Properties

Label 4008.2.a.m.1.8
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $13$
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] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.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: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.25667\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.25667 q^{5} -4.55521 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.25667 q^{5} -4.55521 q^{7} +1.00000 q^{9} -5.23461 q^{11} -1.93555 q^{13} +1.25667 q^{15} +7.59670 q^{17} -4.42731 q^{19} -4.55521 q^{21} +6.61071 q^{23} -3.42078 q^{25} +1.00000 q^{27} +3.73263 q^{29} -1.27190 q^{31} -5.23461 q^{33} -5.72440 q^{35} +6.87153 q^{37} -1.93555 q^{39} +8.71632 q^{41} +5.25563 q^{43} +1.25667 q^{45} -6.55882 q^{47} +13.7500 q^{49} +7.59670 q^{51} +3.93979 q^{53} -6.57819 q^{55} -4.42731 q^{57} +12.3546 q^{59} -0.282357 q^{61} -4.55521 q^{63} -2.43235 q^{65} -8.25155 q^{67} +6.61071 q^{69} -1.66362 q^{71} -3.89727 q^{73} -3.42078 q^{75} +23.8448 q^{77} +6.21505 q^{79} +1.00000 q^{81} +14.1118 q^{83} +9.54655 q^{85} +3.73263 q^{87} -6.93566 q^{89} +8.81684 q^{91} -1.27190 q^{93} -5.56367 q^{95} -3.11931 q^{97} -5.23461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 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
\(5\) 1.25667 0.562000 0.281000 0.959708i \(-0.409334\pi\)
0.281000 + 0.959708i \(0.409334\pi\)
\(6\) 0 0
\(7\) −4.55521 −1.72171 −0.860854 0.508852i \(-0.830070\pi\)
−0.860854 + 0.508852i \(0.830070\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.23461 −1.57829 −0.789147 0.614204i \(-0.789477\pi\)
−0.789147 + 0.614204i \(0.789477\pi\)
\(12\) 0 0
\(13\) −1.93555 −0.536825 −0.268412 0.963304i \(-0.586499\pi\)
−0.268412 + 0.963304i \(0.586499\pi\)
\(14\) 0 0
\(15\) 1.25667 0.324471
\(16\) 0 0
\(17\) 7.59670 1.84247 0.921235 0.389007i \(-0.127182\pi\)
0.921235 + 0.389007i \(0.127182\pi\)
\(18\) 0 0
\(19\) −4.42731 −1.01569 −0.507847 0.861447i \(-0.669558\pi\)
−0.507847 + 0.861447i \(0.669558\pi\)
\(20\) 0 0
\(21\) −4.55521 −0.994029
\(22\) 0 0
\(23\) 6.61071 1.37843 0.689214 0.724558i \(-0.257956\pi\)
0.689214 + 0.724558i \(0.257956\pi\)
\(24\) 0 0
\(25\) −3.42078 −0.684156
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.73263 0.693132 0.346566 0.938026i \(-0.387348\pi\)
0.346566 + 0.938026i \(0.387348\pi\)
\(30\) 0 0
\(31\) −1.27190 −0.228440 −0.114220 0.993455i \(-0.536437\pi\)
−0.114220 + 0.993455i \(0.536437\pi\)
\(32\) 0 0
\(33\) −5.23461 −0.911229
\(34\) 0 0
\(35\) −5.72440 −0.967601
\(36\) 0 0
\(37\) 6.87153 1.12967 0.564836 0.825203i \(-0.308939\pi\)
0.564836 + 0.825203i \(0.308939\pi\)
\(38\) 0 0
\(39\) −1.93555 −0.309936
\(40\) 0 0
\(41\) 8.71632 1.36126 0.680631 0.732627i \(-0.261706\pi\)
0.680631 + 0.732627i \(0.261706\pi\)
\(42\) 0 0
\(43\) 5.25563 0.801476 0.400738 0.916193i \(-0.368754\pi\)
0.400738 + 0.916193i \(0.368754\pi\)
\(44\) 0 0
\(45\) 1.25667 0.187333
\(46\) 0 0
\(47\) −6.55882 −0.956702 −0.478351 0.878169i \(-0.658765\pi\)
−0.478351 + 0.878169i \(0.658765\pi\)
\(48\) 0 0
\(49\) 13.7500 1.96428
\(50\) 0 0
\(51\) 7.59670 1.06375
\(52\) 0 0
\(53\) 3.93979 0.541172 0.270586 0.962696i \(-0.412783\pi\)
0.270586 + 0.962696i \(0.412783\pi\)
\(54\) 0 0
\(55\) −6.57819 −0.887002
\(56\) 0 0
\(57\) −4.42731 −0.586411
\(58\) 0 0
\(59\) 12.3546 1.60844 0.804218 0.594335i \(-0.202584\pi\)
0.804218 + 0.594335i \(0.202584\pi\)
\(60\) 0 0
\(61\) −0.282357 −0.0361522 −0.0180761 0.999837i \(-0.505754\pi\)
−0.0180761 + 0.999837i \(0.505754\pi\)
\(62\) 0 0
\(63\) −4.55521 −0.573903
\(64\) 0 0
\(65\) −2.43235 −0.301696
\(66\) 0 0
\(67\) −8.25155 −1.00809 −0.504043 0.863678i \(-0.668155\pi\)
−0.504043 + 0.863678i \(0.668155\pi\)
\(68\) 0 0
\(69\) 6.61071 0.795836
\(70\) 0 0
\(71\) −1.66362 −0.197435 −0.0987175 0.995115i \(-0.531474\pi\)
−0.0987175 + 0.995115i \(0.531474\pi\)
\(72\) 0 0
\(73\) −3.89727 −0.456141 −0.228071 0.973645i \(-0.573242\pi\)
−0.228071 + 0.973645i \(0.573242\pi\)
\(74\) 0 0
\(75\) −3.42078 −0.394997
\(76\) 0 0
\(77\) 23.8448 2.71736
\(78\) 0 0
\(79\) 6.21505 0.699248 0.349624 0.936890i \(-0.386309\pi\)
0.349624 + 0.936890i \(0.386309\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.1118 1.54898 0.774488 0.632588i \(-0.218007\pi\)
0.774488 + 0.632588i \(0.218007\pi\)
\(84\) 0 0
\(85\) 9.54655 1.03547
\(86\) 0 0
\(87\) 3.73263 0.400180
\(88\) 0 0
\(89\) −6.93566 −0.735178 −0.367589 0.929988i \(-0.619817\pi\)
−0.367589 + 0.929988i \(0.619817\pi\)
\(90\) 0 0
\(91\) 8.81684 0.924256
\(92\) 0 0
\(93\) −1.27190 −0.131890
\(94\) 0 0
\(95\) −5.56367 −0.570820
\(96\) 0 0
\(97\) −3.11931 −0.316718 −0.158359 0.987382i \(-0.550620\pi\)
−0.158359 + 0.987382i \(0.550620\pi\)
\(98\) 0 0
\(99\) −5.23461 −0.526098
\(100\) 0 0
\(101\) 11.8680 1.18091 0.590457 0.807069i \(-0.298948\pi\)
0.590457 + 0.807069i \(0.298948\pi\)
\(102\) 0 0
\(103\) −2.31747 −0.228347 −0.114174 0.993461i \(-0.536422\pi\)
−0.114174 + 0.993461i \(0.536422\pi\)
\(104\) 0 0
\(105\) −5.72440 −0.558645
\(106\) 0 0
\(107\) −1.55158 −0.149997 −0.0749985 0.997184i \(-0.523895\pi\)
−0.0749985 + 0.997184i \(0.523895\pi\)
\(108\) 0 0
\(109\) 5.37197 0.514541 0.257271 0.966339i \(-0.417177\pi\)
0.257271 + 0.966339i \(0.417177\pi\)
\(110\) 0 0
\(111\) 6.87153 0.652217
\(112\) 0 0
\(113\) 17.3036 1.62779 0.813893 0.581014i \(-0.197344\pi\)
0.813893 + 0.581014i \(0.197344\pi\)
\(114\) 0 0
\(115\) 8.30748 0.774677
\(116\) 0 0
\(117\) −1.93555 −0.178942
\(118\) 0 0
\(119\) −34.6046 −3.17220
\(120\) 0 0
\(121\) 16.4012 1.49101
\(122\) 0 0
\(123\) 8.71632 0.785925
\(124\) 0 0
\(125\) −10.5821 −0.946496
\(126\) 0 0
\(127\) 7.88097 0.699323 0.349662 0.936876i \(-0.386297\pi\)
0.349662 + 0.936876i \(0.386297\pi\)
\(128\) 0 0
\(129\) 5.25563 0.462732
\(130\) 0 0
\(131\) 20.2297 1.76748 0.883739 0.467979i \(-0.155018\pi\)
0.883739 + 0.467979i \(0.155018\pi\)
\(132\) 0 0
\(133\) 20.1673 1.74873
\(134\) 0 0
\(135\) 1.25667 0.108157
\(136\) 0 0
\(137\) −3.73853 −0.319404 −0.159702 0.987165i \(-0.551053\pi\)
−0.159702 + 0.987165i \(0.551053\pi\)
\(138\) 0 0
\(139\) 13.1822 1.11810 0.559049 0.829135i \(-0.311167\pi\)
0.559049 + 0.829135i \(0.311167\pi\)
\(140\) 0 0
\(141\) −6.55882 −0.552352
\(142\) 0 0
\(143\) 10.1318 0.847268
\(144\) 0 0
\(145\) 4.69069 0.389540
\(146\) 0 0
\(147\) 13.7500 1.13408
\(148\) 0 0
\(149\) −16.0892 −1.31808 −0.659040 0.752108i \(-0.729037\pi\)
−0.659040 + 0.752108i \(0.729037\pi\)
\(150\) 0 0
\(151\) 19.2676 1.56797 0.783987 0.620778i \(-0.213183\pi\)
0.783987 + 0.620778i \(0.213183\pi\)
\(152\) 0 0
\(153\) 7.59670 0.614156
\(154\) 0 0
\(155\) −1.59836 −0.128383
\(156\) 0 0
\(157\) −8.92368 −0.712187 −0.356094 0.934450i \(-0.615892\pi\)
−0.356094 + 0.934450i \(0.615892\pi\)
\(158\) 0 0
\(159\) 3.93979 0.312446
\(160\) 0 0
\(161\) −30.1132 −2.37325
\(162\) 0 0
\(163\) −2.94760 −0.230874 −0.115437 0.993315i \(-0.536827\pi\)
−0.115437 + 0.993315i \(0.536827\pi\)
\(164\) 0 0
\(165\) −6.57819 −0.512111
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −9.25365 −0.711819
\(170\) 0 0
\(171\) −4.42731 −0.338564
\(172\) 0 0
\(173\) −12.5486 −0.954055 −0.477028 0.878888i \(-0.658286\pi\)
−0.477028 + 0.878888i \(0.658286\pi\)
\(174\) 0 0
\(175\) 15.5824 1.17792
\(176\) 0 0
\(177\) 12.3546 0.928631
\(178\) 0 0
\(179\) 3.07676 0.229968 0.114984 0.993367i \(-0.463318\pi\)
0.114984 + 0.993367i \(0.463318\pi\)
\(180\) 0 0
\(181\) 2.50830 0.186440 0.0932201 0.995646i \(-0.470284\pi\)
0.0932201 + 0.995646i \(0.470284\pi\)
\(182\) 0 0
\(183\) −0.282357 −0.0208725
\(184\) 0 0
\(185\) 8.63525 0.634877
\(186\) 0 0
\(187\) −39.7658 −2.90796
\(188\) 0 0
\(189\) −4.55521 −0.331343
\(190\) 0 0
\(191\) −3.30849 −0.239394 −0.119697 0.992810i \(-0.538192\pi\)
−0.119697 + 0.992810i \(0.538192\pi\)
\(192\) 0 0
\(193\) −18.4096 −1.32515 −0.662577 0.748994i \(-0.730537\pi\)
−0.662577 + 0.748994i \(0.730537\pi\)
\(194\) 0 0
\(195\) −2.43235 −0.174184
\(196\) 0 0
\(197\) −13.0745 −0.931520 −0.465760 0.884911i \(-0.654219\pi\)
−0.465760 + 0.884911i \(0.654219\pi\)
\(198\) 0 0
\(199\) −18.8563 −1.33669 −0.668343 0.743853i \(-0.732996\pi\)
−0.668343 + 0.743853i \(0.732996\pi\)
\(200\) 0 0
\(201\) −8.25155 −0.582019
\(202\) 0 0
\(203\) −17.0029 −1.19337
\(204\) 0 0
\(205\) 10.9536 0.765029
\(206\) 0 0
\(207\) 6.61071 0.459476
\(208\) 0 0
\(209\) 23.1752 1.60306
\(210\) 0 0
\(211\) 7.39616 0.509173 0.254586 0.967050i \(-0.418061\pi\)
0.254586 + 0.967050i \(0.418061\pi\)
\(212\) 0 0
\(213\) −1.66362 −0.113989
\(214\) 0 0
\(215\) 6.60460 0.450430
\(216\) 0 0
\(217\) 5.79377 0.393307
\(218\) 0 0
\(219\) −3.89727 −0.263353
\(220\) 0 0
\(221\) −14.7038 −0.989083
\(222\) 0 0
\(223\) −1.81997 −0.121874 −0.0609371 0.998142i \(-0.519409\pi\)
−0.0609371 + 0.998142i \(0.519409\pi\)
\(224\) 0 0
\(225\) −3.42078 −0.228052
\(226\) 0 0
\(227\) 21.8780 1.45209 0.726045 0.687647i \(-0.241356\pi\)
0.726045 + 0.687647i \(0.241356\pi\)
\(228\) 0 0
\(229\) 1.04962 0.0693609 0.0346804 0.999398i \(-0.488959\pi\)
0.0346804 + 0.999398i \(0.488959\pi\)
\(230\) 0 0
\(231\) 23.8448 1.56887
\(232\) 0 0
\(233\) −17.9898 −1.17855 −0.589274 0.807933i \(-0.700586\pi\)
−0.589274 + 0.807933i \(0.700586\pi\)
\(234\) 0 0
\(235\) −8.24227 −0.537667
\(236\) 0 0
\(237\) 6.21505 0.403711
\(238\) 0 0
\(239\) −27.1655 −1.75719 −0.878596 0.477565i \(-0.841519\pi\)
−0.878596 + 0.477565i \(0.841519\pi\)
\(240\) 0 0
\(241\) 21.7168 1.39890 0.699451 0.714680i \(-0.253428\pi\)
0.699451 + 0.714680i \(0.253428\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 17.2792 1.10393
\(246\) 0 0
\(247\) 8.56927 0.545249
\(248\) 0 0
\(249\) 14.1118 0.894302
\(250\) 0 0
\(251\) 30.4390 1.92130 0.960648 0.277770i \(-0.0895952\pi\)
0.960648 + 0.277770i \(0.0895952\pi\)
\(252\) 0 0
\(253\) −34.6045 −2.17557
\(254\) 0 0
\(255\) 9.54655 0.597828
\(256\) 0 0
\(257\) −23.0748 −1.43937 −0.719684 0.694301i \(-0.755714\pi\)
−0.719684 + 0.694301i \(0.755714\pi\)
\(258\) 0 0
\(259\) −31.3013 −1.94497
\(260\) 0 0
\(261\) 3.73263 0.231044
\(262\) 0 0
\(263\) 21.5164 1.32676 0.663379 0.748284i \(-0.269122\pi\)
0.663379 + 0.748284i \(0.269122\pi\)
\(264\) 0 0
\(265\) 4.95102 0.304139
\(266\) 0 0
\(267\) −6.93566 −0.424455
\(268\) 0 0
\(269\) 16.2955 0.993554 0.496777 0.867878i \(-0.334517\pi\)
0.496777 + 0.867878i \(0.334517\pi\)
\(270\) 0 0
\(271\) −25.9830 −1.57835 −0.789177 0.614166i \(-0.789493\pi\)
−0.789177 + 0.614166i \(0.789493\pi\)
\(272\) 0 0
\(273\) 8.81684 0.533619
\(274\) 0 0
\(275\) 17.9064 1.07980
\(276\) 0 0
\(277\) −8.26377 −0.496522 −0.248261 0.968693i \(-0.579859\pi\)
−0.248261 + 0.968693i \(0.579859\pi\)
\(278\) 0 0
\(279\) −1.27190 −0.0761466
\(280\) 0 0
\(281\) −0.956330 −0.0570499 −0.0285249 0.999593i \(-0.509081\pi\)
−0.0285249 + 0.999593i \(0.509081\pi\)
\(282\) 0 0
\(283\) 7.81438 0.464517 0.232258 0.972654i \(-0.425389\pi\)
0.232258 + 0.972654i \(0.425389\pi\)
\(284\) 0 0
\(285\) −5.56367 −0.329563
\(286\) 0 0
\(287\) −39.7047 −2.34369
\(288\) 0 0
\(289\) 40.7098 2.39469
\(290\) 0 0
\(291\) −3.11931 −0.182857
\(292\) 0 0
\(293\) −18.0569 −1.05490 −0.527448 0.849587i \(-0.676851\pi\)
−0.527448 + 0.849587i \(0.676851\pi\)
\(294\) 0 0
\(295\) 15.5257 0.903942
\(296\) 0 0
\(297\) −5.23461 −0.303743
\(298\) 0 0
\(299\) −12.7954 −0.739974
\(300\) 0 0
\(301\) −23.9405 −1.37991
\(302\) 0 0
\(303\) 11.8680 0.681801
\(304\) 0 0
\(305\) −0.354830 −0.0203175
\(306\) 0 0
\(307\) −26.9495 −1.53809 −0.769046 0.639193i \(-0.779268\pi\)
−0.769046 + 0.639193i \(0.779268\pi\)
\(308\) 0 0
\(309\) −2.31747 −0.131836
\(310\) 0 0
\(311\) −7.51738 −0.426272 −0.213136 0.977023i \(-0.568368\pi\)
−0.213136 + 0.977023i \(0.568368\pi\)
\(312\) 0 0
\(313\) 24.9480 1.41015 0.705073 0.709135i \(-0.250914\pi\)
0.705073 + 0.709135i \(0.250914\pi\)
\(314\) 0 0
\(315\) −5.72440 −0.322534
\(316\) 0 0
\(317\) 28.0071 1.57303 0.786517 0.617569i \(-0.211882\pi\)
0.786517 + 0.617569i \(0.211882\pi\)
\(318\) 0 0
\(319\) −19.5389 −1.09397
\(320\) 0 0
\(321\) −1.55158 −0.0866008
\(322\) 0 0
\(323\) −33.6329 −1.87138
\(324\) 0 0
\(325\) 6.62108 0.367272
\(326\) 0 0
\(327\) 5.37197 0.297071
\(328\) 0 0
\(329\) 29.8768 1.64716
\(330\) 0 0
\(331\) 7.36380 0.404751 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(332\) 0 0
\(333\) 6.87153 0.376558
\(334\) 0 0
\(335\) −10.3695 −0.566545
\(336\) 0 0
\(337\) 12.2355 0.666512 0.333256 0.942836i \(-0.391853\pi\)
0.333256 + 0.942836i \(0.391853\pi\)
\(338\) 0 0
\(339\) 17.3036 0.939803
\(340\) 0 0
\(341\) 6.65789 0.360545
\(342\) 0 0
\(343\) −30.7475 −1.66021
\(344\) 0 0
\(345\) 8.30748 0.447260
\(346\) 0 0
\(347\) 19.3771 1.04022 0.520109 0.854100i \(-0.325891\pi\)
0.520109 + 0.854100i \(0.325891\pi\)
\(348\) 0 0
\(349\) −7.13099 −0.381713 −0.190857 0.981618i \(-0.561127\pi\)
−0.190857 + 0.981618i \(0.561127\pi\)
\(350\) 0 0
\(351\) −1.93555 −0.103312
\(352\) 0 0
\(353\) 10.7743 0.573457 0.286729 0.958012i \(-0.407432\pi\)
0.286729 + 0.958012i \(0.407432\pi\)
\(354\) 0 0
\(355\) −2.09062 −0.110959
\(356\) 0 0
\(357\) −34.6046 −1.83147
\(358\) 0 0
\(359\) −21.8570 −1.15357 −0.576784 0.816896i \(-0.695693\pi\)
−0.576784 + 0.816896i \(0.695693\pi\)
\(360\) 0 0
\(361\) 0.601031 0.0316332
\(362\) 0 0
\(363\) 16.4012 0.860838
\(364\) 0 0
\(365\) −4.89759 −0.256352
\(366\) 0 0
\(367\) −0.0341568 −0.00178297 −0.000891485 1.00000i \(-0.500284\pi\)
−0.000891485 1.00000i \(0.500284\pi\)
\(368\) 0 0
\(369\) 8.71632 0.453754
\(370\) 0 0
\(371\) −17.9466 −0.931740
\(372\) 0 0
\(373\) −4.34690 −0.225074 −0.112537 0.993648i \(-0.535898\pi\)
−0.112537 + 0.993648i \(0.535898\pi\)
\(374\) 0 0
\(375\) −10.5821 −0.546460
\(376\) 0 0
\(377\) −7.22469 −0.372090
\(378\) 0 0
\(379\) 32.2351 1.65581 0.827904 0.560870i \(-0.189533\pi\)
0.827904 + 0.560870i \(0.189533\pi\)
\(380\) 0 0
\(381\) 7.88097 0.403754
\(382\) 0 0
\(383\) 23.6117 1.20650 0.603250 0.797552i \(-0.293872\pi\)
0.603250 + 0.797552i \(0.293872\pi\)
\(384\) 0 0
\(385\) 29.9650 1.52716
\(386\) 0 0
\(387\) 5.25563 0.267159
\(388\) 0 0
\(389\) −31.0376 −1.57367 −0.786834 0.617164i \(-0.788281\pi\)
−0.786834 + 0.617164i \(0.788281\pi\)
\(390\) 0 0
\(391\) 50.2195 2.53971
\(392\) 0 0
\(393\) 20.2297 1.02045
\(394\) 0 0
\(395\) 7.81028 0.392978
\(396\) 0 0
\(397\) 3.95462 0.198476 0.0992382 0.995064i \(-0.468359\pi\)
0.0992382 + 0.995064i \(0.468359\pi\)
\(398\) 0 0
\(399\) 20.1673 1.00963
\(400\) 0 0
\(401\) 19.9560 0.996556 0.498278 0.867017i \(-0.333966\pi\)
0.498278 + 0.867017i \(0.333966\pi\)
\(402\) 0 0
\(403\) 2.46182 0.122632
\(404\) 0 0
\(405\) 1.25667 0.0624445
\(406\) 0 0
\(407\) −35.9698 −1.78296
\(408\) 0 0
\(409\) 38.6158 1.90943 0.954714 0.297524i \(-0.0961607\pi\)
0.954714 + 0.297524i \(0.0961607\pi\)
\(410\) 0 0
\(411\) −3.73853 −0.184408
\(412\) 0 0
\(413\) −56.2780 −2.76926
\(414\) 0 0
\(415\) 17.7340 0.870526
\(416\) 0 0
\(417\) 13.1822 0.645534
\(418\) 0 0
\(419\) 4.87110 0.237969 0.118984 0.992896i \(-0.462036\pi\)
0.118984 + 0.992896i \(0.462036\pi\)
\(420\) 0 0
\(421\) 10.1671 0.495512 0.247756 0.968822i \(-0.420307\pi\)
0.247756 + 0.968822i \(0.420307\pi\)
\(422\) 0 0
\(423\) −6.55882 −0.318901
\(424\) 0 0
\(425\) −25.9866 −1.26054
\(426\) 0 0
\(427\) 1.28620 0.0622435
\(428\) 0 0
\(429\) 10.1318 0.489170
\(430\) 0 0
\(431\) −34.5825 −1.66578 −0.832890 0.553438i \(-0.813315\pi\)
−0.832890 + 0.553438i \(0.813315\pi\)
\(432\) 0 0
\(433\) −30.2208 −1.45232 −0.726160 0.687526i \(-0.758697\pi\)
−0.726160 + 0.687526i \(0.758697\pi\)
\(434\) 0 0
\(435\) 4.69069 0.224901
\(436\) 0 0
\(437\) −29.2676 −1.40006
\(438\) 0 0
\(439\) 3.35552 0.160150 0.0800751 0.996789i \(-0.474484\pi\)
0.0800751 + 0.996789i \(0.474484\pi\)
\(440\) 0 0
\(441\) 13.7500 0.654760
\(442\) 0 0
\(443\) 24.9007 1.18307 0.591533 0.806281i \(-0.298523\pi\)
0.591533 + 0.806281i \(0.298523\pi\)
\(444\) 0 0
\(445\) −8.71584 −0.413170
\(446\) 0 0
\(447\) −16.0892 −0.760993
\(448\) 0 0
\(449\) 23.2157 1.09562 0.547808 0.836604i \(-0.315463\pi\)
0.547808 + 0.836604i \(0.315463\pi\)
\(450\) 0 0
\(451\) −45.6266 −2.14847
\(452\) 0 0
\(453\) 19.2676 0.905270
\(454\) 0 0
\(455\) 11.0799 0.519432
\(456\) 0 0
\(457\) −1.56399 −0.0731602 −0.0365801 0.999331i \(-0.511646\pi\)
−0.0365801 + 0.999331i \(0.511646\pi\)
\(458\) 0 0
\(459\) 7.59670 0.354583
\(460\) 0 0
\(461\) −19.9153 −0.927550 −0.463775 0.885953i \(-0.653505\pi\)
−0.463775 + 0.885953i \(0.653505\pi\)
\(462\) 0 0
\(463\) 26.9153 1.25086 0.625431 0.780280i \(-0.284923\pi\)
0.625431 + 0.780280i \(0.284923\pi\)
\(464\) 0 0
\(465\) −1.59836 −0.0741221
\(466\) 0 0
\(467\) −39.6163 −1.83322 −0.916611 0.399780i \(-0.869087\pi\)
−0.916611 + 0.399780i \(0.869087\pi\)
\(468\) 0 0
\(469\) 37.5875 1.73563
\(470\) 0 0
\(471\) −8.92368 −0.411182
\(472\) 0 0
\(473\) −27.5112 −1.26497
\(474\) 0 0
\(475\) 15.1448 0.694892
\(476\) 0 0
\(477\) 3.93979 0.180391
\(478\) 0 0
\(479\) 38.0876 1.74027 0.870134 0.492815i \(-0.164032\pi\)
0.870134 + 0.492815i \(0.164032\pi\)
\(480\) 0 0
\(481\) −13.3002 −0.606436
\(482\) 0 0
\(483\) −30.1132 −1.37020
\(484\) 0 0
\(485\) −3.91994 −0.177995
\(486\) 0 0
\(487\) −4.70083 −0.213015 −0.106508 0.994312i \(-0.533967\pi\)
−0.106508 + 0.994312i \(0.533967\pi\)
\(488\) 0 0
\(489\) −2.94760 −0.133295
\(490\) 0 0
\(491\) 26.9587 1.21663 0.608315 0.793696i \(-0.291846\pi\)
0.608315 + 0.793696i \(0.291846\pi\)
\(492\) 0 0
\(493\) 28.3556 1.27707
\(494\) 0 0
\(495\) −6.57819 −0.295667
\(496\) 0 0
\(497\) 7.57813 0.339926
\(498\) 0 0
\(499\) −29.4450 −1.31814 −0.659069 0.752082i \(-0.729050\pi\)
−0.659069 + 0.752082i \(0.729050\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 27.2438 1.21474 0.607371 0.794418i \(-0.292224\pi\)
0.607371 + 0.794418i \(0.292224\pi\)
\(504\) 0 0
\(505\) 14.9142 0.663674
\(506\) 0 0
\(507\) −9.25365 −0.410969
\(508\) 0 0
\(509\) 5.63468 0.249753 0.124876 0.992172i \(-0.460147\pi\)
0.124876 + 0.992172i \(0.460147\pi\)
\(510\) 0 0
\(511\) 17.7529 0.785342
\(512\) 0 0
\(513\) −4.42731 −0.195470
\(514\) 0 0
\(515\) −2.91230 −0.128331
\(516\) 0 0
\(517\) 34.3329 1.50996
\(518\) 0 0
\(519\) −12.5486 −0.550824
\(520\) 0 0
\(521\) −6.45179 −0.282658 −0.141329 0.989963i \(-0.545138\pi\)
−0.141329 + 0.989963i \(0.545138\pi\)
\(522\) 0 0
\(523\) 4.82181 0.210843 0.105422 0.994428i \(-0.466381\pi\)
0.105422 + 0.994428i \(0.466381\pi\)
\(524\) 0 0
\(525\) 15.5824 0.680070
\(526\) 0 0
\(527\) −9.66222 −0.420893
\(528\) 0 0
\(529\) 20.7015 0.900063
\(530\) 0 0
\(531\) 12.3546 0.536145
\(532\) 0 0
\(533\) −16.8709 −0.730759
\(534\) 0 0
\(535\) −1.94983 −0.0842984
\(536\) 0 0
\(537\) 3.07676 0.132772
\(538\) 0 0
\(539\) −71.9757 −3.10021
\(540\) 0 0
\(541\) 35.7479 1.53692 0.768461 0.639896i \(-0.221023\pi\)
0.768461 + 0.639896i \(0.221023\pi\)
\(542\) 0 0
\(543\) 2.50830 0.107641
\(544\) 0 0
\(545\) 6.75080 0.289172
\(546\) 0 0
\(547\) 2.22907 0.0953084 0.0476542 0.998864i \(-0.484825\pi\)
0.0476542 + 0.998864i \(0.484825\pi\)
\(548\) 0 0
\(549\) −0.282357 −0.0120507
\(550\) 0 0
\(551\) −16.5255 −0.704009
\(552\) 0 0
\(553\) −28.3109 −1.20390
\(554\) 0 0
\(555\) 8.63525 0.366546
\(556\) 0 0
\(557\) 14.8215 0.628008 0.314004 0.949422i \(-0.398330\pi\)
0.314004 + 0.949422i \(0.398330\pi\)
\(558\) 0 0
\(559\) −10.1725 −0.430252
\(560\) 0 0
\(561\) −39.7658 −1.67891
\(562\) 0 0
\(563\) −21.1799 −0.892627 −0.446314 0.894877i \(-0.647263\pi\)
−0.446314 + 0.894877i \(0.647263\pi\)
\(564\) 0 0
\(565\) 21.7449 0.914817
\(566\) 0 0
\(567\) −4.55521 −0.191301
\(568\) 0 0
\(569\) −7.59884 −0.318560 −0.159280 0.987233i \(-0.550917\pi\)
−0.159280 + 0.987233i \(0.550917\pi\)
\(570\) 0 0
\(571\) −10.3154 −0.431688 −0.215844 0.976428i \(-0.569250\pi\)
−0.215844 + 0.976428i \(0.569250\pi\)
\(572\) 0 0
\(573\) −3.30849 −0.138214
\(574\) 0 0
\(575\) −22.6138 −0.943059
\(576\) 0 0
\(577\) 16.2672 0.677211 0.338605 0.940928i \(-0.390045\pi\)
0.338605 + 0.940928i \(0.390045\pi\)
\(578\) 0 0
\(579\) −18.4096 −0.765078
\(580\) 0 0
\(581\) −64.2825 −2.66689
\(582\) 0 0
\(583\) −20.6233 −0.854129
\(584\) 0 0
\(585\) −2.43235 −0.100565
\(586\) 0 0
\(587\) −26.5861 −1.09733 −0.548664 0.836043i \(-0.684863\pi\)
−0.548664 + 0.836043i \(0.684863\pi\)
\(588\) 0 0
\(589\) 5.63108 0.232025
\(590\) 0 0
\(591\) −13.0745 −0.537813
\(592\) 0 0
\(593\) 11.4438 0.469939 0.234969 0.972003i \(-0.424501\pi\)
0.234969 + 0.972003i \(0.424501\pi\)
\(594\) 0 0
\(595\) −43.4866 −1.78277
\(596\) 0 0
\(597\) −18.8563 −0.771736
\(598\) 0 0
\(599\) 17.6317 0.720410 0.360205 0.932873i \(-0.382707\pi\)
0.360205 + 0.932873i \(0.382707\pi\)
\(600\) 0 0
\(601\) −5.54770 −0.226295 −0.113148 0.993578i \(-0.536093\pi\)
−0.113148 + 0.993578i \(0.536093\pi\)
\(602\) 0 0
\(603\) −8.25155 −0.336029
\(604\) 0 0
\(605\) 20.6109 0.837951
\(606\) 0 0
\(607\) 21.5641 0.875259 0.437630 0.899155i \(-0.355818\pi\)
0.437630 + 0.899155i \(0.355818\pi\)
\(608\) 0 0
\(609\) −17.0029 −0.688993
\(610\) 0 0
\(611\) 12.6949 0.513581
\(612\) 0 0
\(613\) 47.5125 1.91901 0.959506 0.281687i \(-0.0908939\pi\)
0.959506 + 0.281687i \(0.0908939\pi\)
\(614\) 0 0
\(615\) 10.9536 0.441690
\(616\) 0 0
\(617\) −3.79509 −0.152785 −0.0763923 0.997078i \(-0.524340\pi\)
−0.0763923 + 0.997078i \(0.524340\pi\)
\(618\) 0 0
\(619\) 10.0986 0.405898 0.202949 0.979189i \(-0.434947\pi\)
0.202949 + 0.979189i \(0.434947\pi\)
\(620\) 0 0
\(621\) 6.61071 0.265279
\(622\) 0 0
\(623\) 31.5934 1.26576
\(624\) 0 0
\(625\) 3.80561 0.152224
\(626\) 0 0
\(627\) 23.1752 0.925529
\(628\) 0 0
\(629\) 52.2009 2.08139
\(630\) 0 0
\(631\) −30.5310 −1.21542 −0.607709 0.794160i \(-0.707911\pi\)
−0.607709 + 0.794160i \(0.707911\pi\)
\(632\) 0 0
\(633\) 7.39616 0.293971
\(634\) 0 0
\(635\) 9.90379 0.393020
\(636\) 0 0
\(637\) −26.6137 −1.05447
\(638\) 0 0
\(639\) −1.66362 −0.0658117
\(640\) 0 0
\(641\) −3.23785 −0.127887 −0.0639437 0.997954i \(-0.520368\pi\)
−0.0639437 + 0.997954i \(0.520368\pi\)
\(642\) 0 0
\(643\) −30.2108 −1.19140 −0.595698 0.803209i \(-0.703124\pi\)
−0.595698 + 0.803209i \(0.703124\pi\)
\(644\) 0 0
\(645\) 6.60460 0.260056
\(646\) 0 0
\(647\) −31.9382 −1.25562 −0.627810 0.778367i \(-0.716048\pi\)
−0.627810 + 0.778367i \(0.716048\pi\)
\(648\) 0 0
\(649\) −64.6717 −2.53859
\(650\) 0 0
\(651\) 5.79377 0.227076
\(652\) 0 0
\(653\) −21.2765 −0.832613 −0.416307 0.909224i \(-0.636676\pi\)
−0.416307 + 0.909224i \(0.636676\pi\)
\(654\) 0 0
\(655\) 25.4221 0.993324
\(656\) 0 0
\(657\) −3.89727 −0.152047
\(658\) 0 0
\(659\) −20.4493 −0.796593 −0.398297 0.917257i \(-0.630399\pi\)
−0.398297 + 0.917257i \(0.630399\pi\)
\(660\) 0 0
\(661\) 22.9118 0.891165 0.445582 0.895241i \(-0.352997\pi\)
0.445582 + 0.895241i \(0.352997\pi\)
\(662\) 0 0
\(663\) −14.7038 −0.571047
\(664\) 0 0
\(665\) 25.3437 0.982786
\(666\) 0 0
\(667\) 24.6753 0.955432
\(668\) 0 0
\(669\) −1.81997 −0.0703641
\(670\) 0 0
\(671\) 1.47803 0.0570588
\(672\) 0 0
\(673\) 29.8721 1.15149 0.575743 0.817630i \(-0.304713\pi\)
0.575743 + 0.817630i \(0.304713\pi\)
\(674\) 0 0
\(675\) −3.42078 −0.131666
\(676\) 0 0
\(677\) −10.0786 −0.387353 −0.193676 0.981065i \(-0.562041\pi\)
−0.193676 + 0.981065i \(0.562041\pi\)
\(678\) 0 0
\(679\) 14.2091 0.545295
\(680\) 0 0
\(681\) 21.8780 0.838365
\(682\) 0 0
\(683\) 17.1938 0.657901 0.328951 0.944347i \(-0.393305\pi\)
0.328951 + 0.944347i \(0.393305\pi\)
\(684\) 0 0
\(685\) −4.69810 −0.179505
\(686\) 0 0
\(687\) 1.04962 0.0400455
\(688\) 0 0
\(689\) −7.62566 −0.290515
\(690\) 0 0
\(691\) 4.30843 0.163900 0.0819502 0.996636i \(-0.473885\pi\)
0.0819502 + 0.996636i \(0.473885\pi\)
\(692\) 0 0
\(693\) 23.8448 0.905788
\(694\) 0 0
\(695\) 16.5657 0.628371
\(696\) 0 0
\(697\) 66.2153 2.50808
\(698\) 0 0
\(699\) −17.9898 −0.680435
\(700\) 0 0
\(701\) −20.4059 −0.770720 −0.385360 0.922766i \(-0.625923\pi\)
−0.385360 + 0.922766i \(0.625923\pi\)
\(702\) 0 0
\(703\) −30.4224 −1.14740
\(704\) 0 0
\(705\) −8.24227 −0.310422
\(706\) 0 0
\(707\) −54.0614 −2.03319
\(708\) 0 0
\(709\) −8.65515 −0.325051 −0.162525 0.986704i \(-0.551964\pi\)
−0.162525 + 0.986704i \(0.551964\pi\)
\(710\) 0 0
\(711\) 6.21505 0.233083
\(712\) 0 0
\(713\) −8.40815 −0.314888
\(714\) 0 0
\(715\) 12.7324 0.476165
\(716\) 0 0
\(717\) −27.1655 −1.01452
\(718\) 0 0
\(719\) −37.7603 −1.40822 −0.704111 0.710090i \(-0.748654\pi\)
−0.704111 + 0.710090i \(0.748654\pi\)
\(720\) 0 0
\(721\) 10.5566 0.393148
\(722\) 0 0
\(723\) 21.7168 0.807657
\(724\) 0 0
\(725\) −12.7685 −0.474210
\(726\) 0 0
\(727\) 50.5464 1.87466 0.937332 0.348438i \(-0.113288\pi\)
0.937332 + 0.348438i \(0.113288\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 39.9254 1.47669
\(732\) 0 0
\(733\) 45.7422 1.68953 0.844763 0.535140i \(-0.179741\pi\)
0.844763 + 0.535140i \(0.179741\pi\)
\(734\) 0 0
\(735\) 17.2792 0.637352
\(736\) 0 0
\(737\) 43.1936 1.59106
\(738\) 0 0
\(739\) 9.01671 0.331685 0.165842 0.986152i \(-0.446966\pi\)
0.165842 + 0.986152i \(0.446966\pi\)
\(740\) 0 0
\(741\) 8.56927 0.314800
\(742\) 0 0
\(743\) 24.8707 0.912416 0.456208 0.889873i \(-0.349207\pi\)
0.456208 + 0.889873i \(0.349207\pi\)
\(744\) 0 0
\(745\) −20.2188 −0.740761
\(746\) 0 0
\(747\) 14.1118 0.516326
\(748\) 0 0
\(749\) 7.06778 0.258251
\(750\) 0 0
\(751\) −50.0545 −1.82652 −0.913258 0.407382i \(-0.866442\pi\)
−0.913258 + 0.407382i \(0.866442\pi\)
\(752\) 0 0
\(753\) 30.4390 1.10926
\(754\) 0 0
\(755\) 24.2130 0.881201
\(756\) 0 0
\(757\) 6.17487 0.224430 0.112215 0.993684i \(-0.464206\pi\)
0.112215 + 0.993684i \(0.464206\pi\)
\(758\) 0 0
\(759\) −34.6045 −1.25606
\(760\) 0 0
\(761\) −29.9236 −1.08473 −0.542365 0.840143i \(-0.682471\pi\)
−0.542365 + 0.840143i \(0.682471\pi\)
\(762\) 0 0
\(763\) −24.4705 −0.885890
\(764\) 0 0
\(765\) 9.54655 0.345156
\(766\) 0 0
\(767\) −23.9130 −0.863448
\(768\) 0 0
\(769\) −18.6073 −0.670995 −0.335497 0.942041i \(-0.608904\pi\)
−0.335497 + 0.942041i \(0.608904\pi\)
\(770\) 0 0
\(771\) −23.0748 −0.831020
\(772\) 0 0
\(773\) 29.9797 1.07830 0.539148 0.842211i \(-0.318746\pi\)
0.539148 + 0.842211i \(0.318746\pi\)
\(774\) 0 0
\(775\) 4.35088 0.156288
\(776\) 0 0
\(777\) −31.3013 −1.12293
\(778\) 0 0
\(779\) −38.5898 −1.38262
\(780\) 0 0
\(781\) 8.70839 0.311611
\(782\) 0 0
\(783\) 3.73263 0.133393
\(784\) 0 0
\(785\) −11.2141 −0.400250
\(786\) 0 0
\(787\) 37.9894 1.35417 0.677087 0.735903i \(-0.263242\pi\)
0.677087 + 0.735903i \(0.263242\pi\)
\(788\) 0 0
\(789\) 21.5164 0.766004
\(790\) 0 0
\(791\) −78.8216 −2.80257
\(792\) 0 0
\(793\) 0.546517 0.0194074
\(794\) 0 0
\(795\) 4.95102 0.175595
\(796\) 0 0
\(797\) 22.2033 0.786483 0.393242 0.919435i \(-0.371354\pi\)
0.393242 + 0.919435i \(0.371354\pi\)
\(798\) 0 0
\(799\) −49.8253 −1.76269
\(800\) 0 0
\(801\) −6.93566 −0.245059
\(802\) 0 0
\(803\) 20.4007 0.719925
\(804\) 0 0
\(805\) −37.8424 −1.33377
\(806\) 0 0
\(807\) 16.2955 0.573629
\(808\) 0 0
\(809\) −15.6829 −0.551380 −0.275690 0.961247i \(-0.588906\pi\)
−0.275690 + 0.961247i \(0.588906\pi\)
\(810\) 0 0
\(811\) 14.4336 0.506834 0.253417 0.967357i \(-0.418446\pi\)
0.253417 + 0.967357i \(0.418446\pi\)
\(812\) 0 0
\(813\) −25.9830 −0.911263
\(814\) 0 0
\(815\) −3.70417 −0.129751
\(816\) 0 0
\(817\) −23.2683 −0.814054
\(818\) 0 0
\(819\) 8.81684 0.308085
\(820\) 0 0
\(821\) 9.25042 0.322842 0.161421 0.986886i \(-0.448392\pi\)
0.161421 + 0.986886i \(0.448392\pi\)
\(822\) 0 0
\(823\) −15.7006 −0.547290 −0.273645 0.961831i \(-0.588229\pi\)
−0.273645 + 0.961831i \(0.588229\pi\)
\(824\) 0 0
\(825\) 17.9064 0.623422
\(826\) 0 0
\(827\) 30.4073 1.05737 0.528683 0.848819i \(-0.322686\pi\)
0.528683 + 0.848819i \(0.322686\pi\)
\(828\) 0 0
\(829\) −38.7371 −1.34539 −0.672697 0.739918i \(-0.734864\pi\)
−0.672697 + 0.739918i \(0.734864\pi\)
\(830\) 0 0
\(831\) −8.26377 −0.286667
\(832\) 0 0
\(833\) 104.454 3.61913
\(834\) 0 0
\(835\) −1.25667 −0.0434889
\(836\) 0 0
\(837\) −1.27190 −0.0439632
\(838\) 0 0
\(839\) −32.4743 −1.12114 −0.560568 0.828108i \(-0.689417\pi\)
−0.560568 + 0.828108i \(0.689417\pi\)
\(840\) 0 0
\(841\) −15.0675 −0.519568
\(842\) 0 0
\(843\) −0.956330 −0.0329378
\(844\) 0 0
\(845\) −11.6288 −0.400043
\(846\) 0 0
\(847\) −74.7108 −2.56709
\(848\) 0 0
\(849\) 7.81438 0.268189
\(850\) 0 0
\(851\) 45.4257 1.55717
\(852\) 0 0
\(853\) −21.3706 −0.731716 −0.365858 0.930671i \(-0.619224\pi\)
−0.365858 + 0.930671i \(0.619224\pi\)
\(854\) 0 0
\(855\) −5.56367 −0.190273
\(856\) 0 0
\(857\) −38.7853 −1.32488 −0.662440 0.749115i \(-0.730479\pi\)
−0.662440 + 0.749115i \(0.730479\pi\)
\(858\) 0 0
\(859\) 41.2770 1.40835 0.704176 0.710025i \(-0.251317\pi\)
0.704176 + 0.710025i \(0.251317\pi\)
\(860\) 0 0
\(861\) −39.7047 −1.35313
\(862\) 0 0
\(863\) 3.03297 0.103244 0.0516218 0.998667i \(-0.483561\pi\)
0.0516218 + 0.998667i \(0.483561\pi\)
\(864\) 0 0
\(865\) −15.7695 −0.536179
\(866\) 0 0
\(867\) 40.7098 1.38258
\(868\) 0 0
\(869\) −32.5334 −1.10362
\(870\) 0 0
\(871\) 15.9713 0.541166
\(872\) 0 0
\(873\) −3.11931 −0.105573
\(874\) 0 0
\(875\) 48.2039 1.62959
\(876\) 0 0
\(877\) 26.2512 0.886439 0.443220 0.896413i \(-0.353836\pi\)
0.443220 + 0.896413i \(0.353836\pi\)
\(878\) 0 0
\(879\) −18.0569 −0.609045
\(880\) 0 0
\(881\) 3.69895 0.124621 0.0623104 0.998057i \(-0.480153\pi\)
0.0623104 + 0.998057i \(0.480153\pi\)
\(882\) 0 0
\(883\) −47.7365 −1.60646 −0.803230 0.595669i \(-0.796887\pi\)
−0.803230 + 0.595669i \(0.796887\pi\)
\(884\) 0 0
\(885\) 15.5257 0.521891
\(886\) 0 0
\(887\) 1.04390 0.0350508 0.0175254 0.999846i \(-0.494421\pi\)
0.0175254 + 0.999846i \(0.494421\pi\)
\(888\) 0 0
\(889\) −35.8995 −1.20403
\(890\) 0 0
\(891\) −5.23461 −0.175366
\(892\) 0 0
\(893\) 29.0379 0.971716
\(894\) 0 0
\(895\) 3.86647 0.129242
\(896\) 0 0
\(897\) −12.7954 −0.427224
\(898\) 0 0
\(899\) −4.74752 −0.158339
\(900\) 0 0
\(901\) 29.9294 0.997093
\(902\) 0 0
\(903\) −23.9405 −0.796690
\(904\) 0 0
\(905\) 3.15210 0.104779
\(906\) 0 0
\(907\) −15.2974 −0.507943 −0.253971 0.967212i \(-0.581737\pi\)
−0.253971 + 0.967212i \(0.581737\pi\)
\(908\) 0 0
\(909\) 11.8680 0.393638
\(910\) 0 0
\(911\) −33.8233 −1.12062 −0.560308 0.828285i \(-0.689317\pi\)
−0.560308 + 0.828285i \(0.689317\pi\)
\(912\) 0 0
\(913\) −73.8701 −2.44474
\(914\) 0 0
\(915\) −0.354830 −0.0117303
\(916\) 0 0
\(917\) −92.1507 −3.04308
\(918\) 0 0
\(919\) −18.1404 −0.598396 −0.299198 0.954191i \(-0.596719\pi\)
−0.299198 + 0.954191i \(0.596719\pi\)
\(920\) 0 0
\(921\) −26.9495 −0.888018
\(922\) 0 0
\(923\) 3.22001 0.105988
\(924\) 0 0
\(925\) −23.5060 −0.772872
\(926\) 0 0
\(927\) −2.31747 −0.0761158
\(928\) 0 0
\(929\) −49.9152 −1.63767 −0.818833 0.574032i \(-0.805378\pi\)
−0.818833 + 0.574032i \(0.805378\pi\)
\(930\) 0 0
\(931\) −60.8753 −1.99511
\(932\) 0 0
\(933\) −7.51738 −0.246108
\(934\) 0 0
\(935\) −49.9725 −1.63427
\(936\) 0 0
\(937\) −45.2081 −1.47689 −0.738443 0.674316i \(-0.764438\pi\)
−0.738443 + 0.674316i \(0.764438\pi\)
\(938\) 0 0
\(939\) 24.9480 0.814148
\(940\) 0 0
\(941\) 0.416944 0.0135920 0.00679599 0.999977i \(-0.497837\pi\)
0.00679599 + 0.999977i \(0.497837\pi\)
\(942\) 0 0
\(943\) 57.6211 1.87640
\(944\) 0 0
\(945\) −5.72440 −0.186215
\(946\) 0 0
\(947\) −40.5028 −1.31616 −0.658082 0.752946i \(-0.728632\pi\)
−0.658082 + 0.752946i \(0.728632\pi\)
\(948\) 0 0
\(949\) 7.54336 0.244868
\(950\) 0 0
\(951\) 28.0071 0.908192
\(952\) 0 0
\(953\) 46.8908 1.51894 0.759470 0.650542i \(-0.225458\pi\)
0.759470 + 0.650542i \(0.225458\pi\)
\(954\) 0 0
\(955\) −4.15768 −0.134539
\(956\) 0 0
\(957\) −19.5389 −0.631602
\(958\) 0 0
\(959\) 17.0298 0.549920
\(960\) 0 0
\(961\) −29.3823 −0.947815
\(962\) 0 0
\(963\) −1.55158 −0.0499990
\(964\) 0 0
\(965\) −23.1349 −0.744737
\(966\) 0 0
\(967\) −28.5237 −0.917262 −0.458631 0.888627i \(-0.651660\pi\)
−0.458631 + 0.888627i \(0.651660\pi\)
\(968\) 0 0
\(969\) −33.6329 −1.08044
\(970\) 0 0
\(971\) 32.4613 1.04173 0.520867 0.853638i \(-0.325609\pi\)
0.520867 + 0.853638i \(0.325609\pi\)
\(972\) 0 0
\(973\) −60.0476 −1.92504
\(974\) 0 0
\(975\) 6.62108 0.212044
\(976\) 0 0
\(977\) 29.7999 0.953382 0.476691 0.879071i \(-0.341836\pi\)
0.476691 + 0.879071i \(0.341836\pi\)
\(978\) 0 0
\(979\) 36.3055 1.16033
\(980\) 0 0
\(981\) 5.37197 0.171514
\(982\) 0 0
\(983\) −17.3606 −0.553717 −0.276859 0.960911i \(-0.589293\pi\)
−0.276859 + 0.960911i \(0.589293\pi\)
\(984\) 0 0
\(985\) −16.4304 −0.523515
\(986\) 0 0
\(987\) 29.8768 0.950989
\(988\) 0 0
\(989\) 34.7434 1.10478
\(990\) 0 0
\(991\) −6.60573 −0.209838 −0.104919 0.994481i \(-0.533458\pi\)
−0.104919 + 0.994481i \(0.533458\pi\)
\(992\) 0 0
\(993\) 7.36380 0.233683
\(994\) 0 0
\(995\) −23.6961 −0.751218
\(996\) 0 0
\(997\) 0.907767 0.0287493 0.0143746 0.999897i \(-0.495424\pi\)
0.0143746 + 0.999897i \(0.495424\pi\)
\(998\) 0 0
\(999\) 6.87153 0.217406
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 4008.2.a.m.1.8 13
4.3 odd 2 8016.2.a.bg.1.8 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.m.1.8 13 1.1 even 1 trivial
8016.2.a.bg.1.8 13 4.3 odd 2