296\) 0 0
\(297\) 2.07280 0.120276
\(298\) 0 0
\(299\) −25.1204 −1.45275
\(300\) 0 0
\(301\) 4.50511 0.259670
\(302\) 0 0
\(303\) 1.08758 0.0624799
\(304\) 0 0
\(305\) −4.92625 −0.282076
\(306\) 0 0
\(307\) −4.92451 −0.281057 −0.140528 0.990077i \(-0.544880\pi\)
−0.140528 + 0.990077i \(0.544880\pi\)
\(308\) 0 0
\(309\) −8.55080 −0.486438
\(310\) 0 0
\(311\) 7.34593 0.416549 0.208275 0.978070i \(-0.433215\pi\)
0.208275 + 0.978070i \(0.433215\pi\)
\(312\) 0 0
\(313\) 10.3083 0.582657 0.291328 0.956623i \(-0.405903\pi\)
0.291328 + 0.956623i \(0.405903\pi\)
\(314\) 0 0
\(315\) 1.77344 0.0999218
\(316\) 0 0
\(317\) −20.5279 −1.15296 −0.576481 0.817110i \(-0.695575\pi\)
−0.576481 + 0.817110i \(0.695575\pi\)
\(318\) 0 0
\(319\) −18.6365 −1.04344
\(320\) 0 0
\(321\) −1.28422 −0.0716782
\(322\) 0 0
\(323\) 0.251930 0.0140178
\(324\) 0 0
\(325\) 28.0220 1.55438
\(326\) 0 0
\(327\) 2.12627 0.117583
\(328\) 0 0
\(329\) 40.6811 2.24282
\(330\) 0 0
\(331\) −1.88498 −0.103608 −0.0518039 0.998657i \(-0.516497\pi\)
−0.0518039 + 0.998657i \(0.516497\pi\)
\(332\) 0 0
\(333\) 6.31045 0.345810
\(334\) 0 0
\(335\) 2.15302 0.117632
\(336\) 0 0
\(337\) −8.26097 −0.450004 −0.225002 0.974358i \(-0.572239\pi\)
−0.225002 + 0.974358i \(0.572239\pi\)
\(338\) 0 0
\(339\) −9.70039 −0.526853
\(340\) 0 0
\(341\) −14.3349 −0.776277
\(342\) 0 0
\(343\) −16.5548 −0.893877
\(344\) 0 0
\(345\) 1.81226 0.0975687
\(346\) 0 0
\(347\) −31.7422 −1.70401 −0.852004 0.523535i \(-0.824613\pi\)
−0.852004 + 0.523535i \(0.824613\pi\)
\(348\) 0 0
\(349\) 3.31362 0.177374 0.0886870 0.996060i \(-0.471733\pi\)
0.0886870 + 0.996060i \(0.471733\pi\)
\(350\) 0 0
\(351\) 5.80837 0.310028
\(352\) 0 0
\(353\) −32.8238 −1.74704 −0.873518 0.486791i \(-0.838167\pi\)
−0.873518 + 0.486791i \(0.838167\pi\)
\(354\) 0 0
\(355\) 0.192300 0.0102062
\(356\) 0 0
\(357\) −0.317900 −0.0168251
\(358\) 0 0
\(359\) 8.55562 0.451549 0.225774 0.974180i \(-0.427509\pi\)
0.225774 + 0.974180i \(0.427509\pi\)
\(360\) 0 0
\(361\) −7.75103 −0.407949
\(362\) 0 0
\(363\) 6.70349 0.351842
\(364\) 0 0
\(365\) −4.83184 −0.252910
\(366\) 0 0
\(367\) 21.2074 1.10702 0.553508 0.832844i \(-0.313289\pi\)
0.553508 + 0.832844i \(0.313289\pi\)
\(368\) 0 0
\(369\) 4.28666 0.223155
\(370\) 0 0
\(371\) 59.5660 3.09251
\(372\) 0 0
\(373\) −4.30504 −0.222906 −0.111453 0.993770i \(-0.535551\pi\)
−0.111453 + 0.993770i \(0.535551\pi\)
\(374\) 0 0
\(375\) −4.11675 −0.212588
\(376\) 0 0
\(377\) −52.2229 −2.68962
\(378\) 0 0
\(379\) −27.9843 −1.43746 −0.718729 0.695290i \(-0.755276\pi\)
−0.718729 + 0.695290i \(0.755276\pi\)
\(380\) 0 0
\(381\) 7.62835 0.390812
\(382\) 0 0
\(383\) −1.66131 −0.0848890 −0.0424445 0.999099i \(-0.513515\pi\)
−0.0424445 + 0.999099i \(0.513515\pi\)
\(384\) 0 0
\(385\) −3.67598 −0.187345
\(386\) 0 0
\(387\) −1.06448 −0.0541106
\(388\) 0 0
\(389\) −2.71849 −0.137833 −0.0689165 0.997622i \(-0.521954\pi\)
−0.0689165 + 0.997622i \(0.521954\pi\)
\(390\) 0 0
\(391\) −0.324860 −0.0164289
\(392\) 0 0
\(393\) −2.58643 −0.130468
\(394\) 0 0
\(395\) 2.06594 0.103949
\(396\) 0 0
\(397\) −17.0954 −0.857993 −0.428996 0.903306i \(-0.641133\pi\)
−0.428996 + 0.903306i \(0.641133\pi\)
\(398\) 0 0
\(399\) −14.1946 −0.710620
\(400\) 0 0
\(401\) 4.89575 0.244482 0.122241 0.992500i \(-0.460992\pi\)
0.122241 + 0.992500i \(0.460992\pi\)
\(402\) 0 0
\(403\) −40.1690 −2.00096
\(404\) 0 0
\(405\) −0.419033 −0.0208219
\(406\) 0 0
\(407\) −13.0803 −0.648367
\(408\) 0 0
\(409\) 10.6040 0.524332 0.262166 0.965023i \(-0.415563\pi\)
0.262166 + 0.965023i \(0.415563\pi\)
\(410\) 0 0
\(411\) 4.21345 0.207834
\(412\) 0 0
\(413\) −13.5747 −0.667968
\(414\) 0 0
\(415\) 3.78274 0.185688
\(416\) 0 0
\(417\) 7.30842 0.357895
\(418\) 0 0
\(419\) 16.5290 0.807492 0.403746 0.914871i \(-0.367708\pi\)
0.403746 + 0.914871i \(0.367708\pi\)
\(420\) 0 0
\(421\) −4.87460 −0.237573 −0.118787 0.992920i \(-0.537900\pi\)
−0.118787 + 0.992920i \(0.537900\pi\)
\(422\) 0 0
\(423\) −9.61226 −0.467364
\(424\) 0 0
\(425\) 0.362383 0.0175782
\(426\) 0 0
\(427\) −49.7549 −2.40781
\(428\) 0 0
\(429\) −12.0396 −0.581278
\(430\) 0 0
\(431\) −38.9911 −1.87814 −0.939069 0.343730i \(-0.888310\pi\)
−0.939069 + 0.343730i \(0.888310\pi\)
\(432\) 0 0
\(433\) −22.1079 −1.06244 −0.531218 0.847235i \(-0.678266\pi\)
−0.531218 + 0.847235i \(0.678266\pi\)
\(434\) 0 0
\(435\) 3.76751 0.180638
\(436\) 0 0
\(437\) −14.5054 −0.693885
\(438\) 0 0
\(439\) 1.62530 0.0775714 0.0387857 0.999248i \(-0.487651\pi\)
0.0387857 + 0.999248i \(0.487651\pi\)
\(440\) 0 0
\(441\) 10.9116 0.519601
\(442\) 0 0
\(443\) −18.4447 −0.876333 −0.438167 0.898894i \(-0.644372\pi\)
−0.438167 + 0.898894i \(0.644372\pi\)
\(444\) 0 0
\(445\) −5.61306 −0.266085
\(446\) 0 0
\(447\) −8.66285 −0.409739
\(448\) 0 0
\(449\) 24.6369 1.16269 0.581344 0.813658i \(-0.302527\pi\)
0.581344 + 0.813658i \(0.302527\pi\)
\(450\) 0 0
\(451\) −8.88541 −0.418397
\(452\) 0 0
\(453\) −7.01818 −0.329743
\(454\) 0 0
\(455\) −10.3008 −0.482908
\(456\) 0 0
\(457\) 1.21824 0.0569869 0.0284934 0.999594i \(-0.490929\pi\)
0.0284934 + 0.999594i \(0.490929\pi\)
\(458\) 0 0
\(459\) 0.0751145 0.00350604
\(460\) 0 0
\(461\) 38.3645 1.78681 0.893406 0.449250i \(-0.148309\pi\)
0.893406 + 0.449250i \(0.148309\pi\)
\(462\) 0 0
\(463\) 26.0196 1.20923 0.604617 0.796517i \(-0.293326\pi\)
0.604617 + 0.796517i \(0.293326\pi\)
\(464\) 0 0
\(465\) 2.89790 0.134387
\(466\) 0 0
\(467\) 8.06356 0.373137 0.186569 0.982442i \(-0.440263\pi\)
0.186569 + 0.982442i \(0.440263\pi\)
\(468\) 0 0
\(469\) 21.7454 1.00411
\(470\) 0 0
\(471\) −13.1769 −0.607158
\(472\) 0 0
\(473\) 2.20646 0.101453
\(474\) 0 0
\(475\) 16.1808 0.742427
\(476\) 0 0
\(477\) −14.0744 −0.644424
\(478\) 0 0
\(479\) −31.6001 −1.44385 −0.721923 0.691973i \(-0.756742\pi\)
−0.721923 + 0.691973i \(0.756742\pi\)
\(480\) 0 0
\(481\) −36.6534 −1.67125
\(482\) 0 0
\(483\) 18.3037 0.832848
\(484\) 0 0
\(485\) −4.52168 −0.205319
\(486\) 0 0
\(487\) 36.5964 1.65834 0.829170 0.558996i \(-0.188813\pi\)
0.829170 + 0.558996i \(0.188813\pi\)
\(488\) 0 0
\(489\) −8.40709 −0.380182
\(490\) 0 0
\(491\) −18.1135 −0.817451 −0.408726 0.912657i \(-0.634027\pi\)
−0.408726 + 0.912657i \(0.634027\pi\)
\(492\) 0 0
\(493\) −0.675352 −0.0304163
\(494\) 0 0
\(495\) 0.868572 0.0390394
\(496\) 0 0
\(497\) 1.94222 0.0871205
\(498\) 0 0
\(499\) 11.8702 0.531383 0.265691 0.964058i \(-0.414400\pi\)
0.265691 + 0.964058i \(0.414400\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −2.76481 −0.123277 −0.0616384 0.998099i \(-0.519633\pi\)
−0.0616384 + 0.998099i \(0.519633\pi\)
\(504\) 0 0
\(505\) 0.455732 0.0202798
\(506\) 0 0
\(507\) −20.7372 −0.920971
\(508\) 0 0
\(509\) 16.9838 0.752796 0.376398 0.926458i \(-0.377163\pi\)
0.376398 + 0.926458i \(0.377163\pi\)
\(510\) 0 0
\(511\) −48.8014 −2.15885
\(512\) 0 0
\(513\) 3.35395 0.148080
\(514\) 0 0
\(515\) −3.58306 −0.157889
\(516\) 0 0
\(517\) 19.9243 0.876270
\(518\) 0 0
\(519\) −0.397914 −0.0174665
\(520\) 0 0
\(521\) 20.9133 0.916229 0.458115 0.888893i \(-0.348525\pi\)
0.458115 + 0.888893i \(0.348525\pi\)
\(522\) 0 0
\(523\) −8.41272 −0.367862 −0.183931 0.982939i \(-0.558882\pi\)
−0.183931 + 0.982939i \(0.558882\pi\)
\(524\) 0 0
\(525\) −20.4179 −0.891112
\(526\) 0 0
\(527\) −0.519469 −0.0226284
\(528\) 0 0
\(529\) −4.29559 −0.186765
\(530\) 0 0
\(531\) 3.20748 0.139193
\(532\) 0 0
\(533\) −24.8985 −1.07848
\(534\) 0 0
\(535\) −0.538131 −0.0232654
\(536\) 0 0
\(537\) 22.3288 0.963559
\(538\) 0 0
\(539\) −22.6176 −0.974211
\(540\) 0 0
\(541\) −11.9997 −0.515907 −0.257954 0.966157i \(-0.583048\pi\)
−0.257954 + 0.966157i \(0.583048\pi\)
\(542\) 0 0
\(543\) 6.82249 0.292781
\(544\) 0 0
\(545\) 0.890978 0.0381653
\(546\) 0 0
\(547\) −33.6543 −1.43896 −0.719478 0.694516i \(-0.755619\pi\)
−0.719478 + 0.694516i \(0.755619\pi\)
\(548\) 0 0
\(549\) 11.7562 0.501744
\(550\) 0 0
\(551\) −30.1552 −1.28466
\(552\) 0 0
\(553\) 20.8659 0.887310
\(554\) 0 0
\(555\) 2.64428 0.112244
\(556\) 0 0
\(557\) 3.94121 0.166994 0.0834972 0.996508i \(-0.473391\pi\)
0.0834972 + 0.996508i \(0.473391\pi\)
\(558\) 0 0
\(559\) 6.18290 0.261509
\(560\) 0 0
\(561\) −0.155697 −0.00657355
\(562\) 0 0
\(563\) −39.1203 −1.64872 −0.824361 0.566064i \(-0.808466\pi\)
−0.824361 + 0.566064i \(0.808466\pi\)
\(564\) 0 0
\(565\) −4.06478 −0.171007
\(566\) 0 0
\(567\) −4.23221 −0.177736
\(568\) 0 0
\(569\) 30.3792 1.27356 0.636781 0.771045i \(-0.280266\pi\)
0.636781 + 0.771045i \(0.280266\pi\)
\(570\) 0 0
\(571\) 14.1764 0.593263 0.296631 0.954992i \(-0.404137\pi\)
0.296631 + 0.954992i \(0.404137\pi\)
\(572\) 0 0
\(573\) 10.9092 0.455738
\(574\) 0 0
\(575\) −20.8649 −0.870127
\(576\) 0 0
\(577\) 1.21416 0.0505461 0.0252731 0.999681i \(-0.491954\pi\)
0.0252731 + 0.999681i \(0.491954\pi\)
\(578\) 0 0
\(579\) 9.78214 0.406532
\(580\) 0 0
\(581\) 38.2055 1.58503
\(582\) 0 0
\(583\) 29.1735 1.20824
\(584\) 0 0
\(585\) 2.43390 0.100629
\(586\) 0 0
\(587\) 47.1128 1.94455 0.972277 0.233833i \(-0.0751271\pi\)
0.972277 + 0.233833i \(0.0751271\pi\)
\(588\) 0 0
\(589\) −23.1949 −0.955729
\(590\) 0 0
\(591\) −15.2135 −0.625798
\(592\) 0 0
\(593\) −22.1617 −0.910074 −0.455037 0.890473i \(-0.650374\pi\)
−0.455037 + 0.890473i \(0.650374\pi\)
\(594\) 0 0
\(595\) −0.133211 −0.00546111
\(596\) 0 0
\(597\) −21.8623 −0.894763
\(598\) 0 0
\(599\) 21.3831 0.873690 0.436845 0.899537i \(-0.356096\pi\)
0.436845 + 0.899537i \(0.356096\pi\)
\(600\) 0 0
\(601\) 37.6389 1.53532 0.767662 0.640855i \(-0.221420\pi\)
0.767662 + 0.640855i \(0.221420\pi\)
\(602\) 0 0
\(603\) −5.13807 −0.209238
\(604\) 0 0
\(605\) 2.80898 0.114201
\(606\) 0 0
\(607\) −13.2663 −0.538463 −0.269231 0.963076i \(-0.586770\pi\)
−0.269231 + 0.963076i \(0.586770\pi\)
\(608\) 0 0
\(609\) 38.0517 1.54193
\(610\) 0 0
\(611\) 55.8316 2.25870
\(612\) 0 0
\(613\) 28.5709 1.15397 0.576984 0.816756i \(-0.304230\pi\)
0.576984 + 0.816756i \(0.304230\pi\)
\(614\) 0 0
\(615\) 1.79625 0.0724319
\(616\) 0 0
\(617\) 21.5762 0.868627 0.434314 0.900762i \(-0.356991\pi\)
0.434314 + 0.900762i \(0.356991\pi\)
\(618\) 0 0
\(619\) 14.3494 0.576752 0.288376 0.957517i \(-0.406885\pi\)
0.288376 + 0.957517i \(0.406885\pi\)
\(620\) 0 0
\(621\) −4.32486 −0.173551
\(622\) 0 0
\(623\) −56.6917 −2.27130
\(624\) 0 0
\(625\) 22.3970 0.895880
\(626\) 0 0
\(627\) −6.95207 −0.277639
\(628\) 0 0
\(629\) −0.474006 −0.0188999
\(630\) 0 0
\(631\) 28.1153 1.11925 0.559626 0.828745i \(-0.310945\pi\)
0.559626 + 0.828745i \(0.310945\pi\)
\(632\) 0 0
\(633\) −5.76231 −0.229031
\(634\) 0 0
\(635\) 3.19653 0.126850
\(636\) 0 0
\(637\) −63.3788 −2.51116
\(638\) 0 0
\(639\) −0.458914 −0.0181543
\(640\) 0 0
\(641\) −32.1878 −1.27134 −0.635672 0.771959i \(-0.719277\pi\)
−0.635672 + 0.771959i \(0.719277\pi\)
\(642\) 0 0
\(643\) −14.6701 −0.578534 −0.289267 0.957249i \(-0.593411\pi\)
−0.289267 + 0.957249i \(0.593411\pi\)
\(644\) 0 0
\(645\) −0.446052 −0.0175633
\(646\) 0 0
\(647\) 4.94159 0.194274 0.0971370 0.995271i \(-0.469031\pi\)
0.0971370 + 0.995271i \(0.469031\pi\)
\(648\) 0 0
\(649\) −6.64846 −0.260975
\(650\) 0 0
\(651\) 29.2687 1.14713
\(652\) 0 0
\(653\) 15.9200 0.622996 0.311498 0.950247i \(-0.399169\pi\)
0.311498 + 0.950247i \(0.399169\pi\)
\(654\) 0 0
\(655\) −1.08380 −0.0423475
\(656\) 0 0
\(657\) 11.5309 0.449865
\(658\) 0 0
\(659\) −16.4683 −0.641513 −0.320757 0.947162i \(-0.603937\pi\)
−0.320757 + 0.947162i \(0.603937\pi\)
\(660\) 0 0
\(661\) 29.8115 1.15953 0.579767 0.814783i \(-0.303144\pi\)
0.579767 + 0.814783i \(0.303144\pi\)
\(662\) 0 0
\(663\) −0.436293 −0.0169442
\(664\) 0 0
\(665\) −5.94801 −0.230654
\(666\) 0 0
\(667\) 38.8847 1.50562
\(668\) 0 0
\(669\) −13.1968 −0.510217
\(670\) 0 0
\(671\) −24.3683 −0.940730
\(672\) 0 0
\(673\) 20.2024 0.778744 0.389372 0.921081i \(-0.372692\pi\)
0.389372 + 0.921081i \(0.372692\pi\)
\(674\) 0 0
\(675\) 4.82441 0.185692
\(676\) 0 0
\(677\) 4.25637 0.163586 0.0817929 0.996649i \(-0.473935\pi\)
0.0817929 + 0.996649i \(0.473935\pi\)
\(678\) 0 0
\(679\) −45.6688 −1.75261
\(680\) 0 0
\(681\) −22.5195 −0.862948
\(682\) 0 0
\(683\) 37.8779 1.44936 0.724679 0.689087i \(-0.241988\pi\)
0.724679 + 0.689087i \(0.241988\pi\)
\(684\) 0 0
\(685\) 1.76557 0.0674591
\(686\) 0 0
\(687\) −5.47130 −0.208743
\(688\) 0 0
\(689\) 81.7495 3.11441
\(690\) 0 0
\(691\) 43.0240 1.63671 0.818355 0.574713i \(-0.194886\pi\)
0.818355 + 0.574713i \(0.194886\pi\)
\(692\) 0 0
\(693\) 8.77254 0.333241
\(694\) 0 0
\(695\) 3.06247 0.116166
\(696\) 0 0
\(697\) −0.321990 −0.0121963
\(698\) 0 0
\(699\) 21.9451 0.830042
\(700\) 0 0
\(701\) 34.3231 1.29637 0.648184 0.761484i \(-0.275529\pi\)
0.648184 + 0.761484i \(0.275529\pi\)
\(702\) 0 0
\(703\) −21.1649 −0.798250
\(704\) 0 0
\(705\) −4.02785 −0.151698
\(706\) 0 0
\(707\) 4.60288 0.173109
\(708\) 0 0
\(709\) 35.4476 1.33126 0.665631 0.746281i \(-0.268162\pi\)
0.665631 + 0.746281i \(0.268162\pi\)
\(710\) 0 0
\(711\) −4.93027 −0.184899
\(712\) 0 0
\(713\) 29.9094 1.12012
\(714\) 0 0
\(715\) −5.04499 −0.188672
\(716\) 0 0
\(717\) 3.51693 0.131342
\(718\) 0 0
\(719\) 36.9032 1.37626 0.688129 0.725589i \(-0.258432\pi\)
0.688129 + 0.725589i \(0.258432\pi\)
\(720\) 0 0
\(721\) −36.1888 −1.34774
\(722\) 0 0
\(723\) 21.6462 0.805031
\(724\) 0 0
\(725\) −43.3761 −1.61095
\(726\) 0 0
\(727\) 1.47138 0.0545705 0.0272852 0.999628i \(-0.491314\pi\)
0.0272852 + 0.999628i \(0.491314\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.0799579 0.00295735
\(732\) 0 0
\(733\) −21.8906 −0.808548 −0.404274 0.914638i \(-0.632476\pi\)
−0.404274 + 0.914638i \(0.632476\pi\)
\(734\) 0 0
\(735\) 4.57233 0.168653
\(736\) 0 0
\(737\) 10.6502 0.392305
\(738\) 0 0
\(739\) −19.8564 −0.730429 −0.365215 0.930923i \(-0.619004\pi\)
−0.365215 + 0.930923i \(0.619004\pi\)
\(740\) 0 0
\(741\) −19.4810 −0.715652
\(742\) 0 0
\(743\) 44.7189 1.64058 0.820288 0.571950i \(-0.193813\pi\)
0.820288 + 0.571950i \(0.193813\pi\)
\(744\) 0 0
\(745\) −3.63002 −0.132994
\(746\) 0 0
\(747\) −9.02732 −0.330292
\(748\) 0 0
\(749\) −5.43510 −0.198594
\(750\) 0 0
\(751\) 31.9142 1.16457 0.582283 0.812986i \(-0.302160\pi\)
0.582283 + 0.812986i \(0.302160\pi\)
\(752\) 0 0
\(753\) 19.1241 0.696921
\(754\) 0 0
\(755\) −2.94085 −0.107028
\(756\) 0 0
\(757\) 24.7634 0.900039 0.450020 0.893019i \(-0.351417\pi\)
0.450020 + 0.893019i \(0.351417\pi\)
\(758\) 0 0
\(759\) 8.96458 0.325394
\(760\) 0 0
\(761\) 39.2201 1.42173 0.710863 0.703330i \(-0.248304\pi\)
0.710863 + 0.703330i \(0.248304\pi\)
\(762\) 0 0
\(763\) 8.99884 0.325780
\(764\) 0 0
\(765\) 0.0314754 0.00113800
\(766\) 0 0
\(767\) −18.6302 −0.672698
\(768\) 0 0
\(769\) −48.0481 −1.73266 −0.866329 0.499473i \(-0.833527\pi\)
−0.866329 + 0.499473i \(0.833527\pi\)
\(770\) 0 0
\(771\) −27.7536 −0.999522
\(772\) 0 0
\(773\) −54.4339 −1.95785 −0.978925 0.204219i \(-0.934535\pi\)
−0.978925 + 0.204219i \(0.934535\pi\)
\(774\) 0 0
\(775\) −33.3642 −1.19848
\(776\) 0 0
\(777\) 26.7072 0.958114
\(778\) 0 0
\(779\) −14.3772 −0.515118
\(780\) 0 0
\(781\) 0.951237 0.0340380
\(782\) 0 0
\(783\) −8.99097 −0.321311
\(784\) 0 0
\(785\) −5.52154 −0.197072
\(786\) 0 0
\(787\) −26.5959 −0.948042 −0.474021 0.880513i \(-0.657198\pi\)
−0.474021 + 0.880513i \(0.657198\pi\)
\(788\) 0 0
\(789\) 0.701597 0.0249775
\(790\) 0 0
\(791\) −41.0541 −1.45972
\(792\) 0 0
\(793\) −68.2846 −2.42486
\(794\) 0 0
\(795\) −5.89765 −0.209168
\(796\) 0 0
\(797\) −29.0860 −1.03028 −0.515140 0.857106i \(-0.672260\pi\)
−0.515140 + 0.857106i \(0.672260\pi\)
\(798\) 0 0
\(799\) 0.722020 0.0255432
\(800\) 0 0
\(801\) 13.3953 0.473299
\(802\) 0 0
\(803\) −23.9014 −0.843461
\(804\) 0 0
\(805\) 7.66986 0.270327
\(806\) 0 0
\(807\) 10.5331 0.370782
\(808\) 0 0
\(809\) −39.9699 −1.40527 −0.702634 0.711552i \(-0.747993\pi\)
−0.702634 + 0.711552i \(0.747993\pi\)
\(810\) 0 0
\(811\) −13.5883 −0.477149 −0.238574 0.971124i \(-0.576680\pi\)
−0.238574 + 0.971124i \(0.576680\pi\)
\(812\) 0 0
\(813\) −24.9096 −0.873618
\(814\) 0 0
\(815\) −3.52285 −0.123400
\(816\) 0 0
\(817\) 3.57021 0.124906
\(818\) 0 0
\(819\) 24.5823 0.858974
\(820\) 0 0
\(821\) −49.0067 −1.71035 −0.855173 0.518343i \(-0.826549\pi\)
−0.855173 + 0.518343i \(0.826549\pi\)
\(822\) 0 0
\(823\) 44.0006 1.53376 0.766882 0.641788i \(-0.221807\pi\)
0.766882 + 0.641788i \(0.221807\pi\)
\(824\) 0 0
\(825\) −10.0001 −0.348157
\(826\) 0 0
\(827\) −49.9051 −1.73537 −0.867685 0.497114i \(-0.834393\pi\)
−0.867685 + 0.497114i \(0.834393\pi\)
\(828\) 0 0
\(829\) −18.3711 −0.638054 −0.319027 0.947746i \(-0.603356\pi\)
−0.319027 + 0.947746i \(0.603356\pi\)
\(830\) 0 0
\(831\) −15.3100 −0.531099
\(832\) 0 0
\(833\) −0.819621 −0.0283982
\(834\) 0 0
\(835\) 0.419033 0.0145012
\(836\) 0 0
\(837\) −6.91570 −0.239041
\(838\) 0 0
\(839\) 18.8506 0.650795 0.325398 0.945577i \(-0.394502\pi\)
0.325398 + 0.945577i \(0.394502\pi\)
\(840\) 0 0
\(841\) 51.8376 1.78750
\(842\) 0 0
\(843\) −20.2673 −0.698042
\(844\) 0 0
\(845\) −8.68956 −0.298930
\(846\) 0 0
\(847\) 28.3706 0.974825
\(848\) 0 0
\(849\) −29.3380 −1.00688
\(850\) 0 0
\(851\) 27.2918 0.935551
\(852\) 0 0
\(853\) −32.3912 −1.10905 −0.554526 0.832166i \(-0.687100\pi\)
−0.554526 + 0.832166i \(0.687100\pi\)
\(854\) 0 0
\(855\) 1.40541 0.0480641
\(856\) 0 0
\(857\) −38.9280 −1.32975 −0.664877 0.746953i \(-0.731516\pi\)
−0.664877 + 0.746953i \(0.731516\pi\)
\(858\) 0 0
\(859\) −20.0265 −0.683295 −0.341648 0.939828i \(-0.610985\pi\)
−0.341648 + 0.939828i \(0.610985\pi\)
\(860\) 0 0
\(861\) 18.1421 0.618280
\(862\) 0 0
\(863\) −9.32903 −0.317564 −0.158782 0.987314i \(-0.550757\pi\)
−0.158782 + 0.987314i \(0.550757\pi\)
\(864\) 0 0
\(865\) −0.166739 −0.00566930
\(866\) 0 0
\(867\) 16.9944 0.577159
\(868\) 0 0
\(869\) 10.2195 0.346672
\(870\) 0 0
\(871\) 29.8438 1.01122
\(872\) 0 0
\(873\) 10.7908 0.365212
\(874\) 0 0
\(875\) −17.4230 −0.589004
\(876\) 0 0
\(877\) 55.3456 1.86889 0.934444 0.356111i \(-0.115898\pi\)
0.934444 + 0.356111i \(0.115898\pi\)
\(878\) 0 0
\(879\) −8.29085 −0.279644
\(880\) 0 0
\(881\) −16.8218 −0.566740 −0.283370 0.959011i \(-0.591452\pi\)
−0.283370 + 0.959011i \(0.591452\pi\)
\(882\) 0 0
\(883\) −34.5574 −1.16295 −0.581474 0.813565i \(-0.697524\pi\)
−0.581474 + 0.813565i \(0.697524\pi\)
\(884\) 0 0
\(885\) 1.34404 0.0451793
\(886\) 0 0
\(887\) 51.9595 1.74463 0.872314 0.488945i \(-0.162618\pi\)
0.872314 + 0.488945i \(0.162618\pi\)
\(888\) 0 0
\(889\) 32.2848 1.08280
\(890\) 0 0
\(891\) −2.07280 −0.0694415
\(892\) 0 0
\(893\) 32.2390 1.07884
\(894\) 0 0
\(895\) 9.35650 0.312753
\(896\) 0 0
\(897\) 25.1204 0.838746
\(898\) 0 0
\(899\) 62.1788 2.07378
\(900\) 0 0
\(901\) 1.05719 0.0352202
\(902\) 0 0
\(903\) −4.50511 −0.149921
\(904\) 0 0
\(905\) 2.85885 0.0950313
\(906\) 0 0
\(907\) 28.4886 0.945949 0.472975 0.881076i \(-0.343180\pi\)
0.472975 + 0.881076i \(0.343180\pi\)
\(908\) 0 0
\(909\) −1.08758 −0.0360728
\(910\) 0 0
\(911\) 41.6638 1.38038 0.690191 0.723627i \(-0.257527\pi\)
0.690191 + 0.723627i \(0.257527\pi\)
\(912\) 0 0
\(913\) 18.7119 0.619272
\(914\) 0 0
\(915\) 4.92625 0.162857
\(916\) 0 0
\(917\) −10.9463 −0.361479
\(918\) 0 0
\(919\) 51.9148 1.71251 0.856255 0.516553i \(-0.172785\pi\)
0.856255 + 0.516553i \(0.172785\pi\)
\(920\) 0 0
\(921\) 4.92451 0.162268
\(922\) 0 0
\(923\) 2.66554 0.0877374
\(924\) 0 0
\(925\) −30.4442 −1.00100
\(926\) 0 0
\(927\) 8.55080 0.280845
\(928\) 0 0
\(929\) −21.5365 −0.706588 −0.353294 0.935512i \(-0.614939\pi\)
−0.353294 + 0.935512i \(0.614939\pi\)
\(930\) 0 0
\(931\) −36.5970 −1.19942
\(932\) 0 0
\(933\) −7.34593 −0.240495
\(934\) 0 0
\(935\) −0.0652423 −0.00213365
\(936\) 0 0
\(937\) 56.3298 1.84021 0.920107 0.391667i \(-0.128101\pi\)
0.920107 + 0.391667i \(0.128101\pi\)
\(938\) 0 0
\(939\) −10.3083 −0.336397
\(940\) 0 0
\(941\) 14.6528 0.477667 0.238833 0.971061i \(-0.423235\pi\)
0.238833 + 0.971061i \(0.423235\pi\)
\(942\) 0 0
\(943\) 18.5392 0.603720
\(944\) 0 0
\(945\) −1.77344 −0.0576899
\(946\) 0 0
\(947\) −18.7078 −0.607921 −0.303960 0.952685i \(-0.598309\pi\)
−0.303960 + 0.952685i \(0.598309\pi\)
\(948\) 0 0
\(949\) −66.9760 −2.17413
\(950\) 0 0
\(951\) 20.5279 0.665663
\(952\) 0 0
\(953\) −57.5762 −1.86507 −0.932537 0.361074i \(-0.882410\pi\)
−0.932537 + 0.361074i \(0.882410\pi\)
\(954\) 0 0
\(955\) 4.57131 0.147924
\(956\) 0 0
\(957\) 18.6365 0.602433
\(958\) 0 0
\(959\) 17.8322 0.575832
\(960\) 0 0
\(961\) 16.8269 0.542802
\(962\) 0 0
\(963\) 1.28422 0.0413835
\(964\) 0 0
\(965\) 4.09904 0.131953
\(966\) 0 0
\(967\) 46.5575 1.49719 0.748594 0.663028i \(-0.230729\pi\)
0.748594 + 0.663028i \(0.230729\pi\)
\(968\) 0 0
\(969\) −0.251930 −0.00809316
\(970\) 0 0
\(971\) 57.5407 1.84657 0.923284 0.384119i \(-0.125495\pi\)
0.923284 + 0.384119i \(0.125495\pi\)
\(972\) 0 0
\(973\) 30.9308 0.991595
\(974\) 0 0
\(975\) −28.0220 −0.897422
\(976\) 0 0
\(977\) 30.6548 0.980735 0.490367 0.871516i \(-0.336863\pi\)
0.490367 + 0.871516i \(0.336863\pi\)
\(978\) 0 0
\(979\) −27.7658 −0.887397
\(980\) 0 0
\(981\) −2.12627 −0.0678867
\(982\) 0 0
\(983\) 51.5579 1.64444 0.822221 0.569168i \(-0.192735\pi\)
0.822221 + 0.569168i \(0.192735\pi\)
\(984\) 0 0
\(985\) −6.37494 −0.203122
\(986\) 0 0
\(987\) −40.6811 −1.29489
\(988\) 0 0
\(989\) −4.60373 −0.146390
\(990\) 0 0
\(991\) 31.9100 1.01365 0.506827 0.862048i \(-0.330818\pi\)
0.506827 + 0.862048i \(0.330818\pi\)
\(992\) 0 0
\(993\) 1.88498 0.0598180
\(994\) 0 0
\(995\) −9.16101 −0.290424
\(996\) 0 0
\(997\) 12.2665 0.388485 0.194242 0.980954i \(-0.437775\pi\)
0.194242 + 0.980954i \(0.437775\pi\)
\(998\) 0 0
\(999\) −6.31045 −0.199654
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 2004.2.a.c.1.4 9
3.2 odd 2 6012.2.a.i.1.6 9
4.3 odd 2 8016.2.a.bc.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.c.1.4 9 1.1 even 1 trivial
6012.2.a.i.1.6 9 3.2 odd 2
8016.2.a.bc.1.4 9 4.3 odd 2