Properties

Label 8048.2.a.y.1.12
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, 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("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: no (minimal twist has level 4024)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.07783 q^{3} -1.09164 q^{5} -2.08785 q^{7} -1.83829 q^{9} +O(q^{10})\) \(q-1.07783 q^{3} -1.09164 q^{5} -2.08785 q^{7} -1.83829 q^{9} +2.69673 q^{11} +0.315489 q^{13} +1.17660 q^{15} +3.53047 q^{17} -7.12979 q^{19} +2.25034 q^{21} -3.16379 q^{23} -3.80832 q^{25} +5.21484 q^{27} +5.04684 q^{29} +3.99293 q^{31} -2.90661 q^{33} +2.27918 q^{35} -1.32100 q^{37} -0.340043 q^{39} -0.528913 q^{41} -11.6427 q^{43} +2.00675 q^{45} -1.96128 q^{47} -2.64090 q^{49} -3.80524 q^{51} +2.54974 q^{53} -2.94386 q^{55} +7.68467 q^{57} +4.20847 q^{59} -14.7620 q^{61} +3.83806 q^{63} -0.344401 q^{65} +6.34098 q^{67} +3.41002 q^{69} -10.3257 q^{71} +11.4248 q^{73} +4.10471 q^{75} -5.63036 q^{77} -12.9631 q^{79} -0.105823 q^{81} -13.8083 q^{83} -3.85401 q^{85} -5.43962 q^{87} +10.2851 q^{89} -0.658693 q^{91} -4.30369 q^{93} +7.78316 q^{95} +5.53931 q^{97} -4.95737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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
\(3\) −1.07783 −0.622284 −0.311142 0.950364i \(-0.600711\pi\)
−0.311142 + 0.950364i \(0.600711\pi\)
\(4\) 0 0
\(5\) −1.09164 −0.488197 −0.244098 0.969750i \(-0.578492\pi\)
−0.244098 + 0.969750i \(0.578492\pi\)
\(6\) 0 0
\(7\) −2.08785 −0.789131 −0.394566 0.918868i \(-0.629105\pi\)
−0.394566 + 0.918868i \(0.629105\pi\)
\(8\) 0 0
\(9\) −1.83829 −0.612763
\(10\) 0 0
\(11\) 2.69673 0.813095 0.406548 0.913630i \(-0.366733\pi\)
0.406548 + 0.913630i \(0.366733\pi\)
\(12\) 0 0
\(13\) 0.315489 0.0875010 0.0437505 0.999042i \(-0.486069\pi\)
0.0437505 + 0.999042i \(0.486069\pi\)
\(14\) 0 0
\(15\) 1.17660 0.303797
\(16\) 0 0
\(17\) 3.53047 0.856265 0.428132 0.903716i \(-0.359172\pi\)
0.428132 + 0.903716i \(0.359172\pi\)
\(18\) 0 0
\(19\) −7.12979 −1.63569 −0.817843 0.575442i \(-0.804830\pi\)
−0.817843 + 0.575442i \(0.804830\pi\)
\(20\) 0 0
\(21\) 2.25034 0.491064
\(22\) 0 0
\(23\) −3.16379 −0.659697 −0.329848 0.944034i \(-0.606998\pi\)
−0.329848 + 0.944034i \(0.606998\pi\)
\(24\) 0 0
\(25\) −3.80832 −0.761664
\(26\) 0 0
\(27\) 5.21484 1.00360
\(28\) 0 0
\(29\) 5.04684 0.937175 0.468587 0.883417i \(-0.344763\pi\)
0.468587 + 0.883417i \(0.344763\pi\)
\(30\) 0 0
\(31\) 3.99293 0.717152 0.358576 0.933500i \(-0.383262\pi\)
0.358576 + 0.933500i \(0.383262\pi\)
\(32\) 0 0
\(33\) −2.90661 −0.505976
\(34\) 0 0
\(35\) 2.27918 0.385251
\(36\) 0 0
\(37\) −1.32100 −0.217171 −0.108586 0.994087i \(-0.534632\pi\)
−0.108586 + 0.994087i \(0.534632\pi\)
\(38\) 0 0
\(39\) −0.340043 −0.0544504
\(40\) 0 0
\(41\) −0.528913 −0.0826024 −0.0413012 0.999147i \(-0.513150\pi\)
−0.0413012 + 0.999147i \(0.513150\pi\)
\(42\) 0 0
\(43\) −11.6427 −1.77550 −0.887750 0.460326i \(-0.847733\pi\)
−0.887750 + 0.460326i \(0.847733\pi\)
\(44\) 0 0
\(45\) 2.00675 0.299149
\(46\) 0 0
\(47\) −1.96128 −0.286083 −0.143041 0.989717i \(-0.545688\pi\)
−0.143041 + 0.989717i \(0.545688\pi\)
\(48\) 0 0
\(49\) −2.64090 −0.377272
\(50\) 0 0
\(51\) −3.80524 −0.532840
\(52\) 0 0
\(53\) 2.54974 0.350234 0.175117 0.984548i \(-0.443970\pi\)
0.175117 + 0.984548i \(0.443970\pi\)
\(54\) 0 0
\(55\) −2.94386 −0.396950
\(56\) 0 0
\(57\) 7.68467 1.01786
\(58\) 0 0
\(59\) 4.20847 0.547896 0.273948 0.961745i \(-0.411670\pi\)
0.273948 + 0.961745i \(0.411670\pi\)
\(60\) 0 0
\(61\) −14.7620 −1.89009 −0.945043 0.326946i \(-0.893981\pi\)
−0.945043 + 0.326946i \(0.893981\pi\)
\(62\) 0 0
\(63\) 3.83806 0.483551
\(64\) 0 0
\(65\) −0.344401 −0.0427177
\(66\) 0 0
\(67\) 6.34098 0.774674 0.387337 0.921938i \(-0.373395\pi\)
0.387337 + 0.921938i \(0.373395\pi\)
\(68\) 0 0
\(69\) 3.41002 0.410518
\(70\) 0 0
\(71\) −10.3257 −1.22544 −0.612720 0.790300i \(-0.709925\pi\)
−0.612720 + 0.790300i \(0.709925\pi\)
\(72\) 0 0
\(73\) 11.4248 1.33717 0.668587 0.743634i \(-0.266899\pi\)
0.668587 + 0.743634i \(0.266899\pi\)
\(74\) 0 0
\(75\) 4.10471 0.473971
\(76\) 0 0
\(77\) −5.63036 −0.641639
\(78\) 0 0
\(79\) −12.9631 −1.45846 −0.729229 0.684270i \(-0.760121\pi\)
−0.729229 + 0.684270i \(0.760121\pi\)
\(80\) 0 0
\(81\) −0.105823 −0.0117581
\(82\) 0 0
\(83\) −13.8083 −1.51566 −0.757831 0.652451i \(-0.773741\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(84\) 0 0
\(85\) −3.85401 −0.418026
\(86\) 0 0
\(87\) −5.43962 −0.583188
\(88\) 0 0
\(89\) 10.2851 1.09022 0.545110 0.838365i \(-0.316488\pi\)
0.545110 + 0.838365i \(0.316488\pi\)
\(90\) 0 0
\(91\) −0.658693 −0.0690498
\(92\) 0 0
\(93\) −4.30369 −0.446272
\(94\) 0 0
\(95\) 7.78316 0.798536
\(96\) 0 0
\(97\) 5.53931 0.562432 0.281216 0.959644i \(-0.409262\pi\)
0.281216 + 0.959644i \(0.409262\pi\)
\(98\) 0 0
\(99\) −4.95737 −0.498235
\(100\) 0 0
\(101\) 11.3127 1.12565 0.562825 0.826576i \(-0.309714\pi\)
0.562825 + 0.826576i \(0.309714\pi\)
\(102\) 0 0
\(103\) 10.3946 1.02421 0.512106 0.858922i \(-0.328865\pi\)
0.512106 + 0.858922i \(0.328865\pi\)
\(104\) 0 0
\(105\) −2.45656 −0.239736
\(106\) 0 0
\(107\) −13.4790 −1.30307 −0.651533 0.758620i \(-0.725874\pi\)
−0.651533 + 0.758620i \(0.725874\pi\)
\(108\) 0 0
\(109\) −4.03302 −0.386293 −0.193146 0.981170i \(-0.561869\pi\)
−0.193146 + 0.981170i \(0.561869\pi\)
\(110\) 0 0
\(111\) 1.42381 0.135142
\(112\) 0 0
\(113\) 8.93417 0.840456 0.420228 0.907419i \(-0.361950\pi\)
0.420228 + 0.907419i \(0.361950\pi\)
\(114\) 0 0
\(115\) 3.45373 0.322062
\(116\) 0 0
\(117\) −0.579960 −0.0536174
\(118\) 0 0
\(119\) −7.37108 −0.675706
\(120\) 0 0
\(121\) −3.72764 −0.338876
\(122\) 0 0
\(123\) 0.570077 0.0514021
\(124\) 0 0
\(125\) 9.61552 0.860038
\(126\) 0 0
\(127\) −16.2938 −1.44584 −0.722920 0.690932i \(-0.757201\pi\)
−0.722920 + 0.690932i \(0.757201\pi\)
\(128\) 0 0
\(129\) 12.5488 1.10486
\(130\) 0 0
\(131\) −0.452067 −0.0394972 −0.0197486 0.999805i \(-0.506287\pi\)
−0.0197486 + 0.999805i \(0.506287\pi\)
\(132\) 0 0
\(133\) 14.8859 1.29077
\(134\) 0 0
\(135\) −5.69273 −0.489952
\(136\) 0 0
\(137\) −0.0610484 −0.00521572 −0.00260786 0.999997i \(-0.500830\pi\)
−0.00260786 + 0.999997i \(0.500830\pi\)
\(138\) 0 0
\(139\) −0.756749 −0.0641866 −0.0320933 0.999485i \(-0.510217\pi\)
−0.0320933 + 0.999485i \(0.510217\pi\)
\(140\) 0 0
\(141\) 2.11392 0.178024
\(142\) 0 0
\(143\) 0.850790 0.0711466
\(144\) 0 0
\(145\) −5.50934 −0.457526
\(146\) 0 0
\(147\) 2.84643 0.234770
\(148\) 0 0
\(149\) 14.5148 1.18910 0.594549 0.804060i \(-0.297331\pi\)
0.594549 + 0.804060i \(0.297331\pi\)
\(150\) 0 0
\(151\) 0.746583 0.0607561 0.0303780 0.999538i \(-0.490329\pi\)
0.0303780 + 0.999538i \(0.490329\pi\)
\(152\) 0 0
\(153\) −6.49003 −0.524688
\(154\) 0 0
\(155\) −4.35885 −0.350111
\(156\) 0 0
\(157\) −7.21886 −0.576128 −0.288064 0.957611i \(-0.593012\pi\)
−0.288064 + 0.957611i \(0.593012\pi\)
\(158\) 0 0
\(159\) −2.74818 −0.217945
\(160\) 0 0
\(161\) 6.60551 0.520587
\(162\) 0 0
\(163\) −22.5224 −1.76409 −0.882045 0.471166i \(-0.843833\pi\)
−0.882045 + 0.471166i \(0.843833\pi\)
\(164\) 0 0
\(165\) 3.17297 0.247016
\(166\) 0 0
\(167\) 18.7352 1.44977 0.724887 0.688867i \(-0.241892\pi\)
0.724887 + 0.688867i \(0.241892\pi\)
\(168\) 0 0
\(169\) −12.9005 −0.992344
\(170\) 0 0
\(171\) 13.1066 1.00229
\(172\) 0 0
\(173\) −4.74517 −0.360768 −0.180384 0.983596i \(-0.557734\pi\)
−0.180384 + 0.983596i \(0.557734\pi\)
\(174\) 0 0
\(175\) 7.95119 0.601053
\(176\) 0 0
\(177\) −4.53600 −0.340947
\(178\) 0 0
\(179\) 24.6303 1.84095 0.920476 0.390799i \(-0.127801\pi\)
0.920476 + 0.390799i \(0.127801\pi\)
\(180\) 0 0
\(181\) 20.6193 1.53262 0.766309 0.642472i \(-0.222091\pi\)
0.766309 + 0.642472i \(0.222091\pi\)
\(182\) 0 0
\(183\) 15.9109 1.17617
\(184\) 0 0
\(185\) 1.44206 0.106022
\(186\) 0 0
\(187\) 9.52073 0.696225
\(188\) 0 0
\(189\) −10.8878 −0.791969
\(190\) 0 0
\(191\) −9.84208 −0.712148 −0.356074 0.934458i \(-0.615885\pi\)
−0.356074 + 0.934458i \(0.615885\pi\)
\(192\) 0 0
\(193\) −20.8828 −1.50318 −0.751589 0.659632i \(-0.770712\pi\)
−0.751589 + 0.659632i \(0.770712\pi\)
\(194\) 0 0
\(195\) 0.371204 0.0265825
\(196\) 0 0
\(197\) 10.6869 0.761412 0.380706 0.924696i \(-0.375681\pi\)
0.380706 + 0.924696i \(0.375681\pi\)
\(198\) 0 0
\(199\) 7.79192 0.552355 0.276177 0.961107i \(-0.410932\pi\)
0.276177 + 0.961107i \(0.410932\pi\)
\(200\) 0 0
\(201\) −6.83448 −0.482067
\(202\) 0 0
\(203\) −10.5370 −0.739554
\(204\) 0 0
\(205\) 0.577383 0.0403262
\(206\) 0 0
\(207\) 5.81597 0.404238
\(208\) 0 0
\(209\) −19.2271 −1.32997
\(210\) 0 0
\(211\) 27.2693 1.87730 0.938649 0.344874i \(-0.112078\pi\)
0.938649 + 0.344874i \(0.112078\pi\)
\(212\) 0 0
\(213\) 11.1294 0.762571
\(214\) 0 0
\(215\) 12.7097 0.866793
\(216\) 0 0
\(217\) −8.33663 −0.565927
\(218\) 0 0
\(219\) −12.3140 −0.832102
\(220\) 0 0
\(221\) 1.11383 0.0749240
\(222\) 0 0
\(223\) −16.4941 −1.10453 −0.552265 0.833669i \(-0.686236\pi\)
−0.552265 + 0.833669i \(0.686236\pi\)
\(224\) 0 0
\(225\) 7.00080 0.466720
\(226\) 0 0
\(227\) −0.235622 −0.0156388 −0.00781940 0.999969i \(-0.502489\pi\)
−0.00781940 + 0.999969i \(0.502489\pi\)
\(228\) 0 0
\(229\) 19.6417 1.29796 0.648981 0.760805i \(-0.275195\pi\)
0.648981 + 0.760805i \(0.275195\pi\)
\(230\) 0 0
\(231\) 6.06855 0.399281
\(232\) 0 0
\(233\) 24.4561 1.60217 0.801087 0.598548i \(-0.204255\pi\)
0.801087 + 0.598548i \(0.204255\pi\)
\(234\) 0 0
\(235\) 2.14102 0.139665
\(236\) 0 0
\(237\) 13.9719 0.907574
\(238\) 0 0
\(239\) 11.5322 0.745956 0.372978 0.927840i \(-0.378337\pi\)
0.372978 + 0.927840i \(0.378337\pi\)
\(240\) 0 0
\(241\) −23.2970 −1.50069 −0.750345 0.661047i \(-0.770112\pi\)
−0.750345 + 0.661047i \(0.770112\pi\)
\(242\) 0 0
\(243\) −15.5305 −0.996279
\(244\) 0 0
\(245\) 2.88291 0.184183
\(246\) 0 0
\(247\) −2.24937 −0.143124
\(248\) 0 0
\(249\) 14.8830 0.943172
\(250\) 0 0
\(251\) −17.6645 −1.11497 −0.557487 0.830186i \(-0.688234\pi\)
−0.557487 + 0.830186i \(0.688234\pi\)
\(252\) 0 0
\(253\) −8.53190 −0.536396
\(254\) 0 0
\(255\) 4.15395 0.260130
\(256\) 0 0
\(257\) 27.1780 1.69532 0.847659 0.530542i \(-0.178012\pi\)
0.847659 + 0.530542i \(0.178012\pi\)
\(258\) 0 0
\(259\) 2.75805 0.171377
\(260\) 0 0
\(261\) −9.27755 −0.574266
\(262\) 0 0
\(263\) 9.44008 0.582100 0.291050 0.956708i \(-0.405995\pi\)
0.291050 + 0.956708i \(0.405995\pi\)
\(264\) 0 0
\(265\) −2.78341 −0.170983
\(266\) 0 0
\(267\) −11.0856 −0.678426
\(268\) 0 0
\(269\) 18.6112 1.13474 0.567371 0.823462i \(-0.307961\pi\)
0.567371 + 0.823462i \(0.307961\pi\)
\(270\) 0 0
\(271\) −17.7573 −1.07868 −0.539341 0.842088i \(-0.681327\pi\)
−0.539341 + 0.842088i \(0.681327\pi\)
\(272\) 0 0
\(273\) 0.709957 0.0429685
\(274\) 0 0
\(275\) −10.2700 −0.619305
\(276\) 0 0
\(277\) 21.4657 1.28975 0.644874 0.764289i \(-0.276910\pi\)
0.644874 + 0.764289i \(0.276910\pi\)
\(278\) 0 0
\(279\) −7.34017 −0.439445
\(280\) 0 0
\(281\) −26.2281 −1.56464 −0.782318 0.622880i \(-0.785963\pi\)
−0.782318 + 0.622880i \(0.785963\pi\)
\(282\) 0 0
\(283\) 15.7796 0.938000 0.469000 0.883198i \(-0.344614\pi\)
0.469000 + 0.883198i \(0.344614\pi\)
\(284\) 0 0
\(285\) −8.38890 −0.496916
\(286\) 0 0
\(287\) 1.10429 0.0651841
\(288\) 0 0
\(289\) −4.53578 −0.266810
\(290\) 0 0
\(291\) −5.97042 −0.349992
\(292\) 0 0
\(293\) −2.48083 −0.144932 −0.0724659 0.997371i \(-0.523087\pi\)
−0.0724659 + 0.997371i \(0.523087\pi\)
\(294\) 0 0
\(295\) −4.59414 −0.267481
\(296\) 0 0
\(297\) 14.0630 0.816019
\(298\) 0 0
\(299\) −0.998143 −0.0577241
\(300\) 0 0
\(301\) 24.3082 1.40110
\(302\) 0 0
\(303\) −12.1931 −0.700474
\(304\) 0 0
\(305\) 16.1148 0.922734
\(306\) 0 0
\(307\) 27.1739 1.55090 0.775448 0.631411i \(-0.217524\pi\)
0.775448 + 0.631411i \(0.217524\pi\)
\(308\) 0 0
\(309\) −11.2036 −0.637350
\(310\) 0 0
\(311\) 7.61285 0.431685 0.215843 0.976428i \(-0.430750\pi\)
0.215843 + 0.976428i \(0.430750\pi\)
\(312\) 0 0
\(313\) −16.4041 −0.927215 −0.463608 0.886041i \(-0.653445\pi\)
−0.463608 + 0.886041i \(0.653445\pi\)
\(314\) 0 0
\(315\) −4.18979 −0.236068
\(316\) 0 0
\(317\) 17.3242 0.973025 0.486512 0.873674i \(-0.338269\pi\)
0.486512 + 0.873674i \(0.338269\pi\)
\(318\) 0 0
\(319\) 13.6100 0.762012
\(320\) 0 0
\(321\) 14.5281 0.810877
\(322\) 0 0
\(323\) −25.1715 −1.40058
\(324\) 0 0
\(325\) −1.20148 −0.0666463
\(326\) 0 0
\(327\) 4.34689 0.240384
\(328\) 0 0
\(329\) 4.09486 0.225757
\(330\) 0 0
\(331\) 31.7355 1.74434 0.872170 0.489203i \(-0.162712\pi\)
0.872170 + 0.489203i \(0.162712\pi\)
\(332\) 0 0
\(333\) 2.42838 0.133075
\(334\) 0 0
\(335\) −6.92207 −0.378193
\(336\) 0 0
\(337\) −1.54463 −0.0841412 −0.0420706 0.999115i \(-0.513395\pi\)
−0.0420706 + 0.999115i \(0.513395\pi\)
\(338\) 0 0
\(339\) −9.62949 −0.523002
\(340\) 0 0
\(341\) 10.7679 0.583113
\(342\) 0 0
\(343\) 20.1287 1.08685
\(344\) 0 0
\(345\) −3.72252 −0.200414
\(346\) 0 0
\(347\) −0.697526 −0.0374452 −0.0187226 0.999825i \(-0.505960\pi\)
−0.0187226 + 0.999825i \(0.505960\pi\)
\(348\) 0 0
\(349\) 23.1003 1.23653 0.618265 0.785970i \(-0.287836\pi\)
0.618265 + 0.785970i \(0.287836\pi\)
\(350\) 0 0
\(351\) 1.64522 0.0878156
\(352\) 0 0
\(353\) 8.46525 0.450560 0.225280 0.974294i \(-0.427670\pi\)
0.225280 + 0.974294i \(0.427670\pi\)
\(354\) 0 0
\(355\) 11.2720 0.598256
\(356\) 0 0
\(357\) 7.94474 0.420480
\(358\) 0 0
\(359\) 19.9446 1.05264 0.526319 0.850287i \(-0.323572\pi\)
0.526319 + 0.850287i \(0.323572\pi\)
\(360\) 0 0
\(361\) 31.8339 1.67547
\(362\) 0 0
\(363\) 4.01775 0.210877
\(364\) 0 0
\(365\) −12.4718 −0.652804
\(366\) 0 0
\(367\) 11.0930 0.579052 0.289526 0.957170i \(-0.406502\pi\)
0.289526 + 0.957170i \(0.406502\pi\)
\(368\) 0 0
\(369\) 0.972296 0.0506157
\(370\) 0 0
\(371\) −5.32347 −0.276381
\(372\) 0 0
\(373\) −4.19976 −0.217455 −0.108728 0.994072i \(-0.534678\pi\)
−0.108728 + 0.994072i \(0.534678\pi\)
\(374\) 0 0
\(375\) −10.3639 −0.535188
\(376\) 0 0
\(377\) 1.59222 0.0820037
\(378\) 0 0
\(379\) 5.71111 0.293360 0.146680 0.989184i \(-0.453141\pi\)
0.146680 + 0.989184i \(0.453141\pi\)
\(380\) 0 0
\(381\) 17.5619 0.899723
\(382\) 0 0
\(383\) 17.3126 0.884633 0.442316 0.896859i \(-0.354157\pi\)
0.442316 + 0.896859i \(0.354157\pi\)
\(384\) 0 0
\(385\) 6.14633 0.313246
\(386\) 0 0
\(387\) 21.4027 1.08796
\(388\) 0 0
\(389\) 27.8613 1.41262 0.706311 0.707901i \(-0.250358\pi\)
0.706311 + 0.707901i \(0.250358\pi\)
\(390\) 0 0
\(391\) −11.1697 −0.564875
\(392\) 0 0
\(393\) 0.487249 0.0245785
\(394\) 0 0
\(395\) 14.1510 0.712014
\(396\) 0 0
\(397\) −30.3468 −1.52306 −0.761531 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(398\) 0 0
\(399\) −16.0444 −0.803225
\(400\) 0 0
\(401\) 21.6813 1.08271 0.541355 0.840794i \(-0.317911\pi\)
0.541355 + 0.840794i \(0.317911\pi\)
\(402\) 0 0
\(403\) 1.25973 0.0627515
\(404\) 0 0
\(405\) 0.115521 0.00574026
\(406\) 0 0
\(407\) −3.56239 −0.176581
\(408\) 0 0
\(409\) −20.9451 −1.03567 −0.517834 0.855481i \(-0.673261\pi\)
−0.517834 + 0.855481i \(0.673261\pi\)
\(410\) 0 0
\(411\) 0.0657996 0.00324566
\(412\) 0 0
\(413\) −8.78663 −0.432362
\(414\) 0 0
\(415\) 15.0737 0.739941
\(416\) 0 0
\(417\) 0.815644 0.0399423
\(418\) 0 0
\(419\) 2.85144 0.139302 0.0696509 0.997571i \(-0.477811\pi\)
0.0696509 + 0.997571i \(0.477811\pi\)
\(420\) 0 0
\(421\) 23.1212 1.12686 0.563430 0.826164i \(-0.309481\pi\)
0.563430 + 0.826164i \(0.309481\pi\)
\(422\) 0 0
\(423\) 3.60541 0.175301
\(424\) 0 0
\(425\) −13.4452 −0.652186
\(426\) 0 0
\(427\) 30.8209 1.49153
\(428\) 0 0
\(429\) −0.917004 −0.0442734
\(430\) 0 0
\(431\) −14.2031 −0.684138 −0.342069 0.939675i \(-0.611128\pi\)
−0.342069 + 0.939675i \(0.611128\pi\)
\(432\) 0 0
\(433\) 24.5386 1.17925 0.589624 0.807678i \(-0.299276\pi\)
0.589624 + 0.807678i \(0.299276\pi\)
\(434\) 0 0
\(435\) 5.93811 0.284711
\(436\) 0 0
\(437\) 22.5572 1.07906
\(438\) 0 0
\(439\) 26.3497 1.25760 0.628800 0.777567i \(-0.283546\pi\)
0.628800 + 0.777567i \(0.283546\pi\)
\(440\) 0 0
\(441\) 4.85474 0.231178
\(442\) 0 0
\(443\) 18.0001 0.855209 0.427604 0.903966i \(-0.359358\pi\)
0.427604 + 0.903966i \(0.359358\pi\)
\(444\) 0 0
\(445\) −11.2276 −0.532242
\(446\) 0 0
\(447\) −15.6444 −0.739956
\(448\) 0 0
\(449\) −20.4589 −0.965517 −0.482758 0.875754i \(-0.660365\pi\)
−0.482758 + 0.875754i \(0.660365\pi\)
\(450\) 0 0
\(451\) −1.42634 −0.0671636
\(452\) 0 0
\(453\) −0.804687 −0.0378075
\(454\) 0 0
\(455\) 0.719056 0.0337099
\(456\) 0 0
\(457\) 4.03988 0.188978 0.0944888 0.995526i \(-0.469878\pi\)
0.0944888 + 0.995526i \(0.469878\pi\)
\(458\) 0 0
\(459\) 18.4108 0.859344
\(460\) 0 0
\(461\) −31.2568 −1.45577 −0.727887 0.685697i \(-0.759498\pi\)
−0.727887 + 0.685697i \(0.759498\pi\)
\(462\) 0 0
\(463\) −27.7833 −1.29120 −0.645600 0.763676i \(-0.723393\pi\)
−0.645600 + 0.763676i \(0.723393\pi\)
\(464\) 0 0
\(465\) 4.69809 0.217868
\(466\) 0 0
\(467\) −9.91012 −0.458586 −0.229293 0.973357i \(-0.573641\pi\)
−0.229293 + 0.973357i \(0.573641\pi\)
\(468\) 0 0
\(469\) −13.2390 −0.611320
\(470\) 0 0
\(471\) 7.78069 0.358515
\(472\) 0 0
\(473\) −31.3973 −1.44365
\(474\) 0 0
\(475\) 27.1525 1.24584
\(476\) 0 0
\(477\) −4.68717 −0.214611
\(478\) 0 0
\(479\) −17.8938 −0.817589 −0.408795 0.912626i \(-0.634051\pi\)
−0.408795 + 0.912626i \(0.634051\pi\)
\(480\) 0 0
\(481\) −0.416762 −0.0190027
\(482\) 0 0
\(483\) −7.11960 −0.323953
\(484\) 0 0
\(485\) −6.04694 −0.274577
\(486\) 0 0
\(487\) 29.2727 1.32647 0.663237 0.748409i \(-0.269182\pi\)
0.663237 + 0.748409i \(0.269182\pi\)
\(488\) 0 0
\(489\) 24.2752 1.09776
\(490\) 0 0
\(491\) 36.3888 1.64221 0.821103 0.570780i \(-0.193359\pi\)
0.821103 + 0.570780i \(0.193359\pi\)
\(492\) 0 0
\(493\) 17.8177 0.802470
\(494\) 0 0
\(495\) 5.41167 0.243236
\(496\) 0 0
\(497\) 21.5586 0.967033
\(498\) 0 0
\(499\) −19.5482 −0.875100 −0.437550 0.899194i \(-0.644154\pi\)
−0.437550 + 0.899194i \(0.644154\pi\)
\(500\) 0 0
\(501\) −20.1933 −0.902171
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −12.3494 −0.549539
\(506\) 0 0
\(507\) 13.9045 0.617519
\(508\) 0 0
\(509\) −35.6532 −1.58030 −0.790149 0.612915i \(-0.789997\pi\)
−0.790149 + 0.612915i \(0.789997\pi\)
\(510\) 0 0
\(511\) −23.8533 −1.05521
\(512\) 0 0
\(513\) −37.1807 −1.64157
\(514\) 0 0
\(515\) −11.3472 −0.500017
\(516\) 0 0
\(517\) −5.28905 −0.232612
\(518\) 0 0
\(519\) 5.11447 0.224500
\(520\) 0 0
\(521\) −19.7841 −0.866755 −0.433378 0.901212i \(-0.642678\pi\)
−0.433378 + 0.901212i \(0.642678\pi\)
\(522\) 0 0
\(523\) −30.9566 −1.35364 −0.676819 0.736149i \(-0.736642\pi\)
−0.676819 + 0.736149i \(0.736642\pi\)
\(524\) 0 0
\(525\) −8.57000 −0.374025
\(526\) 0 0
\(527\) 14.0969 0.614072
\(528\) 0 0
\(529\) −12.9904 −0.564800
\(530\) 0 0
\(531\) −7.73638 −0.335730
\(532\) 0 0
\(533\) −0.166866 −0.00722779
\(534\) 0 0
\(535\) 14.7143 0.636153
\(536\) 0 0
\(537\) −26.5472 −1.14559
\(538\) 0 0
\(539\) −7.12180 −0.306758
\(540\) 0 0
\(541\) −18.7017 −0.804050 −0.402025 0.915629i \(-0.631694\pi\)
−0.402025 + 0.915629i \(0.631694\pi\)
\(542\) 0 0
\(543\) −22.2240 −0.953723
\(544\) 0 0
\(545\) 4.40260 0.188587
\(546\) 0 0
\(547\) 19.5786 0.837120 0.418560 0.908189i \(-0.362535\pi\)
0.418560 + 0.908189i \(0.362535\pi\)
\(548\) 0 0
\(549\) 27.1369 1.15818
\(550\) 0 0
\(551\) −35.9829 −1.53292
\(552\) 0 0
\(553\) 27.0649 1.15092
\(554\) 0 0
\(555\) −1.55429 −0.0659760
\(556\) 0 0
\(557\) 26.7877 1.13503 0.567516 0.823362i \(-0.307904\pi\)
0.567516 + 0.823362i \(0.307904\pi\)
\(558\) 0 0
\(559\) −3.67316 −0.155358
\(560\) 0 0
\(561\) −10.2617 −0.433249
\(562\) 0 0
\(563\) −37.8793 −1.59642 −0.798211 0.602378i \(-0.794220\pi\)
−0.798211 + 0.602378i \(0.794220\pi\)
\(564\) 0 0
\(565\) −9.75290 −0.410308
\(566\) 0 0
\(567\) 0.220942 0.00927868
\(568\) 0 0
\(569\) −33.9553 −1.42348 −0.711740 0.702443i \(-0.752093\pi\)
−0.711740 + 0.702443i \(0.752093\pi\)
\(570\) 0 0
\(571\) −20.8686 −0.873324 −0.436662 0.899626i \(-0.643839\pi\)
−0.436662 + 0.899626i \(0.643839\pi\)
\(572\) 0 0
\(573\) 10.6081 0.443158
\(574\) 0 0
\(575\) 12.0487 0.502467
\(576\) 0 0
\(577\) 36.3441 1.51303 0.756513 0.653978i \(-0.226901\pi\)
0.756513 + 0.653978i \(0.226901\pi\)
\(578\) 0 0
\(579\) 22.5081 0.935403
\(580\) 0 0
\(581\) 28.8297 1.19606
\(582\) 0 0
\(583\) 6.87598 0.284774
\(584\) 0 0
\(585\) 0.633108 0.0261758
\(586\) 0 0
\(587\) 9.30158 0.383917 0.191959 0.981403i \(-0.438516\pi\)
0.191959 + 0.981403i \(0.438516\pi\)
\(588\) 0 0
\(589\) −28.4688 −1.17304
\(590\) 0 0
\(591\) −11.5187 −0.473814
\(592\) 0 0
\(593\) −31.0040 −1.27318 −0.636592 0.771201i \(-0.719656\pi\)
−0.636592 + 0.771201i \(0.719656\pi\)
\(594\) 0 0
\(595\) 8.04657 0.329877
\(596\) 0 0
\(597\) −8.39834 −0.343721
\(598\) 0 0
\(599\) 41.3474 1.68941 0.844705 0.535231i \(-0.179776\pi\)
0.844705 + 0.535231i \(0.179776\pi\)
\(600\) 0 0
\(601\) −16.0347 −0.654071 −0.327035 0.945012i \(-0.606050\pi\)
−0.327035 + 0.945012i \(0.606050\pi\)
\(602\) 0 0
\(603\) −11.6566 −0.474692
\(604\) 0 0
\(605\) 4.06924 0.165438
\(606\) 0 0
\(607\) −23.6655 −0.960552 −0.480276 0.877117i \(-0.659463\pi\)
−0.480276 + 0.877117i \(0.659463\pi\)
\(608\) 0 0
\(609\) 11.3571 0.460212
\(610\) 0 0
\(611\) −0.618764 −0.0250325
\(612\) 0 0
\(613\) 31.4762 1.27131 0.635657 0.771972i \(-0.280729\pi\)
0.635657 + 0.771972i \(0.280729\pi\)
\(614\) 0 0
\(615\) −0.622319 −0.0250943
\(616\) 0 0
\(617\) 48.3544 1.94668 0.973338 0.229376i \(-0.0736685\pi\)
0.973338 + 0.229376i \(0.0736685\pi\)
\(618\) 0 0
\(619\) 15.4928 0.622709 0.311355 0.950294i \(-0.399217\pi\)
0.311355 + 0.950294i \(0.399217\pi\)
\(620\) 0 0
\(621\) −16.4987 −0.662069
\(622\) 0 0
\(623\) −21.4737 −0.860327
\(624\) 0 0
\(625\) 8.54491 0.341796
\(626\) 0 0
\(627\) 20.7235 0.827617
\(628\) 0 0
\(629\) −4.66376 −0.185956
\(630\) 0 0
\(631\) 15.5340 0.618400 0.309200 0.950997i \(-0.399939\pi\)
0.309200 + 0.950997i \(0.399939\pi\)
\(632\) 0 0
\(633\) −29.3916 −1.16821
\(634\) 0 0
\(635\) 17.7870 0.705854
\(636\) 0 0
\(637\) −0.833176 −0.0330116
\(638\) 0 0
\(639\) 18.9817 0.750905
\(640\) 0 0
\(641\) −28.2557 −1.11603 −0.558017 0.829829i \(-0.688438\pi\)
−0.558017 + 0.829829i \(0.688438\pi\)
\(642\) 0 0
\(643\) 35.9951 1.41951 0.709754 0.704449i \(-0.248806\pi\)
0.709754 + 0.704449i \(0.248806\pi\)
\(644\) 0 0
\(645\) −13.6988 −0.539391
\(646\) 0 0
\(647\) 15.1203 0.594441 0.297220 0.954809i \(-0.403940\pi\)
0.297220 + 0.954809i \(0.403940\pi\)
\(648\) 0 0
\(649\) 11.3491 0.445492
\(650\) 0 0
\(651\) 8.98544 0.352167
\(652\) 0 0
\(653\) 38.3334 1.50010 0.750052 0.661379i \(-0.230029\pi\)
0.750052 + 0.661379i \(0.230029\pi\)
\(654\) 0 0
\(655\) 0.493494 0.0192824
\(656\) 0 0
\(657\) −21.0021 −0.819371
\(658\) 0 0
\(659\) −15.8517 −0.617496 −0.308748 0.951144i \(-0.599910\pi\)
−0.308748 + 0.951144i \(0.599910\pi\)
\(660\) 0 0
\(661\) 7.53797 0.293193 0.146597 0.989196i \(-0.453168\pi\)
0.146597 + 0.989196i \(0.453168\pi\)
\(662\) 0 0
\(663\) −1.20051 −0.0466240
\(664\) 0 0
\(665\) −16.2500 −0.630150
\(666\) 0 0
\(667\) −15.9672 −0.618251
\(668\) 0 0
\(669\) 17.7778 0.687330
\(670\) 0 0
\(671\) −39.8093 −1.53682
\(672\) 0 0
\(673\) −5.74054 −0.221282 −0.110641 0.993860i \(-0.535290\pi\)
−0.110641 + 0.993860i \(0.535290\pi\)
\(674\) 0 0
\(675\) −19.8598 −0.764403
\(676\) 0 0
\(677\) 20.2200 0.777118 0.388559 0.921424i \(-0.372973\pi\)
0.388559 + 0.921424i \(0.372973\pi\)
\(678\) 0 0
\(679\) −11.5652 −0.443833
\(680\) 0 0
\(681\) 0.253960 0.00973177
\(682\) 0 0
\(683\) −16.1656 −0.618559 −0.309279 0.950971i \(-0.600088\pi\)
−0.309279 + 0.950971i \(0.600088\pi\)
\(684\) 0 0
\(685\) 0.0666430 0.00254630
\(686\) 0 0
\(687\) −21.1704 −0.807700
\(688\) 0 0
\(689\) 0.804417 0.0306458
\(690\) 0 0
\(691\) −23.7362 −0.902968 −0.451484 0.892279i \(-0.649105\pi\)
−0.451484 + 0.892279i \(0.649105\pi\)
\(692\) 0 0
\(693\) 10.3502 0.393173
\(694\) 0 0
\(695\) 0.826098 0.0313357
\(696\) 0 0
\(697\) −1.86731 −0.0707295
\(698\) 0 0
\(699\) −26.3595 −0.997007
\(700\) 0 0
\(701\) −38.1029 −1.43913 −0.719563 0.694427i \(-0.755658\pi\)
−0.719563 + 0.694427i \(0.755658\pi\)
\(702\) 0 0
\(703\) 9.41846 0.355224
\(704\) 0 0
\(705\) −2.30764 −0.0869109
\(706\) 0 0
\(707\) −23.6191 −0.888286
\(708\) 0 0
\(709\) 47.4945 1.78370 0.891848 0.452336i \(-0.149409\pi\)
0.891848 + 0.452336i \(0.149409\pi\)
\(710\) 0 0
\(711\) 23.8299 0.893689
\(712\) 0 0
\(713\) −12.6328 −0.473103
\(714\) 0 0
\(715\) −0.928756 −0.0347335
\(716\) 0 0
\(717\) −12.4297 −0.464196
\(718\) 0 0
\(719\) 42.3751 1.58033 0.790163 0.612897i \(-0.209996\pi\)
0.790163 + 0.612897i \(0.209996\pi\)
\(720\) 0 0
\(721\) −21.7024 −0.808238
\(722\) 0 0
\(723\) 25.1101 0.933854
\(724\) 0 0
\(725\) −19.2200 −0.713812
\(726\) 0 0
\(727\) −20.3476 −0.754651 −0.377326 0.926081i \(-0.623156\pi\)
−0.377326 + 0.926081i \(0.623156\pi\)
\(728\) 0 0
\(729\) 17.0566 0.631726
\(730\) 0 0
\(731\) −41.1043 −1.52030
\(732\) 0 0
\(733\) −21.2441 −0.784668 −0.392334 0.919823i \(-0.628332\pi\)
−0.392334 + 0.919823i \(0.628332\pi\)
\(734\) 0 0
\(735\) −3.10728 −0.114614
\(736\) 0 0
\(737\) 17.0999 0.629884
\(738\) 0 0
\(739\) 36.1913 1.33132 0.665659 0.746256i \(-0.268150\pi\)
0.665659 + 0.746256i \(0.268150\pi\)
\(740\) 0 0
\(741\) 2.42443 0.0890637
\(742\) 0 0
\(743\) 29.8356 1.09456 0.547281 0.836949i \(-0.315663\pi\)
0.547281 + 0.836949i \(0.315663\pi\)
\(744\) 0 0
\(745\) −15.8449 −0.580513
\(746\) 0 0
\(747\) 25.3837 0.928742
\(748\) 0 0
\(749\) 28.1421 1.02829
\(750\) 0 0
\(751\) 17.7038 0.646019 0.323010 0.946396i \(-0.395305\pi\)
0.323010 + 0.946396i \(0.395305\pi\)
\(752\) 0 0
\(753\) 19.0393 0.693829
\(754\) 0 0
\(755\) −0.815001 −0.0296609
\(756\) 0 0
\(757\) −0.818007 −0.0297310 −0.0148655 0.999890i \(-0.504732\pi\)
−0.0148655 + 0.999890i \(0.504732\pi\)
\(758\) 0 0
\(759\) 9.19591 0.333791
\(760\) 0 0
\(761\) −12.8127 −0.464459 −0.232229 0.972661i \(-0.574602\pi\)
−0.232229 + 0.972661i \(0.574602\pi\)
\(762\) 0 0
\(763\) 8.42032 0.304836
\(764\) 0 0
\(765\) 7.08478 0.256151
\(766\) 0 0
\(767\) 1.32773 0.0479414
\(768\) 0 0
\(769\) 1.17687 0.0424389 0.0212194 0.999775i \(-0.493245\pi\)
0.0212194 + 0.999775i \(0.493245\pi\)
\(770\) 0 0
\(771\) −29.2932 −1.05497
\(772\) 0 0
\(773\) 10.2684 0.369329 0.184664 0.982802i \(-0.440880\pi\)
0.184664 + 0.982802i \(0.440880\pi\)
\(774\) 0 0
\(775\) −15.2064 −0.546229
\(776\) 0 0
\(777\) −2.97270 −0.106645
\(778\) 0 0
\(779\) 3.77104 0.135111
\(780\) 0 0
\(781\) −27.8457 −0.996399
\(782\) 0 0
\(783\) 26.3185 0.940545
\(784\) 0 0
\(785\) 7.88041 0.281264
\(786\) 0 0
\(787\) −17.5512 −0.625633 −0.312816 0.949814i \(-0.601272\pi\)
−0.312816 + 0.949814i \(0.601272\pi\)
\(788\) 0 0
\(789\) −10.1748 −0.362231
\(790\) 0 0
\(791\) −18.6532 −0.663230
\(792\) 0 0
\(793\) −4.65727 −0.165384
\(794\) 0 0
\(795\) 3.00003 0.106400
\(796\) 0 0
\(797\) 20.6806 0.732546 0.366273 0.930507i \(-0.380634\pi\)
0.366273 + 0.930507i \(0.380634\pi\)
\(798\) 0 0
\(799\) −6.92425 −0.244962
\(800\) 0 0
\(801\) −18.9070 −0.668047
\(802\) 0 0
\(803\) 30.8097 1.08725
\(804\) 0 0
\(805\) −7.21085 −0.254149
\(806\) 0 0
\(807\) −20.0596 −0.706131
\(808\) 0 0
\(809\) −15.1849 −0.533871 −0.266936 0.963714i \(-0.586011\pi\)
−0.266936 + 0.963714i \(0.586011\pi\)
\(810\) 0 0
\(811\) 3.41204 0.119813 0.0599064 0.998204i \(-0.480920\pi\)
0.0599064 + 0.998204i \(0.480920\pi\)
\(812\) 0 0
\(813\) 19.1393 0.671246
\(814\) 0 0
\(815\) 24.5864 0.861222
\(816\) 0 0
\(817\) 83.0102 2.90416
\(818\) 0 0
\(819\) 1.21087 0.0423112
\(820\) 0 0
\(821\) 24.7846 0.864990 0.432495 0.901636i \(-0.357633\pi\)
0.432495 + 0.901636i \(0.357633\pi\)
\(822\) 0 0
\(823\) 27.9361 0.973793 0.486896 0.873460i \(-0.338129\pi\)
0.486896 + 0.873460i \(0.338129\pi\)
\(824\) 0 0
\(825\) 11.0693 0.385384
\(826\) 0 0
\(827\) 39.9686 1.38984 0.694922 0.719085i \(-0.255439\pi\)
0.694922 + 0.719085i \(0.255439\pi\)
\(828\) 0 0
\(829\) 8.18764 0.284368 0.142184 0.989840i \(-0.454587\pi\)
0.142184 + 0.989840i \(0.454587\pi\)
\(830\) 0 0
\(831\) −23.1363 −0.802589
\(832\) 0 0
\(833\) −9.32362 −0.323044
\(834\) 0 0
\(835\) −20.4521 −0.707775
\(836\) 0 0
\(837\) 20.8225 0.719731
\(838\) 0 0
\(839\) 34.9548 1.20678 0.603388 0.797448i \(-0.293817\pi\)
0.603388 + 0.797448i \(0.293817\pi\)
\(840\) 0 0
\(841\) −3.52940 −0.121703
\(842\) 0 0
\(843\) 28.2693 0.973647
\(844\) 0 0
\(845\) 14.0827 0.484459
\(846\) 0 0
\(847\) 7.78274 0.267418
\(848\) 0 0
\(849\) −17.0077 −0.583702
\(850\) 0 0
\(851\) 4.17938 0.143267
\(852\) 0 0
\(853\) 6.87862 0.235519 0.117760 0.993042i \(-0.462429\pi\)
0.117760 + 0.993042i \(0.462429\pi\)
\(854\) 0 0
\(855\) −14.3077 −0.489313
\(856\) 0 0
\(857\) −6.03666 −0.206208 −0.103104 0.994671i \(-0.532877\pi\)
−0.103104 + 0.994671i \(0.532877\pi\)
\(858\) 0 0
\(859\) 40.9708 1.39791 0.698953 0.715168i \(-0.253650\pi\)
0.698953 + 0.715168i \(0.253650\pi\)
\(860\) 0 0
\(861\) −1.19023 −0.0405630
\(862\) 0 0
\(863\) 37.8792 1.28942 0.644712 0.764426i \(-0.276977\pi\)
0.644712 + 0.764426i \(0.276977\pi\)
\(864\) 0 0
\(865\) 5.18002 0.176126
\(866\) 0 0
\(867\) 4.88878 0.166032
\(868\) 0 0
\(869\) −34.9579 −1.18586
\(870\) 0 0
\(871\) 2.00051 0.0677847
\(872\) 0 0
\(873\) −10.1829 −0.344638
\(874\) 0 0
\(875\) −20.0757 −0.678683
\(876\) 0 0
\(877\) −0.111887 −0.00377816 −0.00188908 0.999998i \(-0.500601\pi\)
−0.00188908 + 0.999998i \(0.500601\pi\)
\(878\) 0 0
\(879\) 2.67391 0.0901887
\(880\) 0 0
\(881\) −54.6495 −1.84119 −0.920595 0.390519i \(-0.872296\pi\)
−0.920595 + 0.390519i \(0.872296\pi\)
\(882\) 0 0
\(883\) 54.6962 1.84067 0.920336 0.391128i \(-0.127915\pi\)
0.920336 + 0.391128i \(0.127915\pi\)
\(884\) 0 0
\(885\) 4.95168 0.166449
\(886\) 0 0
\(887\) −53.3624 −1.79173 −0.895867 0.444322i \(-0.853445\pi\)
−0.895867 + 0.444322i \(0.853445\pi\)
\(888\) 0 0
\(889\) 34.0189 1.14096
\(890\) 0 0
\(891\) −0.285376 −0.00956045
\(892\) 0 0
\(893\) 13.9835 0.467941
\(894\) 0 0
\(895\) −26.8874 −0.898747
\(896\) 0 0
\(897\) 1.07583 0.0359208
\(898\) 0 0
\(899\) 20.1517 0.672097
\(900\) 0 0
\(901\) 9.00180 0.299893
\(902\) 0 0
\(903\) −26.2001 −0.871883
\(904\) 0 0
\(905\) −22.5088 −0.748219
\(906\) 0 0
\(907\) 5.24736 0.174236 0.0871179 0.996198i \(-0.472234\pi\)
0.0871179 + 0.996198i \(0.472234\pi\)
\(908\) 0 0
\(909\) −20.7959 −0.689757
\(910\) 0 0
\(911\) 32.8403 1.08805 0.544023 0.839070i \(-0.316900\pi\)
0.544023 + 0.839070i \(0.316900\pi\)
\(912\) 0 0
\(913\) −37.2374 −1.23238
\(914\) 0 0
\(915\) −17.3690 −0.574202
\(916\) 0 0
\(917\) 0.943845 0.0311685
\(918\) 0 0
\(919\) −52.6044 −1.73526 −0.867630 0.497211i \(-0.834358\pi\)
−0.867630 + 0.497211i \(0.834358\pi\)
\(920\) 0 0
\(921\) −29.2888 −0.965097
\(922\) 0 0
\(923\) −3.25766 −0.107227
\(924\) 0 0
\(925\) 5.03080 0.165412
\(926\) 0 0
\(927\) −19.1083 −0.627600
\(928\) 0 0
\(929\) −3.31771 −0.108850 −0.0544252 0.998518i \(-0.517333\pi\)
−0.0544252 + 0.998518i \(0.517333\pi\)
\(930\) 0 0
\(931\) 18.8291 0.617097
\(932\) 0 0
\(933\) −8.20534 −0.268631
\(934\) 0 0
\(935\) −10.3932 −0.339895
\(936\) 0 0
\(937\) −35.1170 −1.14722 −0.573612 0.819127i \(-0.694458\pi\)
−0.573612 + 0.819127i \(0.694458\pi\)
\(938\) 0 0
\(939\) 17.6808 0.576991
\(940\) 0 0
\(941\) 29.8064 0.971661 0.485831 0.874053i \(-0.338517\pi\)
0.485831 + 0.874053i \(0.338517\pi\)
\(942\) 0 0
\(943\) 1.67337 0.0544925
\(944\) 0 0
\(945\) 11.8855 0.386637
\(946\) 0 0
\(947\) 22.6189 0.735015 0.367508 0.930020i \(-0.380211\pi\)
0.367508 + 0.930020i \(0.380211\pi\)
\(948\) 0 0
\(949\) 3.60441 0.117004
\(950\) 0 0
\(951\) −18.6725 −0.605497
\(952\) 0 0
\(953\) −5.46365 −0.176985 −0.0884926 0.996077i \(-0.528205\pi\)
−0.0884926 + 0.996077i \(0.528205\pi\)
\(954\) 0 0
\(955\) 10.7440 0.347668
\(956\) 0 0
\(957\) −14.6692 −0.474188
\(958\) 0 0
\(959\) 0.127460 0.00411589
\(960\) 0 0
\(961\) −15.0565 −0.485693
\(962\) 0 0
\(963\) 24.7784 0.798471
\(964\) 0 0
\(965\) 22.7965 0.733846
\(966\) 0 0
\(967\) −58.7962 −1.89076 −0.945379 0.325974i \(-0.894308\pi\)
−0.945379 + 0.325974i \(0.894308\pi\)
\(968\) 0 0
\(969\) 27.1305 0.871558
\(970\) 0 0
\(971\) −6.30246 −0.202256 −0.101128 0.994873i \(-0.532245\pi\)
−0.101128 + 0.994873i \(0.532245\pi\)
\(972\) 0 0
\(973\) 1.57998 0.0506517
\(974\) 0 0
\(975\) 1.29499 0.0414729
\(976\) 0 0
\(977\) 22.5507 0.721461 0.360730 0.932670i \(-0.382527\pi\)
0.360730 + 0.932670i \(0.382527\pi\)
\(978\) 0 0
\(979\) 27.7362 0.886452
\(980\) 0 0
\(981\) 7.41385 0.236706
\(982\) 0 0
\(983\) −40.0002 −1.27581 −0.637904 0.770116i \(-0.720198\pi\)
−0.637904 + 0.770116i \(0.720198\pi\)
\(984\) 0 0
\(985\) −11.6663 −0.371719
\(986\) 0 0
\(987\) −4.41355 −0.140485
\(988\) 0 0
\(989\) 36.8352 1.17129
\(990\) 0 0
\(991\) −5.53861 −0.175940 −0.0879699 0.996123i \(-0.528038\pi\)
−0.0879699 + 0.996123i \(0.528038\pi\)
\(992\) 0 0
\(993\) −34.2053 −1.08547
\(994\) 0 0
\(995\) −8.50598 −0.269658
\(996\) 0 0
\(997\) 56.4989 1.78934 0.894669 0.446729i \(-0.147411\pi\)
0.894669 + 0.446729i \(0.147411\pi\)
\(998\) 0 0
\(999\) −6.88881 −0.217952
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 8048.2.a.y.1.12 33
4.3 odd 2 4024.2.a.f.1.22 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.f.1.22 33 4.3 odd 2
8048.2.a.y.1.12 33 1.1 even 1 trivial