Properties

Label 4024.2.a.f.1.2
Level $4024$
Weight $2$
Character 4024.1
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, 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("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.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: \(32.1318017734\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 4024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.20822 q^{3} +2.09579 q^{5} -1.61648 q^{7} +7.29270 q^{9} +O(q^{10})\) \(q-3.20822 q^{3} +2.09579 q^{5} -1.61648 q^{7} +7.29270 q^{9} -3.47682 q^{11} -2.62727 q^{13} -6.72378 q^{15} -3.87015 q^{17} -4.93031 q^{19} +5.18602 q^{21} -2.23385 q^{23} -0.607646 q^{25} -13.7719 q^{27} +8.77456 q^{29} +3.42683 q^{31} +11.1544 q^{33} -3.38781 q^{35} +1.38186 q^{37} +8.42886 q^{39} -0.103622 q^{41} +7.46770 q^{43} +15.2840 q^{45} -10.4671 q^{47} -4.38700 q^{49} +12.4163 q^{51} +2.46619 q^{53} -7.28670 q^{55} +15.8175 q^{57} +5.54740 q^{59} -14.5678 q^{61} -11.7885 q^{63} -5.50621 q^{65} +12.8462 q^{67} +7.16668 q^{69} -7.44695 q^{71} -11.8578 q^{73} +1.94946 q^{75} +5.62020 q^{77} +15.1803 q^{79} +22.3054 q^{81} -5.59809 q^{83} -8.11105 q^{85} -28.1508 q^{87} -0.398585 q^{89} +4.24692 q^{91} -10.9940 q^{93} -10.3329 q^{95} -5.67735 q^{97} -25.3554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 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
\(3\) −3.20822 −1.85227 −0.926135 0.377193i \(-0.876889\pi\)
−0.926135 + 0.377193i \(0.876889\pi\)
\(4\) 0 0
\(5\) 2.09579 0.937268 0.468634 0.883392i \(-0.344746\pi\)
0.468634 + 0.883392i \(0.344746\pi\)
\(6\) 0 0
\(7\) −1.61648 −0.610971 −0.305486 0.952197i \(-0.598819\pi\)
−0.305486 + 0.952197i \(0.598819\pi\)
\(8\) 0 0
\(9\) 7.29270 2.43090
\(10\) 0 0
\(11\) −3.47682 −1.04830 −0.524150 0.851626i \(-0.675617\pi\)
−0.524150 + 0.851626i \(0.675617\pi\)
\(12\) 0 0
\(13\) −2.62727 −0.728673 −0.364337 0.931267i \(-0.618704\pi\)
−0.364337 + 0.931267i \(0.618704\pi\)
\(14\) 0 0
\(15\) −6.72378 −1.73607
\(16\) 0 0
\(17\) −3.87015 −0.938650 −0.469325 0.883025i \(-0.655503\pi\)
−0.469325 + 0.883025i \(0.655503\pi\)
\(18\) 0 0
\(19\) −4.93031 −1.13109 −0.565545 0.824717i \(-0.691334\pi\)
−0.565545 + 0.824717i \(0.691334\pi\)
\(20\) 0 0
\(21\) 5.18602 1.13168
\(22\) 0 0
\(23\) −2.23385 −0.465789 −0.232895 0.972502i \(-0.574820\pi\)
−0.232895 + 0.972502i \(0.574820\pi\)
\(24\) 0 0
\(25\) −0.607646 −0.121529
\(26\) 0 0
\(27\) −13.7719 −2.65041
\(28\) 0 0
\(29\) 8.77456 1.62939 0.814697 0.579886i \(-0.196903\pi\)
0.814697 + 0.579886i \(0.196903\pi\)
\(30\) 0 0
\(31\) 3.42683 0.615477 0.307738 0.951471i \(-0.400428\pi\)
0.307738 + 0.951471i \(0.400428\pi\)
\(32\) 0 0
\(33\) 11.1544 1.94173
\(34\) 0 0
\(35\) −3.38781 −0.572644
\(36\) 0 0
\(37\) 1.38186 0.227177 0.113589 0.993528i \(-0.463765\pi\)
0.113589 + 0.993528i \(0.463765\pi\)
\(38\) 0 0
\(39\) 8.42886 1.34970
\(40\) 0 0
\(41\) −0.103622 −0.0161830 −0.00809148 0.999967i \(-0.502576\pi\)
−0.00809148 + 0.999967i \(0.502576\pi\)
\(42\) 0 0
\(43\) 7.46770 1.13881 0.569407 0.822056i \(-0.307173\pi\)
0.569407 + 0.822056i \(0.307173\pi\)
\(44\) 0 0
\(45\) 15.2840 2.27840
\(46\) 0 0
\(47\) −10.4671 −1.52678 −0.763392 0.645936i \(-0.776467\pi\)
−0.763392 + 0.645936i \(0.776467\pi\)
\(48\) 0 0
\(49\) −4.38700 −0.626714
\(50\) 0 0
\(51\) 12.4163 1.73863
\(52\) 0 0
\(53\) 2.46619 0.338757 0.169379 0.985551i \(-0.445824\pi\)
0.169379 + 0.985551i \(0.445824\pi\)
\(54\) 0 0
\(55\) −7.28670 −0.982538
\(56\) 0 0
\(57\) 15.8175 2.09508
\(58\) 0 0
\(59\) 5.54740 0.722209 0.361105 0.932525i \(-0.382400\pi\)
0.361105 + 0.932525i \(0.382400\pi\)
\(60\) 0 0
\(61\) −14.5678 −1.86522 −0.932608 0.360891i \(-0.882473\pi\)
−0.932608 + 0.360891i \(0.882473\pi\)
\(62\) 0 0
\(63\) −11.7885 −1.48521
\(64\) 0 0
\(65\) −5.50621 −0.682962
\(66\) 0 0
\(67\) 12.8462 1.56942 0.784710 0.619864i \(-0.212812\pi\)
0.784710 + 0.619864i \(0.212812\pi\)
\(68\) 0 0
\(69\) 7.16668 0.862767
\(70\) 0 0
\(71\) −7.44695 −0.883791 −0.441895 0.897067i \(-0.645694\pi\)
−0.441895 + 0.897067i \(0.645694\pi\)
\(72\) 0 0
\(73\) −11.8578 −1.38785 −0.693927 0.720045i \(-0.744121\pi\)
−0.693927 + 0.720045i \(0.744121\pi\)
\(74\) 0 0
\(75\) 1.94946 0.225105
\(76\) 0 0
\(77\) 5.62020 0.640481
\(78\) 0 0
\(79\) 15.1803 1.70791 0.853957 0.520343i \(-0.174196\pi\)
0.853957 + 0.520343i \(0.174196\pi\)
\(80\) 0 0
\(81\) 22.3054 2.47838
\(82\) 0 0
\(83\) −5.59809 −0.614471 −0.307235 0.951634i \(-0.599404\pi\)
−0.307235 + 0.951634i \(0.599404\pi\)
\(84\) 0 0
\(85\) −8.11105 −0.879767
\(86\) 0 0
\(87\) −28.1508 −3.01808
\(88\) 0 0
\(89\) −0.398585 −0.0422500 −0.0211250 0.999777i \(-0.506725\pi\)
−0.0211250 + 0.999777i \(0.506725\pi\)
\(90\) 0 0
\(91\) 4.24692 0.445198
\(92\) 0 0
\(93\) −10.9940 −1.14003
\(94\) 0 0
\(95\) −10.3329 −1.06013
\(96\) 0 0
\(97\) −5.67735 −0.576447 −0.288224 0.957563i \(-0.593065\pi\)
−0.288224 + 0.957563i \(0.593065\pi\)
\(98\) 0 0
\(99\) −25.3554 −2.54831
\(100\) 0 0
\(101\) −5.19860 −0.517281 −0.258640 0.965974i \(-0.583274\pi\)
−0.258640 + 0.965974i \(0.583274\pi\)
\(102\) 0 0
\(103\) 1.83961 0.181262 0.0906310 0.995885i \(-0.471112\pi\)
0.0906310 + 0.995885i \(0.471112\pi\)
\(104\) 0 0
\(105\) 10.8688 1.06069
\(106\) 0 0
\(107\) 11.1507 1.07798 0.538991 0.842312i \(-0.318806\pi\)
0.538991 + 0.842312i \(0.318806\pi\)
\(108\) 0 0
\(109\) 15.3151 1.46692 0.733462 0.679731i \(-0.237903\pi\)
0.733462 + 0.679731i \(0.237903\pi\)
\(110\) 0 0
\(111\) −4.43333 −0.420793
\(112\) 0 0
\(113\) −2.47904 −0.233208 −0.116604 0.993178i \(-0.537201\pi\)
−0.116604 + 0.993178i \(0.537201\pi\)
\(114\) 0 0
\(115\) −4.68168 −0.436569
\(116\) 0 0
\(117\) −19.1599 −1.77133
\(118\) 0 0
\(119\) 6.25602 0.573488
\(120\) 0 0
\(121\) 1.08826 0.0989329
\(122\) 0 0
\(123\) 0.332441 0.0299752
\(124\) 0 0
\(125\) −11.7525 −1.05117
\(126\) 0 0
\(127\) −17.8868 −1.58719 −0.793597 0.608444i \(-0.791794\pi\)
−0.793597 + 0.608444i \(0.791794\pi\)
\(128\) 0 0
\(129\) −23.9581 −2.10939
\(130\) 0 0
\(131\) −5.83716 −0.509995 −0.254998 0.966942i \(-0.582075\pi\)
−0.254998 + 0.966942i \(0.582075\pi\)
\(132\) 0 0
\(133\) 7.96973 0.691063
\(134\) 0 0
\(135\) −28.8632 −2.48415
\(136\) 0 0
\(137\) 19.5744 1.67235 0.836175 0.548463i \(-0.184787\pi\)
0.836175 + 0.548463i \(0.184787\pi\)
\(138\) 0 0
\(139\) 15.4351 1.30919 0.654593 0.755982i \(-0.272840\pi\)
0.654593 + 0.755982i \(0.272840\pi\)
\(140\) 0 0
\(141\) 33.5808 2.82801
\(142\) 0 0
\(143\) 9.13453 0.763868
\(144\) 0 0
\(145\) 18.3897 1.52718
\(146\) 0 0
\(147\) 14.0745 1.16084
\(148\) 0 0
\(149\) 20.9215 1.71396 0.856980 0.515350i \(-0.172338\pi\)
0.856980 + 0.515350i \(0.172338\pi\)
\(150\) 0 0
\(151\) −2.09501 −0.170489 −0.0852446 0.996360i \(-0.527167\pi\)
−0.0852446 + 0.996360i \(0.527167\pi\)
\(152\) 0 0
\(153\) −28.2239 −2.28177
\(154\) 0 0
\(155\) 7.18193 0.576866
\(156\) 0 0
\(157\) 8.40896 0.671108 0.335554 0.942021i \(-0.391076\pi\)
0.335554 + 0.942021i \(0.391076\pi\)
\(158\) 0 0
\(159\) −7.91209 −0.627470
\(160\) 0 0
\(161\) 3.61096 0.284584
\(162\) 0 0
\(163\) −14.3484 −1.12385 −0.561925 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(164\) 0 0
\(165\) 23.3774 1.81992
\(166\) 0 0
\(167\) 2.46761 0.190950 0.0954749 0.995432i \(-0.469563\pi\)
0.0954749 + 0.995432i \(0.469563\pi\)
\(168\) 0 0
\(169\) −6.09746 −0.469036
\(170\) 0 0
\(171\) −35.9553 −2.74957
\(172\) 0 0
\(173\) −10.1780 −0.773816 −0.386908 0.922118i \(-0.626457\pi\)
−0.386908 + 0.922118i \(0.626457\pi\)
\(174\) 0 0
\(175\) 0.982247 0.0742509
\(176\) 0 0
\(177\) −17.7973 −1.33773
\(178\) 0 0
\(179\) 3.10429 0.232026 0.116013 0.993248i \(-0.462989\pi\)
0.116013 + 0.993248i \(0.462989\pi\)
\(180\) 0 0
\(181\) 17.4282 1.29543 0.647713 0.761884i \(-0.275726\pi\)
0.647713 + 0.761884i \(0.275726\pi\)
\(182\) 0 0
\(183\) 46.7368 3.45488
\(184\) 0 0
\(185\) 2.89610 0.212926
\(186\) 0 0
\(187\) 13.4558 0.983987
\(188\) 0 0
\(189\) 22.2621 1.61933
\(190\) 0 0
\(191\) −7.42790 −0.537464 −0.268732 0.963215i \(-0.586605\pi\)
−0.268732 + 0.963215i \(0.586605\pi\)
\(192\) 0 0
\(193\) 7.27423 0.523611 0.261805 0.965121i \(-0.415682\pi\)
0.261805 + 0.965121i \(0.415682\pi\)
\(194\) 0 0
\(195\) 17.6652 1.26503
\(196\) 0 0
\(197\) −11.9919 −0.854390 −0.427195 0.904160i \(-0.640498\pi\)
−0.427195 + 0.904160i \(0.640498\pi\)
\(198\) 0 0
\(199\) 25.2366 1.78897 0.894487 0.447093i \(-0.147541\pi\)
0.894487 + 0.447093i \(0.147541\pi\)
\(200\) 0 0
\(201\) −41.2136 −2.90699
\(202\) 0 0
\(203\) −14.1839 −0.995514
\(204\) 0 0
\(205\) −0.217169 −0.0151678
\(206\) 0 0
\(207\) −16.2908 −1.13229
\(208\) 0 0
\(209\) 17.1418 1.18572
\(210\) 0 0
\(211\) 9.08568 0.625484 0.312742 0.949838i \(-0.398753\pi\)
0.312742 + 0.949838i \(0.398753\pi\)
\(212\) 0 0
\(213\) 23.8915 1.63702
\(214\) 0 0
\(215\) 15.6508 1.06737
\(216\) 0 0
\(217\) −5.53939 −0.376039
\(218\) 0 0
\(219\) 38.0426 2.57068
\(220\) 0 0
\(221\) 10.1679 0.683969
\(222\) 0 0
\(223\) −8.97094 −0.600738 −0.300369 0.953823i \(-0.597110\pi\)
−0.300369 + 0.953823i \(0.597110\pi\)
\(224\) 0 0
\(225\) −4.43138 −0.295425
\(226\) 0 0
\(227\) 15.6988 1.04197 0.520984 0.853566i \(-0.325565\pi\)
0.520984 + 0.853566i \(0.325565\pi\)
\(228\) 0 0
\(229\) 21.6432 1.43023 0.715113 0.699009i \(-0.246375\pi\)
0.715113 + 0.699009i \(0.246375\pi\)
\(230\) 0 0
\(231\) −18.0309 −1.18634
\(232\) 0 0
\(233\) 6.18830 0.405409 0.202705 0.979240i \(-0.435027\pi\)
0.202705 + 0.979240i \(0.435027\pi\)
\(234\) 0 0
\(235\) −21.9369 −1.43101
\(236\) 0 0
\(237\) −48.7017 −3.16352
\(238\) 0 0
\(239\) 15.0352 0.972546 0.486273 0.873807i \(-0.338356\pi\)
0.486273 + 0.873807i \(0.338356\pi\)
\(240\) 0 0
\(241\) 15.4752 0.996845 0.498423 0.866934i \(-0.333913\pi\)
0.498423 + 0.866934i \(0.333913\pi\)
\(242\) 0 0
\(243\) −30.2449 −1.94021
\(244\) 0 0
\(245\) −9.19425 −0.587399
\(246\) 0 0
\(247\) 12.9532 0.824195
\(248\) 0 0
\(249\) 17.9599 1.13816
\(250\) 0 0
\(251\) 12.5171 0.790070 0.395035 0.918666i \(-0.370732\pi\)
0.395035 + 0.918666i \(0.370732\pi\)
\(252\) 0 0
\(253\) 7.76668 0.488287
\(254\) 0 0
\(255\) 26.0221 1.62956
\(256\) 0 0
\(257\) 27.0901 1.68984 0.844918 0.534896i \(-0.179649\pi\)
0.844918 + 0.534896i \(0.179649\pi\)
\(258\) 0 0
\(259\) −2.23375 −0.138799
\(260\) 0 0
\(261\) 63.9903 3.96090
\(262\) 0 0
\(263\) 19.0150 1.17251 0.586257 0.810125i \(-0.300601\pi\)
0.586257 + 0.810125i \(0.300601\pi\)
\(264\) 0 0
\(265\) 5.16863 0.317506
\(266\) 0 0
\(267\) 1.27875 0.0782583
\(268\) 0 0
\(269\) −20.6523 −1.25920 −0.629598 0.776921i \(-0.716780\pi\)
−0.629598 + 0.776921i \(0.716780\pi\)
\(270\) 0 0
\(271\) −8.67779 −0.527138 −0.263569 0.964641i \(-0.584900\pi\)
−0.263569 + 0.964641i \(0.584900\pi\)
\(272\) 0 0
\(273\) −13.6251 −0.824627
\(274\) 0 0
\(275\) 2.11267 0.127399
\(276\) 0 0
\(277\) 7.17921 0.431357 0.215678 0.976464i \(-0.430804\pi\)
0.215678 + 0.976464i \(0.430804\pi\)
\(278\) 0 0
\(279\) 24.9908 1.49616
\(280\) 0 0
\(281\) 0.0419627 0.00250328 0.00125164 0.999999i \(-0.499602\pi\)
0.00125164 + 0.999999i \(0.499602\pi\)
\(282\) 0 0
\(283\) 15.4083 0.915927 0.457964 0.888971i \(-0.348579\pi\)
0.457964 + 0.888971i \(0.348579\pi\)
\(284\) 0 0
\(285\) 33.1503 1.96365
\(286\) 0 0
\(287\) 0.167502 0.00988733
\(288\) 0 0
\(289\) −2.02191 −0.118936
\(290\) 0 0
\(291\) 18.2142 1.06774
\(292\) 0 0
\(293\) −4.84061 −0.282792 −0.141396 0.989953i \(-0.545159\pi\)
−0.141396 + 0.989953i \(0.545159\pi\)
\(294\) 0 0
\(295\) 11.6262 0.676904
\(296\) 0 0
\(297\) 47.8826 2.77843
\(298\) 0 0
\(299\) 5.86891 0.339408
\(300\) 0 0
\(301\) −12.0714 −0.695783
\(302\) 0 0
\(303\) 16.6783 0.958143
\(304\) 0 0
\(305\) −30.5311 −1.74821
\(306\) 0 0
\(307\) 18.1532 1.03606 0.518030 0.855363i \(-0.326666\pi\)
0.518030 + 0.855363i \(0.326666\pi\)
\(308\) 0 0
\(309\) −5.90188 −0.335746
\(310\) 0 0
\(311\) 17.4671 0.990468 0.495234 0.868760i \(-0.335082\pi\)
0.495234 + 0.868760i \(0.335082\pi\)
\(312\) 0 0
\(313\) −12.6243 −0.713568 −0.356784 0.934187i \(-0.616127\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(314\) 0 0
\(315\) −24.7063 −1.39204
\(316\) 0 0
\(317\) −27.7096 −1.55633 −0.778164 0.628061i \(-0.783849\pi\)
−0.778164 + 0.628061i \(0.783849\pi\)
\(318\) 0 0
\(319\) −30.5075 −1.70809
\(320\) 0 0
\(321\) −35.7740 −1.99671
\(322\) 0 0
\(323\) 19.0810 1.06170
\(324\) 0 0
\(325\) 1.59645 0.0885551
\(326\) 0 0
\(327\) −49.1344 −2.71714
\(328\) 0 0
\(329\) 16.9198 0.932821
\(330\) 0 0
\(331\) −16.3622 −0.899347 −0.449674 0.893193i \(-0.648460\pi\)
−0.449674 + 0.893193i \(0.648460\pi\)
\(332\) 0 0
\(333\) 10.0775 0.552245
\(334\) 0 0
\(335\) 26.9231 1.47097
\(336\) 0 0
\(337\) −12.4560 −0.678523 −0.339261 0.940692i \(-0.610177\pi\)
−0.339261 + 0.940692i \(0.610177\pi\)
\(338\) 0 0
\(339\) 7.95330 0.431964
\(340\) 0 0
\(341\) −11.9145 −0.645204
\(342\) 0 0
\(343\) 18.4068 0.993876
\(344\) 0 0
\(345\) 15.0199 0.808643
\(346\) 0 0
\(347\) −33.6467 −1.80625 −0.903126 0.429376i \(-0.858734\pi\)
−0.903126 + 0.429376i \(0.858734\pi\)
\(348\) 0 0
\(349\) 13.1391 0.703319 0.351659 0.936128i \(-0.385618\pi\)
0.351659 + 0.936128i \(0.385618\pi\)
\(350\) 0 0
\(351\) 36.1826 1.93128
\(352\) 0 0
\(353\) 34.9486 1.86013 0.930063 0.367400i \(-0.119752\pi\)
0.930063 + 0.367400i \(0.119752\pi\)
\(354\) 0 0
\(355\) −15.6073 −0.828349
\(356\) 0 0
\(357\) −20.0707 −1.06225
\(358\) 0 0
\(359\) −17.3635 −0.916410 −0.458205 0.888847i \(-0.651507\pi\)
−0.458205 + 0.888847i \(0.651507\pi\)
\(360\) 0 0
\(361\) 5.30792 0.279364
\(362\) 0 0
\(363\) −3.49139 −0.183250
\(364\) 0 0
\(365\) −24.8516 −1.30079
\(366\) 0 0
\(367\) 3.15685 0.164786 0.0823931 0.996600i \(-0.473744\pi\)
0.0823931 + 0.996600i \(0.473744\pi\)
\(368\) 0 0
\(369\) −0.755681 −0.0393392
\(370\) 0 0
\(371\) −3.98654 −0.206971
\(372\) 0 0
\(373\) 0.224251 0.0116113 0.00580564 0.999983i \(-0.498152\pi\)
0.00580564 + 0.999983i \(0.498152\pi\)
\(374\) 0 0
\(375\) 37.7046 1.94706
\(376\) 0 0
\(377\) −23.0531 −1.18730
\(378\) 0 0
\(379\) 24.9179 1.27995 0.639974 0.768396i \(-0.278945\pi\)
0.639974 + 0.768396i \(0.278945\pi\)
\(380\) 0 0
\(381\) 57.3848 2.93991
\(382\) 0 0
\(383\) −24.9691 −1.27586 −0.637931 0.770093i \(-0.720210\pi\)
−0.637931 + 0.770093i \(0.720210\pi\)
\(384\) 0 0
\(385\) 11.7788 0.600302
\(386\) 0 0
\(387\) 54.4597 2.76834
\(388\) 0 0
\(389\) −3.63420 −0.184261 −0.0921307 0.995747i \(-0.529368\pi\)
−0.0921307 + 0.995747i \(0.529368\pi\)
\(390\) 0 0
\(391\) 8.64533 0.437213
\(392\) 0 0
\(393\) 18.7269 0.944648
\(394\) 0 0
\(395\) 31.8147 1.60077
\(396\) 0 0
\(397\) 4.76527 0.239162 0.119581 0.992824i \(-0.461845\pi\)
0.119581 + 0.992824i \(0.461845\pi\)
\(398\) 0 0
\(399\) −25.5687 −1.28004
\(400\) 0 0
\(401\) 28.9525 1.44582 0.722909 0.690944i \(-0.242805\pi\)
0.722909 + 0.690944i \(0.242805\pi\)
\(402\) 0 0
\(403\) −9.00320 −0.448481
\(404\) 0 0
\(405\) 46.7475 2.32290
\(406\) 0 0
\(407\) −4.80449 −0.238150
\(408\) 0 0
\(409\) 9.95291 0.492140 0.246070 0.969252i \(-0.420861\pi\)
0.246070 + 0.969252i \(0.420861\pi\)
\(410\) 0 0
\(411\) −62.7989 −3.09764
\(412\) 0 0
\(413\) −8.96724 −0.441249
\(414\) 0 0
\(415\) −11.7325 −0.575924
\(416\) 0 0
\(417\) −49.5192 −2.42496
\(418\) 0 0
\(419\) −9.94987 −0.486083 −0.243041 0.970016i \(-0.578145\pi\)
−0.243041 + 0.970016i \(0.578145\pi\)
\(420\) 0 0
\(421\) 4.02506 0.196170 0.0980848 0.995178i \(-0.468728\pi\)
0.0980848 + 0.995178i \(0.468728\pi\)
\(422\) 0 0
\(423\) −76.3335 −3.71146
\(424\) 0 0
\(425\) 2.35168 0.114073
\(426\) 0 0
\(427\) 23.5485 1.13959
\(428\) 0 0
\(429\) −29.3056 −1.41489
\(430\) 0 0
\(431\) 9.83643 0.473804 0.236902 0.971534i \(-0.423868\pi\)
0.236902 + 0.971534i \(0.423868\pi\)
\(432\) 0 0
\(433\) −29.1021 −1.39856 −0.699280 0.714848i \(-0.746496\pi\)
−0.699280 + 0.714848i \(0.746496\pi\)
\(434\) 0 0
\(435\) −58.9982 −2.82875
\(436\) 0 0
\(437\) 11.0135 0.526849
\(438\) 0 0
\(439\) −18.8345 −0.898922 −0.449461 0.893300i \(-0.648384\pi\)
−0.449461 + 0.893300i \(0.648384\pi\)
\(440\) 0 0
\(441\) −31.9931 −1.52348
\(442\) 0 0
\(443\) −31.2229 −1.48345 −0.741723 0.670706i \(-0.765991\pi\)
−0.741723 + 0.670706i \(0.765991\pi\)
\(444\) 0 0
\(445\) −0.835353 −0.0395995
\(446\) 0 0
\(447\) −67.1210 −3.17471
\(448\) 0 0
\(449\) −7.76706 −0.366550 −0.183275 0.983062i \(-0.558670\pi\)
−0.183275 + 0.983062i \(0.558670\pi\)
\(450\) 0 0
\(451\) 0.360273 0.0169646
\(452\) 0 0
\(453\) 6.72125 0.315792
\(454\) 0 0
\(455\) 8.90067 0.417270
\(456\) 0 0
\(457\) 14.8235 0.693416 0.346708 0.937973i \(-0.387299\pi\)
0.346708 + 0.937973i \(0.387299\pi\)
\(458\) 0 0
\(459\) 53.2996 2.48781
\(460\) 0 0
\(461\) 15.6348 0.728184 0.364092 0.931363i \(-0.381379\pi\)
0.364092 + 0.931363i \(0.381379\pi\)
\(462\) 0 0
\(463\) −23.1410 −1.07545 −0.537727 0.843119i \(-0.680717\pi\)
−0.537727 + 0.843119i \(0.680717\pi\)
\(464\) 0 0
\(465\) −23.0412 −1.06851
\(466\) 0 0
\(467\) 23.4147 1.08350 0.541751 0.840539i \(-0.317762\pi\)
0.541751 + 0.840539i \(0.317762\pi\)
\(468\) 0 0
\(469\) −20.7657 −0.958870
\(470\) 0 0
\(471\) −26.9778 −1.24307
\(472\) 0 0
\(473\) −25.9638 −1.19382
\(474\) 0 0
\(475\) 2.99588 0.137460
\(476\) 0 0
\(477\) 17.9852 0.823485
\(478\) 0 0
\(479\) −14.8541 −0.678700 −0.339350 0.940660i \(-0.610207\pi\)
−0.339350 + 0.940660i \(0.610207\pi\)
\(480\) 0 0
\(481\) −3.63053 −0.165538
\(482\) 0 0
\(483\) −11.5848 −0.527126
\(484\) 0 0
\(485\) −11.8986 −0.540285
\(486\) 0 0
\(487\) 15.4569 0.700417 0.350209 0.936672i \(-0.386111\pi\)
0.350209 + 0.936672i \(0.386111\pi\)
\(488\) 0 0
\(489\) 46.0327 2.08167
\(490\) 0 0
\(491\) 42.8923 1.93570 0.967851 0.251526i \(-0.0809323\pi\)
0.967851 + 0.251526i \(0.0809323\pi\)
\(492\) 0 0
\(493\) −33.9589 −1.52943
\(494\) 0 0
\(495\) −53.1397 −2.38845
\(496\) 0 0
\(497\) 12.0378 0.539971
\(498\) 0 0
\(499\) 3.89126 0.174197 0.0870984 0.996200i \(-0.472241\pi\)
0.0870984 + 0.996200i \(0.472241\pi\)
\(500\) 0 0
\(501\) −7.91666 −0.353690
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) −10.8952 −0.484830
\(506\) 0 0
\(507\) 19.5620 0.868780
\(508\) 0 0
\(509\) 23.6610 1.04875 0.524377 0.851486i \(-0.324298\pi\)
0.524377 + 0.851486i \(0.324298\pi\)
\(510\) 0 0
\(511\) 19.1679 0.847939
\(512\) 0 0
\(513\) 67.8999 2.99786
\(514\) 0 0
\(515\) 3.85544 0.169891
\(516\) 0 0
\(517\) 36.3922 1.60053
\(518\) 0 0
\(519\) 32.6532 1.43332
\(520\) 0 0
\(521\) 27.3018 1.19611 0.598057 0.801454i \(-0.295940\pi\)
0.598057 + 0.801454i \(0.295940\pi\)
\(522\) 0 0
\(523\) 27.1525 1.18730 0.593648 0.804725i \(-0.297687\pi\)
0.593648 + 0.804725i \(0.297687\pi\)
\(524\) 0 0
\(525\) −3.15127 −0.137533
\(526\) 0 0
\(527\) −13.2624 −0.577717
\(528\) 0 0
\(529\) −18.0099 −0.783041
\(530\) 0 0
\(531\) 40.4555 1.75562
\(532\) 0 0
\(533\) 0.272242 0.0117921
\(534\) 0 0
\(535\) 23.3696 1.01036
\(536\) 0 0
\(537\) −9.95927 −0.429774
\(538\) 0 0
\(539\) 15.2528 0.656984
\(540\) 0 0
\(541\) −1.15273 −0.0495597 −0.0247798 0.999693i \(-0.507888\pi\)
−0.0247798 + 0.999693i \(0.507888\pi\)
\(542\) 0 0
\(543\) −55.9135 −2.39948
\(544\) 0 0
\(545\) 32.0974 1.37490
\(546\) 0 0
\(547\) −30.0762 −1.28596 −0.642982 0.765881i \(-0.722303\pi\)
−0.642982 + 0.765881i \(0.722303\pi\)
\(548\) 0 0
\(549\) −106.239 −4.53415
\(550\) 0 0
\(551\) −43.2613 −1.84299
\(552\) 0 0
\(553\) −24.5386 −1.04349
\(554\) 0 0
\(555\) −9.29135 −0.394396
\(556\) 0 0
\(557\) 10.2479 0.434216 0.217108 0.976148i \(-0.430338\pi\)
0.217108 + 0.976148i \(0.430338\pi\)
\(558\) 0 0
\(559\) −19.6197 −0.829823
\(560\) 0 0
\(561\) −43.1693 −1.82261
\(562\) 0 0
\(563\) 35.3886 1.49145 0.745726 0.666253i \(-0.232103\pi\)
0.745726 + 0.666253i \(0.232103\pi\)
\(564\) 0 0
\(565\) −5.19555 −0.218578
\(566\) 0 0
\(567\) −36.0562 −1.51422
\(568\) 0 0
\(569\) −1.97851 −0.0829435 −0.0414718 0.999140i \(-0.513205\pi\)
−0.0414718 + 0.999140i \(0.513205\pi\)
\(570\) 0 0
\(571\) 34.0964 1.42689 0.713444 0.700712i \(-0.247134\pi\)
0.713444 + 0.700712i \(0.247134\pi\)
\(572\) 0 0
\(573\) 23.8304 0.995528
\(574\) 0 0
\(575\) 1.35739 0.0566070
\(576\) 0 0
\(577\) −5.83972 −0.243111 −0.121555 0.992585i \(-0.538788\pi\)
−0.121555 + 0.992585i \(0.538788\pi\)
\(578\) 0 0
\(579\) −23.3374 −0.969868
\(580\) 0 0
\(581\) 9.04920 0.375424
\(582\) 0 0
\(583\) −8.57450 −0.355119
\(584\) 0 0
\(585\) −40.1552 −1.66021
\(586\) 0 0
\(587\) 10.1983 0.420929 0.210464 0.977602i \(-0.432502\pi\)
0.210464 + 0.977602i \(0.432502\pi\)
\(588\) 0 0
\(589\) −16.8953 −0.696159
\(590\) 0 0
\(591\) 38.4728 1.58256
\(592\) 0 0
\(593\) 24.0738 0.988591 0.494295 0.869294i \(-0.335426\pi\)
0.494295 + 0.869294i \(0.335426\pi\)
\(594\) 0 0
\(595\) 13.1113 0.537512
\(596\) 0 0
\(597\) −80.9647 −3.31366
\(598\) 0 0
\(599\) 2.84785 0.116360 0.0581801 0.998306i \(-0.481470\pi\)
0.0581801 + 0.998306i \(0.481470\pi\)
\(600\) 0 0
\(601\) 40.9345 1.66975 0.834876 0.550439i \(-0.185539\pi\)
0.834876 + 0.550439i \(0.185539\pi\)
\(602\) 0 0
\(603\) 93.6839 3.81510
\(604\) 0 0
\(605\) 2.28077 0.0927266
\(606\) 0 0
\(607\) −1.17608 −0.0477358 −0.0238679 0.999715i \(-0.507598\pi\)
−0.0238679 + 0.999715i \(0.507598\pi\)
\(608\) 0 0
\(609\) 45.5051 1.84396
\(610\) 0 0
\(611\) 27.4999 1.11253
\(612\) 0 0
\(613\) 28.3582 1.14538 0.572688 0.819773i \(-0.305901\pi\)
0.572688 + 0.819773i \(0.305901\pi\)
\(614\) 0 0
\(615\) 0.696728 0.0280948
\(616\) 0 0
\(617\) 24.7927 0.998115 0.499058 0.866569i \(-0.333680\pi\)
0.499058 + 0.866569i \(0.333680\pi\)
\(618\) 0 0
\(619\) −1.07443 −0.0431851 −0.0215926 0.999767i \(-0.506874\pi\)
−0.0215926 + 0.999767i \(0.506874\pi\)
\(620\) 0 0
\(621\) 30.7644 1.23453
\(622\) 0 0
\(623\) 0.644304 0.0258135
\(624\) 0 0
\(625\) −21.5925 −0.863701
\(626\) 0 0
\(627\) −54.9947 −2.19628
\(628\) 0 0
\(629\) −5.34803 −0.213240
\(630\) 0 0
\(631\) 33.2677 1.32437 0.662183 0.749342i \(-0.269630\pi\)
0.662183 + 0.749342i \(0.269630\pi\)
\(632\) 0 0
\(633\) −29.1489 −1.15856
\(634\) 0 0
\(635\) −37.4870 −1.48763
\(636\) 0 0
\(637\) 11.5258 0.456670
\(638\) 0 0
\(639\) −54.3084 −2.14841
\(640\) 0 0
\(641\) 18.3713 0.725622 0.362811 0.931863i \(-0.381817\pi\)
0.362811 + 0.931863i \(0.381817\pi\)
\(642\) 0 0
\(643\) 4.75300 0.187440 0.0937201 0.995599i \(-0.470124\pi\)
0.0937201 + 0.995599i \(0.470124\pi\)
\(644\) 0 0
\(645\) −50.2112 −1.97706
\(646\) 0 0
\(647\) −32.0508 −1.26005 −0.630024 0.776576i \(-0.716955\pi\)
−0.630024 + 0.776576i \(0.716955\pi\)
\(648\) 0 0
\(649\) −19.2873 −0.757092
\(650\) 0 0
\(651\) 17.7716 0.696525
\(652\) 0 0
\(653\) 31.3032 1.22499 0.612494 0.790475i \(-0.290166\pi\)
0.612494 + 0.790475i \(0.290166\pi\)
\(654\) 0 0
\(655\) −12.2335 −0.478002
\(656\) 0 0
\(657\) −86.4756 −3.37373
\(658\) 0 0
\(659\) 14.5190 0.565579 0.282789 0.959182i \(-0.408740\pi\)
0.282789 + 0.959182i \(0.408740\pi\)
\(660\) 0 0
\(661\) −42.4437 −1.65087 −0.825434 0.564498i \(-0.809070\pi\)
−0.825434 + 0.564498i \(0.809070\pi\)
\(662\) 0 0
\(663\) −32.6210 −1.26689
\(664\) 0 0
\(665\) 16.7029 0.647711
\(666\) 0 0
\(667\) −19.6010 −0.758954
\(668\) 0 0
\(669\) 28.7808 1.11273
\(670\) 0 0
\(671\) 50.6496 1.95531
\(672\) 0 0
\(673\) 0.425190 0.0163899 0.00819493 0.999966i \(-0.497391\pi\)
0.00819493 + 0.999966i \(0.497391\pi\)
\(674\) 0 0
\(675\) 8.36847 0.322103
\(676\) 0 0
\(677\) −30.9147 −1.18815 −0.594074 0.804410i \(-0.702481\pi\)
−0.594074 + 0.804410i \(0.702481\pi\)
\(678\) 0 0
\(679\) 9.17731 0.352193
\(680\) 0 0
\(681\) −50.3654 −1.93001
\(682\) 0 0
\(683\) −9.85610 −0.377133 −0.188567 0.982060i \(-0.560384\pi\)
−0.188567 + 0.982060i \(0.560384\pi\)
\(684\) 0 0
\(685\) 41.0238 1.56744
\(686\) 0 0
\(687\) −69.4364 −2.64916
\(688\) 0 0
\(689\) −6.47934 −0.246843
\(690\) 0 0
\(691\) 33.8382 1.28727 0.643633 0.765334i \(-0.277426\pi\)
0.643633 + 0.765334i \(0.277426\pi\)
\(692\) 0 0
\(693\) 40.9864 1.55695
\(694\) 0 0
\(695\) 32.3487 1.22706
\(696\) 0 0
\(697\) 0.401031 0.0151901
\(698\) 0 0
\(699\) −19.8535 −0.750927
\(700\) 0 0
\(701\) −26.3540 −0.995376 −0.497688 0.867356i \(-0.665817\pi\)
−0.497688 + 0.867356i \(0.665817\pi\)
\(702\) 0 0
\(703\) −6.81302 −0.256958
\(704\) 0 0
\(705\) 70.3785 2.65061
\(706\) 0 0
\(707\) 8.40343 0.316044
\(708\) 0 0
\(709\) −22.0631 −0.828596 −0.414298 0.910141i \(-0.635973\pi\)
−0.414298 + 0.910141i \(0.635973\pi\)
\(710\) 0 0
\(711\) 110.705 4.15177
\(712\) 0 0
\(713\) −7.65501 −0.286682
\(714\) 0 0
\(715\) 19.1441 0.715949
\(716\) 0 0
\(717\) −48.2363 −1.80142
\(718\) 0 0
\(719\) 0.335174 0.0124999 0.00624994 0.999980i \(-0.498011\pi\)
0.00624994 + 0.999980i \(0.498011\pi\)
\(720\) 0 0
\(721\) −2.97369 −0.110746
\(722\) 0 0
\(723\) −49.6479 −1.84643
\(724\) 0 0
\(725\) −5.33183 −0.198019
\(726\) 0 0
\(727\) −39.3516 −1.45947 −0.729736 0.683729i \(-0.760357\pi\)
−0.729736 + 0.683729i \(0.760357\pi\)
\(728\) 0 0
\(729\) 30.1161 1.11541
\(730\) 0 0
\(731\) −28.9012 −1.06895
\(732\) 0 0
\(733\) −20.3715 −0.752437 −0.376219 0.926531i \(-0.622776\pi\)
−0.376219 + 0.926531i \(0.622776\pi\)
\(734\) 0 0
\(735\) 29.4972 1.08802
\(736\) 0 0
\(737\) −44.6641 −1.64522
\(738\) 0 0
\(739\) −6.37625 −0.234554 −0.117277 0.993099i \(-0.537417\pi\)
−0.117277 + 0.993099i \(0.537417\pi\)
\(740\) 0 0
\(741\) −41.5569 −1.52663
\(742\) 0 0
\(743\) 16.5912 0.608671 0.304335 0.952565i \(-0.401566\pi\)
0.304335 + 0.952565i \(0.401566\pi\)
\(744\) 0 0
\(745\) 43.8472 1.60644
\(746\) 0 0
\(747\) −40.8252 −1.49372
\(748\) 0 0
\(749\) −18.0249 −0.658615
\(750\) 0 0
\(751\) 10.0477 0.366647 0.183324 0.983053i \(-0.441314\pi\)
0.183324 + 0.983053i \(0.441314\pi\)
\(752\) 0 0
\(753\) −40.1575 −1.46342
\(754\) 0 0
\(755\) −4.39070 −0.159794
\(756\) 0 0
\(757\) 24.2007 0.879588 0.439794 0.898099i \(-0.355051\pi\)
0.439794 + 0.898099i \(0.355051\pi\)
\(758\) 0 0
\(759\) −24.9172 −0.904438
\(760\) 0 0
\(761\) −53.4090 −1.93608 −0.968038 0.250805i \(-0.919305\pi\)
−0.968038 + 0.250805i \(0.919305\pi\)
\(762\) 0 0
\(763\) −24.7566 −0.896248
\(764\) 0 0
\(765\) −59.1515 −2.13863
\(766\) 0 0
\(767\) −14.5745 −0.526255
\(768\) 0 0
\(769\) −41.4710 −1.49548 −0.747741 0.663991i \(-0.768861\pi\)
−0.747741 + 0.663991i \(0.768861\pi\)
\(770\) 0 0
\(771\) −86.9112 −3.13003
\(772\) 0 0
\(773\) 0.112722 0.00405433 0.00202717 0.999998i \(-0.499355\pi\)
0.00202717 + 0.999998i \(0.499355\pi\)
\(774\) 0 0
\(775\) −2.08230 −0.0747984
\(776\) 0 0
\(777\) 7.16638 0.257093
\(778\) 0 0
\(779\) 0.510886 0.0183044
\(780\) 0 0
\(781\) 25.8917 0.926478
\(782\) 0 0
\(783\) −120.843 −4.31857
\(784\) 0 0
\(785\) 17.6235 0.629008
\(786\) 0 0
\(787\) −9.29851 −0.331456 −0.165728 0.986172i \(-0.552997\pi\)
−0.165728 + 0.986172i \(0.552997\pi\)
\(788\) 0 0
\(789\) −61.0044 −2.17181
\(790\) 0 0
\(791\) 4.00731 0.142483
\(792\) 0 0
\(793\) 38.2735 1.35913
\(794\) 0 0
\(795\) −16.5821 −0.588107
\(796\) 0 0
\(797\) 39.3907 1.39529 0.697645 0.716444i \(-0.254231\pi\)
0.697645 + 0.716444i \(0.254231\pi\)
\(798\) 0 0
\(799\) 40.5093 1.43312
\(800\) 0 0
\(801\) −2.90676 −0.102705
\(802\) 0 0
\(803\) 41.2275 1.45489
\(804\) 0 0
\(805\) 7.56784 0.266731
\(806\) 0 0
\(807\) 66.2573 2.33237
\(808\) 0 0
\(809\) −0.151497 −0.00532634 −0.00266317 0.999996i \(-0.500848\pi\)
−0.00266317 + 0.999996i \(0.500848\pi\)
\(810\) 0 0
\(811\) −0.359399 −0.0126202 −0.00631010 0.999980i \(-0.502009\pi\)
−0.00631010 + 0.999980i \(0.502009\pi\)
\(812\) 0 0
\(813\) 27.8403 0.976402
\(814\) 0 0
\(815\) −30.0712 −1.05335
\(816\) 0 0
\(817\) −36.8181 −1.28810
\(818\) 0 0
\(819\) 30.9715 1.08223
\(820\) 0 0
\(821\) −27.4715 −0.958761 −0.479381 0.877607i \(-0.659139\pi\)
−0.479381 + 0.877607i \(0.659139\pi\)
\(822\) 0 0
\(823\) 6.90142 0.240568 0.120284 0.992740i \(-0.461619\pi\)
0.120284 + 0.992740i \(0.461619\pi\)
\(824\) 0 0
\(825\) −6.77793 −0.235977
\(826\) 0 0
\(827\) −52.0396 −1.80960 −0.904798 0.425842i \(-0.859978\pi\)
−0.904798 + 0.425842i \(0.859978\pi\)
\(828\) 0 0
\(829\) −13.6045 −0.472504 −0.236252 0.971692i \(-0.575919\pi\)
−0.236252 + 0.971692i \(0.575919\pi\)
\(830\) 0 0
\(831\) −23.0325 −0.798989
\(832\) 0 0
\(833\) 16.9784 0.588265
\(834\) 0 0
\(835\) 5.17161 0.178971
\(836\) 0 0
\(837\) −47.1941 −1.63127
\(838\) 0 0
\(839\) 28.3083 0.977310 0.488655 0.872477i \(-0.337488\pi\)
0.488655 + 0.872477i \(0.337488\pi\)
\(840\) 0 0
\(841\) 47.9929 1.65493
\(842\) 0 0
\(843\) −0.134626 −0.00463675
\(844\) 0 0
\(845\) −12.7790 −0.439612
\(846\) 0 0
\(847\) −1.75915 −0.0604452
\(848\) 0 0
\(849\) −49.4332 −1.69654
\(850\) 0 0
\(851\) −3.08687 −0.105817
\(852\) 0 0
\(853\) 51.5592 1.76535 0.882677 0.469979i \(-0.155739\pi\)
0.882677 + 0.469979i \(0.155739\pi\)
\(854\) 0 0
\(855\) −75.3548 −2.57708
\(856\) 0 0
\(857\) −5.94196 −0.202974 −0.101487 0.994837i \(-0.532360\pi\)
−0.101487 + 0.994837i \(0.532360\pi\)
\(858\) 0 0
\(859\) −43.3698 −1.47976 −0.739880 0.672739i \(-0.765118\pi\)
−0.739880 + 0.672739i \(0.765118\pi\)
\(860\) 0 0
\(861\) −0.537384 −0.0183140
\(862\) 0 0
\(863\) −10.6702 −0.363218 −0.181609 0.983371i \(-0.558131\pi\)
−0.181609 + 0.983371i \(0.558131\pi\)
\(864\) 0 0
\(865\) −21.3309 −0.725273
\(866\) 0 0
\(867\) 6.48672 0.220301
\(868\) 0 0
\(869\) −52.7791 −1.79041
\(870\) 0 0
\(871\) −33.7505 −1.14359
\(872\) 0 0
\(873\) −41.4032 −1.40129
\(874\) 0 0
\(875\) 18.9976 0.642237
\(876\) 0 0
\(877\) −7.89159 −0.266480 −0.133240 0.991084i \(-0.542538\pi\)
−0.133240 + 0.991084i \(0.542538\pi\)
\(878\) 0 0
\(879\) 15.5298 0.523806
\(880\) 0 0
\(881\) 4.45742 0.150174 0.0750871 0.997177i \(-0.476077\pi\)
0.0750871 + 0.997177i \(0.476077\pi\)
\(882\) 0 0
\(883\) −32.7520 −1.10219 −0.551096 0.834442i \(-0.685790\pi\)
−0.551096 + 0.834442i \(0.685790\pi\)
\(884\) 0 0
\(885\) −37.2995 −1.25381
\(886\) 0 0
\(887\) 16.1312 0.541633 0.270816 0.962631i \(-0.412706\pi\)
0.270816 + 0.962631i \(0.412706\pi\)
\(888\) 0 0
\(889\) 28.9136 0.969730
\(890\) 0 0
\(891\) −77.5518 −2.59808
\(892\) 0 0
\(893\) 51.6060 1.72693
\(894\) 0 0
\(895\) 6.50596 0.217470
\(896\) 0 0
\(897\) −18.8288 −0.628675
\(898\) 0 0
\(899\) 30.0689 1.00285
\(900\) 0 0
\(901\) −9.54454 −0.317975
\(902\) 0 0
\(903\) 38.7277 1.28878
\(904\) 0 0
\(905\) 36.5259 1.21416
\(906\) 0 0
\(907\) 17.0929 0.567560 0.283780 0.958889i \(-0.408412\pi\)
0.283780 + 0.958889i \(0.408412\pi\)
\(908\) 0 0
\(909\) −37.9119 −1.25746
\(910\) 0 0
\(911\) −10.8125 −0.358235 −0.179117 0.983828i \(-0.557324\pi\)
−0.179117 + 0.983828i \(0.557324\pi\)
\(912\) 0 0
\(913\) 19.4636 0.644150
\(914\) 0 0
\(915\) 97.9507 3.23815
\(916\) 0 0
\(917\) 9.43564 0.311592
\(918\) 0 0
\(919\) −59.0595 −1.94819 −0.974097 0.226132i \(-0.927392\pi\)
−0.974097 + 0.226132i \(0.927392\pi\)
\(920\) 0 0
\(921\) −58.2396 −1.91906
\(922\) 0 0
\(923\) 19.5651 0.643995
\(924\) 0 0
\(925\) −0.839685 −0.0276087
\(926\) 0 0
\(927\) 13.4157 0.440630
\(928\) 0 0
\(929\) −46.9487 −1.54034 −0.770168 0.637841i \(-0.779828\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(930\) 0 0
\(931\) 21.6292 0.708870
\(932\) 0 0
\(933\) −56.0384 −1.83461
\(934\) 0 0
\(935\) 28.2006 0.922259
\(936\) 0 0
\(937\) −41.2362 −1.34713 −0.673564 0.739129i \(-0.735237\pi\)
−0.673564 + 0.739129i \(0.735237\pi\)
\(938\) 0 0
\(939\) 40.5016 1.32172
\(940\) 0 0
\(941\) −16.1514 −0.526519 −0.263259 0.964725i \(-0.584798\pi\)
−0.263259 + 0.964725i \(0.584798\pi\)
\(942\) 0 0
\(943\) 0.231475 0.00753785
\(944\) 0 0
\(945\) 46.6567 1.51774
\(946\) 0 0
\(947\) −46.3126 −1.50496 −0.752478 0.658618i \(-0.771142\pi\)
−0.752478 + 0.658618i \(0.771142\pi\)
\(948\) 0 0
\(949\) 31.1537 1.01129
\(950\) 0 0
\(951\) 88.8987 2.88274
\(952\) 0 0
\(953\) 60.4403 1.95785 0.978926 0.204214i \(-0.0654637\pi\)
0.978926 + 0.204214i \(0.0654637\pi\)
\(954\) 0 0
\(955\) −15.5673 −0.503748
\(956\) 0 0
\(957\) 97.8750 3.16385
\(958\) 0 0
\(959\) −31.6415 −1.02176
\(960\) 0 0
\(961\) −19.2568 −0.621188
\(962\) 0 0
\(963\) 81.3189 2.62046
\(964\) 0 0
\(965\) 15.2453 0.490763
\(966\) 0 0
\(967\) −15.3919 −0.494970 −0.247485 0.968892i \(-0.579604\pi\)
−0.247485 + 0.968892i \(0.579604\pi\)
\(968\) 0 0
\(969\) −61.2163 −1.96655
\(970\) 0 0
\(971\) −12.2488 −0.393083 −0.196542 0.980495i \(-0.562971\pi\)
−0.196542 + 0.980495i \(0.562971\pi\)
\(972\) 0 0
\(973\) −24.9505 −0.799875
\(974\) 0 0
\(975\) −5.12177 −0.164028
\(976\) 0 0
\(977\) −2.75271 −0.0880669 −0.0440335 0.999030i \(-0.514021\pi\)
−0.0440335 + 0.999030i \(0.514021\pi\)
\(978\) 0 0
\(979\) 1.38581 0.0442906
\(980\) 0 0
\(981\) 111.689 3.56594
\(982\) 0 0
\(983\) 1.47820 0.0471474 0.0235737 0.999722i \(-0.492496\pi\)
0.0235737 + 0.999722i \(0.492496\pi\)
\(984\) 0 0
\(985\) −25.1326 −0.800792
\(986\) 0 0
\(987\) −54.2826 −1.72784
\(988\) 0 0
\(989\) −16.6817 −0.530447
\(990\) 0 0
\(991\) 13.5421 0.430178 0.215089 0.976594i \(-0.430996\pi\)
0.215089 + 0.976594i \(0.430996\pi\)
\(992\) 0 0
\(993\) 52.4936 1.66583
\(994\) 0 0
\(995\) 52.8907 1.67675
\(996\) 0 0
\(997\) 27.2611 0.863368 0.431684 0.902025i \(-0.357920\pi\)
0.431684 + 0.902025i \(0.357920\pi\)
\(998\) 0 0
\(999\) −19.0310 −0.602113
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 4024.2.a.f.1.2 33
4.3 odd 2 8048.2.a.y.1.32 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.f.1.2 33 1.1 even 1 trivial
8048.2.a.y.1.32 33 4.3 odd 2