Properties

Label 6008.2.a.d.1.3
Level $6008$
Weight $2$
Character 6008.1
Self dual yes
Analytic conductor $47.974$
Analytic rank $0$
Dimension $49$
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] = [6008,2,Mod(1,6008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, 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("6008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6008.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: \(47.9741215344\)
Analytic rank: \(0\)
Dimension: \(49\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 6008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.90482 q^{3} -3.52511 q^{5} +3.26846 q^{7} +5.43799 q^{9} +O(q^{10})\) \(q-2.90482 q^{3} -3.52511 q^{5} +3.26846 q^{7} +5.43799 q^{9} -4.64760 q^{11} -0.0450431 q^{13} +10.2398 q^{15} -5.09573 q^{17} +2.40802 q^{19} -9.49429 q^{21} -1.86095 q^{23} +7.42643 q^{25} -7.08191 q^{27} -1.84468 q^{29} -3.76774 q^{31} +13.5005 q^{33} -11.5217 q^{35} +8.39868 q^{37} +0.130842 q^{39} +3.06429 q^{41} -7.51254 q^{43} -19.1695 q^{45} +8.33898 q^{47} +3.68283 q^{49} +14.8022 q^{51} -10.2994 q^{53} +16.3833 q^{55} -6.99488 q^{57} -2.35032 q^{59} -3.15473 q^{61} +17.7738 q^{63} +0.158782 q^{65} -6.89025 q^{67} +5.40574 q^{69} -0.257052 q^{71} -7.34373 q^{73} -21.5725 q^{75} -15.1905 q^{77} -8.04528 q^{79} +4.25773 q^{81} +4.16290 q^{83} +17.9630 q^{85} +5.35846 q^{87} -4.10488 q^{89} -0.147221 q^{91} +10.9446 q^{93} -8.48856 q^{95} -7.27541 q^{97} -25.2736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 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
\(3\) −2.90482 −1.67710 −0.838550 0.544825i \(-0.816596\pi\)
−0.838550 + 0.544825i \(0.816596\pi\)
\(4\) 0 0
\(5\) −3.52511 −1.57648 −0.788239 0.615369i \(-0.789007\pi\)
−0.788239 + 0.615369i \(0.789007\pi\)
\(6\) 0 0
\(7\) 3.26846 1.23536 0.617681 0.786429i \(-0.288072\pi\)
0.617681 + 0.786429i \(0.288072\pi\)
\(8\) 0 0
\(9\) 5.43799 1.81266
\(10\) 0 0
\(11\) −4.64760 −1.40131 −0.700653 0.713502i \(-0.747108\pi\)
−0.700653 + 0.713502i \(0.747108\pi\)
\(12\) 0 0
\(13\) −0.0450431 −0.0124927 −0.00624635 0.999980i \(-0.501988\pi\)
−0.00624635 + 0.999980i \(0.501988\pi\)
\(14\) 0 0
\(15\) 10.2398 2.64391
\(16\) 0 0
\(17\) −5.09573 −1.23590 −0.617948 0.786219i \(-0.712036\pi\)
−0.617948 + 0.786219i \(0.712036\pi\)
\(18\) 0 0
\(19\) 2.40802 0.552439 0.276219 0.961095i \(-0.410918\pi\)
0.276219 + 0.961095i \(0.410918\pi\)
\(20\) 0 0
\(21\) −9.49429 −2.07182
\(22\) 0 0
\(23\) −1.86095 −0.388036 −0.194018 0.980998i \(-0.562152\pi\)
−0.194018 + 0.980998i \(0.562152\pi\)
\(24\) 0 0
\(25\) 7.42643 1.48529
\(26\) 0 0
\(27\) −7.08191 −1.36291
\(28\) 0 0
\(29\) −1.84468 −0.342548 −0.171274 0.985223i \(-0.554788\pi\)
−0.171274 + 0.985223i \(0.554788\pi\)
\(30\) 0 0
\(31\) −3.76774 −0.676705 −0.338353 0.941019i \(-0.609870\pi\)
−0.338353 + 0.941019i \(0.609870\pi\)
\(32\) 0 0
\(33\) 13.5005 2.35013
\(34\) 0 0
\(35\) −11.5217 −1.94752
\(36\) 0 0
\(37\) 8.39868 1.38073 0.690367 0.723459i \(-0.257449\pi\)
0.690367 + 0.723459i \(0.257449\pi\)
\(38\) 0 0
\(39\) 0.130842 0.0209515
\(40\) 0 0
\(41\) 3.06429 0.478561 0.239281 0.970950i \(-0.423088\pi\)
0.239281 + 0.970950i \(0.423088\pi\)
\(42\) 0 0
\(43\) −7.51254 −1.14565 −0.572826 0.819677i \(-0.694153\pi\)
−0.572826 + 0.819677i \(0.694153\pi\)
\(44\) 0 0
\(45\) −19.1695 −2.85762
\(46\) 0 0
\(47\) 8.33898 1.21637 0.608183 0.793797i \(-0.291899\pi\)
0.608183 + 0.793797i \(0.291899\pi\)
\(48\) 0 0
\(49\) 3.68283 0.526119
\(50\) 0 0
\(51\) 14.8022 2.07272
\(52\) 0 0
\(53\) −10.2994 −1.41473 −0.707365 0.706848i \(-0.750116\pi\)
−0.707365 + 0.706848i \(0.750116\pi\)
\(54\) 0 0
\(55\) 16.3833 2.20913
\(56\) 0 0
\(57\) −6.99488 −0.926495
\(58\) 0 0
\(59\) −2.35032 −0.305985 −0.152993 0.988227i \(-0.548891\pi\)
−0.152993 + 0.988227i \(0.548891\pi\)
\(60\) 0 0
\(61\) −3.15473 −0.403922 −0.201961 0.979394i \(-0.564731\pi\)
−0.201961 + 0.979394i \(0.564731\pi\)
\(62\) 0 0
\(63\) 17.7738 2.23929
\(64\) 0 0
\(65\) 0.158782 0.0196945
\(66\) 0 0
\(67\) −6.89025 −0.841778 −0.420889 0.907112i \(-0.638282\pi\)
−0.420889 + 0.907112i \(0.638282\pi\)
\(68\) 0 0
\(69\) 5.40574 0.650775
\(70\) 0 0
\(71\) −0.257052 −0.0305065 −0.0152532 0.999884i \(-0.504855\pi\)
−0.0152532 + 0.999884i \(0.504855\pi\)
\(72\) 0 0
\(73\) −7.34373 −0.859519 −0.429759 0.902943i \(-0.641402\pi\)
−0.429759 + 0.902943i \(0.641402\pi\)
\(74\) 0 0
\(75\) −21.5725 −2.49097
\(76\) 0 0
\(77\) −15.1905 −1.73112
\(78\) 0 0
\(79\) −8.04528 −0.905165 −0.452582 0.891723i \(-0.649497\pi\)
−0.452582 + 0.891723i \(0.649497\pi\)
\(80\) 0 0
\(81\) 4.25773 0.473081
\(82\) 0 0
\(83\) 4.16290 0.456938 0.228469 0.973551i \(-0.426628\pi\)
0.228469 + 0.973551i \(0.426628\pi\)
\(84\) 0 0
\(85\) 17.9630 1.94836
\(86\) 0 0
\(87\) 5.35846 0.574487
\(88\) 0 0
\(89\) −4.10488 −0.435117 −0.217558 0.976047i \(-0.569809\pi\)
−0.217558 + 0.976047i \(0.569809\pi\)
\(90\) 0 0
\(91\) −0.147221 −0.0154330
\(92\) 0 0
\(93\) 10.9446 1.13490
\(94\) 0 0
\(95\) −8.48856 −0.870908
\(96\) 0 0
\(97\) −7.27541 −0.738706 −0.369353 0.929289i \(-0.620421\pi\)
−0.369353 + 0.929289i \(0.620421\pi\)
\(98\) 0 0
\(99\) −25.2736 −2.54009
\(100\) 0 0
\(101\) −15.1900 −1.51147 −0.755733 0.654880i \(-0.772719\pi\)
−0.755733 + 0.654880i \(0.772719\pi\)
\(102\) 0 0
\(103\) 1.19234 0.117485 0.0587424 0.998273i \(-0.481291\pi\)
0.0587424 + 0.998273i \(0.481291\pi\)
\(104\) 0 0
\(105\) 33.4685 3.26619
\(106\) 0 0
\(107\) −13.3674 −1.29227 −0.646137 0.763222i \(-0.723616\pi\)
−0.646137 + 0.763222i \(0.723616\pi\)
\(108\) 0 0
\(109\) −11.2854 −1.08094 −0.540472 0.841362i \(-0.681754\pi\)
−0.540472 + 0.841362i \(0.681754\pi\)
\(110\) 0 0
\(111\) −24.3967 −2.31563
\(112\) 0 0
\(113\) −4.36482 −0.410607 −0.205304 0.978698i \(-0.565818\pi\)
−0.205304 + 0.978698i \(0.565818\pi\)
\(114\) 0 0
\(115\) 6.56008 0.611730
\(116\) 0 0
\(117\) −0.244944 −0.0226450
\(118\) 0 0
\(119\) −16.6552 −1.52678
\(120\) 0 0
\(121\) 10.6002 0.963657
\(122\) 0 0
\(123\) −8.90120 −0.802595
\(124\) 0 0
\(125\) −8.55344 −0.765043
\(126\) 0 0
\(127\) −1.38340 −0.122757 −0.0613784 0.998115i \(-0.519550\pi\)
−0.0613784 + 0.998115i \(0.519550\pi\)
\(128\) 0 0
\(129\) 21.8226 1.92137
\(130\) 0 0
\(131\) 8.57506 0.749207 0.374603 0.927185i \(-0.377779\pi\)
0.374603 + 0.927185i \(0.377779\pi\)
\(132\) 0 0
\(133\) 7.87053 0.682462
\(134\) 0 0
\(135\) 24.9646 2.14861
\(136\) 0 0
\(137\) −7.35232 −0.628151 −0.314076 0.949398i \(-0.601695\pi\)
−0.314076 + 0.949398i \(0.601695\pi\)
\(138\) 0 0
\(139\) 9.46826 0.803088 0.401544 0.915840i \(-0.368474\pi\)
0.401544 + 0.915840i \(0.368474\pi\)
\(140\) 0 0
\(141\) −24.2233 −2.03997
\(142\) 0 0
\(143\) 0.209342 0.0175061
\(144\) 0 0
\(145\) 6.50270 0.540020
\(146\) 0 0
\(147\) −10.6980 −0.882353
\(148\) 0 0
\(149\) −12.2018 −0.999611 −0.499805 0.866138i \(-0.666595\pi\)
−0.499805 + 0.866138i \(0.666595\pi\)
\(150\) 0 0
\(151\) 19.8095 1.61207 0.806036 0.591867i \(-0.201609\pi\)
0.806036 + 0.591867i \(0.201609\pi\)
\(152\) 0 0
\(153\) −27.7105 −2.24026
\(154\) 0 0
\(155\) 13.2817 1.06681
\(156\) 0 0
\(157\) −20.8024 −1.66021 −0.830107 0.557604i \(-0.811721\pi\)
−0.830107 + 0.557604i \(0.811721\pi\)
\(158\) 0 0
\(159\) 29.9179 2.37264
\(160\) 0 0
\(161\) −6.08246 −0.479365
\(162\) 0 0
\(163\) −2.72139 −0.213156 −0.106578 0.994304i \(-0.533989\pi\)
−0.106578 + 0.994304i \(0.533989\pi\)
\(164\) 0 0
\(165\) −47.5907 −3.70493
\(166\) 0 0
\(167\) 20.1603 1.56005 0.780026 0.625748i \(-0.215206\pi\)
0.780026 + 0.625748i \(0.215206\pi\)
\(168\) 0 0
\(169\) −12.9980 −0.999844
\(170\) 0 0
\(171\) 13.0948 1.00138
\(172\) 0 0
\(173\) 21.5453 1.63806 0.819029 0.573752i \(-0.194512\pi\)
0.819029 + 0.573752i \(0.194512\pi\)
\(174\) 0 0
\(175\) 24.2730 1.83487
\(176\) 0 0
\(177\) 6.82725 0.513168
\(178\) 0 0
\(179\) −3.69574 −0.276233 −0.138116 0.990416i \(-0.544105\pi\)
−0.138116 + 0.990416i \(0.544105\pi\)
\(180\) 0 0
\(181\) 7.18186 0.533823 0.266912 0.963721i \(-0.413997\pi\)
0.266912 + 0.963721i \(0.413997\pi\)
\(182\) 0 0
\(183\) 9.16392 0.677417
\(184\) 0 0
\(185\) −29.6063 −2.17670
\(186\) 0 0
\(187\) 23.6829 1.73187
\(188\) 0 0
\(189\) −23.1469 −1.68369
\(190\) 0 0
\(191\) 8.01302 0.579802 0.289901 0.957057i \(-0.406378\pi\)
0.289901 + 0.957057i \(0.406378\pi\)
\(192\) 0 0
\(193\) 7.26269 0.522780 0.261390 0.965233i \(-0.415819\pi\)
0.261390 + 0.965233i \(0.415819\pi\)
\(194\) 0 0
\(195\) −0.461233 −0.0330296
\(196\) 0 0
\(197\) 3.29301 0.234617 0.117309 0.993095i \(-0.462573\pi\)
0.117309 + 0.993095i \(0.462573\pi\)
\(198\) 0 0
\(199\) −5.31505 −0.376774 −0.188387 0.982095i \(-0.560326\pi\)
−0.188387 + 0.982095i \(0.560326\pi\)
\(200\) 0 0
\(201\) 20.0150 1.41175
\(202\) 0 0
\(203\) −6.02926 −0.423171
\(204\) 0 0
\(205\) −10.8020 −0.754442
\(206\) 0 0
\(207\) −10.1198 −0.703378
\(208\) 0 0
\(209\) −11.1915 −0.774135
\(210\) 0 0
\(211\) 14.7288 1.01397 0.506985 0.861955i \(-0.330760\pi\)
0.506985 + 0.861955i \(0.330760\pi\)
\(212\) 0 0
\(213\) 0.746691 0.0511624
\(214\) 0 0
\(215\) 26.4826 1.80610
\(216\) 0 0
\(217\) −12.3147 −0.835976
\(218\) 0 0
\(219\) 21.3322 1.44150
\(220\) 0 0
\(221\) 0.229527 0.0154397
\(222\) 0 0
\(223\) 7.17168 0.480251 0.240126 0.970742i \(-0.422811\pi\)
0.240126 + 0.970742i \(0.422811\pi\)
\(224\) 0 0
\(225\) 40.3848 2.69232
\(226\) 0 0
\(227\) −17.3297 −1.15021 −0.575107 0.818079i \(-0.695039\pi\)
−0.575107 + 0.818079i \(0.695039\pi\)
\(228\) 0 0
\(229\) −1.11565 −0.0737241 −0.0368621 0.999320i \(-0.511736\pi\)
−0.0368621 + 0.999320i \(0.511736\pi\)
\(230\) 0 0
\(231\) 44.1257 2.90326
\(232\) 0 0
\(233\) −10.9785 −0.719228 −0.359614 0.933101i \(-0.617092\pi\)
−0.359614 + 0.933101i \(0.617092\pi\)
\(234\) 0 0
\(235\) −29.3959 −1.91758
\(236\) 0 0
\(237\) 23.3701 1.51805
\(238\) 0 0
\(239\) −2.70638 −0.175061 −0.0875307 0.996162i \(-0.527898\pi\)
−0.0875307 + 0.996162i \(0.527898\pi\)
\(240\) 0 0
\(241\) 19.4495 1.25285 0.626427 0.779480i \(-0.284517\pi\)
0.626427 + 0.779480i \(0.284517\pi\)
\(242\) 0 0
\(243\) 8.87779 0.569510
\(244\) 0 0
\(245\) −12.9824 −0.829415
\(246\) 0 0
\(247\) −0.108465 −0.00690145
\(248\) 0 0
\(249\) −12.0925 −0.766330
\(250\) 0 0
\(251\) 30.1612 1.90376 0.951880 0.306472i \(-0.0991486\pi\)
0.951880 + 0.306472i \(0.0991486\pi\)
\(252\) 0 0
\(253\) 8.64898 0.543757
\(254\) 0 0
\(255\) −52.1794 −3.26760
\(256\) 0 0
\(257\) −27.6829 −1.72681 −0.863405 0.504511i \(-0.831673\pi\)
−0.863405 + 0.504511i \(0.831673\pi\)
\(258\) 0 0
\(259\) 27.4507 1.70571
\(260\) 0 0
\(261\) −10.0313 −0.620924
\(262\) 0 0
\(263\) −13.5189 −0.833612 −0.416806 0.908995i \(-0.636851\pi\)
−0.416806 + 0.908995i \(0.636851\pi\)
\(264\) 0 0
\(265\) 36.3065 2.23029
\(266\) 0 0
\(267\) 11.9239 0.729734
\(268\) 0 0
\(269\) 9.19352 0.560539 0.280269 0.959921i \(-0.409576\pi\)
0.280269 + 0.959921i \(0.409576\pi\)
\(270\) 0 0
\(271\) 27.8199 1.68994 0.844968 0.534817i \(-0.179619\pi\)
0.844968 + 0.534817i \(0.179619\pi\)
\(272\) 0 0
\(273\) 0.427652 0.0258827
\(274\) 0 0
\(275\) −34.5151 −2.08134
\(276\) 0 0
\(277\) 26.4555 1.58956 0.794778 0.606900i \(-0.207587\pi\)
0.794778 + 0.606900i \(0.207587\pi\)
\(278\) 0 0
\(279\) −20.4889 −1.22664
\(280\) 0 0
\(281\) −18.5284 −1.10531 −0.552657 0.833409i \(-0.686386\pi\)
−0.552657 + 0.833409i \(0.686386\pi\)
\(282\) 0 0
\(283\) 20.5921 1.22407 0.612037 0.790829i \(-0.290350\pi\)
0.612037 + 0.790829i \(0.290350\pi\)
\(284\) 0 0
\(285\) 24.6578 1.46060
\(286\) 0 0
\(287\) 10.0155 0.591196
\(288\) 0 0
\(289\) 8.96646 0.527439
\(290\) 0 0
\(291\) 21.1338 1.23888
\(292\) 0 0
\(293\) 22.1527 1.29417 0.647086 0.762417i \(-0.275987\pi\)
0.647086 + 0.762417i \(0.275987\pi\)
\(294\) 0 0
\(295\) 8.28514 0.482379
\(296\) 0 0
\(297\) 32.9139 1.90986
\(298\) 0 0
\(299\) 0.0838231 0.00484762
\(300\) 0 0
\(301\) −24.5544 −1.41529
\(302\) 0 0
\(303\) 44.1244 2.53488
\(304\) 0 0
\(305\) 11.1208 0.636774
\(306\) 0 0
\(307\) 1.00131 0.0571477 0.0285739 0.999592i \(-0.490903\pi\)
0.0285739 + 0.999592i \(0.490903\pi\)
\(308\) 0 0
\(309\) −3.46354 −0.197034
\(310\) 0 0
\(311\) −13.6042 −0.771426 −0.385713 0.922619i \(-0.626044\pi\)
−0.385713 + 0.922619i \(0.626044\pi\)
\(312\) 0 0
\(313\) −2.02662 −0.114551 −0.0572757 0.998358i \(-0.518241\pi\)
−0.0572757 + 0.998358i \(0.518241\pi\)
\(314\) 0 0
\(315\) −62.6548 −3.53020
\(316\) 0 0
\(317\) 1.59531 0.0896018 0.0448009 0.998996i \(-0.485735\pi\)
0.0448009 + 0.998996i \(0.485735\pi\)
\(318\) 0 0
\(319\) 8.57334 0.480015
\(320\) 0 0
\(321\) 38.8299 2.16727
\(322\) 0 0
\(323\) −12.2706 −0.682757
\(324\) 0 0
\(325\) −0.334509 −0.0185552
\(326\) 0 0
\(327\) 32.7820 1.81285
\(328\) 0 0
\(329\) 27.2556 1.50265
\(330\) 0 0
\(331\) −14.0047 −0.769771 −0.384885 0.922964i \(-0.625759\pi\)
−0.384885 + 0.922964i \(0.625759\pi\)
\(332\) 0 0
\(333\) 45.6719 2.50280
\(334\) 0 0
\(335\) 24.2889 1.32705
\(336\) 0 0
\(337\) 8.50883 0.463505 0.231753 0.972775i \(-0.425554\pi\)
0.231753 + 0.972775i \(0.425554\pi\)
\(338\) 0 0
\(339\) 12.6790 0.688629
\(340\) 0 0
\(341\) 17.5109 0.948271
\(342\) 0 0
\(343\) −10.8420 −0.585415
\(344\) 0 0
\(345\) −19.0559 −1.02593
\(346\) 0 0
\(347\) 36.0884 1.93733 0.968663 0.248380i \(-0.0798982\pi\)
0.968663 + 0.248380i \(0.0798982\pi\)
\(348\) 0 0
\(349\) −3.88702 −0.208067 −0.104034 0.994574i \(-0.533175\pi\)
−0.104034 + 0.994574i \(0.533175\pi\)
\(350\) 0 0
\(351\) 0.318991 0.0170265
\(352\) 0 0
\(353\) 3.59663 0.191429 0.0957147 0.995409i \(-0.469486\pi\)
0.0957147 + 0.995409i \(0.469486\pi\)
\(354\) 0 0
\(355\) 0.906138 0.0480928
\(356\) 0 0
\(357\) 48.3803 2.56056
\(358\) 0 0
\(359\) −30.1587 −1.59172 −0.795859 0.605483i \(-0.792980\pi\)
−0.795859 + 0.605483i \(0.792980\pi\)
\(360\) 0 0
\(361\) −13.2014 −0.694811
\(362\) 0 0
\(363\) −30.7918 −1.61615
\(364\) 0 0
\(365\) 25.8875 1.35501
\(366\) 0 0
\(367\) 30.1574 1.57421 0.787103 0.616822i \(-0.211580\pi\)
0.787103 + 0.616822i \(0.211580\pi\)
\(368\) 0 0
\(369\) 16.6635 0.867470
\(370\) 0 0
\(371\) −33.6631 −1.74770
\(372\) 0 0
\(373\) 18.3686 0.951090 0.475545 0.879691i \(-0.342251\pi\)
0.475545 + 0.879691i \(0.342251\pi\)
\(374\) 0 0
\(375\) 24.8462 1.28305
\(376\) 0 0
\(377\) 0.0830900 0.00427935
\(378\) 0 0
\(379\) −29.0624 −1.49283 −0.746417 0.665478i \(-0.768228\pi\)
−0.746417 + 0.665478i \(0.768228\pi\)
\(380\) 0 0
\(381\) 4.01853 0.205875
\(382\) 0 0
\(383\) −13.0563 −0.667147 −0.333574 0.942724i \(-0.608255\pi\)
−0.333574 + 0.942724i \(0.608255\pi\)
\(384\) 0 0
\(385\) 53.5483 2.72907
\(386\) 0 0
\(387\) −40.8531 −2.07668
\(388\) 0 0
\(389\) −12.3178 −0.624539 −0.312270 0.949993i \(-0.601089\pi\)
−0.312270 + 0.949993i \(0.601089\pi\)
\(390\) 0 0
\(391\) 9.48292 0.479572
\(392\) 0 0
\(393\) −24.9090 −1.25649
\(394\) 0 0
\(395\) 28.3605 1.42697
\(396\) 0 0
\(397\) −8.08564 −0.405807 −0.202903 0.979199i \(-0.565038\pi\)
−0.202903 + 0.979199i \(0.565038\pi\)
\(398\) 0 0
\(399\) −22.8625 −1.14456
\(400\) 0 0
\(401\) 1.80321 0.0900482 0.0450241 0.998986i \(-0.485664\pi\)
0.0450241 + 0.998986i \(0.485664\pi\)
\(402\) 0 0
\(403\) 0.169710 0.00845388
\(404\) 0 0
\(405\) −15.0090 −0.745803
\(406\) 0 0
\(407\) −39.0337 −1.93483
\(408\) 0 0
\(409\) −36.4847 −1.80405 −0.902025 0.431683i \(-0.857920\pi\)
−0.902025 + 0.431683i \(0.857920\pi\)
\(410\) 0 0
\(411\) 21.3572 1.05347
\(412\) 0 0
\(413\) −7.68192 −0.378002
\(414\) 0 0
\(415\) −14.6747 −0.720353
\(416\) 0 0
\(417\) −27.5036 −1.34686
\(418\) 0 0
\(419\) −35.4041 −1.72960 −0.864801 0.502115i \(-0.832555\pi\)
−0.864801 + 0.502115i \(0.832555\pi\)
\(420\) 0 0
\(421\) −13.3305 −0.649689 −0.324844 0.945767i \(-0.605312\pi\)
−0.324844 + 0.945767i \(0.605312\pi\)
\(422\) 0 0
\(423\) 45.3473 2.20486
\(424\) 0 0
\(425\) −37.8431 −1.83566
\(426\) 0 0
\(427\) −10.3111 −0.498989
\(428\) 0 0
\(429\) −0.608102 −0.0293595
\(430\) 0 0
\(431\) 33.2799 1.60304 0.801519 0.597969i \(-0.204025\pi\)
0.801519 + 0.597969i \(0.204025\pi\)
\(432\) 0 0
\(433\) 12.0058 0.576962 0.288481 0.957486i \(-0.406850\pi\)
0.288481 + 0.957486i \(0.406850\pi\)
\(434\) 0 0
\(435\) −18.8892 −0.905667
\(436\) 0 0
\(437\) −4.48122 −0.214366
\(438\) 0 0
\(439\) 25.6186 1.22271 0.611355 0.791357i \(-0.290625\pi\)
0.611355 + 0.791357i \(0.290625\pi\)
\(440\) 0 0
\(441\) 20.0272 0.953675
\(442\) 0 0
\(443\) −1.67532 −0.0795968 −0.0397984 0.999208i \(-0.512672\pi\)
−0.0397984 + 0.999208i \(0.512672\pi\)
\(444\) 0 0
\(445\) 14.4702 0.685952
\(446\) 0 0
\(447\) 35.4441 1.67645
\(448\) 0 0
\(449\) 12.7593 0.602148 0.301074 0.953601i \(-0.402655\pi\)
0.301074 + 0.953601i \(0.402655\pi\)
\(450\) 0 0
\(451\) −14.2416 −0.670610
\(452\) 0 0
\(453\) −57.5429 −2.70360
\(454\) 0 0
\(455\) 0.518973 0.0243298
\(456\) 0 0
\(457\) 27.2083 1.27275 0.636376 0.771379i \(-0.280433\pi\)
0.636376 + 0.771379i \(0.280433\pi\)
\(458\) 0 0
\(459\) 36.0875 1.68442
\(460\) 0 0
\(461\) 28.7155 1.33741 0.668706 0.743527i \(-0.266848\pi\)
0.668706 + 0.743527i \(0.266848\pi\)
\(462\) 0 0
\(463\) 24.6857 1.14724 0.573621 0.819121i \(-0.305538\pi\)
0.573621 + 0.819121i \(0.305538\pi\)
\(464\) 0 0
\(465\) −38.5810 −1.78915
\(466\) 0 0
\(467\) 30.8583 1.42795 0.713975 0.700171i \(-0.246893\pi\)
0.713975 + 0.700171i \(0.246893\pi\)
\(468\) 0 0
\(469\) −22.5205 −1.03990
\(470\) 0 0
\(471\) 60.4273 2.78434
\(472\) 0 0
\(473\) 34.9153 1.60541
\(474\) 0 0
\(475\) 17.8830 0.820530
\(476\) 0 0
\(477\) −56.0079 −2.56443
\(478\) 0 0
\(479\) 33.3513 1.52386 0.761929 0.647660i \(-0.224252\pi\)
0.761929 + 0.647660i \(0.224252\pi\)
\(480\) 0 0
\(481\) −0.378302 −0.0172491
\(482\) 0 0
\(483\) 17.6684 0.803942
\(484\) 0 0
\(485\) 25.6467 1.16455
\(486\) 0 0
\(487\) −27.4361 −1.24325 −0.621624 0.783316i \(-0.713527\pi\)
−0.621624 + 0.783316i \(0.713527\pi\)
\(488\) 0 0
\(489\) 7.90515 0.357483
\(490\) 0 0
\(491\) 35.6578 1.60921 0.804607 0.593807i \(-0.202376\pi\)
0.804607 + 0.593807i \(0.202376\pi\)
\(492\) 0 0
\(493\) 9.39998 0.423354
\(494\) 0 0
\(495\) 89.0923 4.00440
\(496\) 0 0
\(497\) −0.840165 −0.0376865
\(498\) 0 0
\(499\) 7.08748 0.317279 0.158640 0.987337i \(-0.449289\pi\)
0.158640 + 0.987337i \(0.449289\pi\)
\(500\) 0 0
\(501\) −58.5621 −2.61636
\(502\) 0 0
\(503\) 28.9768 1.29201 0.646006 0.763332i \(-0.276438\pi\)
0.646006 + 0.763332i \(0.276438\pi\)
\(504\) 0 0
\(505\) 53.5467 2.38280
\(506\) 0 0
\(507\) 37.7568 1.67684
\(508\) 0 0
\(509\) −1.75520 −0.0777978 −0.0388989 0.999243i \(-0.512385\pi\)
−0.0388989 + 0.999243i \(0.512385\pi\)
\(510\) 0 0
\(511\) −24.0027 −1.06182
\(512\) 0 0
\(513\) −17.0534 −0.752927
\(514\) 0 0
\(515\) −4.20314 −0.185212
\(516\) 0 0
\(517\) −38.7563 −1.70450
\(518\) 0 0
\(519\) −62.5852 −2.74719
\(520\) 0 0
\(521\) 30.6076 1.34094 0.670472 0.741935i \(-0.266092\pi\)
0.670472 + 0.741935i \(0.266092\pi\)
\(522\) 0 0
\(523\) 28.6993 1.25493 0.627465 0.778644i \(-0.284092\pi\)
0.627465 + 0.778644i \(0.284092\pi\)
\(524\) 0 0
\(525\) −70.5087 −3.07725
\(526\) 0 0
\(527\) 19.1994 0.836337
\(528\) 0 0
\(529\) −19.5368 −0.849428
\(530\) 0 0
\(531\) −12.7810 −0.554648
\(532\) 0 0
\(533\) −0.138025 −0.00597852
\(534\) 0 0
\(535\) 47.1216 2.03724
\(536\) 0 0
\(537\) 10.7355 0.463270
\(538\) 0 0
\(539\) −17.1163 −0.737253
\(540\) 0 0
\(541\) 7.12633 0.306385 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(542\) 0 0
\(543\) −20.8620 −0.895275
\(544\) 0 0
\(545\) 39.7823 1.70409
\(546\) 0 0
\(547\) −7.08335 −0.302862 −0.151431 0.988468i \(-0.548388\pi\)
−0.151431 + 0.988468i \(0.548388\pi\)
\(548\) 0 0
\(549\) −17.1554 −0.732173
\(550\) 0 0
\(551\) −4.44203 −0.189237
\(552\) 0 0
\(553\) −26.2957 −1.11821
\(554\) 0 0
\(555\) 86.0010 3.65054
\(556\) 0 0
\(557\) −16.5666 −0.701951 −0.350976 0.936385i \(-0.614150\pi\)
−0.350976 + 0.936385i \(0.614150\pi\)
\(558\) 0 0
\(559\) 0.338388 0.0143123
\(560\) 0 0
\(561\) −68.7947 −2.90451
\(562\) 0 0
\(563\) 0.909317 0.0383231 0.0191616 0.999816i \(-0.493900\pi\)
0.0191616 + 0.999816i \(0.493900\pi\)
\(564\) 0 0
\(565\) 15.3865 0.647314
\(566\) 0 0
\(567\) 13.9162 0.584427
\(568\) 0 0
\(569\) −6.63794 −0.278277 −0.139138 0.990273i \(-0.544433\pi\)
−0.139138 + 0.990273i \(0.544433\pi\)
\(570\) 0 0
\(571\) 23.2806 0.974263 0.487131 0.873329i \(-0.338043\pi\)
0.487131 + 0.873329i \(0.338043\pi\)
\(572\) 0 0
\(573\) −23.2764 −0.972385
\(574\) 0 0
\(575\) −13.8202 −0.576344
\(576\) 0 0
\(577\) 1.44482 0.0601485 0.0300742 0.999548i \(-0.490426\pi\)
0.0300742 + 0.999548i \(0.490426\pi\)
\(578\) 0 0
\(579\) −21.0968 −0.876754
\(580\) 0 0
\(581\) 13.6063 0.564484
\(582\) 0 0
\(583\) 47.8675 1.98247
\(584\) 0 0
\(585\) 0.863454 0.0356994
\(586\) 0 0
\(587\) −13.2366 −0.546334 −0.273167 0.961967i \(-0.588071\pi\)
−0.273167 + 0.961967i \(0.588071\pi\)
\(588\) 0 0
\(589\) −9.07280 −0.373838
\(590\) 0 0
\(591\) −9.56561 −0.393477
\(592\) 0 0
\(593\) −40.3145 −1.65552 −0.827758 0.561085i \(-0.810384\pi\)
−0.827758 + 0.561085i \(0.810384\pi\)
\(594\) 0 0
\(595\) 58.7114 2.40693
\(596\) 0 0
\(597\) 15.4393 0.631887
\(598\) 0 0
\(599\) −34.6587 −1.41612 −0.708058 0.706155i \(-0.750428\pi\)
−0.708058 + 0.706155i \(0.750428\pi\)
\(600\) 0 0
\(601\) 35.9913 1.46812 0.734059 0.679086i \(-0.237624\pi\)
0.734059 + 0.679086i \(0.237624\pi\)
\(602\) 0 0
\(603\) −37.4691 −1.52586
\(604\) 0 0
\(605\) −37.3670 −1.51918
\(606\) 0 0
\(607\) −31.3487 −1.27241 −0.636203 0.771522i \(-0.719496\pi\)
−0.636203 + 0.771522i \(0.719496\pi\)
\(608\) 0 0
\(609\) 17.5139 0.709700
\(610\) 0 0
\(611\) −0.375613 −0.0151957
\(612\) 0 0
\(613\) 29.1047 1.17553 0.587764 0.809033i \(-0.300009\pi\)
0.587764 + 0.809033i \(0.300009\pi\)
\(614\) 0 0
\(615\) 31.3778 1.26527
\(616\) 0 0
\(617\) 21.1416 0.851129 0.425565 0.904928i \(-0.360076\pi\)
0.425565 + 0.904928i \(0.360076\pi\)
\(618\) 0 0
\(619\) −20.7431 −0.833737 −0.416868 0.908967i \(-0.636872\pi\)
−0.416868 + 0.908967i \(0.636872\pi\)
\(620\) 0 0
\(621\) 13.1791 0.528860
\(622\) 0 0
\(623\) −13.4166 −0.537526
\(624\) 0 0
\(625\) −6.98029 −0.279212
\(626\) 0 0
\(627\) 32.5094 1.29830
\(628\) 0 0
\(629\) −42.7974 −1.70644
\(630\) 0 0
\(631\) −18.6048 −0.740645 −0.370323 0.928903i \(-0.620753\pi\)
−0.370323 + 0.928903i \(0.620753\pi\)
\(632\) 0 0
\(633\) −42.7845 −1.70053
\(634\) 0 0
\(635\) 4.87664 0.193524
\(636\) 0 0
\(637\) −0.165886 −0.00657264
\(638\) 0 0
\(639\) −1.39785 −0.0552979
\(640\) 0 0
\(641\) −0.403031 −0.0159188 −0.00795938 0.999968i \(-0.502534\pi\)
−0.00795938 + 0.999968i \(0.502534\pi\)
\(642\) 0 0
\(643\) −3.77077 −0.148705 −0.0743523 0.997232i \(-0.523689\pi\)
−0.0743523 + 0.997232i \(0.523689\pi\)
\(644\) 0 0
\(645\) −76.9271 −3.02900
\(646\) 0 0
\(647\) −8.56734 −0.336817 −0.168408 0.985717i \(-0.553863\pi\)
−0.168408 + 0.985717i \(0.553863\pi\)
\(648\) 0 0
\(649\) 10.9233 0.428779
\(650\) 0 0
\(651\) 35.7720 1.40201
\(652\) 0 0
\(653\) 18.9154 0.740215 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(654\) 0 0
\(655\) −30.2281 −1.18111
\(656\) 0 0
\(657\) −39.9351 −1.55802
\(658\) 0 0
\(659\) −1.31655 −0.0512856 −0.0256428 0.999671i \(-0.508163\pi\)
−0.0256428 + 0.999671i \(0.508163\pi\)
\(660\) 0 0
\(661\) −0.986571 −0.0383732 −0.0191866 0.999816i \(-0.506108\pi\)
−0.0191866 + 0.999816i \(0.506108\pi\)
\(662\) 0 0
\(663\) −0.666736 −0.0258939
\(664\) 0 0
\(665\) −27.7445 −1.07589
\(666\) 0 0
\(667\) 3.43286 0.132921
\(668\) 0 0
\(669\) −20.8324 −0.805429
\(670\) 0 0
\(671\) 14.6619 0.566018
\(672\) 0 0
\(673\) 34.2168 1.31896 0.659480 0.751722i \(-0.270777\pi\)
0.659480 + 0.751722i \(0.270777\pi\)
\(674\) 0 0
\(675\) −52.5933 −2.02432
\(676\) 0 0
\(677\) −17.9512 −0.689922 −0.344961 0.938617i \(-0.612108\pi\)
−0.344961 + 0.938617i \(0.612108\pi\)
\(678\) 0 0
\(679\) −23.7794 −0.912569
\(680\) 0 0
\(681\) 50.3397 1.92902
\(682\) 0 0
\(683\) −4.19391 −0.160476 −0.0802378 0.996776i \(-0.525568\pi\)
−0.0802378 + 0.996776i \(0.525568\pi\)
\(684\) 0 0
\(685\) 25.9178 0.990267
\(686\) 0 0
\(687\) 3.24076 0.123643
\(688\) 0 0
\(689\) 0.463916 0.0176738
\(690\) 0 0
\(691\) 35.1107 1.33567 0.667837 0.744307i \(-0.267220\pi\)
0.667837 + 0.744307i \(0.267220\pi\)
\(692\) 0 0
\(693\) −82.6058 −3.13793
\(694\) 0 0
\(695\) −33.3767 −1.26605
\(696\) 0 0
\(697\) −15.6148 −0.591452
\(698\) 0 0
\(699\) 31.8907 1.20622
\(700\) 0 0
\(701\) −22.1149 −0.835269 −0.417634 0.908615i \(-0.637141\pi\)
−0.417634 + 0.908615i \(0.637141\pi\)
\(702\) 0 0
\(703\) 20.2242 0.762771
\(704\) 0 0
\(705\) 85.3897 3.21596
\(706\) 0 0
\(707\) −49.6481 −1.86721
\(708\) 0 0
\(709\) −10.6284 −0.399157 −0.199579 0.979882i \(-0.563957\pi\)
−0.199579 + 0.979882i \(0.563957\pi\)
\(710\) 0 0
\(711\) −43.7501 −1.64076
\(712\) 0 0
\(713\) 7.01159 0.262586
\(714\) 0 0
\(715\) −0.737956 −0.0275980
\(716\) 0 0
\(717\) 7.86156 0.293595
\(718\) 0 0
\(719\) 24.9050 0.928798 0.464399 0.885626i \(-0.346270\pi\)
0.464399 + 0.885626i \(0.346270\pi\)
\(720\) 0 0
\(721\) 3.89712 0.145136
\(722\) 0 0
\(723\) −56.4974 −2.10116
\(724\) 0 0
\(725\) −13.6994 −0.508782
\(726\) 0 0
\(727\) −28.2334 −1.04712 −0.523560 0.851989i \(-0.675396\pi\)
−0.523560 + 0.851989i \(0.675396\pi\)
\(728\) 0 0
\(729\) −38.5616 −1.42821
\(730\) 0 0
\(731\) 38.2819 1.41591
\(732\) 0 0
\(733\) 23.8385 0.880494 0.440247 0.897877i \(-0.354891\pi\)
0.440247 + 0.897877i \(0.354891\pi\)
\(734\) 0 0
\(735\) 37.7115 1.39101
\(736\) 0 0
\(737\) 32.0232 1.17959
\(738\) 0 0
\(739\) −45.0849 −1.65847 −0.829237 0.558897i \(-0.811225\pi\)
−0.829237 + 0.558897i \(0.811225\pi\)
\(740\) 0 0
\(741\) 0.315071 0.0115744
\(742\) 0 0
\(743\) 38.9612 1.42935 0.714673 0.699458i \(-0.246575\pi\)
0.714673 + 0.699458i \(0.246575\pi\)
\(744\) 0 0
\(745\) 43.0128 1.57587
\(746\) 0 0
\(747\) 22.6378 0.828274
\(748\) 0 0
\(749\) −43.6908 −1.59643
\(750\) 0 0
\(751\) 1.00000 0.0364905
\(752\) 0 0
\(753\) −87.6129 −3.19279
\(754\) 0 0
\(755\) −69.8306 −2.54140
\(756\) 0 0
\(757\) −51.1309 −1.85839 −0.929193 0.369596i \(-0.879496\pi\)
−0.929193 + 0.369596i \(0.879496\pi\)
\(758\) 0 0
\(759\) −25.1237 −0.911934
\(760\) 0 0
\(761\) −42.8507 −1.55334 −0.776669 0.629909i \(-0.783092\pi\)
−0.776669 + 0.629909i \(0.783092\pi\)
\(762\) 0 0
\(763\) −36.8859 −1.33536
\(764\) 0 0
\(765\) 97.6827 3.53172
\(766\) 0 0
\(767\) 0.105865 0.00382258
\(768\) 0 0
\(769\) 14.1534 0.510383 0.255192 0.966891i \(-0.417861\pi\)
0.255192 + 0.966891i \(0.417861\pi\)
\(770\) 0 0
\(771\) 80.4138 2.89603
\(772\) 0 0
\(773\) −30.5119 −1.09744 −0.548719 0.836007i \(-0.684884\pi\)
−0.548719 + 0.836007i \(0.684884\pi\)
\(774\) 0 0
\(775\) −27.9808 −1.00510
\(776\) 0 0
\(777\) −79.7395 −2.86064
\(778\) 0 0
\(779\) 7.37888 0.264376
\(780\) 0 0
\(781\) 1.19468 0.0427489
\(782\) 0 0
\(783\) 13.0639 0.466864
\(784\) 0 0
\(785\) 73.3309 2.61729
\(786\) 0 0
\(787\) −30.6189 −1.09145 −0.545724 0.837965i \(-0.683745\pi\)
−0.545724 + 0.837965i \(0.683745\pi\)
\(788\) 0 0
\(789\) 39.2701 1.39805
\(790\) 0 0
\(791\) −14.2662 −0.507249
\(792\) 0 0
\(793\) 0.142099 0.00504607
\(794\) 0 0
\(795\) −105.464 −3.74042
\(796\) 0 0
\(797\) −26.7125 −0.946205 −0.473102 0.881007i \(-0.656866\pi\)
−0.473102 + 0.881007i \(0.656866\pi\)
\(798\) 0 0
\(799\) −42.4932 −1.50330
\(800\) 0 0
\(801\) −22.3223 −0.788719
\(802\) 0 0
\(803\) 34.1308 1.20445
\(804\) 0 0
\(805\) 21.4413 0.755708
\(806\) 0 0
\(807\) −26.7055 −0.940079
\(808\) 0 0
\(809\) 42.2289 1.48469 0.742344 0.670019i \(-0.233714\pi\)
0.742344 + 0.670019i \(0.233714\pi\)
\(810\) 0 0
\(811\) 0.145146 0.00509676 0.00254838 0.999997i \(-0.499189\pi\)
0.00254838 + 0.999997i \(0.499189\pi\)
\(812\) 0 0
\(813\) −80.8117 −2.83419
\(814\) 0 0
\(815\) 9.59321 0.336035
\(816\) 0 0
\(817\) −18.0904 −0.632903
\(818\) 0 0
\(819\) −0.800588 −0.0279748
\(820\) 0 0
\(821\) 12.8853 0.449699 0.224849 0.974394i \(-0.427811\pi\)
0.224849 + 0.974394i \(0.427811\pi\)
\(822\) 0 0
\(823\) 36.4979 1.27224 0.636119 0.771591i \(-0.280539\pi\)
0.636119 + 0.771591i \(0.280539\pi\)
\(824\) 0 0
\(825\) 100.260 3.49061
\(826\) 0 0
\(827\) 42.0933 1.46373 0.731865 0.681450i \(-0.238650\pi\)
0.731865 + 0.681450i \(0.238650\pi\)
\(828\) 0 0
\(829\) 0.364644 0.0126646 0.00633231 0.999980i \(-0.497984\pi\)
0.00633231 + 0.999980i \(0.497984\pi\)
\(830\) 0 0
\(831\) −76.8485 −2.66584
\(832\) 0 0
\(833\) −18.7667 −0.650228
\(834\) 0 0
\(835\) −71.0674 −2.45939
\(836\) 0 0
\(837\) 26.6828 0.922292
\(838\) 0 0
\(839\) −44.1110 −1.52288 −0.761441 0.648235i \(-0.775508\pi\)
−0.761441 + 0.648235i \(0.775508\pi\)
\(840\) 0 0
\(841\) −25.5972 −0.882661
\(842\) 0 0
\(843\) 53.8218 1.85372
\(844\) 0 0
\(845\) 45.8193 1.57623
\(846\) 0 0
\(847\) 34.6464 1.19046
\(848\) 0 0
\(849\) −59.8164 −2.05289
\(850\) 0 0
\(851\) −15.6296 −0.535774
\(852\) 0 0
\(853\) −36.0779 −1.23528 −0.617642 0.786459i \(-0.711912\pi\)
−0.617642 + 0.786459i \(0.711912\pi\)
\(854\) 0 0
\(855\) −46.1607 −1.57866
\(856\) 0 0
\(857\) −35.9596 −1.22836 −0.614178 0.789167i \(-0.710512\pi\)
−0.614178 + 0.789167i \(0.710512\pi\)
\(858\) 0 0
\(859\) 10.2381 0.349320 0.174660 0.984629i \(-0.444117\pi\)
0.174660 + 0.984629i \(0.444117\pi\)
\(860\) 0 0
\(861\) −29.0932 −0.991495
\(862\) 0 0
\(863\) 7.23190 0.246177 0.123088 0.992396i \(-0.460720\pi\)
0.123088 + 0.992396i \(0.460720\pi\)
\(864\) 0 0
\(865\) −75.9496 −2.58237
\(866\) 0 0
\(867\) −26.0460 −0.884567
\(868\) 0 0
\(869\) 37.3913 1.26841
\(870\) 0 0
\(871\) 0.310358 0.0105161
\(872\) 0 0
\(873\) −39.5636 −1.33902
\(874\) 0 0
\(875\) −27.9566 −0.945105
\(876\) 0 0
\(877\) 12.7751 0.431384 0.215692 0.976461i \(-0.430799\pi\)
0.215692 + 0.976461i \(0.430799\pi\)
\(878\) 0 0
\(879\) −64.3495 −2.17046
\(880\) 0 0
\(881\) −16.3509 −0.550877 −0.275438 0.961319i \(-0.588823\pi\)
−0.275438 + 0.961319i \(0.588823\pi\)
\(882\) 0 0
\(883\) −1.72647 −0.0581004 −0.0290502 0.999578i \(-0.509248\pi\)
−0.0290502 + 0.999578i \(0.509248\pi\)
\(884\) 0 0
\(885\) −24.0668 −0.808998
\(886\) 0 0
\(887\) 0.878493 0.0294969 0.0147485 0.999891i \(-0.495305\pi\)
0.0147485 + 0.999891i \(0.495305\pi\)
\(888\) 0 0
\(889\) −4.52159 −0.151649
\(890\) 0 0
\(891\) −19.7883 −0.662931
\(892\) 0 0
\(893\) 20.0805 0.671968
\(894\) 0 0
\(895\) 13.0279 0.435475
\(896\) 0 0
\(897\) −0.243491 −0.00812993
\(898\) 0 0
\(899\) 6.95026 0.231804
\(900\) 0 0
\(901\) 52.4829 1.74846
\(902\) 0 0
\(903\) 71.3263 2.37359
\(904\) 0 0
\(905\) −25.3169 −0.841561
\(906\) 0 0
\(907\) 31.1660 1.03485 0.517426 0.855728i \(-0.326890\pi\)
0.517426 + 0.855728i \(0.326890\pi\)
\(908\) 0 0
\(909\) −82.6033 −2.73978
\(910\) 0 0
\(911\) −40.5934 −1.34492 −0.672459 0.740134i \(-0.734762\pi\)
−0.672459 + 0.740134i \(0.734762\pi\)
\(912\) 0 0
\(913\) −19.3475 −0.640310
\(914\) 0 0
\(915\) −32.3039 −1.06793
\(916\) 0 0
\(917\) 28.0273 0.925541
\(918\) 0 0
\(919\) 53.3539 1.75998 0.879992 0.474989i \(-0.157548\pi\)
0.879992 + 0.474989i \(0.157548\pi\)
\(920\) 0 0
\(921\) −2.90862 −0.0958424
\(922\) 0 0
\(923\) 0.0115784 0.000381108 0
\(924\) 0 0
\(925\) 62.3722 2.05079
\(926\) 0 0
\(927\) 6.48393 0.212960
\(928\) 0 0
\(929\) 32.1988 1.05641 0.528205 0.849117i \(-0.322865\pi\)
0.528205 + 0.849117i \(0.322865\pi\)
\(930\) 0 0
\(931\) 8.86834 0.290648
\(932\) 0 0
\(933\) 39.5179 1.29376
\(934\) 0 0
\(935\) −83.4850 −2.73025
\(936\) 0 0
\(937\) 51.7175 1.68954 0.844768 0.535133i \(-0.179739\pi\)
0.844768 + 0.535133i \(0.179739\pi\)
\(938\) 0 0
\(939\) 5.88697 0.192114
\(940\) 0 0
\(941\) −4.72993 −0.154191 −0.0770956 0.997024i \(-0.524565\pi\)
−0.0770956 + 0.997024i \(0.524565\pi\)
\(942\) 0 0
\(943\) −5.70250 −0.185699
\(944\) 0 0
\(945\) 81.5956 2.65431
\(946\) 0 0
\(947\) 5.94001 0.193024 0.0965122 0.995332i \(-0.469231\pi\)
0.0965122 + 0.995332i \(0.469231\pi\)
\(948\) 0 0
\(949\) 0.330784 0.0107377
\(950\) 0 0
\(951\) −4.63410 −0.150271
\(952\) 0 0
\(953\) −0.894420 −0.0289731 −0.0144865 0.999895i \(-0.504611\pi\)
−0.0144865 + 0.999895i \(0.504611\pi\)
\(954\) 0 0
\(955\) −28.2468 −0.914045
\(956\) 0 0
\(957\) −24.9040 −0.805032
\(958\) 0 0
\(959\) −24.0308 −0.775994
\(960\) 0 0
\(961\) −16.8042 −0.542070
\(962\) 0 0
\(963\) −72.6916 −2.34246
\(964\) 0 0
\(965\) −25.6018 −0.824152
\(966\) 0 0
\(967\) 28.6760 0.922157 0.461078 0.887359i \(-0.347463\pi\)
0.461078 + 0.887359i \(0.347463\pi\)
\(968\) 0 0
\(969\) 35.6440 1.14505
\(970\) 0 0
\(971\) 12.8305 0.411751 0.205875 0.978578i \(-0.433996\pi\)
0.205875 + 0.978578i \(0.433996\pi\)
\(972\) 0 0
\(973\) 30.9466 0.992104
\(974\) 0 0
\(975\) 0.971689 0.0311190
\(976\) 0 0
\(977\) 53.2446 1.70344 0.851722 0.523994i \(-0.175559\pi\)
0.851722 + 0.523994i \(0.175559\pi\)
\(978\) 0 0
\(979\) 19.0779 0.609731
\(980\) 0 0
\(981\) −61.3698 −1.95939
\(982\) 0 0
\(983\) 12.3312 0.393306 0.196653 0.980473i \(-0.436993\pi\)
0.196653 + 0.980473i \(0.436993\pi\)
\(984\) 0 0
\(985\) −11.6082 −0.369869
\(986\) 0 0
\(987\) −79.1727 −2.52010
\(988\) 0 0
\(989\) 13.9805 0.444554
\(990\) 0 0
\(991\) −53.4595 −1.69820 −0.849099 0.528233i \(-0.822855\pi\)
−0.849099 + 0.528233i \(0.822855\pi\)
\(992\) 0 0
\(993\) 40.6813 1.29098
\(994\) 0 0
\(995\) 18.7362 0.593976
\(996\) 0 0
\(997\) 45.7533 1.44902 0.724511 0.689264i \(-0.242066\pi\)
0.724511 + 0.689264i \(0.242066\pi\)
\(998\) 0 0
\(999\) −59.4787 −1.88182
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 6008.2.a.d.1.3 49
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6008.2.a.d.1.3 49 1.1 even 1 trivial