Properties

Label 4008.2.a.h.1.4
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 \) 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 \(-0.938106\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.938106 q^{5} +4.96672 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.938106 q^{5} +4.96672 q^{7} +1.00000 q^{9} +3.79099 q^{11} -5.41916 q^{13} +0.938106 q^{15} -3.64027 q^{17} +4.92607 q^{19} -4.96672 q^{21} -3.96672 q^{23} -4.11996 q^{25} -1.00000 q^{27} -10.2334 q^{29} -9.29712 q^{31} -3.79099 q^{33} -4.65931 q^{35} +2.89525 q^{37} +5.41916 q^{39} -5.61453 q^{41} +0.934214 q^{43} -0.938106 q^{45} -9.64800 q^{47} +17.6683 q^{49} +3.64027 q^{51} +0.782015 q^{53} -3.55635 q^{55} -4.92607 q^{57} +15.0714 q^{59} -12.5501 q^{61} +4.96672 q^{63} +5.08375 q^{65} +7.96283 q^{67} +3.96672 q^{69} -8.53856 q^{71} -5.77746 q^{73} +4.11996 q^{75} +18.8288 q^{77} -0.994199 q^{79} +1.00000 q^{81} +2.08014 q^{83} +3.41496 q^{85} +10.2334 q^{87} +9.46795 q^{89} -26.9154 q^{91} +9.29712 q^{93} -4.62117 q^{95} +4.16825 q^{97} +3.79099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - q^{7} + 8 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} + 9 q^{23} + 6 q^{25} - 8 q^{27} + 17 q^{29} - 23 q^{31} + 3 q^{33} - 15 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 2 q^{43} - 34 q^{47} + 5 q^{49} + 7 q^{51} + 12 q^{53} - 7 q^{55} - 16 q^{59} - 2 q^{61} - q^{63} - 14 q^{65} + 21 q^{67} - 9 q^{69} - 29 q^{71} - 38 q^{73} - 6 q^{75} + 20 q^{77} - 12 q^{79} + 8 q^{81} - 32 q^{83} + 23 q^{85} - 17 q^{87} + 11 q^{89} - 5 q^{91} + 23 q^{93} - 67 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(5\) −0.938106 −0.419534 −0.209767 0.977751i \(-0.567271\pi\)
−0.209767 + 0.977751i \(0.567271\pi\)
\(6\) 0 0
\(7\) 4.96672 1.87724 0.938621 0.344949i \(-0.112104\pi\)
0.938621 + 0.344949i \(0.112104\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.79099 1.14303 0.571513 0.820593i \(-0.306357\pi\)
0.571513 + 0.820593i \(0.306357\pi\)
\(12\) 0 0
\(13\) −5.41916 −1.50300 −0.751502 0.659730i \(-0.770670\pi\)
−0.751502 + 0.659730i \(0.770670\pi\)
\(14\) 0 0
\(15\) 0.938106 0.242218
\(16\) 0 0
\(17\) −3.64027 −0.882895 −0.441448 0.897287i \(-0.645535\pi\)
−0.441448 + 0.897287i \(0.645535\pi\)
\(18\) 0 0
\(19\) 4.92607 1.13012 0.565059 0.825051i \(-0.308853\pi\)
0.565059 + 0.825051i \(0.308853\pi\)
\(20\) 0 0
\(21\) −4.96672 −1.08383
\(22\) 0 0
\(23\) −3.96672 −0.827118 −0.413559 0.910477i \(-0.635714\pi\)
−0.413559 + 0.910477i \(0.635714\pi\)
\(24\) 0 0
\(25\) −4.11996 −0.823991
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.2334 −1.90030 −0.950151 0.311791i \(-0.899071\pi\)
−0.950151 + 0.311791i \(0.899071\pi\)
\(30\) 0 0
\(31\) −9.29712 −1.66981 −0.834906 0.550393i \(-0.814478\pi\)
−0.834906 + 0.550393i \(0.814478\pi\)
\(32\) 0 0
\(33\) −3.79099 −0.659926
\(34\) 0 0
\(35\) −4.65931 −0.787567
\(36\) 0 0
\(37\) 2.89525 0.475976 0.237988 0.971268i \(-0.423512\pi\)
0.237988 + 0.971268i \(0.423512\pi\)
\(38\) 0 0
\(39\) 5.41916 0.867760
\(40\) 0 0
\(41\) −5.61453 −0.876842 −0.438421 0.898770i \(-0.644462\pi\)
−0.438421 + 0.898770i \(0.644462\pi\)
\(42\) 0 0
\(43\) 0.934214 0.142466 0.0712332 0.997460i \(-0.477307\pi\)
0.0712332 + 0.997460i \(0.477307\pi\)
\(44\) 0 0
\(45\) −0.938106 −0.139845
\(46\) 0 0
\(47\) −9.64800 −1.40731 −0.703653 0.710544i \(-0.748449\pi\)
−0.703653 + 0.710544i \(0.748449\pi\)
\(48\) 0 0
\(49\) 17.6683 2.52404
\(50\) 0 0
\(51\) 3.64027 0.509740
\(52\) 0 0
\(53\) 0.782015 0.107418 0.0537090 0.998557i \(-0.482896\pi\)
0.0537090 + 0.998557i \(0.482896\pi\)
\(54\) 0 0
\(55\) −3.55635 −0.479538
\(56\) 0 0
\(57\) −4.92607 −0.652474
\(58\) 0 0
\(59\) 15.0714 1.96214 0.981068 0.193665i \(-0.0620373\pi\)
0.981068 + 0.193665i \(0.0620373\pi\)
\(60\) 0 0
\(61\) −12.5501 −1.60687 −0.803436 0.595391i \(-0.796997\pi\)
−0.803436 + 0.595391i \(0.796997\pi\)
\(62\) 0 0
\(63\) 4.96672 0.625748
\(64\) 0 0
\(65\) 5.08375 0.630561
\(66\) 0 0
\(67\) 7.96283 0.972814 0.486407 0.873732i \(-0.338307\pi\)
0.486407 + 0.873732i \(0.338307\pi\)
\(68\) 0 0
\(69\) 3.96672 0.477537
\(70\) 0 0
\(71\) −8.53856 −1.01334 −0.506670 0.862140i \(-0.669124\pi\)
−0.506670 + 0.862140i \(0.669124\pi\)
\(72\) 0 0
\(73\) −5.77746 −0.676200 −0.338100 0.941110i \(-0.609784\pi\)
−0.338100 + 0.941110i \(0.609784\pi\)
\(74\) 0 0
\(75\) 4.11996 0.475732
\(76\) 0 0
\(77\) 18.8288 2.14574
\(78\) 0 0
\(79\) −0.994199 −0.111856 −0.0559281 0.998435i \(-0.517812\pi\)
−0.0559281 + 0.998435i \(0.517812\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.08014 0.228325 0.114162 0.993462i \(-0.463582\pi\)
0.114162 + 0.993462i \(0.463582\pi\)
\(84\) 0 0
\(85\) 3.41496 0.370404
\(86\) 0 0
\(87\) 10.2334 1.09714
\(88\) 0 0
\(89\) 9.46795 1.00360 0.501801 0.864983i \(-0.332671\pi\)
0.501801 + 0.864983i \(0.332671\pi\)
\(90\) 0 0
\(91\) −26.9154 −2.82150
\(92\) 0 0
\(93\) 9.29712 0.964066
\(94\) 0 0
\(95\) −4.62117 −0.474122
\(96\) 0 0
\(97\) 4.16825 0.423221 0.211611 0.977354i \(-0.432129\pi\)
0.211611 + 0.977354i \(0.432129\pi\)
\(98\) 0 0
\(99\) 3.79099 0.381009
\(100\) 0 0
\(101\) −7.91578 −0.787649 −0.393825 0.919186i \(-0.628848\pi\)
−0.393825 + 0.919186i \(0.628848\pi\)
\(102\) 0 0
\(103\) −4.67646 −0.460786 −0.230393 0.973098i \(-0.574001\pi\)
−0.230393 + 0.973098i \(0.574001\pi\)
\(104\) 0 0
\(105\) 4.65931 0.454702
\(106\) 0 0
\(107\) −11.6949 −1.13059 −0.565293 0.824890i \(-0.691237\pi\)
−0.565293 + 0.824890i \(0.691237\pi\)
\(108\) 0 0
\(109\) 15.9966 1.53220 0.766100 0.642721i \(-0.222195\pi\)
0.766100 + 0.642721i \(0.222195\pi\)
\(110\) 0 0
\(111\) −2.89525 −0.274805
\(112\) 0 0
\(113\) −6.39876 −0.601944 −0.300972 0.953633i \(-0.597311\pi\)
−0.300972 + 0.953633i \(0.597311\pi\)
\(114\) 0 0
\(115\) 3.72120 0.347004
\(116\) 0 0
\(117\) −5.41916 −0.501002
\(118\) 0 0
\(119\) −18.0802 −1.65741
\(120\) 0 0
\(121\) 3.37159 0.306508
\(122\) 0 0
\(123\) 5.61453 0.506245
\(124\) 0 0
\(125\) 8.55549 0.765226
\(126\) 0 0
\(127\) −2.89284 −0.256698 −0.128349 0.991729i \(-0.540968\pi\)
−0.128349 + 0.991729i \(0.540968\pi\)
\(128\) 0 0
\(129\) −0.934214 −0.0822530
\(130\) 0 0
\(131\) −8.96987 −0.783701 −0.391850 0.920029i \(-0.628165\pi\)
−0.391850 + 0.920029i \(0.628165\pi\)
\(132\) 0 0
\(133\) 24.4664 2.12151
\(134\) 0 0
\(135\) 0.938106 0.0807393
\(136\) 0 0
\(137\) −17.6172 −1.50514 −0.752570 0.658512i \(-0.771186\pi\)
−0.752570 + 0.658512i \(0.771186\pi\)
\(138\) 0 0
\(139\) −10.9504 −0.928803 −0.464402 0.885625i \(-0.653731\pi\)
−0.464402 + 0.885625i \(0.653731\pi\)
\(140\) 0 0
\(141\) 9.64800 0.812508
\(142\) 0 0
\(143\) −20.5440 −1.71797
\(144\) 0 0
\(145\) 9.60005 0.797241
\(146\) 0 0
\(147\) −17.6683 −1.45726
\(148\) 0 0
\(149\) −10.7367 −0.879581 −0.439790 0.898100i \(-0.644947\pi\)
−0.439790 + 0.898100i \(0.644947\pi\)
\(150\) 0 0
\(151\) −0.776575 −0.0631968 −0.0315984 0.999501i \(-0.510060\pi\)
−0.0315984 + 0.999501i \(0.510060\pi\)
\(152\) 0 0
\(153\) −3.64027 −0.294298
\(154\) 0 0
\(155\) 8.72168 0.700542
\(156\) 0 0
\(157\) −7.39402 −0.590107 −0.295054 0.955481i \(-0.595338\pi\)
−0.295054 + 0.955481i \(0.595338\pi\)
\(158\) 0 0
\(159\) −0.782015 −0.0620178
\(160\) 0 0
\(161\) −19.7016 −1.55270
\(162\) 0 0
\(163\) 16.8717 1.32149 0.660746 0.750609i \(-0.270240\pi\)
0.660746 + 0.750609i \(0.270240\pi\)
\(164\) 0 0
\(165\) 3.55635 0.276861
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 16.3673 1.25902
\(170\) 0 0
\(171\) 4.92607 0.376706
\(172\) 0 0
\(173\) −18.4626 −1.40368 −0.701842 0.712333i \(-0.747639\pi\)
−0.701842 + 0.712333i \(0.747639\pi\)
\(174\) 0 0
\(175\) −20.4627 −1.54683
\(176\) 0 0
\(177\) −15.0714 −1.13284
\(178\) 0 0
\(179\) 14.1981 1.06122 0.530609 0.847617i \(-0.321963\pi\)
0.530609 + 0.847617i \(0.321963\pi\)
\(180\) 0 0
\(181\) 18.3089 1.36089 0.680443 0.732801i \(-0.261787\pi\)
0.680443 + 0.732801i \(0.261787\pi\)
\(182\) 0 0
\(183\) 12.5501 0.927728
\(184\) 0 0
\(185\) −2.71605 −0.199688
\(186\) 0 0
\(187\) −13.8002 −1.00917
\(188\) 0 0
\(189\) −4.96672 −0.361276
\(190\) 0 0
\(191\) 13.4667 0.974413 0.487206 0.873287i \(-0.338016\pi\)
0.487206 + 0.873287i \(0.338016\pi\)
\(192\) 0 0
\(193\) 13.6449 0.982184 0.491092 0.871108i \(-0.336598\pi\)
0.491092 + 0.871108i \(0.336598\pi\)
\(194\) 0 0
\(195\) −5.08375 −0.364055
\(196\) 0 0
\(197\) 20.9789 1.49468 0.747342 0.664439i \(-0.231330\pi\)
0.747342 + 0.664439i \(0.231330\pi\)
\(198\) 0 0
\(199\) −13.0296 −0.923647 −0.461823 0.886972i \(-0.652805\pi\)
−0.461823 + 0.886972i \(0.652805\pi\)
\(200\) 0 0
\(201\) −7.96283 −0.561654
\(202\) 0 0
\(203\) −50.8266 −3.56733
\(204\) 0 0
\(205\) 5.26702 0.367865
\(206\) 0 0
\(207\) −3.96672 −0.275706
\(208\) 0 0
\(209\) 18.6747 1.29175
\(210\) 0 0
\(211\) 18.2109 1.25369 0.626845 0.779144i \(-0.284346\pi\)
0.626845 + 0.779144i \(0.284346\pi\)
\(212\) 0 0
\(213\) 8.53856 0.585052
\(214\) 0 0
\(215\) −0.876392 −0.0597694
\(216\) 0 0
\(217\) −46.1762 −3.13464
\(218\) 0 0
\(219\) 5.77746 0.390404
\(220\) 0 0
\(221\) 19.7272 1.32700
\(222\) 0 0
\(223\) 16.9336 1.13396 0.566978 0.823733i \(-0.308112\pi\)
0.566978 + 0.823733i \(0.308112\pi\)
\(224\) 0 0
\(225\) −4.11996 −0.274664
\(226\) 0 0
\(227\) 3.66081 0.242976 0.121488 0.992593i \(-0.461233\pi\)
0.121488 + 0.992593i \(0.461233\pi\)
\(228\) 0 0
\(229\) −23.7283 −1.56801 −0.784004 0.620756i \(-0.786826\pi\)
−0.784004 + 0.620756i \(0.786826\pi\)
\(230\) 0 0
\(231\) −18.8288 −1.23884
\(232\) 0 0
\(233\) −12.4226 −0.813834 −0.406917 0.913465i \(-0.633396\pi\)
−0.406917 + 0.913465i \(0.633396\pi\)
\(234\) 0 0
\(235\) 9.05085 0.590412
\(236\) 0 0
\(237\) 0.994199 0.0645802
\(238\) 0 0
\(239\) −21.7982 −1.41001 −0.705003 0.709204i \(-0.749055\pi\)
−0.705003 + 0.709204i \(0.749055\pi\)
\(240\) 0 0
\(241\) 5.02354 0.323594 0.161797 0.986824i \(-0.448271\pi\)
0.161797 + 0.986824i \(0.448271\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −16.5747 −1.05892
\(246\) 0 0
\(247\) −26.6952 −1.69857
\(248\) 0 0
\(249\) −2.08014 −0.131823
\(250\) 0 0
\(251\) −10.2662 −0.647999 −0.324000 0.946057i \(-0.605028\pi\)
−0.324000 + 0.946057i \(0.605028\pi\)
\(252\) 0 0
\(253\) −15.0378 −0.945417
\(254\) 0 0
\(255\) −3.41496 −0.213853
\(256\) 0 0
\(257\) −11.9857 −0.747650 −0.373825 0.927499i \(-0.621954\pi\)
−0.373825 + 0.927499i \(0.621954\pi\)
\(258\) 0 0
\(259\) 14.3799 0.893522
\(260\) 0 0
\(261\) −10.2334 −0.633434
\(262\) 0 0
\(263\) −26.9764 −1.66344 −0.831719 0.555197i \(-0.812643\pi\)
−0.831719 + 0.555197i \(0.812643\pi\)
\(264\) 0 0
\(265\) −0.733613 −0.0450655
\(266\) 0 0
\(267\) −9.46795 −0.579429
\(268\) 0 0
\(269\) 29.3337 1.78851 0.894254 0.447559i \(-0.147707\pi\)
0.894254 + 0.447559i \(0.147707\pi\)
\(270\) 0 0
\(271\) 11.4504 0.695564 0.347782 0.937575i \(-0.386935\pi\)
0.347782 + 0.937575i \(0.386935\pi\)
\(272\) 0 0
\(273\) 26.9154 1.62900
\(274\) 0 0
\(275\) −15.6187 −0.941844
\(276\) 0 0
\(277\) 2.10038 0.126199 0.0630997 0.998007i \(-0.479901\pi\)
0.0630997 + 0.998007i \(0.479901\pi\)
\(278\) 0 0
\(279\) −9.29712 −0.556604
\(280\) 0 0
\(281\) −2.37932 −0.141938 −0.0709691 0.997479i \(-0.522609\pi\)
−0.0709691 + 0.997479i \(0.522609\pi\)
\(282\) 0 0
\(283\) −28.0104 −1.66504 −0.832522 0.553991i \(-0.813104\pi\)
−0.832522 + 0.553991i \(0.813104\pi\)
\(284\) 0 0
\(285\) 4.62117 0.273735
\(286\) 0 0
\(287\) −27.8858 −1.64605
\(288\) 0 0
\(289\) −3.74843 −0.220496
\(290\) 0 0
\(291\) −4.16825 −0.244347
\(292\) 0 0
\(293\) −6.26096 −0.365769 −0.182885 0.983134i \(-0.558543\pi\)
−0.182885 + 0.983134i \(0.558543\pi\)
\(294\) 0 0
\(295\) −14.1386 −0.823182
\(296\) 0 0
\(297\) −3.79099 −0.219975
\(298\) 0 0
\(299\) 21.4963 1.24316
\(300\) 0 0
\(301\) 4.63998 0.267444
\(302\) 0 0
\(303\) 7.91578 0.454749
\(304\) 0 0
\(305\) 11.7733 0.674137
\(306\) 0 0
\(307\) −29.9131 −1.70723 −0.853617 0.520902i \(-0.825596\pi\)
−0.853617 + 0.520902i \(0.825596\pi\)
\(308\) 0 0
\(309\) 4.67646 0.266035
\(310\) 0 0
\(311\) 16.6641 0.944933 0.472467 0.881349i \(-0.343364\pi\)
0.472467 + 0.881349i \(0.343364\pi\)
\(312\) 0 0
\(313\) 1.82677 0.103255 0.0516276 0.998666i \(-0.483559\pi\)
0.0516276 + 0.998666i \(0.483559\pi\)
\(314\) 0 0
\(315\) −4.65931 −0.262522
\(316\) 0 0
\(317\) −6.80617 −0.382273 −0.191136 0.981563i \(-0.561217\pi\)
−0.191136 + 0.981563i \(0.561217\pi\)
\(318\) 0 0
\(319\) −38.7948 −2.17209
\(320\) 0 0
\(321\) 11.6949 0.652745
\(322\) 0 0
\(323\) −17.9322 −0.997776
\(324\) 0 0
\(325\) 22.3267 1.23846
\(326\) 0 0
\(327\) −15.9966 −0.884616
\(328\) 0 0
\(329\) −47.9189 −2.64185
\(330\) 0 0
\(331\) 33.7166 1.85323 0.926615 0.376011i \(-0.122705\pi\)
0.926615 + 0.376011i \(0.122705\pi\)
\(332\) 0 0
\(333\) 2.89525 0.158659
\(334\) 0 0
\(335\) −7.46997 −0.408128
\(336\) 0 0
\(337\) −30.7686 −1.67607 −0.838036 0.545615i \(-0.816296\pi\)
−0.838036 + 0.545615i \(0.816296\pi\)
\(338\) 0 0
\(339\) 6.39876 0.347533
\(340\) 0 0
\(341\) −35.2453 −1.90864
\(342\) 0 0
\(343\) 52.9863 2.86099
\(344\) 0 0
\(345\) −3.72120 −0.200343
\(346\) 0 0
\(347\) 5.35902 0.287687 0.143844 0.989600i \(-0.454054\pi\)
0.143844 + 0.989600i \(0.454054\pi\)
\(348\) 0 0
\(349\) −17.0047 −0.910239 −0.455119 0.890430i \(-0.650403\pi\)
−0.455119 + 0.890430i \(0.650403\pi\)
\(350\) 0 0
\(351\) 5.41916 0.289253
\(352\) 0 0
\(353\) 11.9678 0.636984 0.318492 0.947926i \(-0.396824\pi\)
0.318492 + 0.947926i \(0.396824\pi\)
\(354\) 0 0
\(355\) 8.01007 0.425130
\(356\) 0 0
\(357\) 18.0802 0.956906
\(358\) 0 0
\(359\) −10.0766 −0.531823 −0.265912 0.963997i \(-0.585673\pi\)
−0.265912 + 0.963997i \(0.585673\pi\)
\(360\) 0 0
\(361\) 5.26615 0.277166
\(362\) 0 0
\(363\) −3.37159 −0.176963
\(364\) 0 0
\(365\) 5.41987 0.283689
\(366\) 0 0
\(367\) 34.9226 1.82295 0.911473 0.411360i \(-0.134946\pi\)
0.911473 + 0.411360i \(0.134946\pi\)
\(368\) 0 0
\(369\) −5.61453 −0.292281
\(370\) 0 0
\(371\) 3.88405 0.201650
\(372\) 0 0
\(373\) −25.7963 −1.33568 −0.667842 0.744303i \(-0.732782\pi\)
−0.667842 + 0.744303i \(0.732782\pi\)
\(374\) 0 0
\(375\) −8.55549 −0.441803
\(376\) 0 0
\(377\) 55.4566 2.85616
\(378\) 0 0
\(379\) 10.7412 0.551736 0.275868 0.961196i \(-0.411035\pi\)
0.275868 + 0.961196i \(0.411035\pi\)
\(380\) 0 0
\(381\) 2.89284 0.148205
\(382\) 0 0
\(383\) −30.2013 −1.54322 −0.771608 0.636098i \(-0.780547\pi\)
−0.771608 + 0.636098i \(0.780547\pi\)
\(384\) 0 0
\(385\) −17.6634 −0.900209
\(386\) 0 0
\(387\) 0.934214 0.0474888
\(388\) 0 0
\(389\) −3.09465 −0.156905 −0.0784525 0.996918i \(-0.524998\pi\)
−0.0784525 + 0.996918i \(0.524998\pi\)
\(390\) 0 0
\(391\) 14.4399 0.730258
\(392\) 0 0
\(393\) 8.96987 0.452470
\(394\) 0 0
\(395\) 0.932664 0.0469274
\(396\) 0 0
\(397\) −8.86237 −0.444790 −0.222395 0.974957i \(-0.571387\pi\)
−0.222395 + 0.974957i \(0.571387\pi\)
\(398\) 0 0
\(399\) −24.4664 −1.22485
\(400\) 0 0
\(401\) −14.5081 −0.724499 −0.362249 0.932081i \(-0.617991\pi\)
−0.362249 + 0.932081i \(0.617991\pi\)
\(402\) 0 0
\(403\) 50.3826 2.50973
\(404\) 0 0
\(405\) −0.938106 −0.0466149
\(406\) 0 0
\(407\) 10.9759 0.544053
\(408\) 0 0
\(409\) −31.7417 −1.56953 −0.784763 0.619796i \(-0.787215\pi\)
−0.784763 + 0.619796i \(0.787215\pi\)
\(410\) 0 0
\(411\) 17.6172 0.868993
\(412\) 0 0
\(413\) 74.8556 3.68340
\(414\) 0 0
\(415\) −1.95139 −0.0957899
\(416\) 0 0
\(417\) 10.9504 0.536245
\(418\) 0 0
\(419\) 2.47217 0.120773 0.0603866 0.998175i \(-0.480767\pi\)
0.0603866 + 0.998175i \(0.480767\pi\)
\(420\) 0 0
\(421\) 35.7214 1.74096 0.870478 0.492207i \(-0.163810\pi\)
0.870478 + 0.492207i \(0.163810\pi\)
\(422\) 0 0
\(423\) −9.64800 −0.469102
\(424\) 0 0
\(425\) 14.9978 0.727498
\(426\) 0 0
\(427\) −62.3327 −3.01649
\(428\) 0 0
\(429\) 20.5440 0.991872
\(430\) 0 0
\(431\) −21.1608 −1.01928 −0.509639 0.860388i \(-0.670221\pi\)
−0.509639 + 0.860388i \(0.670221\pi\)
\(432\) 0 0
\(433\) 9.61648 0.462138 0.231069 0.972937i \(-0.425778\pi\)
0.231069 + 0.972937i \(0.425778\pi\)
\(434\) 0 0
\(435\) −9.60005 −0.460287
\(436\) 0 0
\(437\) −19.5403 −0.934740
\(438\) 0 0
\(439\) 9.51986 0.454358 0.227179 0.973853i \(-0.427050\pi\)
0.227179 + 0.973853i \(0.427050\pi\)
\(440\) 0 0
\(441\) 17.6683 0.841347
\(442\) 0 0
\(443\) 16.5948 0.788441 0.394220 0.919016i \(-0.371015\pi\)
0.394220 + 0.919016i \(0.371015\pi\)
\(444\) 0 0
\(445\) −8.88194 −0.421045
\(446\) 0 0
\(447\) 10.7367 0.507826
\(448\) 0 0
\(449\) −12.7360 −0.601050 −0.300525 0.953774i \(-0.597162\pi\)
−0.300525 + 0.953774i \(0.597162\pi\)
\(450\) 0 0
\(451\) −21.2846 −1.00225
\(452\) 0 0
\(453\) 0.776575 0.0364867
\(454\) 0 0
\(455\) 25.2495 1.18372
\(456\) 0 0
\(457\) 22.9605 1.07405 0.537024 0.843567i \(-0.319548\pi\)
0.537024 + 0.843567i \(0.319548\pi\)
\(458\) 0 0
\(459\) 3.64027 0.169913
\(460\) 0 0
\(461\) 25.7347 1.19859 0.599293 0.800530i \(-0.295449\pi\)
0.599293 + 0.800530i \(0.295449\pi\)
\(462\) 0 0
\(463\) 39.3604 1.82923 0.914616 0.404323i \(-0.132493\pi\)
0.914616 + 0.404323i \(0.132493\pi\)
\(464\) 0 0
\(465\) −8.72168 −0.404458
\(466\) 0 0
\(467\) 14.9555 0.692058 0.346029 0.938224i \(-0.387530\pi\)
0.346029 + 0.938224i \(0.387530\pi\)
\(468\) 0 0
\(469\) 39.5491 1.82621
\(470\) 0 0
\(471\) 7.39402 0.340699
\(472\) 0 0
\(473\) 3.54160 0.162843
\(474\) 0 0
\(475\) −20.2952 −0.931207
\(476\) 0 0
\(477\) 0.782015 0.0358060
\(478\) 0 0
\(479\) 0.494235 0.0225822 0.0112911 0.999936i \(-0.496406\pi\)
0.0112911 + 0.999936i \(0.496406\pi\)
\(480\) 0 0
\(481\) −15.6898 −0.715394
\(482\) 0 0
\(483\) 19.7016 0.896452
\(484\) 0 0
\(485\) −3.91026 −0.177556
\(486\) 0 0
\(487\) −1.27382 −0.0577222 −0.0288611 0.999583i \(-0.509188\pi\)
−0.0288611 + 0.999583i \(0.509188\pi\)
\(488\) 0 0
\(489\) −16.8717 −0.762964
\(490\) 0 0
\(491\) −3.45413 −0.155883 −0.0779415 0.996958i \(-0.524835\pi\)
−0.0779415 + 0.996958i \(0.524835\pi\)
\(492\) 0 0
\(493\) 37.2525 1.67777
\(494\) 0 0
\(495\) −3.55635 −0.159846
\(496\) 0 0
\(497\) −42.4086 −1.90229
\(498\) 0 0
\(499\) 14.0025 0.626839 0.313420 0.949615i \(-0.398525\pi\)
0.313420 + 0.949615i \(0.398525\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −16.5706 −0.738848 −0.369424 0.929261i \(-0.620445\pi\)
−0.369424 + 0.929261i \(0.620445\pi\)
\(504\) 0 0
\(505\) 7.42584 0.330445
\(506\) 0 0
\(507\) −16.3673 −0.726897
\(508\) 0 0
\(509\) 23.9733 1.06260 0.531299 0.847184i \(-0.321704\pi\)
0.531299 + 0.847184i \(0.321704\pi\)
\(510\) 0 0
\(511\) −28.6950 −1.26939
\(512\) 0 0
\(513\) −4.92607 −0.217491
\(514\) 0 0
\(515\) 4.38702 0.193315
\(516\) 0 0
\(517\) −36.5754 −1.60859
\(518\) 0 0
\(519\) 18.4626 0.810417
\(520\) 0 0
\(521\) 7.77526 0.340640 0.170320 0.985389i \(-0.445520\pi\)
0.170320 + 0.985389i \(0.445520\pi\)
\(522\) 0 0
\(523\) −35.1796 −1.53830 −0.769149 0.639069i \(-0.779320\pi\)
−0.769149 + 0.639069i \(0.779320\pi\)
\(524\) 0 0
\(525\) 20.4627 0.893064
\(526\) 0 0
\(527\) 33.8440 1.47427
\(528\) 0 0
\(529\) −7.26515 −0.315876
\(530\) 0 0
\(531\) 15.0714 0.654045
\(532\) 0 0
\(533\) 30.4260 1.31790
\(534\) 0 0
\(535\) 10.9710 0.474319
\(536\) 0 0
\(537\) −14.1981 −0.612694
\(538\) 0 0
\(539\) 66.9802 2.88504
\(540\) 0 0
\(541\) 32.5670 1.40017 0.700083 0.714062i \(-0.253146\pi\)
0.700083 + 0.714062i \(0.253146\pi\)
\(542\) 0 0
\(543\) −18.3089 −0.785708
\(544\) 0 0
\(545\) −15.0065 −0.642810
\(546\) 0 0
\(547\) 8.59032 0.367296 0.183648 0.982992i \(-0.441209\pi\)
0.183648 + 0.982992i \(0.441209\pi\)
\(548\) 0 0
\(549\) −12.5501 −0.535624
\(550\) 0 0
\(551\) −50.4106 −2.14756
\(552\) 0 0
\(553\) −4.93791 −0.209981
\(554\) 0 0
\(555\) 2.71605 0.115290
\(556\) 0 0
\(557\) −3.86029 −0.163566 −0.0817830 0.996650i \(-0.526061\pi\)
−0.0817830 + 0.996650i \(0.526061\pi\)
\(558\) 0 0
\(559\) −5.06266 −0.214128
\(560\) 0 0
\(561\) 13.8002 0.582646
\(562\) 0 0
\(563\) 15.8475 0.667891 0.333946 0.942592i \(-0.391620\pi\)
0.333946 + 0.942592i \(0.391620\pi\)
\(564\) 0 0
\(565\) 6.00271 0.252536
\(566\) 0 0
\(567\) 4.96672 0.208583
\(568\) 0 0
\(569\) 31.8436 1.33496 0.667478 0.744630i \(-0.267374\pi\)
0.667478 + 0.744630i \(0.267374\pi\)
\(570\) 0 0
\(571\) −34.7301 −1.45341 −0.726706 0.686949i \(-0.758950\pi\)
−0.726706 + 0.686949i \(0.758950\pi\)
\(572\) 0 0
\(573\) −13.4667 −0.562577
\(574\) 0 0
\(575\) 16.3427 0.681538
\(576\) 0 0
\(577\) −20.7968 −0.865783 −0.432892 0.901446i \(-0.642507\pi\)
−0.432892 + 0.901446i \(0.642507\pi\)
\(578\) 0 0
\(579\) −13.6449 −0.567064
\(580\) 0 0
\(581\) 10.3315 0.428621
\(582\) 0 0
\(583\) 2.96461 0.122782
\(584\) 0 0
\(585\) 5.08375 0.210187
\(586\) 0 0
\(587\) −33.8987 −1.39915 −0.699574 0.714560i \(-0.746627\pi\)
−0.699574 + 0.714560i \(0.746627\pi\)
\(588\) 0 0
\(589\) −45.7982 −1.88708
\(590\) 0 0
\(591\) −20.9789 −0.862956
\(592\) 0 0
\(593\) −25.7423 −1.05711 −0.528554 0.848900i \(-0.677266\pi\)
−0.528554 + 0.848900i \(0.677266\pi\)
\(594\) 0 0
\(595\) 16.9611 0.695339
\(596\) 0 0
\(597\) 13.0296 0.533268
\(598\) 0 0
\(599\) −3.85022 −0.157316 −0.0786578 0.996902i \(-0.525063\pi\)
−0.0786578 + 0.996902i \(0.525063\pi\)
\(600\) 0 0
\(601\) −2.44645 −0.0997930 −0.0498965 0.998754i \(-0.515889\pi\)
−0.0498965 + 0.998754i \(0.515889\pi\)
\(602\) 0 0
\(603\) 7.96283 0.324271
\(604\) 0 0
\(605\) −3.16291 −0.128590
\(606\) 0 0
\(607\) 47.3391 1.92143 0.960717 0.277529i \(-0.0895156\pi\)
0.960717 + 0.277529i \(0.0895156\pi\)
\(608\) 0 0
\(609\) 50.8266 2.05960
\(610\) 0 0
\(611\) 52.2841 2.11519
\(612\) 0 0
\(613\) 24.1587 0.975762 0.487881 0.872910i \(-0.337770\pi\)
0.487881 + 0.872910i \(0.337770\pi\)
\(614\) 0 0
\(615\) −5.26702 −0.212387
\(616\) 0 0
\(617\) 28.6706 1.15424 0.577118 0.816661i \(-0.304177\pi\)
0.577118 + 0.816661i \(0.304177\pi\)
\(618\) 0 0
\(619\) −8.44432 −0.339406 −0.169703 0.985495i \(-0.554281\pi\)
−0.169703 + 0.985495i \(0.554281\pi\)
\(620\) 0 0
\(621\) 3.96672 0.159179
\(622\) 0 0
\(623\) 47.0247 1.88400
\(624\) 0 0
\(625\) 12.5738 0.502953
\(626\) 0 0
\(627\) −18.6747 −0.745794
\(628\) 0 0
\(629\) −10.5395 −0.420237
\(630\) 0 0
\(631\) 16.9953 0.676574 0.338287 0.941043i \(-0.390153\pi\)
0.338287 + 0.941043i \(0.390153\pi\)
\(632\) 0 0
\(633\) −18.2109 −0.723819
\(634\) 0 0
\(635\) 2.71379 0.107694
\(636\) 0 0
\(637\) −95.7473 −3.79364
\(638\) 0 0
\(639\) −8.53856 −0.337780
\(640\) 0 0
\(641\) −5.16277 −0.203917 −0.101959 0.994789i \(-0.532511\pi\)
−0.101959 + 0.994789i \(0.532511\pi\)
\(642\) 0 0
\(643\) 21.2816 0.839266 0.419633 0.907694i \(-0.362159\pi\)
0.419633 + 0.907694i \(0.362159\pi\)
\(644\) 0 0
\(645\) 0.876392 0.0345079
\(646\) 0 0
\(647\) 16.0273 0.630099 0.315049 0.949075i \(-0.397979\pi\)
0.315049 + 0.949075i \(0.397979\pi\)
\(648\) 0 0
\(649\) 57.1357 2.24277
\(650\) 0 0
\(651\) 46.1762 1.80979
\(652\) 0 0
\(653\) 0.626015 0.0244979 0.0122489 0.999925i \(-0.496101\pi\)
0.0122489 + 0.999925i \(0.496101\pi\)
\(654\) 0 0
\(655\) 8.41468 0.328789
\(656\) 0 0
\(657\) −5.77746 −0.225400
\(658\) 0 0
\(659\) 12.5268 0.487974 0.243987 0.969778i \(-0.421545\pi\)
0.243987 + 0.969778i \(0.421545\pi\)
\(660\) 0 0
\(661\) −42.7520 −1.66286 −0.831431 0.555628i \(-0.812478\pi\)
−0.831431 + 0.555628i \(0.812478\pi\)
\(662\) 0 0
\(663\) −19.7272 −0.766142
\(664\) 0 0
\(665\) −22.9521 −0.890043
\(666\) 0 0
\(667\) 40.5932 1.57177
\(668\) 0 0
\(669\) −16.9336 −0.654690
\(670\) 0 0
\(671\) −47.5772 −1.83670
\(672\) 0 0
\(673\) −36.4562 −1.40528 −0.702641 0.711545i \(-0.747996\pi\)
−0.702641 + 0.711545i \(0.747996\pi\)
\(674\) 0 0
\(675\) 4.11996 0.158577
\(676\) 0 0
\(677\) −13.1589 −0.505738 −0.252869 0.967501i \(-0.581374\pi\)
−0.252869 + 0.967501i \(0.581374\pi\)
\(678\) 0 0
\(679\) 20.7025 0.794489
\(680\) 0 0
\(681\) −3.66081 −0.140282
\(682\) 0 0
\(683\) −26.9825 −1.03246 −0.516228 0.856451i \(-0.672664\pi\)
−0.516228 + 0.856451i \(0.672664\pi\)
\(684\) 0 0
\(685\) 16.5268 0.631457
\(686\) 0 0
\(687\) 23.7283 0.905289
\(688\) 0 0
\(689\) −4.23786 −0.161450
\(690\) 0 0
\(691\) −48.8616 −1.85878 −0.929391 0.369098i \(-0.879667\pi\)
−0.929391 + 0.369098i \(0.879667\pi\)
\(692\) 0 0
\(693\) 18.8288 0.715246
\(694\) 0 0
\(695\) 10.2727 0.389664
\(696\) 0 0
\(697\) 20.4384 0.774160
\(698\) 0 0
\(699\) 12.4226 0.469867
\(700\) 0 0
\(701\) −4.66374 −0.176147 −0.0880735 0.996114i \(-0.528071\pi\)
−0.0880735 + 0.996114i \(0.528071\pi\)
\(702\) 0 0
\(703\) 14.2622 0.537909
\(704\) 0 0
\(705\) −9.05085 −0.340875
\(706\) 0 0
\(707\) −39.3154 −1.47861
\(708\) 0 0
\(709\) −16.0634 −0.603272 −0.301636 0.953423i \(-0.597533\pi\)
−0.301636 + 0.953423i \(0.597533\pi\)
\(710\) 0 0
\(711\) −0.994199 −0.0372854
\(712\) 0 0
\(713\) 36.8790 1.38113
\(714\) 0 0
\(715\) 19.2724 0.720748
\(716\) 0 0
\(717\) 21.7982 0.814068
\(718\) 0 0
\(719\) 3.91597 0.146041 0.0730204 0.997330i \(-0.476736\pi\)
0.0730204 + 0.997330i \(0.476736\pi\)
\(720\) 0 0
\(721\) −23.2267 −0.865007
\(722\) 0 0
\(723\) −5.02354 −0.186827
\(724\) 0 0
\(725\) 42.1613 1.56583
\(726\) 0 0
\(727\) −0.871641 −0.0323274 −0.0161637 0.999869i \(-0.505145\pi\)
−0.0161637 + 0.999869i \(0.505145\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.40079 −0.125783
\(732\) 0 0
\(733\) 20.8543 0.770270 0.385135 0.922860i \(-0.374155\pi\)
0.385135 + 0.922860i \(0.374155\pi\)
\(734\) 0 0
\(735\) 16.5747 0.611368
\(736\) 0 0
\(737\) 30.1870 1.11195
\(738\) 0 0
\(739\) −20.8564 −0.767217 −0.383608 0.923496i \(-0.625319\pi\)
−0.383608 + 0.923496i \(0.625319\pi\)
\(740\) 0 0
\(741\) 26.6952 0.980671
\(742\) 0 0
\(743\) −7.26706 −0.266603 −0.133301 0.991076i \(-0.542558\pi\)
−0.133301 + 0.991076i \(0.542558\pi\)
\(744\) 0 0
\(745\) 10.0721 0.369014
\(746\) 0 0
\(747\) 2.08014 0.0761082
\(748\) 0 0
\(749\) −58.0852 −2.12239
\(750\) 0 0
\(751\) 15.7011 0.572943 0.286471 0.958089i \(-0.407518\pi\)
0.286471 + 0.958089i \(0.407518\pi\)
\(752\) 0 0
\(753\) 10.2662 0.374122
\(754\) 0 0
\(755\) 0.728510 0.0265132
\(756\) 0 0
\(757\) −31.4685 −1.14374 −0.571870 0.820344i \(-0.693782\pi\)
−0.571870 + 0.820344i \(0.693782\pi\)
\(758\) 0 0
\(759\) 15.0378 0.545837
\(760\) 0 0
\(761\) −10.6417 −0.385760 −0.192880 0.981222i \(-0.561783\pi\)
−0.192880 + 0.981222i \(0.561783\pi\)
\(762\) 0 0
\(763\) 79.4508 2.87631
\(764\) 0 0
\(765\) 3.41496 0.123468
\(766\) 0 0
\(767\) −81.6746 −2.94910
\(768\) 0 0
\(769\) 21.4970 0.775201 0.387601 0.921827i \(-0.373304\pi\)
0.387601 + 0.921827i \(0.373304\pi\)
\(770\) 0 0
\(771\) 11.9857 0.431656
\(772\) 0 0
\(773\) 24.0847 0.866267 0.433134 0.901330i \(-0.357408\pi\)
0.433134 + 0.901330i \(0.357408\pi\)
\(774\) 0 0
\(775\) 38.3037 1.37591
\(776\) 0 0
\(777\) −14.3799 −0.515875
\(778\) 0 0
\(779\) −27.6576 −0.990935
\(780\) 0 0
\(781\) −32.3696 −1.15827
\(782\) 0 0
\(783\) 10.2334 0.365713
\(784\) 0 0
\(785\) 6.93638 0.247570
\(786\) 0 0
\(787\) −5.23370 −0.186561 −0.0932807 0.995640i \(-0.529735\pi\)
−0.0932807 + 0.995640i \(0.529735\pi\)
\(788\) 0 0
\(789\) 26.9764 0.960386
\(790\) 0 0
\(791\) −31.7808 −1.13000
\(792\) 0 0
\(793\) 68.0109 2.41514
\(794\) 0 0
\(795\) 0.733613 0.0260186
\(796\) 0 0
\(797\) −39.8380 −1.41113 −0.705567 0.708643i \(-0.749308\pi\)
−0.705567 + 0.708643i \(0.749308\pi\)
\(798\) 0 0
\(799\) 35.1213 1.24250
\(800\) 0 0
\(801\) 9.46795 0.334534
\(802\) 0 0
\(803\) −21.9023 −0.772915
\(804\) 0 0
\(805\) 18.4822 0.651410
\(806\) 0 0
\(807\) −29.3337 −1.03260
\(808\) 0 0
\(809\) −32.0976 −1.12849 −0.564245 0.825607i \(-0.690833\pi\)
−0.564245 + 0.825607i \(0.690833\pi\)
\(810\) 0 0
\(811\) 9.75780 0.342643 0.171321 0.985215i \(-0.445196\pi\)
0.171321 + 0.985215i \(0.445196\pi\)
\(812\) 0 0
\(813\) −11.4504 −0.401584
\(814\) 0 0
\(815\) −15.8274 −0.554411
\(816\) 0 0
\(817\) 4.60200 0.161004
\(818\) 0 0
\(819\) −26.9154 −0.940502
\(820\) 0 0
\(821\) −22.3362 −0.779539 −0.389770 0.920912i \(-0.627445\pi\)
−0.389770 + 0.920912i \(0.627445\pi\)
\(822\) 0 0
\(823\) −11.2312 −0.391496 −0.195748 0.980654i \(-0.562714\pi\)
−0.195748 + 0.980654i \(0.562714\pi\)
\(824\) 0 0
\(825\) 15.6187 0.543774
\(826\) 0 0
\(827\) −23.4558 −0.815639 −0.407819 0.913063i \(-0.633711\pi\)
−0.407819 + 0.913063i \(0.633711\pi\)
\(828\) 0 0
\(829\) −15.4725 −0.537381 −0.268690 0.963227i \(-0.586591\pi\)
−0.268690 + 0.963227i \(0.586591\pi\)
\(830\) 0 0
\(831\) −2.10038 −0.0728613
\(832\) 0 0
\(833\) −64.3173 −2.22846
\(834\) 0 0
\(835\) −0.938106 −0.0324645
\(836\) 0 0
\(837\) 9.29712 0.321355
\(838\) 0 0
\(839\) −1.61128 −0.0556276 −0.0278138 0.999613i \(-0.508855\pi\)
−0.0278138 + 0.999613i \(0.508855\pi\)
\(840\) 0 0
\(841\) 75.7232 2.61115
\(842\) 0 0
\(843\) 2.37932 0.0819481
\(844\) 0 0
\(845\) −15.3543 −0.528203
\(846\) 0 0
\(847\) 16.7457 0.575390
\(848\) 0 0
\(849\) 28.0104 0.961314
\(850\) 0 0
\(851\) −11.4846 −0.393688
\(852\) 0 0
\(853\) −3.22499 −0.110421 −0.0552107 0.998475i \(-0.517583\pi\)
−0.0552107 + 0.998475i \(0.517583\pi\)
\(854\) 0 0
\(855\) −4.62117 −0.158041
\(856\) 0 0
\(857\) −1.31541 −0.0449334 −0.0224667 0.999748i \(-0.507152\pi\)
−0.0224667 + 0.999748i \(0.507152\pi\)
\(858\) 0 0
\(859\) 2.03496 0.0694320 0.0347160 0.999397i \(-0.488947\pi\)
0.0347160 + 0.999397i \(0.488947\pi\)
\(860\) 0 0
\(861\) 27.8858 0.950345
\(862\) 0 0
\(863\) −26.2917 −0.894981 −0.447490 0.894289i \(-0.647682\pi\)
−0.447490 + 0.894289i \(0.647682\pi\)
\(864\) 0 0
\(865\) 17.3198 0.588893
\(866\) 0 0
\(867\) 3.74843 0.127303
\(868\) 0 0
\(869\) −3.76900 −0.127854
\(870\) 0 0
\(871\) −43.1518 −1.46214
\(872\) 0 0
\(873\) 4.16825 0.141074
\(874\) 0 0
\(875\) 42.4927 1.43651
\(876\) 0 0
\(877\) 3.99857 0.135022 0.0675111 0.997719i \(-0.478494\pi\)
0.0675111 + 0.997719i \(0.478494\pi\)
\(878\) 0 0
\(879\) 6.26096 0.211177
\(880\) 0 0
\(881\) 36.8878 1.24278 0.621391 0.783501i \(-0.286568\pi\)
0.621391 + 0.783501i \(0.286568\pi\)
\(882\) 0 0
\(883\) 20.5716 0.692290 0.346145 0.938181i \(-0.387491\pi\)
0.346145 + 0.938181i \(0.387491\pi\)
\(884\) 0 0
\(885\) 14.1386 0.475264
\(886\) 0 0
\(887\) −33.4419 −1.12287 −0.561435 0.827521i \(-0.689750\pi\)
−0.561435 + 0.827521i \(0.689750\pi\)
\(888\) 0 0
\(889\) −14.3679 −0.481885
\(890\) 0 0
\(891\) 3.79099 0.127003
\(892\) 0 0
\(893\) −47.5267 −1.59042
\(894\) 0 0
\(895\) −13.3193 −0.445217
\(896\) 0 0
\(897\) −21.4963 −0.717740
\(898\) 0 0
\(899\) 95.1415 3.17315
\(900\) 0 0
\(901\) −2.84675 −0.0948388
\(902\) 0 0
\(903\) −4.63998 −0.154409
\(904\) 0 0
\(905\) −17.1756 −0.570938
\(906\) 0 0
\(907\) 43.4819 1.44379 0.721897 0.692001i \(-0.243270\pi\)
0.721897 + 0.692001i \(0.243270\pi\)
\(908\) 0 0
\(909\) −7.91578 −0.262550
\(910\) 0 0
\(911\) 7.34014 0.243190 0.121595 0.992580i \(-0.461199\pi\)
0.121595 + 0.992580i \(0.461199\pi\)
\(912\) 0 0
\(913\) 7.88577 0.260981
\(914\) 0 0
\(915\) −11.7733 −0.389213
\(916\) 0 0
\(917\) −44.5508 −1.47120
\(918\) 0 0
\(919\) 10.1239 0.333957 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(920\) 0 0
\(921\) 29.9131 0.985671
\(922\) 0 0
\(923\) 46.2718 1.52306
\(924\) 0 0
\(925\) −11.9283 −0.392200
\(926\) 0 0
\(927\) −4.67646 −0.153595
\(928\) 0 0
\(929\) 10.3148 0.338419 0.169210 0.985580i \(-0.445879\pi\)
0.169210 + 0.985580i \(0.445879\pi\)
\(930\) 0 0
\(931\) 87.0352 2.85246
\(932\) 0 0
\(933\) −16.6641 −0.545557
\(934\) 0 0
\(935\) 12.9461 0.423382
\(936\) 0 0
\(937\) −15.4046 −0.503247 −0.251623 0.967825i \(-0.580964\pi\)
−0.251623 + 0.967825i \(0.580964\pi\)
\(938\) 0 0
\(939\) −1.82677 −0.0596144
\(940\) 0 0
\(941\) 50.3822 1.64241 0.821207 0.570630i \(-0.193301\pi\)
0.821207 + 0.570630i \(0.193301\pi\)
\(942\) 0 0
\(943\) 22.2712 0.725252
\(944\) 0 0
\(945\) 4.65931 0.151567
\(946\) 0 0
\(947\) 19.6479 0.638472 0.319236 0.947675i \(-0.396574\pi\)
0.319236 + 0.947675i \(0.396574\pi\)
\(948\) 0 0
\(949\) 31.3090 1.01633
\(950\) 0 0
\(951\) 6.80617 0.220705
\(952\) 0 0
\(953\) −4.92780 −0.159627 −0.0798136 0.996810i \(-0.525432\pi\)
−0.0798136 + 0.996810i \(0.525432\pi\)
\(954\) 0 0
\(955\) −12.6331 −0.408799
\(956\) 0 0
\(957\) 38.7948 1.25406
\(958\) 0 0
\(959\) −87.4998 −2.82551
\(960\) 0 0
\(961\) 55.4364 1.78827
\(962\) 0 0
\(963\) −11.6949 −0.376862
\(964\) 0 0
\(965\) −12.8004 −0.412059
\(966\) 0 0
\(967\) 24.1654 0.777107 0.388554 0.921426i \(-0.372975\pi\)
0.388554 + 0.921426i \(0.372975\pi\)
\(968\) 0 0
\(969\) 17.9322 0.576066
\(970\) 0 0
\(971\) −19.9672 −0.640780 −0.320390 0.947286i \(-0.603814\pi\)
−0.320390 + 0.947286i \(0.603814\pi\)
\(972\) 0 0
\(973\) −54.3877 −1.74359
\(974\) 0 0
\(975\) −22.3267 −0.715027
\(976\) 0 0
\(977\) 52.2739 1.67239 0.836195 0.548432i \(-0.184775\pi\)
0.836195 + 0.548432i \(0.184775\pi\)
\(978\) 0 0
\(979\) 35.8929 1.14714
\(980\) 0 0
\(981\) 15.9966 0.510734
\(982\) 0 0
\(983\) 26.5159 0.845726 0.422863 0.906194i \(-0.361025\pi\)
0.422863 + 0.906194i \(0.361025\pi\)
\(984\) 0 0
\(985\) −19.6804 −0.627070
\(986\) 0 0
\(987\) 47.9189 1.52527
\(988\) 0 0
\(989\) −3.70576 −0.117836
\(990\) 0 0
\(991\) −23.6727 −0.751988 −0.375994 0.926622i \(-0.622699\pi\)
−0.375994 + 0.926622i \(0.622699\pi\)
\(992\) 0 0
\(993\) −33.7166 −1.06996
\(994\) 0 0
\(995\) 12.2232 0.387501
\(996\) 0 0
\(997\) −29.5729 −0.936584 −0.468292 0.883574i \(-0.655131\pi\)
−0.468292 + 0.883574i \(0.655131\pi\)
\(998\) 0 0
\(999\) −2.89525 −0.0916016
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.h.1.4 8
4.3 odd 2 8016.2.a.y.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.h.1.4 8 1.1 even 1 trivial
8016.2.a.y.1.4 8 4.3 odd 2