Properties

Label 4012.2.a.j.1.6
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $0$
Dimension $21$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.58435 q^{3} +2.56834 q^{5} +3.40444 q^{7} -0.489844 q^{9} +O(q^{10})\) \(q-1.58435 q^{3} +2.56834 q^{5} +3.40444 q^{7} -0.489844 q^{9} +2.40161 q^{11} +2.00961 q^{13} -4.06915 q^{15} +1.00000 q^{17} +3.89035 q^{19} -5.39382 q^{21} -3.11645 q^{23} +1.59639 q^{25} +5.52912 q^{27} +5.87609 q^{29} +3.36059 q^{31} -3.80498 q^{33} +8.74378 q^{35} +4.17469 q^{37} -3.18392 q^{39} -2.04299 q^{41} -4.64231 q^{43} -1.25809 q^{45} -4.53066 q^{47} +4.59024 q^{49} -1.58435 q^{51} +4.71284 q^{53} +6.16815 q^{55} -6.16367 q^{57} +1.00000 q^{59} -6.32381 q^{61} -1.66765 q^{63} +5.16138 q^{65} +9.35564 q^{67} +4.93753 q^{69} -0.403000 q^{71} -0.0127794 q^{73} -2.52924 q^{75} +8.17613 q^{77} -2.39913 q^{79} -7.29052 q^{81} +10.2601 q^{83} +2.56834 q^{85} -9.30977 q^{87} +13.1627 q^{89} +6.84161 q^{91} -5.32433 q^{93} +9.99177 q^{95} -18.2364 q^{97} -1.17641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 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
\(3\) −1.58435 −0.914723 −0.457362 0.889281i \(-0.651205\pi\)
−0.457362 + 0.889281i \(0.651205\pi\)
\(4\) 0 0
\(5\) 2.56834 1.14860 0.574299 0.818646i \(-0.305275\pi\)
0.574299 + 0.818646i \(0.305275\pi\)
\(6\) 0 0
\(7\) 3.40444 1.28676 0.643379 0.765547i \(-0.277532\pi\)
0.643379 + 0.765547i \(0.277532\pi\)
\(8\) 0 0
\(9\) −0.489844 −0.163281
\(10\) 0 0
\(11\) 2.40161 0.724111 0.362056 0.932156i \(-0.382075\pi\)
0.362056 + 0.932156i \(0.382075\pi\)
\(12\) 0 0
\(13\) 2.00961 0.557366 0.278683 0.960383i \(-0.410102\pi\)
0.278683 + 0.960383i \(0.410102\pi\)
\(14\) 0 0
\(15\) −4.06915 −1.05065
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 3.89035 0.892509 0.446254 0.894906i \(-0.352758\pi\)
0.446254 + 0.894906i \(0.352758\pi\)
\(20\) 0 0
\(21\) −5.39382 −1.17703
\(22\) 0 0
\(23\) −3.11645 −0.649824 −0.324912 0.945744i \(-0.605335\pi\)
−0.324912 + 0.945744i \(0.605335\pi\)
\(24\) 0 0
\(25\) 1.59639 0.319278
\(26\) 0 0
\(27\) 5.52912 1.06408
\(28\) 0 0
\(29\) 5.87609 1.09116 0.545581 0.838058i \(-0.316309\pi\)
0.545581 + 0.838058i \(0.316309\pi\)
\(30\) 0 0
\(31\) 3.36059 0.603579 0.301790 0.953375i \(-0.402416\pi\)
0.301790 + 0.953375i \(0.402416\pi\)
\(32\) 0 0
\(33\) −3.80498 −0.662361
\(34\) 0 0
\(35\) 8.74378 1.47797
\(36\) 0 0
\(37\) 4.17469 0.686315 0.343158 0.939278i \(-0.388504\pi\)
0.343158 + 0.939278i \(0.388504\pi\)
\(38\) 0 0
\(39\) −3.18392 −0.509836
\(40\) 0 0
\(41\) −2.04299 −0.319061 −0.159531 0.987193i \(-0.550998\pi\)
−0.159531 + 0.987193i \(0.550998\pi\)
\(42\) 0 0
\(43\) −4.64231 −0.707946 −0.353973 0.935256i \(-0.615169\pi\)
−0.353973 + 0.935256i \(0.615169\pi\)
\(44\) 0 0
\(45\) −1.25809 −0.187545
\(46\) 0 0
\(47\) −4.53066 −0.660865 −0.330432 0.943830i \(-0.607195\pi\)
−0.330432 + 0.943830i \(0.607195\pi\)
\(48\) 0 0
\(49\) 4.59024 0.655749
\(50\) 0 0
\(51\) −1.58435 −0.221853
\(52\) 0 0
\(53\) 4.71284 0.647359 0.323679 0.946167i \(-0.395080\pi\)
0.323679 + 0.946167i \(0.395080\pi\)
\(54\) 0 0
\(55\) 6.16815 0.831713
\(56\) 0 0
\(57\) −6.16367 −0.816398
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) −6.32381 −0.809681 −0.404840 0.914387i \(-0.632673\pi\)
−0.404840 + 0.914387i \(0.632673\pi\)
\(62\) 0 0
\(63\) −1.66765 −0.210104
\(64\) 0 0
\(65\) 5.16138 0.640190
\(66\) 0 0
\(67\) 9.35564 1.14297 0.571487 0.820611i \(-0.306367\pi\)
0.571487 + 0.820611i \(0.306367\pi\)
\(68\) 0 0
\(69\) 4.93753 0.594409
\(70\) 0 0
\(71\) −0.403000 −0.0478273 −0.0239136 0.999714i \(-0.507613\pi\)
−0.0239136 + 0.999714i \(0.507613\pi\)
\(72\) 0 0
\(73\) −0.0127794 −0.00149572 −0.000747860 1.00000i \(-0.500238\pi\)
−0.000747860 1.00000i \(0.500238\pi\)
\(74\) 0 0
\(75\) −2.52924 −0.292051
\(76\) 0 0
\(77\) 8.17613 0.931757
\(78\) 0 0
\(79\) −2.39913 −0.269923 −0.134961 0.990851i \(-0.543091\pi\)
−0.134961 + 0.990851i \(0.543091\pi\)
\(80\) 0 0
\(81\) −7.29052 −0.810058
\(82\) 0 0
\(83\) 10.2601 1.12619 0.563097 0.826391i \(-0.309610\pi\)
0.563097 + 0.826391i \(0.309610\pi\)
\(84\) 0 0
\(85\) 2.56834 0.278576
\(86\) 0 0
\(87\) −9.30977 −0.998112
\(88\) 0 0
\(89\) 13.1627 1.39524 0.697622 0.716466i \(-0.254242\pi\)
0.697622 + 0.716466i \(0.254242\pi\)
\(90\) 0 0
\(91\) 6.84161 0.717196
\(92\) 0 0
\(93\) −5.32433 −0.552108
\(94\) 0 0
\(95\) 9.99177 1.02513
\(96\) 0 0
\(97\) −18.2364 −1.85163 −0.925815 0.377977i \(-0.876620\pi\)
−0.925815 + 0.377977i \(0.876620\pi\)
\(98\) 0 0
\(99\) −1.17641 −0.118234
\(100\) 0 0
\(101\) −6.28016 −0.624899 −0.312450 0.949934i \(-0.601150\pi\)
−0.312450 + 0.949934i \(0.601150\pi\)
\(102\) 0 0
\(103\) −2.85117 −0.280934 −0.140467 0.990085i \(-0.544860\pi\)
−0.140467 + 0.990085i \(0.544860\pi\)
\(104\) 0 0
\(105\) −13.8532 −1.35193
\(106\) 0 0
\(107\) 4.09649 0.396022 0.198011 0.980200i \(-0.436552\pi\)
0.198011 + 0.980200i \(0.436552\pi\)
\(108\) 0 0
\(109\) −16.9504 −1.62355 −0.811775 0.583971i \(-0.801498\pi\)
−0.811775 + 0.583971i \(0.801498\pi\)
\(110\) 0 0
\(111\) −6.61416 −0.627788
\(112\) 0 0
\(113\) 8.25142 0.776228 0.388114 0.921611i \(-0.373127\pi\)
0.388114 + 0.921611i \(0.373127\pi\)
\(114\) 0 0
\(115\) −8.00411 −0.746387
\(116\) 0 0
\(117\) −0.984397 −0.0910075
\(118\) 0 0
\(119\) 3.40444 0.312085
\(120\) 0 0
\(121\) −5.23229 −0.475663
\(122\) 0 0
\(123\) 3.23680 0.291853
\(124\) 0 0
\(125\) −8.74164 −0.781876
\(126\) 0 0
\(127\) −17.5059 −1.55340 −0.776699 0.629871i \(-0.783108\pi\)
−0.776699 + 0.629871i \(0.783108\pi\)
\(128\) 0 0
\(129\) 7.35503 0.647574
\(130\) 0 0
\(131\) −10.9296 −0.954921 −0.477460 0.878653i \(-0.658443\pi\)
−0.477460 + 0.878653i \(0.658443\pi\)
\(132\) 0 0
\(133\) 13.2445 1.14844
\(134\) 0 0
\(135\) 14.2007 1.22220
\(136\) 0 0
\(137\) −15.8932 −1.35784 −0.678922 0.734210i \(-0.737553\pi\)
−0.678922 + 0.734210i \(0.737553\pi\)
\(138\) 0 0
\(139\) 3.02838 0.256863 0.128432 0.991718i \(-0.459006\pi\)
0.128432 + 0.991718i \(0.459006\pi\)
\(140\) 0 0
\(141\) 7.17814 0.604508
\(142\) 0 0
\(143\) 4.82630 0.403595
\(144\) 0 0
\(145\) 15.0918 1.25331
\(146\) 0 0
\(147\) −7.27253 −0.599829
\(148\) 0 0
\(149\) −8.61304 −0.705608 −0.352804 0.935697i \(-0.614772\pi\)
−0.352804 + 0.935697i \(0.614772\pi\)
\(150\) 0 0
\(151\) 8.90735 0.724869 0.362435 0.932009i \(-0.381946\pi\)
0.362435 + 0.932009i \(0.381946\pi\)
\(152\) 0 0
\(153\) −0.489844 −0.0396015
\(154\) 0 0
\(155\) 8.63114 0.693270
\(156\) 0 0
\(157\) 10.7149 0.855144 0.427572 0.903981i \(-0.359369\pi\)
0.427572 + 0.903981i \(0.359369\pi\)
\(158\) 0 0
\(159\) −7.46678 −0.592154
\(160\) 0 0
\(161\) −10.6098 −0.836167
\(162\) 0 0
\(163\) 24.5447 1.92249 0.961244 0.275698i \(-0.0889091\pi\)
0.961244 + 0.275698i \(0.0889091\pi\)
\(164\) 0 0
\(165\) −9.77249 −0.760787
\(166\) 0 0
\(167\) 2.34988 0.181840 0.0909198 0.995858i \(-0.471019\pi\)
0.0909198 + 0.995858i \(0.471019\pi\)
\(168\) 0 0
\(169\) −8.96146 −0.689343
\(170\) 0 0
\(171\) −1.90567 −0.145730
\(172\) 0 0
\(173\) −4.19331 −0.318812 −0.159406 0.987213i \(-0.550958\pi\)
−0.159406 + 0.987213i \(0.550958\pi\)
\(174\) 0 0
\(175\) 5.43482 0.410834
\(176\) 0 0
\(177\) −1.58435 −0.119087
\(178\) 0 0
\(179\) 3.81077 0.284831 0.142415 0.989807i \(-0.454513\pi\)
0.142415 + 0.989807i \(0.454513\pi\)
\(180\) 0 0
\(181\) −14.6218 −1.08683 −0.543414 0.839465i \(-0.682869\pi\)
−0.543414 + 0.839465i \(0.682869\pi\)
\(182\) 0 0
\(183\) 10.0191 0.740634
\(184\) 0 0
\(185\) 10.7220 0.788300
\(186\) 0 0
\(187\) 2.40161 0.175623
\(188\) 0 0
\(189\) 18.8236 1.36922
\(190\) 0 0
\(191\) 21.5203 1.55715 0.778576 0.627550i \(-0.215942\pi\)
0.778576 + 0.627550i \(0.215942\pi\)
\(192\) 0 0
\(193\) −4.06466 −0.292580 −0.146290 0.989242i \(-0.546733\pi\)
−0.146290 + 0.989242i \(0.546733\pi\)
\(194\) 0 0
\(195\) −8.17741 −0.585597
\(196\) 0 0
\(197\) −2.46476 −0.175607 −0.0878035 0.996138i \(-0.527985\pi\)
−0.0878035 + 0.996138i \(0.527985\pi\)
\(198\) 0 0
\(199\) 23.5614 1.67022 0.835110 0.550083i \(-0.185404\pi\)
0.835110 + 0.550083i \(0.185404\pi\)
\(200\) 0 0
\(201\) −14.8226 −1.04550
\(202\) 0 0
\(203\) 20.0048 1.40406
\(204\) 0 0
\(205\) −5.24710 −0.366473
\(206\) 0 0
\(207\) 1.52657 0.106104
\(208\) 0 0
\(209\) 9.34310 0.646276
\(210\) 0 0
\(211\) 15.3754 1.05849 0.529244 0.848470i \(-0.322476\pi\)
0.529244 + 0.848470i \(0.322476\pi\)
\(212\) 0 0
\(213\) 0.638491 0.0437487
\(214\) 0 0
\(215\) −11.9231 −0.813145
\(216\) 0 0
\(217\) 11.4409 0.776661
\(218\) 0 0
\(219\) 0.0202471 0.00136817
\(220\) 0 0
\(221\) 2.00961 0.135181
\(222\) 0 0
\(223\) 23.2116 1.55437 0.777183 0.629275i \(-0.216648\pi\)
0.777183 + 0.629275i \(0.216648\pi\)
\(224\) 0 0
\(225\) −0.781982 −0.0521322
\(226\) 0 0
\(227\) −7.58029 −0.503122 −0.251561 0.967841i \(-0.580944\pi\)
−0.251561 + 0.967841i \(0.580944\pi\)
\(228\) 0 0
\(229\) 22.1259 1.46212 0.731062 0.682311i \(-0.239025\pi\)
0.731062 + 0.682311i \(0.239025\pi\)
\(230\) 0 0
\(231\) −12.9538 −0.852299
\(232\) 0 0
\(233\) 4.66227 0.305436 0.152718 0.988270i \(-0.451197\pi\)
0.152718 + 0.988270i \(0.451197\pi\)
\(234\) 0 0
\(235\) −11.6363 −0.759068
\(236\) 0 0
\(237\) 3.80105 0.246905
\(238\) 0 0
\(239\) 10.0323 0.648934 0.324467 0.945897i \(-0.394815\pi\)
0.324467 + 0.945897i \(0.394815\pi\)
\(240\) 0 0
\(241\) 7.49399 0.482731 0.241365 0.970434i \(-0.422405\pi\)
0.241365 + 0.970434i \(0.422405\pi\)
\(242\) 0 0
\(243\) −5.03666 −0.323102
\(244\) 0 0
\(245\) 11.7893 0.753192
\(246\) 0 0
\(247\) 7.81811 0.497454
\(248\) 0 0
\(249\) −16.2556 −1.03016
\(250\) 0 0
\(251\) −12.3707 −0.780833 −0.390416 0.920638i \(-0.627669\pi\)
−0.390416 + 0.920638i \(0.627669\pi\)
\(252\) 0 0
\(253\) −7.48447 −0.470545
\(254\) 0 0
\(255\) −4.06915 −0.254820
\(256\) 0 0
\(257\) 17.5081 1.09213 0.546064 0.837744i \(-0.316126\pi\)
0.546064 + 0.837744i \(0.316126\pi\)
\(258\) 0 0
\(259\) 14.2125 0.883122
\(260\) 0 0
\(261\) −2.87837 −0.178166
\(262\) 0 0
\(263\) −1.94371 −0.119855 −0.0599273 0.998203i \(-0.519087\pi\)
−0.0599273 + 0.998203i \(0.519087\pi\)
\(264\) 0 0
\(265\) 12.1042 0.743555
\(266\) 0 0
\(267\) −20.8543 −1.27626
\(268\) 0 0
\(269\) 5.14731 0.313837 0.156919 0.987612i \(-0.449844\pi\)
0.156919 + 0.987612i \(0.449844\pi\)
\(270\) 0 0
\(271\) −13.9856 −0.849566 −0.424783 0.905295i \(-0.639650\pi\)
−0.424783 + 0.905295i \(0.639650\pi\)
\(272\) 0 0
\(273\) −10.8395 −0.656036
\(274\) 0 0
\(275\) 3.83390 0.231193
\(276\) 0 0
\(277\) −18.9058 −1.13594 −0.567969 0.823050i \(-0.692271\pi\)
−0.567969 + 0.823050i \(0.692271\pi\)
\(278\) 0 0
\(279\) −1.64616 −0.0985532
\(280\) 0 0
\(281\) 22.8762 1.36468 0.682341 0.731034i \(-0.260962\pi\)
0.682341 + 0.731034i \(0.260962\pi\)
\(282\) 0 0
\(283\) 29.9870 1.78254 0.891271 0.453471i \(-0.149815\pi\)
0.891271 + 0.453471i \(0.149815\pi\)
\(284\) 0 0
\(285\) −15.8304 −0.937714
\(286\) 0 0
\(287\) −6.95524 −0.410555
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 28.8928 1.69373
\(292\) 0 0
\(293\) 11.1458 0.651142 0.325571 0.945518i \(-0.394444\pi\)
0.325571 + 0.945518i \(0.394444\pi\)
\(294\) 0 0
\(295\) 2.56834 0.149535
\(296\) 0 0
\(297\) 13.2788 0.770513
\(298\) 0 0
\(299\) −6.26285 −0.362190
\(300\) 0 0
\(301\) −15.8045 −0.910956
\(302\) 0 0
\(303\) 9.94995 0.571610
\(304\) 0 0
\(305\) −16.2417 −0.929998
\(306\) 0 0
\(307\) −10.0152 −0.571598 −0.285799 0.958290i \(-0.592259\pi\)
−0.285799 + 0.958290i \(0.592259\pi\)
\(308\) 0 0
\(309\) 4.51725 0.256977
\(310\) 0 0
\(311\) −1.84863 −0.104826 −0.0524131 0.998625i \(-0.516691\pi\)
−0.0524131 + 0.998625i \(0.516691\pi\)
\(312\) 0 0
\(313\) −24.3405 −1.37581 −0.687903 0.725803i \(-0.741469\pi\)
−0.687903 + 0.725803i \(0.741469\pi\)
\(314\) 0 0
\(315\) −4.28309 −0.241325
\(316\) 0 0
\(317\) 14.5395 0.816618 0.408309 0.912844i \(-0.366119\pi\)
0.408309 + 0.912844i \(0.366119\pi\)
\(318\) 0 0
\(319\) 14.1120 0.790123
\(320\) 0 0
\(321\) −6.49026 −0.362251
\(322\) 0 0
\(323\) 3.89035 0.216465
\(324\) 0 0
\(325\) 3.20813 0.177955
\(326\) 0 0
\(327\) 26.8552 1.48510
\(328\) 0 0
\(329\) −15.4244 −0.850374
\(330\) 0 0
\(331\) 32.8651 1.80643 0.903215 0.429188i \(-0.141200\pi\)
0.903215 + 0.429188i \(0.141200\pi\)
\(332\) 0 0
\(333\) −2.04495 −0.112062
\(334\) 0 0
\(335\) 24.0285 1.31282
\(336\) 0 0
\(337\) 8.00139 0.435864 0.217932 0.975964i \(-0.430069\pi\)
0.217932 + 0.975964i \(0.430069\pi\)
\(338\) 0 0
\(339\) −13.0731 −0.710034
\(340\) 0 0
\(341\) 8.07080 0.437058
\(342\) 0 0
\(343\) −8.20389 −0.442969
\(344\) 0 0
\(345\) 12.6813 0.682737
\(346\) 0 0
\(347\) 5.18570 0.278383 0.139191 0.990265i \(-0.455550\pi\)
0.139191 + 0.990265i \(0.455550\pi\)
\(348\) 0 0
\(349\) −6.66895 −0.356981 −0.178490 0.983942i \(-0.557121\pi\)
−0.178490 + 0.983942i \(0.557121\pi\)
\(350\) 0 0
\(351\) 11.1114 0.593082
\(352\) 0 0
\(353\) −12.0198 −0.639750 −0.319875 0.947460i \(-0.603641\pi\)
−0.319875 + 0.947460i \(0.603641\pi\)
\(354\) 0 0
\(355\) −1.03504 −0.0549343
\(356\) 0 0
\(357\) −5.39382 −0.285471
\(358\) 0 0
\(359\) 25.1915 1.32956 0.664780 0.747040i \(-0.268525\pi\)
0.664780 + 0.747040i \(0.268525\pi\)
\(360\) 0 0
\(361\) −3.86514 −0.203428
\(362\) 0 0
\(363\) 8.28977 0.435100
\(364\) 0 0
\(365\) −0.0328220 −0.00171798
\(366\) 0 0
\(367\) −28.5663 −1.49115 −0.745576 0.666421i \(-0.767825\pi\)
−0.745576 + 0.666421i \(0.767825\pi\)
\(368\) 0 0
\(369\) 1.00075 0.0520967
\(370\) 0 0
\(371\) 16.0446 0.832994
\(372\) 0 0
\(373\) −8.63027 −0.446859 −0.223429 0.974720i \(-0.571725\pi\)
−0.223429 + 0.974720i \(0.571725\pi\)
\(374\) 0 0
\(375\) 13.8498 0.715200
\(376\) 0 0
\(377\) 11.8087 0.608177
\(378\) 0 0
\(379\) −10.3382 −0.531039 −0.265519 0.964106i \(-0.585543\pi\)
−0.265519 + 0.964106i \(0.585543\pi\)
\(380\) 0 0
\(381\) 27.7354 1.42093
\(382\) 0 0
\(383\) 31.0322 1.58567 0.792837 0.609434i \(-0.208603\pi\)
0.792837 + 0.609434i \(0.208603\pi\)
\(384\) 0 0
\(385\) 20.9991 1.07021
\(386\) 0 0
\(387\) 2.27401 0.115594
\(388\) 0 0
\(389\) 15.2059 0.770970 0.385485 0.922714i \(-0.374034\pi\)
0.385485 + 0.922714i \(0.374034\pi\)
\(390\) 0 0
\(391\) −3.11645 −0.157605
\(392\) 0 0
\(393\) 17.3162 0.873488
\(394\) 0 0
\(395\) −6.16179 −0.310033
\(396\) 0 0
\(397\) −13.9656 −0.700913 −0.350457 0.936579i \(-0.613974\pi\)
−0.350457 + 0.936579i \(0.613974\pi\)
\(398\) 0 0
\(399\) −20.9839 −1.05051
\(400\) 0 0
\(401\) −4.37465 −0.218459 −0.109230 0.994017i \(-0.534838\pi\)
−0.109230 + 0.994017i \(0.534838\pi\)
\(402\) 0 0
\(403\) 6.75347 0.336415
\(404\) 0 0
\(405\) −18.7246 −0.930431
\(406\) 0 0
\(407\) 10.0260 0.496968
\(408\) 0 0
\(409\) −38.3304 −1.89532 −0.947658 0.319287i \(-0.896557\pi\)
−0.947658 + 0.319287i \(0.896557\pi\)
\(410\) 0 0
\(411\) 25.1803 1.24205
\(412\) 0 0
\(413\) 3.40444 0.167522
\(414\) 0 0
\(415\) 26.3515 1.29355
\(416\) 0 0
\(417\) −4.79800 −0.234959
\(418\) 0 0
\(419\) 5.43916 0.265720 0.132860 0.991135i \(-0.457584\pi\)
0.132860 + 0.991135i \(0.457584\pi\)
\(420\) 0 0
\(421\) −7.49826 −0.365443 −0.182721 0.983165i \(-0.558491\pi\)
−0.182721 + 0.983165i \(0.558491\pi\)
\(422\) 0 0
\(423\) 2.21932 0.107907
\(424\) 0 0
\(425\) 1.59639 0.0774363
\(426\) 0 0
\(427\) −21.5291 −1.04186
\(428\) 0 0
\(429\) −7.64653 −0.369178
\(430\) 0 0
\(431\) −4.92380 −0.237171 −0.118586 0.992944i \(-0.537836\pi\)
−0.118586 + 0.992944i \(0.537836\pi\)
\(432\) 0 0
\(433\) 18.1257 0.871066 0.435533 0.900173i \(-0.356560\pi\)
0.435533 + 0.900173i \(0.356560\pi\)
\(434\) 0 0
\(435\) −23.9107 −1.14643
\(436\) 0 0
\(437\) −12.1241 −0.579974
\(438\) 0 0
\(439\) −9.03914 −0.431414 −0.215707 0.976458i \(-0.569206\pi\)
−0.215707 + 0.976458i \(0.569206\pi\)
\(440\) 0 0
\(441\) −2.24850 −0.107072
\(442\) 0 0
\(443\) −32.7692 −1.55691 −0.778456 0.627699i \(-0.783997\pi\)
−0.778456 + 0.627699i \(0.783997\pi\)
\(444\) 0 0
\(445\) 33.8063 1.60257
\(446\) 0 0
\(447\) 13.6461 0.645436
\(448\) 0 0
\(449\) 13.1349 0.619874 0.309937 0.950757i \(-0.399692\pi\)
0.309937 + 0.950757i \(0.399692\pi\)
\(450\) 0 0
\(451\) −4.90645 −0.231036
\(452\) 0 0
\(453\) −14.1123 −0.663055
\(454\) 0 0
\(455\) 17.5716 0.823770
\(456\) 0 0
\(457\) 15.8721 0.742467 0.371233 0.928540i \(-0.378935\pi\)
0.371233 + 0.928540i \(0.378935\pi\)
\(458\) 0 0
\(459\) 5.52912 0.258077
\(460\) 0 0
\(461\) 1.28169 0.0596942 0.0298471 0.999554i \(-0.490498\pi\)
0.0298471 + 0.999554i \(0.490498\pi\)
\(462\) 0 0
\(463\) 15.1278 0.703048 0.351524 0.936179i \(-0.385664\pi\)
0.351524 + 0.936179i \(0.385664\pi\)
\(464\) 0 0
\(465\) −13.6747 −0.634150
\(466\) 0 0
\(467\) −10.8136 −0.500396 −0.250198 0.968195i \(-0.580496\pi\)
−0.250198 + 0.968195i \(0.580496\pi\)
\(468\) 0 0
\(469\) 31.8508 1.47073
\(470\) 0 0
\(471\) −16.9762 −0.782220
\(472\) 0 0
\(473\) −11.1490 −0.512631
\(474\) 0 0
\(475\) 6.21053 0.284958
\(476\) 0 0
\(477\) −2.30856 −0.105702
\(478\) 0 0
\(479\) 15.6620 0.715617 0.357808 0.933795i \(-0.383524\pi\)
0.357808 + 0.933795i \(0.383524\pi\)
\(480\) 0 0
\(481\) 8.38951 0.382529
\(482\) 0 0
\(483\) 16.8096 0.764861
\(484\) 0 0
\(485\) −46.8374 −2.12678
\(486\) 0 0
\(487\) −1.25701 −0.0569605 −0.0284802 0.999594i \(-0.509067\pi\)
−0.0284802 + 0.999594i \(0.509067\pi\)
\(488\) 0 0
\(489\) −38.8873 −1.75854
\(490\) 0 0
\(491\) −5.47839 −0.247236 −0.123618 0.992330i \(-0.539450\pi\)
−0.123618 + 0.992330i \(0.539450\pi\)
\(492\) 0 0
\(493\) 5.87609 0.264646
\(494\) 0 0
\(495\) −3.02143 −0.135803
\(496\) 0 0
\(497\) −1.37199 −0.0615422
\(498\) 0 0
\(499\) −14.0957 −0.631008 −0.315504 0.948924i \(-0.602174\pi\)
−0.315504 + 0.948924i \(0.602174\pi\)
\(500\) 0 0
\(501\) −3.72303 −0.166333
\(502\) 0 0
\(503\) −6.90002 −0.307657 −0.153828 0.988098i \(-0.549160\pi\)
−0.153828 + 0.988098i \(0.549160\pi\)
\(504\) 0 0
\(505\) −16.1296 −0.717758
\(506\) 0 0
\(507\) 14.1981 0.630558
\(508\) 0 0
\(509\) −44.2392 −1.96087 −0.980435 0.196843i \(-0.936931\pi\)
−0.980435 + 0.196843i \(0.936931\pi\)
\(510\) 0 0
\(511\) −0.0435069 −0.00192463
\(512\) 0 0
\(513\) 21.5103 0.949701
\(514\) 0 0
\(515\) −7.32279 −0.322681
\(516\) 0 0
\(517\) −10.8809 −0.478540
\(518\) 0 0
\(519\) 6.64366 0.291624
\(520\) 0 0
\(521\) −27.1948 −1.19142 −0.595712 0.803198i \(-0.703130\pi\)
−0.595712 + 0.803198i \(0.703130\pi\)
\(522\) 0 0
\(523\) −15.5632 −0.680530 −0.340265 0.940330i \(-0.610517\pi\)
−0.340265 + 0.940330i \(0.610517\pi\)
\(524\) 0 0
\(525\) −8.61065 −0.375799
\(526\) 0 0
\(527\) 3.36059 0.146389
\(528\) 0 0
\(529\) −13.2878 −0.577729
\(530\) 0 0
\(531\) −0.489844 −0.0212574
\(532\) 0 0
\(533\) −4.10562 −0.177834
\(534\) 0 0
\(535\) 10.5212 0.454871
\(536\) 0 0
\(537\) −6.03759 −0.260541
\(538\) 0 0
\(539\) 11.0239 0.474835
\(540\) 0 0
\(541\) 13.0833 0.562495 0.281247 0.959635i \(-0.409252\pi\)
0.281247 + 0.959635i \(0.409252\pi\)
\(542\) 0 0
\(543\) 23.1660 0.994147
\(544\) 0 0
\(545\) −43.5343 −1.86481
\(546\) 0 0
\(547\) −29.6539 −1.26791 −0.633954 0.773370i \(-0.718569\pi\)
−0.633954 + 0.773370i \(0.718569\pi\)
\(548\) 0 0
\(549\) 3.09768 0.132206
\(550\) 0 0
\(551\) 22.8601 0.973872
\(552\) 0 0
\(553\) −8.16770 −0.347326
\(554\) 0 0
\(555\) −16.9874 −0.721077
\(556\) 0 0
\(557\) 21.1689 0.896957 0.448478 0.893794i \(-0.351966\pi\)
0.448478 + 0.893794i \(0.351966\pi\)
\(558\) 0 0
\(559\) −9.32925 −0.394585
\(560\) 0 0
\(561\) −3.80498 −0.160646
\(562\) 0 0
\(563\) −11.4916 −0.484315 −0.242158 0.970237i \(-0.577855\pi\)
−0.242158 + 0.970237i \(0.577855\pi\)
\(564\) 0 0
\(565\) 21.1925 0.891574
\(566\) 0 0
\(567\) −24.8202 −1.04235
\(568\) 0 0
\(569\) −0.321730 −0.0134876 −0.00674381 0.999977i \(-0.502147\pi\)
−0.00674381 + 0.999977i \(0.502147\pi\)
\(570\) 0 0
\(571\) −7.79349 −0.326148 −0.163074 0.986614i \(-0.552141\pi\)
−0.163074 + 0.986614i \(0.552141\pi\)
\(572\) 0 0
\(573\) −34.0956 −1.42436
\(574\) 0 0
\(575\) −4.97507 −0.207475
\(576\) 0 0
\(577\) −2.37064 −0.0986910 −0.0493455 0.998782i \(-0.515714\pi\)
−0.0493455 + 0.998782i \(0.515714\pi\)
\(578\) 0 0
\(579\) 6.43983 0.267630
\(580\) 0 0
\(581\) 34.9300 1.44914
\(582\) 0 0
\(583\) 11.3184 0.468760
\(584\) 0 0
\(585\) −2.52827 −0.104531
\(586\) 0 0
\(587\) −20.8757 −0.861633 −0.430817 0.902440i \(-0.641774\pi\)
−0.430817 + 0.902440i \(0.641774\pi\)
\(588\) 0 0
\(589\) 13.0739 0.538699
\(590\) 0 0
\(591\) 3.90504 0.160632
\(592\) 0 0
\(593\) −5.80793 −0.238503 −0.119252 0.992864i \(-0.538049\pi\)
−0.119252 + 0.992864i \(0.538049\pi\)
\(594\) 0 0
\(595\) 8.74378 0.358460
\(596\) 0 0
\(597\) −37.3294 −1.52779
\(598\) 0 0
\(599\) 10.2881 0.420359 0.210180 0.977663i \(-0.432595\pi\)
0.210180 + 0.977663i \(0.432595\pi\)
\(600\) 0 0
\(601\) −18.1095 −0.738702 −0.369351 0.929290i \(-0.620420\pi\)
−0.369351 + 0.929290i \(0.620420\pi\)
\(602\) 0 0
\(603\) −4.58280 −0.186626
\(604\) 0 0
\(605\) −13.4383 −0.546346
\(606\) 0 0
\(607\) 46.2725 1.87814 0.939071 0.343723i \(-0.111688\pi\)
0.939071 + 0.343723i \(0.111688\pi\)
\(608\) 0 0
\(609\) −31.6946 −1.28433
\(610\) 0 0
\(611\) −9.10487 −0.368344
\(612\) 0 0
\(613\) −3.41794 −0.138049 −0.0690246 0.997615i \(-0.521989\pi\)
−0.0690246 + 0.997615i \(0.521989\pi\)
\(614\) 0 0
\(615\) 8.31323 0.335222
\(616\) 0 0
\(617\) 32.9031 1.32463 0.662315 0.749225i \(-0.269574\pi\)
0.662315 + 0.749225i \(0.269574\pi\)
\(618\) 0 0
\(619\) 34.6187 1.39144 0.695721 0.718312i \(-0.255085\pi\)
0.695721 + 0.718312i \(0.255085\pi\)
\(620\) 0 0
\(621\) −17.2312 −0.691465
\(622\) 0 0
\(623\) 44.8117 1.79534
\(624\) 0 0
\(625\) −30.4335 −1.21734
\(626\) 0 0
\(627\) −14.8027 −0.591163
\(628\) 0 0
\(629\) 4.17469 0.166456
\(630\) 0 0
\(631\) 21.5945 0.859663 0.429832 0.902909i \(-0.358573\pi\)
0.429832 + 0.902909i \(0.358573\pi\)
\(632\) 0 0
\(633\) −24.3600 −0.968223
\(634\) 0 0
\(635\) −44.9612 −1.78423
\(636\) 0 0
\(637\) 9.22460 0.365492
\(638\) 0 0
\(639\) 0.197407 0.00780930
\(640\) 0 0
\(641\) −37.2341 −1.47066 −0.735329 0.677710i \(-0.762972\pi\)
−0.735329 + 0.677710i \(0.762972\pi\)
\(642\) 0 0
\(643\) 29.3056 1.15570 0.577851 0.816143i \(-0.303892\pi\)
0.577851 + 0.816143i \(0.303892\pi\)
\(644\) 0 0
\(645\) 18.8903 0.743803
\(646\) 0 0
\(647\) 8.82457 0.346930 0.173465 0.984840i \(-0.444504\pi\)
0.173465 + 0.984840i \(0.444504\pi\)
\(648\) 0 0
\(649\) 2.40161 0.0942713
\(650\) 0 0
\(651\) −18.1264 −0.710430
\(652\) 0 0
\(653\) −8.86857 −0.347054 −0.173527 0.984829i \(-0.555516\pi\)
−0.173527 + 0.984829i \(0.555516\pi\)
\(654\) 0 0
\(655\) −28.0709 −1.09682
\(656\) 0 0
\(657\) 0.00625993 0.000244223 0
\(658\) 0 0
\(659\) −0.571452 −0.0222606 −0.0111303 0.999938i \(-0.503543\pi\)
−0.0111303 + 0.999938i \(0.503543\pi\)
\(660\) 0 0
\(661\) −19.4476 −0.756423 −0.378211 0.925719i \(-0.623461\pi\)
−0.378211 + 0.925719i \(0.623461\pi\)
\(662\) 0 0
\(663\) −3.18392 −0.123653
\(664\) 0 0
\(665\) 34.0164 1.31910
\(666\) 0 0
\(667\) −18.3125 −0.709064
\(668\) 0 0
\(669\) −36.7753 −1.42181
\(670\) 0 0
\(671\) −15.1873 −0.586299
\(672\) 0 0
\(673\) 2.70899 0.104424 0.0522119 0.998636i \(-0.483373\pi\)
0.0522119 + 0.998636i \(0.483373\pi\)
\(674\) 0 0
\(675\) 8.82664 0.339738
\(676\) 0 0
\(677\) −17.4927 −0.672301 −0.336150 0.941808i \(-0.609125\pi\)
−0.336150 + 0.941808i \(0.609125\pi\)
\(678\) 0 0
\(679\) −62.0849 −2.38260
\(680\) 0 0
\(681\) 12.0098 0.460217
\(682\) 0 0
\(683\) −22.4587 −0.859358 −0.429679 0.902982i \(-0.641373\pi\)
−0.429679 + 0.902982i \(0.641373\pi\)
\(684\) 0 0
\(685\) −40.8191 −1.55962
\(686\) 0 0
\(687\) −35.0552 −1.33744
\(688\) 0 0
\(689\) 9.47098 0.360816
\(690\) 0 0
\(691\) −10.8796 −0.413879 −0.206940 0.978354i \(-0.566350\pi\)
−0.206940 + 0.978354i \(0.566350\pi\)
\(692\) 0 0
\(693\) −4.00503 −0.152138
\(694\) 0 0
\(695\) 7.77791 0.295033
\(696\) 0 0
\(697\) −2.04299 −0.0773837
\(698\) 0 0
\(699\) −7.38666 −0.279389
\(700\) 0 0
\(701\) 6.11500 0.230960 0.115480 0.993310i \(-0.463159\pi\)
0.115480 + 0.993310i \(0.463159\pi\)
\(702\) 0 0
\(703\) 16.2410 0.612542
\(704\) 0 0
\(705\) 18.4359 0.694337
\(706\) 0 0
\(707\) −21.3805 −0.804095
\(708\) 0 0
\(709\) 24.4801 0.919371 0.459685 0.888082i \(-0.347962\pi\)
0.459685 + 0.888082i \(0.347962\pi\)
\(710\) 0 0
\(711\) 1.17520 0.0440734
\(712\) 0 0
\(713\) −10.4731 −0.392220
\(714\) 0 0
\(715\) 12.3956 0.463569
\(716\) 0 0
\(717\) −15.8946 −0.593595
\(718\) 0 0
\(719\) −16.2546 −0.606193 −0.303096 0.952960i \(-0.598020\pi\)
−0.303096 + 0.952960i \(0.598020\pi\)
\(720\) 0 0
\(721\) −9.70666 −0.361495
\(722\) 0 0
\(723\) −11.8731 −0.441565
\(724\) 0 0
\(725\) 9.38053 0.348384
\(726\) 0 0
\(727\) −0.147626 −0.00547514 −0.00273757 0.999996i \(-0.500871\pi\)
−0.00273757 + 0.999996i \(0.500871\pi\)
\(728\) 0 0
\(729\) 29.8514 1.10561
\(730\) 0 0
\(731\) −4.64231 −0.171702
\(732\) 0 0
\(733\) −28.1477 −1.03966 −0.519829 0.854270i \(-0.674005\pi\)
−0.519829 + 0.854270i \(0.674005\pi\)
\(734\) 0 0
\(735\) −18.6784 −0.688962
\(736\) 0 0
\(737\) 22.4686 0.827640
\(738\) 0 0
\(739\) −19.0872 −0.702134 −0.351067 0.936350i \(-0.614181\pi\)
−0.351067 + 0.936350i \(0.614181\pi\)
\(740\) 0 0
\(741\) −12.3866 −0.455033
\(742\) 0 0
\(743\) −15.8807 −0.582608 −0.291304 0.956631i \(-0.594089\pi\)
−0.291304 + 0.956631i \(0.594089\pi\)
\(744\) 0 0
\(745\) −22.1213 −0.810460
\(746\) 0 0
\(747\) −5.02586 −0.183887
\(748\) 0 0
\(749\) 13.9463 0.509585
\(750\) 0 0
\(751\) −15.9315 −0.581348 −0.290674 0.956822i \(-0.593880\pi\)
−0.290674 + 0.956822i \(0.593880\pi\)
\(752\) 0 0
\(753\) 19.5995 0.714246
\(754\) 0 0
\(755\) 22.8771 0.832584
\(756\) 0 0
\(757\) −2.90616 −0.105626 −0.0528131 0.998604i \(-0.516819\pi\)
−0.0528131 + 0.998604i \(0.516819\pi\)
\(758\) 0 0
\(759\) 11.8580 0.430418
\(760\) 0 0
\(761\) −26.6016 −0.964306 −0.482153 0.876087i \(-0.660145\pi\)
−0.482153 + 0.876087i \(0.660145\pi\)
\(762\) 0 0
\(763\) −57.7065 −2.08912
\(764\) 0 0
\(765\) −1.25809 −0.0454863
\(766\) 0 0
\(767\) 2.00961 0.0725629
\(768\) 0 0
\(769\) −14.2361 −0.513367 −0.256684 0.966496i \(-0.582630\pi\)
−0.256684 + 0.966496i \(0.582630\pi\)
\(770\) 0 0
\(771\) −27.7390 −0.998995
\(772\) 0 0
\(773\) 24.2010 0.870451 0.435226 0.900321i \(-0.356669\pi\)
0.435226 + 0.900321i \(0.356669\pi\)
\(774\) 0 0
\(775\) 5.36481 0.192710
\(776\) 0 0
\(777\) −22.5175 −0.807812
\(778\) 0 0
\(779\) −7.94795 −0.284765
\(780\) 0 0
\(781\) −0.967846 −0.0346323
\(782\) 0 0
\(783\) 32.4896 1.16108
\(784\) 0 0
\(785\) 27.5196 0.982217
\(786\) 0 0
\(787\) −11.9610 −0.426362 −0.213181 0.977013i \(-0.568382\pi\)
−0.213181 + 0.977013i \(0.568382\pi\)
\(788\) 0 0
\(789\) 3.07952 0.109634
\(790\) 0 0
\(791\) 28.0915 0.998818
\(792\) 0 0
\(793\) −12.7084 −0.451289
\(794\) 0 0
\(795\) −19.1773 −0.680147
\(796\) 0 0
\(797\) 23.7814 0.842379 0.421189 0.906973i \(-0.361613\pi\)
0.421189 + 0.906973i \(0.361613\pi\)
\(798\) 0 0
\(799\) −4.53066 −0.160283
\(800\) 0 0
\(801\) −6.44767 −0.227817
\(802\) 0 0
\(803\) −0.0306912 −0.00108307
\(804\) 0 0
\(805\) −27.2495 −0.960420
\(806\) 0 0
\(807\) −8.15513 −0.287074
\(808\) 0 0
\(809\) 35.8493 1.26039 0.630197 0.776435i \(-0.282974\pi\)
0.630197 + 0.776435i \(0.282974\pi\)
\(810\) 0 0
\(811\) −11.2838 −0.396228 −0.198114 0.980179i \(-0.563482\pi\)
−0.198114 + 0.980179i \(0.563482\pi\)
\(812\) 0 0
\(813\) 22.1581 0.777118
\(814\) 0 0
\(815\) 63.0392 2.20817
\(816\) 0 0
\(817\) −18.0602 −0.631848
\(818\) 0 0
\(819\) −3.35132 −0.117105
\(820\) 0 0
\(821\) 44.6727 1.55909 0.779545 0.626347i \(-0.215451\pi\)
0.779545 + 0.626347i \(0.215451\pi\)
\(822\) 0 0
\(823\) 5.50943 0.192047 0.0960233 0.995379i \(-0.469388\pi\)
0.0960233 + 0.995379i \(0.469388\pi\)
\(824\) 0 0
\(825\) −6.07423 −0.211478
\(826\) 0 0
\(827\) −45.0548 −1.56671 −0.783355 0.621575i \(-0.786493\pi\)
−0.783355 + 0.621575i \(0.786493\pi\)
\(828\) 0 0
\(829\) 8.79544 0.305478 0.152739 0.988267i \(-0.451191\pi\)
0.152739 + 0.988267i \(0.451191\pi\)
\(830\) 0 0
\(831\) 29.9533 1.03907
\(832\) 0 0
\(833\) 4.59024 0.159042
\(834\) 0 0
\(835\) 6.03531 0.208861
\(836\) 0 0
\(837\) 18.5811 0.642257
\(838\) 0 0
\(839\) −36.0845 −1.24578 −0.622888 0.782311i \(-0.714041\pi\)
−0.622888 + 0.782311i \(0.714041\pi\)
\(840\) 0 0
\(841\) 5.52843 0.190635
\(842\) 0 0
\(843\) −36.2439 −1.24831
\(844\) 0 0
\(845\) −23.0161 −0.791778
\(846\) 0 0
\(847\) −17.8130 −0.612064
\(848\) 0 0
\(849\) −47.5098 −1.63053
\(850\) 0 0
\(851\) −13.0102 −0.445984
\(852\) 0 0
\(853\) 51.4019 1.75997 0.879984 0.475003i \(-0.157553\pi\)
0.879984 + 0.475003i \(0.157553\pi\)
\(854\) 0 0
\(855\) −4.89441 −0.167385
\(856\) 0 0
\(857\) 30.1662 1.03046 0.515229 0.857052i \(-0.327707\pi\)
0.515229 + 0.857052i \(0.327707\pi\)
\(858\) 0 0
\(859\) −53.7376 −1.83350 −0.916752 0.399456i \(-0.869199\pi\)
−0.916752 + 0.399456i \(0.869199\pi\)
\(860\) 0 0
\(861\) 11.0195 0.375544
\(862\) 0 0
\(863\) −52.3451 −1.78185 −0.890924 0.454152i \(-0.849942\pi\)
−0.890924 + 0.454152i \(0.849942\pi\)
\(864\) 0 0
\(865\) −10.7699 −0.366187
\(866\) 0 0
\(867\) −1.58435 −0.0538073
\(868\) 0 0
\(869\) −5.76176 −0.195454
\(870\) 0 0
\(871\) 18.8012 0.637055
\(872\) 0 0
\(873\) 8.93301 0.302337
\(874\) 0 0
\(875\) −29.7604 −1.00609
\(876\) 0 0
\(877\) −55.9527 −1.88939 −0.944694 0.327952i \(-0.893642\pi\)
−0.944694 + 0.327952i \(0.893642\pi\)
\(878\) 0 0
\(879\) −17.6587 −0.595615
\(880\) 0 0
\(881\) 9.22755 0.310884 0.155442 0.987845i \(-0.450320\pi\)
0.155442 + 0.987845i \(0.450320\pi\)
\(882\) 0 0
\(883\) 2.84405 0.0957100 0.0478550 0.998854i \(-0.484761\pi\)
0.0478550 + 0.998854i \(0.484761\pi\)
\(884\) 0 0
\(885\) −4.06915 −0.136783
\(886\) 0 0
\(887\) −19.9105 −0.668529 −0.334265 0.942479i \(-0.608488\pi\)
−0.334265 + 0.942479i \(0.608488\pi\)
\(888\) 0 0
\(889\) −59.5979 −1.99885
\(890\) 0 0
\(891\) −17.5090 −0.586572
\(892\) 0 0
\(893\) −17.6259 −0.589827
\(894\) 0 0
\(895\) 9.78738 0.327156
\(896\) 0 0
\(897\) 9.92253 0.331304
\(898\) 0 0
\(899\) 19.7471 0.658603
\(900\) 0 0
\(901\) 4.71284 0.157008
\(902\) 0 0
\(903\) 25.0398 0.833272
\(904\) 0 0
\(905\) −37.5538 −1.24833
\(906\) 0 0
\(907\) −21.1675 −0.702856 −0.351428 0.936215i \(-0.614304\pi\)
−0.351428 + 0.936215i \(0.614304\pi\)
\(908\) 0 0
\(909\) 3.07630 0.102034
\(910\) 0 0
\(911\) 34.2961 1.13628 0.568141 0.822932i \(-0.307663\pi\)
0.568141 + 0.822932i \(0.307663\pi\)
\(912\) 0 0
\(913\) 24.6408 0.815490
\(914\) 0 0
\(915\) 25.7325 0.850691
\(916\) 0 0
\(917\) −37.2091 −1.22875
\(918\) 0 0
\(919\) −36.8275 −1.21483 −0.607414 0.794385i \(-0.707793\pi\)
−0.607414 + 0.794385i \(0.707793\pi\)
\(920\) 0 0
\(921\) 15.8676 0.522854
\(922\) 0 0
\(923\) −0.809873 −0.0266573
\(924\) 0 0
\(925\) 6.66444 0.219125
\(926\) 0 0
\(927\) 1.39663 0.0458713
\(928\) 0 0
\(929\) −17.4432 −0.572292 −0.286146 0.958186i \(-0.592374\pi\)
−0.286146 + 0.958186i \(0.592374\pi\)
\(930\) 0 0
\(931\) 17.8577 0.585261
\(932\) 0 0
\(933\) 2.92887 0.0958869
\(934\) 0 0
\(935\) 6.16815 0.201720
\(936\) 0 0
\(937\) −35.1147 −1.14715 −0.573574 0.819154i \(-0.694444\pi\)
−0.573574 + 0.819154i \(0.694444\pi\)
\(938\) 0 0
\(939\) 38.5638 1.25848
\(940\) 0 0
\(941\) −61.2631 −1.99712 −0.998560 0.0536430i \(-0.982917\pi\)
−0.998560 + 0.0536430i \(0.982917\pi\)
\(942\) 0 0
\(943\) 6.36687 0.207334
\(944\) 0 0
\(945\) 48.3455 1.57268
\(946\) 0 0
\(947\) −8.78619 −0.285513 −0.142756 0.989758i \(-0.545597\pi\)
−0.142756 + 0.989758i \(0.545597\pi\)
\(948\) 0 0
\(949\) −0.0256817 −0.000833664 0
\(950\) 0 0
\(951\) −23.0356 −0.746980
\(952\) 0 0
\(953\) 47.3483 1.53376 0.766880 0.641790i \(-0.221808\pi\)
0.766880 + 0.641790i \(0.221808\pi\)
\(954\) 0 0
\(955\) 55.2715 1.78854
\(956\) 0 0
\(957\) −22.3584 −0.722744
\(958\) 0 0
\(959\) −54.1074 −1.74722
\(960\) 0 0
\(961\) −19.7065 −0.635692
\(962\) 0 0
\(963\) −2.00664 −0.0646631
\(964\) 0 0
\(965\) −10.4394 −0.336057
\(966\) 0 0
\(967\) 25.4753 0.819229 0.409615 0.912259i \(-0.365663\pi\)
0.409615 + 0.912259i \(0.365663\pi\)
\(968\) 0 0
\(969\) −6.16367 −0.198006
\(970\) 0 0
\(971\) −35.9071 −1.15231 −0.576157 0.817339i \(-0.695448\pi\)
−0.576157 + 0.817339i \(0.695448\pi\)
\(972\) 0 0
\(973\) 10.3099 0.330521
\(974\) 0 0
\(975\) −5.08279 −0.162779
\(976\) 0 0
\(977\) 6.56433 0.210012 0.105006 0.994472i \(-0.466514\pi\)
0.105006 + 0.994472i \(0.466514\pi\)
\(978\) 0 0
\(979\) 31.6116 1.01031
\(980\) 0 0
\(981\) 8.30303 0.265095
\(982\) 0 0
\(983\) 7.05582 0.225046 0.112523 0.993649i \(-0.464107\pi\)
0.112523 + 0.993649i \(0.464107\pi\)
\(984\) 0 0
\(985\) −6.33035 −0.201702
\(986\) 0 0
\(987\) 24.4376 0.777857
\(988\) 0 0
\(989\) 14.4675 0.460040
\(990\) 0 0
\(991\) −25.2437 −0.801894 −0.400947 0.916101i \(-0.631319\pi\)
−0.400947 + 0.916101i \(0.631319\pi\)
\(992\) 0 0
\(993\) −52.0698 −1.65238
\(994\) 0 0
\(995\) 60.5137 1.91841
\(996\) 0 0
\(997\) 43.9829 1.39295 0.696477 0.717580i \(-0.254750\pi\)
0.696477 + 0.717580i \(0.254750\pi\)
\(998\) 0 0
\(999\) 23.0824 0.730294
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 4012.2.a.j.1.6 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.j.1.6 21 1.1 even 1 trivial