Properties

Label 4024.2.a.f.1.16
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 1.16
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)
 
\(f(q)\) \(=\) \(q-0.224625 q^{3} -2.01216 q^{5} +1.98253 q^{7} -2.94954 q^{9} +O(q^{10})\) \(q-0.224625 q^{3} -2.01216 q^{5} +1.98253 q^{7} -2.94954 q^{9} +5.16477 q^{11} +4.72296 q^{13} +0.451981 q^{15} +3.77632 q^{17} +1.20339 q^{19} -0.445326 q^{21} -1.63312 q^{23} -0.951215 q^{25} +1.33641 q^{27} -8.63975 q^{29} +6.56407 q^{31} -1.16013 q^{33} -3.98917 q^{35} +1.73314 q^{37} -1.06089 q^{39} +0.973128 q^{41} -6.75379 q^{43} +5.93495 q^{45} +11.9538 q^{47} -3.06956 q^{49} -0.848254 q^{51} -8.24121 q^{53} -10.3923 q^{55} -0.270310 q^{57} +11.2397 q^{59} +1.56229 q^{61} -5.84757 q^{63} -9.50335 q^{65} -8.49996 q^{67} +0.366839 q^{69} -0.0603593 q^{71} -3.47965 q^{73} +0.213666 q^{75} +10.2393 q^{77} -6.07716 q^{79} +8.54844 q^{81} +14.2254 q^{83} -7.59855 q^{85} +1.94070 q^{87} +0.498711 q^{89} +9.36343 q^{91} -1.47445 q^{93} -2.42141 q^{95} -8.81572 q^{97} -15.2337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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
\(3\) −0.224625 −0.129687 −0.0648435 0.997895i \(-0.520655\pi\)
−0.0648435 + 0.997895i \(0.520655\pi\)
\(4\) 0 0
\(5\) −2.01216 −0.899865 −0.449933 0.893063i \(-0.648552\pi\)
−0.449933 + 0.893063i \(0.648552\pi\)
\(6\) 0 0
\(7\) 1.98253 0.749327 0.374664 0.927161i \(-0.377758\pi\)
0.374664 + 0.927161i \(0.377758\pi\)
\(8\) 0 0
\(9\) −2.94954 −0.983181
\(10\) 0 0
\(11\) 5.16477 1.55724 0.778618 0.627498i \(-0.215921\pi\)
0.778618 + 0.627498i \(0.215921\pi\)
\(12\) 0 0
\(13\) 4.72296 1.30991 0.654957 0.755666i \(-0.272687\pi\)
0.654957 + 0.755666i \(0.272687\pi\)
\(14\) 0 0
\(15\) 0.451981 0.116701
\(16\) 0 0
\(17\) 3.77632 0.915891 0.457946 0.888980i \(-0.348585\pi\)
0.457946 + 0.888980i \(0.348585\pi\)
\(18\) 0 0
\(19\) 1.20339 0.276076 0.138038 0.990427i \(-0.455920\pi\)
0.138038 + 0.990427i \(0.455920\pi\)
\(20\) 0 0
\(21\) −0.445326 −0.0971780
\(22\) 0 0
\(23\) −1.63312 −0.340529 −0.170265 0.985398i \(-0.554462\pi\)
−0.170265 + 0.985398i \(0.554462\pi\)
\(24\) 0 0
\(25\) −0.951215 −0.190243
\(26\) 0 0
\(27\) 1.33641 0.257193
\(28\) 0 0
\(29\) −8.63975 −1.60436 −0.802181 0.597081i \(-0.796327\pi\)
−0.802181 + 0.597081i \(0.796327\pi\)
\(30\) 0 0
\(31\) 6.56407 1.17894 0.589471 0.807789i \(-0.299336\pi\)
0.589471 + 0.807789i \(0.299336\pi\)
\(32\) 0 0
\(33\) −1.16013 −0.201953
\(34\) 0 0
\(35\) −3.98917 −0.674293
\(36\) 0 0
\(37\) 1.73314 0.284926 0.142463 0.989800i \(-0.454498\pi\)
0.142463 + 0.989800i \(0.454498\pi\)
\(38\) 0 0
\(39\) −1.06089 −0.169879
\(40\) 0 0
\(41\) 0.973128 0.151977 0.0759885 0.997109i \(-0.475789\pi\)
0.0759885 + 0.997109i \(0.475789\pi\)
\(42\) 0 0
\(43\) −6.75379 −1.02994 −0.514972 0.857207i \(-0.672198\pi\)
−0.514972 + 0.857207i \(0.672198\pi\)
\(44\) 0 0
\(45\) 5.93495 0.884730
\(46\) 0 0
\(47\) 11.9538 1.74364 0.871818 0.489830i \(-0.162941\pi\)
0.871818 + 0.489830i \(0.162941\pi\)
\(48\) 0 0
\(49\) −3.06956 −0.438509
\(50\) 0 0
\(51\) −0.848254 −0.118779
\(52\) 0 0
\(53\) −8.24121 −1.13202 −0.566009 0.824399i \(-0.691513\pi\)
−0.566009 + 0.824399i \(0.691513\pi\)
\(54\) 0 0
\(55\) −10.3923 −1.40130
\(56\) 0 0
\(57\) −0.270310 −0.0358035
\(58\) 0 0
\(59\) 11.2397 1.46328 0.731642 0.681689i \(-0.238754\pi\)
0.731642 + 0.681689i \(0.238754\pi\)
\(60\) 0 0
\(61\) 1.56229 0.200030 0.100015 0.994986i \(-0.468111\pi\)
0.100015 + 0.994986i \(0.468111\pi\)
\(62\) 0 0
\(63\) −5.84757 −0.736724
\(64\) 0 0
\(65\) −9.50335 −1.17875
\(66\) 0 0
\(67\) −8.49996 −1.03843 −0.519217 0.854642i \(-0.673777\pi\)
−0.519217 + 0.854642i \(0.673777\pi\)
\(68\) 0 0
\(69\) 0.366839 0.0441622
\(70\) 0 0
\(71\) −0.0603593 −0.00716333 −0.00358166 0.999994i \(-0.501140\pi\)
−0.00358166 + 0.999994i \(0.501140\pi\)
\(72\) 0 0
\(73\) −3.47965 −0.407262 −0.203631 0.979048i \(-0.565274\pi\)
−0.203631 + 0.979048i \(0.565274\pi\)
\(74\) 0 0
\(75\) 0.213666 0.0246720
\(76\) 0 0
\(77\) 10.2393 1.16688
\(78\) 0 0
\(79\) −6.07716 −0.683734 −0.341867 0.939748i \(-0.611059\pi\)
−0.341867 + 0.939748i \(0.611059\pi\)
\(80\) 0 0
\(81\) 8.54844 0.949827
\(82\) 0 0
\(83\) 14.2254 1.56144 0.780719 0.624883i \(-0.214853\pi\)
0.780719 + 0.624883i \(0.214853\pi\)
\(84\) 0 0
\(85\) −7.59855 −0.824179
\(86\) 0 0
\(87\) 1.94070 0.208065
\(88\) 0 0
\(89\) 0.498711 0.0528633 0.0264317 0.999651i \(-0.491586\pi\)
0.0264317 + 0.999651i \(0.491586\pi\)
\(90\) 0 0
\(91\) 9.36343 0.981554
\(92\) 0 0
\(93\) −1.47445 −0.152894
\(94\) 0 0
\(95\) −2.42141 −0.248431
\(96\) 0 0
\(97\) −8.81572 −0.895100 −0.447550 0.894259i \(-0.647703\pi\)
−0.447550 + 0.894259i \(0.647703\pi\)
\(98\) 0 0
\(99\) −15.2337 −1.53105
\(100\) 0 0
\(101\) 7.76484 0.772630 0.386315 0.922367i \(-0.373748\pi\)
0.386315 + 0.922367i \(0.373748\pi\)
\(102\) 0 0
\(103\) −3.42751 −0.337723 −0.168861 0.985640i \(-0.554009\pi\)
−0.168861 + 0.985640i \(0.554009\pi\)
\(104\) 0 0
\(105\) 0.896066 0.0874471
\(106\) 0 0
\(107\) 4.05457 0.391970 0.195985 0.980607i \(-0.437210\pi\)
0.195985 + 0.980607i \(0.437210\pi\)
\(108\) 0 0
\(109\) 3.19030 0.305576 0.152788 0.988259i \(-0.451175\pi\)
0.152788 + 0.988259i \(0.451175\pi\)
\(110\) 0 0
\(111\) −0.389305 −0.0369512
\(112\) 0 0
\(113\) 7.25475 0.682469 0.341235 0.939978i \(-0.389155\pi\)
0.341235 + 0.939978i \(0.389155\pi\)
\(114\) 0 0
\(115\) 3.28610 0.306430
\(116\) 0 0
\(117\) −13.9306 −1.28788
\(118\) 0 0
\(119\) 7.48667 0.686302
\(120\) 0 0
\(121\) 15.6748 1.42498
\(122\) 0 0
\(123\) −0.218588 −0.0197095
\(124\) 0 0
\(125\) 11.9748 1.07106
\(126\) 0 0
\(127\) 10.4834 0.930249 0.465125 0.885245i \(-0.346010\pi\)
0.465125 + 0.885245i \(0.346010\pi\)
\(128\) 0 0
\(129\) 1.51707 0.133570
\(130\) 0 0
\(131\) 18.5971 1.62484 0.812419 0.583074i \(-0.198150\pi\)
0.812419 + 0.583074i \(0.198150\pi\)
\(132\) 0 0
\(133\) 2.38575 0.206871
\(134\) 0 0
\(135\) −2.68908 −0.231439
\(136\) 0 0
\(137\) 7.85356 0.670975 0.335488 0.942045i \(-0.391099\pi\)
0.335488 + 0.942045i \(0.391099\pi\)
\(138\) 0 0
\(139\) −12.9436 −1.09786 −0.548930 0.835868i \(-0.684965\pi\)
−0.548930 + 0.835868i \(0.684965\pi\)
\(140\) 0 0
\(141\) −2.68511 −0.226127
\(142\) 0 0
\(143\) 24.3930 2.03984
\(144\) 0 0
\(145\) 17.3846 1.44371
\(146\) 0 0
\(147\) 0.689499 0.0568689
\(148\) 0 0
\(149\) 15.4675 1.26715 0.633574 0.773682i \(-0.281587\pi\)
0.633574 + 0.773682i \(0.281587\pi\)
\(150\) 0 0
\(151\) 6.76326 0.550387 0.275193 0.961389i \(-0.411258\pi\)
0.275193 + 0.961389i \(0.411258\pi\)
\(152\) 0 0
\(153\) −11.1384 −0.900487
\(154\) 0 0
\(155\) −13.2080 −1.06089
\(156\) 0 0
\(157\) 24.3594 1.94409 0.972044 0.234797i \(-0.0754425\pi\)
0.972044 + 0.234797i \(0.0754425\pi\)
\(158\) 0 0
\(159\) 1.85118 0.146808
\(160\) 0 0
\(161\) −3.23771 −0.255168
\(162\) 0 0
\(163\) 0.132237 0.0103576 0.00517879 0.999987i \(-0.498352\pi\)
0.00517879 + 0.999987i \(0.498352\pi\)
\(164\) 0 0
\(165\) 2.33437 0.181731
\(166\) 0 0
\(167\) 9.44130 0.730590 0.365295 0.930892i \(-0.380968\pi\)
0.365295 + 0.930892i \(0.380968\pi\)
\(168\) 0 0
\(169\) 9.30636 0.715873
\(170\) 0 0
\(171\) −3.54944 −0.271433
\(172\) 0 0
\(173\) 21.5044 1.63495 0.817474 0.575965i \(-0.195374\pi\)
0.817474 + 0.575965i \(0.195374\pi\)
\(174\) 0 0
\(175\) −1.88581 −0.142554
\(176\) 0 0
\(177\) −2.52471 −0.189769
\(178\) 0 0
\(179\) −10.9909 −0.821496 −0.410748 0.911749i \(-0.634732\pi\)
−0.410748 + 0.911749i \(0.634732\pi\)
\(180\) 0 0
\(181\) 0.150153 0.0111608 0.00558038 0.999984i \(-0.498224\pi\)
0.00558038 + 0.999984i \(0.498224\pi\)
\(182\) 0 0
\(183\) −0.350928 −0.0259413
\(184\) 0 0
\(185\) −3.48735 −0.256395
\(186\) 0 0
\(187\) 19.5038 1.42626
\(188\) 0 0
\(189\) 2.64949 0.192722
\(190\) 0 0
\(191\) −14.1390 −1.02307 −0.511533 0.859264i \(-0.670922\pi\)
−0.511533 + 0.859264i \(0.670922\pi\)
\(192\) 0 0
\(193\) −7.84115 −0.564419 −0.282209 0.959353i \(-0.591067\pi\)
−0.282209 + 0.959353i \(0.591067\pi\)
\(194\) 0 0
\(195\) 2.13469 0.152868
\(196\) 0 0
\(197\) −5.24987 −0.374038 −0.187019 0.982356i \(-0.559883\pi\)
−0.187019 + 0.982356i \(0.559883\pi\)
\(198\) 0 0
\(199\) −24.6850 −1.74987 −0.874936 0.484239i \(-0.839096\pi\)
−0.874936 + 0.484239i \(0.839096\pi\)
\(200\) 0 0
\(201\) 1.90930 0.134672
\(202\) 0 0
\(203\) −17.1286 −1.20219
\(204\) 0 0
\(205\) −1.95809 −0.136759
\(206\) 0 0
\(207\) 4.81696 0.334802
\(208\) 0 0
\(209\) 6.21521 0.429915
\(210\) 0 0
\(211\) 6.54286 0.450429 0.225214 0.974309i \(-0.427692\pi\)
0.225214 + 0.974309i \(0.427692\pi\)
\(212\) 0 0
\(213\) 0.0135582 0.000928991 0
\(214\) 0 0
\(215\) 13.5897 0.926810
\(216\) 0 0
\(217\) 13.0135 0.883413
\(218\) 0 0
\(219\) 0.781615 0.0528167
\(220\) 0 0
\(221\) 17.8354 1.19974
\(222\) 0 0
\(223\) 14.7331 0.986603 0.493302 0.869858i \(-0.335790\pi\)
0.493302 + 0.869858i \(0.335790\pi\)
\(224\) 0 0
\(225\) 2.80565 0.187043
\(226\) 0 0
\(227\) −21.2222 −1.40857 −0.704284 0.709918i \(-0.748732\pi\)
−0.704284 + 0.709918i \(0.748732\pi\)
\(228\) 0 0
\(229\) −11.9943 −0.792606 −0.396303 0.918120i \(-0.629707\pi\)
−0.396303 + 0.918120i \(0.629707\pi\)
\(230\) 0 0
\(231\) −2.30000 −0.151329
\(232\) 0 0
\(233\) 21.6787 1.42022 0.710108 0.704092i \(-0.248646\pi\)
0.710108 + 0.704092i \(0.248646\pi\)
\(234\) 0 0
\(235\) −24.0529 −1.56904
\(236\) 0 0
\(237\) 1.36508 0.0886714
\(238\) 0 0
\(239\) −4.50056 −0.291117 −0.145558 0.989350i \(-0.546498\pi\)
−0.145558 + 0.989350i \(0.546498\pi\)
\(240\) 0 0
\(241\) 28.1175 1.81121 0.905605 0.424122i \(-0.139417\pi\)
0.905605 + 0.424122i \(0.139417\pi\)
\(242\) 0 0
\(243\) −5.92943 −0.380373
\(244\) 0 0
\(245\) 6.17645 0.394599
\(246\) 0 0
\(247\) 5.68355 0.361635
\(248\) 0 0
\(249\) −3.19537 −0.202498
\(250\) 0 0
\(251\) −15.1946 −0.959073 −0.479536 0.877522i \(-0.659195\pi\)
−0.479536 + 0.877522i \(0.659195\pi\)
\(252\) 0 0
\(253\) −8.43468 −0.530284
\(254\) 0 0
\(255\) 1.70682 0.106885
\(256\) 0 0
\(257\) 15.1876 0.947376 0.473688 0.880693i \(-0.342922\pi\)
0.473688 + 0.880693i \(0.342922\pi\)
\(258\) 0 0
\(259\) 3.43600 0.213503
\(260\) 0 0
\(261\) 25.4833 1.57738
\(262\) 0 0
\(263\) 1.30783 0.0806442 0.0403221 0.999187i \(-0.487162\pi\)
0.0403221 + 0.999187i \(0.487162\pi\)
\(264\) 0 0
\(265\) 16.5826 1.01866
\(266\) 0 0
\(267\) −0.112023 −0.00685569
\(268\) 0 0
\(269\) 9.71242 0.592177 0.296088 0.955161i \(-0.404318\pi\)
0.296088 + 0.955161i \(0.404318\pi\)
\(270\) 0 0
\(271\) −3.72847 −0.226489 −0.113244 0.993567i \(-0.536124\pi\)
−0.113244 + 0.993567i \(0.536124\pi\)
\(272\) 0 0
\(273\) −2.10326 −0.127295
\(274\) 0 0
\(275\) −4.91280 −0.296253
\(276\) 0 0
\(277\) 7.54205 0.453158 0.226579 0.973993i \(-0.427246\pi\)
0.226579 + 0.973993i \(0.427246\pi\)
\(278\) 0 0
\(279\) −19.3610 −1.15911
\(280\) 0 0
\(281\) −32.1053 −1.91524 −0.957621 0.288033i \(-0.906999\pi\)
−0.957621 + 0.288033i \(0.906999\pi\)
\(282\) 0 0
\(283\) −5.00605 −0.297579 −0.148789 0.988869i \(-0.547538\pi\)
−0.148789 + 0.988869i \(0.547538\pi\)
\(284\) 0 0
\(285\) 0.543907 0.0322183
\(286\) 0 0
\(287\) 1.92926 0.113881
\(288\) 0 0
\(289\) −2.73943 −0.161143
\(290\) 0 0
\(291\) 1.98023 0.116083
\(292\) 0 0
\(293\) −17.7890 −1.03924 −0.519621 0.854397i \(-0.673927\pi\)
−0.519621 + 0.854397i \(0.673927\pi\)
\(294\) 0 0
\(295\) −22.6161 −1.31676
\(296\) 0 0
\(297\) 6.90227 0.400510
\(298\) 0 0
\(299\) −7.71316 −0.446064
\(300\) 0 0
\(301\) −13.3896 −0.771764
\(302\) 0 0
\(303\) −1.74417 −0.100200
\(304\) 0 0
\(305\) −3.14357 −0.180000
\(306\) 0 0
\(307\) 18.6908 1.06674 0.533371 0.845881i \(-0.320925\pi\)
0.533371 + 0.845881i \(0.320925\pi\)
\(308\) 0 0
\(309\) 0.769904 0.0437983
\(310\) 0 0
\(311\) −3.96417 −0.224787 −0.112394 0.993664i \(-0.535852\pi\)
−0.112394 + 0.993664i \(0.535852\pi\)
\(312\) 0 0
\(313\) 23.3388 1.31919 0.659593 0.751623i \(-0.270729\pi\)
0.659593 + 0.751623i \(0.270729\pi\)
\(314\) 0 0
\(315\) 11.7662 0.662953
\(316\) 0 0
\(317\) −1.46135 −0.0820778 −0.0410389 0.999158i \(-0.513067\pi\)
−0.0410389 + 0.999158i \(0.513067\pi\)
\(318\) 0 0
\(319\) −44.6223 −2.49837
\(320\) 0 0
\(321\) −0.910756 −0.0508334
\(322\) 0 0
\(323\) 4.54437 0.252855
\(324\) 0 0
\(325\) −4.49255 −0.249202
\(326\) 0 0
\(327\) −0.716621 −0.0396292
\(328\) 0 0
\(329\) 23.6987 1.30655
\(330\) 0 0
\(331\) −12.9449 −0.711515 −0.355758 0.934578i \(-0.615777\pi\)
−0.355758 + 0.934578i \(0.615777\pi\)
\(332\) 0 0
\(333\) −5.11197 −0.280134
\(334\) 0 0
\(335\) 17.1033 0.934451
\(336\) 0 0
\(337\) 14.6080 0.795748 0.397874 0.917440i \(-0.369748\pi\)
0.397874 + 0.917440i \(0.369748\pi\)
\(338\) 0 0
\(339\) −1.62960 −0.0885075
\(340\) 0 0
\(341\) 33.9019 1.83589
\(342\) 0 0
\(343\) −19.9632 −1.07791
\(344\) 0 0
\(345\) −0.738138 −0.0397400
\(346\) 0 0
\(347\) 17.3246 0.930035 0.465018 0.885301i \(-0.346048\pi\)
0.465018 + 0.885301i \(0.346048\pi\)
\(348\) 0 0
\(349\) 22.8502 1.22314 0.611572 0.791189i \(-0.290538\pi\)
0.611572 + 0.791189i \(0.290538\pi\)
\(350\) 0 0
\(351\) 6.31183 0.336901
\(352\) 0 0
\(353\) −17.7276 −0.943546 −0.471773 0.881720i \(-0.656386\pi\)
−0.471773 + 0.881720i \(0.656386\pi\)
\(354\) 0 0
\(355\) 0.121452 0.00644603
\(356\) 0 0
\(357\) −1.68169 −0.0890045
\(358\) 0 0
\(359\) −21.0884 −1.11300 −0.556501 0.830847i \(-0.687857\pi\)
−0.556501 + 0.830847i \(0.687857\pi\)
\(360\) 0 0
\(361\) −17.5519 −0.923782
\(362\) 0 0
\(363\) −3.52095 −0.184802
\(364\) 0 0
\(365\) 7.00161 0.366481
\(366\) 0 0
\(367\) 21.3511 1.11452 0.557259 0.830339i \(-0.311853\pi\)
0.557259 + 0.830339i \(0.311853\pi\)
\(368\) 0 0
\(369\) −2.87028 −0.149421
\(370\) 0 0
\(371\) −16.3385 −0.848252
\(372\) 0 0
\(373\) 15.7653 0.816297 0.408149 0.912916i \(-0.366175\pi\)
0.408149 + 0.912916i \(0.366175\pi\)
\(374\) 0 0
\(375\) −2.68983 −0.138902
\(376\) 0 0
\(377\) −40.8052 −2.10157
\(378\) 0 0
\(379\) 16.1382 0.828963 0.414482 0.910058i \(-0.363963\pi\)
0.414482 + 0.910058i \(0.363963\pi\)
\(380\) 0 0
\(381\) −2.35482 −0.120641
\(382\) 0 0
\(383\) −6.23886 −0.318791 −0.159395 0.987215i \(-0.550954\pi\)
−0.159395 + 0.987215i \(0.550954\pi\)
\(384\) 0 0
\(385\) −20.6031 −1.05003
\(386\) 0 0
\(387\) 19.9206 1.01262
\(388\) 0 0
\(389\) 25.0716 1.27118 0.635591 0.772026i \(-0.280756\pi\)
0.635591 + 0.772026i \(0.280756\pi\)
\(390\) 0 0
\(391\) −6.16718 −0.311888
\(392\) 0 0
\(393\) −4.17737 −0.210721
\(394\) 0 0
\(395\) 12.2282 0.615268
\(396\) 0 0
\(397\) −0.0124853 −0.000626621 0 −0.000313311 1.00000i \(-0.500100\pi\)
−0.000313311 1.00000i \(0.500100\pi\)
\(398\) 0 0
\(399\) −0.535899 −0.0268285
\(400\) 0 0
\(401\) −15.4353 −0.770803 −0.385402 0.922749i \(-0.625937\pi\)
−0.385402 + 0.922749i \(0.625937\pi\)
\(402\) 0 0
\(403\) 31.0019 1.54431
\(404\) 0 0
\(405\) −17.2008 −0.854716
\(406\) 0 0
\(407\) 8.95125 0.443697
\(408\) 0 0
\(409\) 25.9121 1.28127 0.640635 0.767846i \(-0.278671\pi\)
0.640635 + 0.767846i \(0.278671\pi\)
\(410\) 0 0
\(411\) −1.76410 −0.0870168
\(412\) 0 0
\(413\) 22.2831 1.09648
\(414\) 0 0
\(415\) −28.6237 −1.40508
\(416\) 0 0
\(417\) 2.90745 0.142378
\(418\) 0 0
\(419\) 34.8981 1.70488 0.852441 0.522823i \(-0.175121\pi\)
0.852441 + 0.522823i \(0.175121\pi\)
\(420\) 0 0
\(421\) −20.6385 −1.00586 −0.502929 0.864328i \(-0.667744\pi\)
−0.502929 + 0.864328i \(0.667744\pi\)
\(422\) 0 0
\(423\) −35.2581 −1.71431
\(424\) 0 0
\(425\) −3.59209 −0.174242
\(426\) 0 0
\(427\) 3.09729 0.149888
\(428\) 0 0
\(429\) −5.47927 −0.264541
\(430\) 0 0
\(431\) 29.0615 1.39984 0.699922 0.714219i \(-0.253218\pi\)
0.699922 + 0.714219i \(0.253218\pi\)
\(432\) 0 0
\(433\) −36.2641 −1.74274 −0.871372 0.490624i \(-0.836769\pi\)
−0.871372 + 0.490624i \(0.836769\pi\)
\(434\) 0 0
\(435\) −3.90500 −0.187230
\(436\) 0 0
\(437\) −1.96527 −0.0940118
\(438\) 0 0
\(439\) 3.54940 0.169404 0.0847019 0.996406i \(-0.473006\pi\)
0.0847019 + 0.996406i \(0.473006\pi\)
\(440\) 0 0
\(441\) 9.05381 0.431134
\(442\) 0 0
\(443\) 12.2692 0.582926 0.291463 0.956582i \(-0.405858\pi\)
0.291463 + 0.956582i \(0.405858\pi\)
\(444\) 0 0
\(445\) −1.00349 −0.0475698
\(446\) 0 0
\(447\) −3.47439 −0.164333
\(448\) 0 0
\(449\) −11.3872 −0.537393 −0.268697 0.963225i \(-0.586593\pi\)
−0.268697 + 0.963225i \(0.586593\pi\)
\(450\) 0 0
\(451\) 5.02598 0.236664
\(452\) 0 0
\(453\) −1.51920 −0.0713780
\(454\) 0 0
\(455\) −18.8407 −0.883266
\(456\) 0 0
\(457\) −36.2701 −1.69664 −0.848322 0.529481i \(-0.822387\pi\)
−0.848322 + 0.529481i \(0.822387\pi\)
\(458\) 0 0
\(459\) 5.04672 0.235561
\(460\) 0 0
\(461\) 11.6996 0.544906 0.272453 0.962169i \(-0.412165\pi\)
0.272453 + 0.962169i \(0.412165\pi\)
\(462\) 0 0
\(463\) 27.7477 1.28954 0.644772 0.764375i \(-0.276952\pi\)
0.644772 + 0.764375i \(0.276952\pi\)
\(464\) 0 0
\(465\) 2.96683 0.137584
\(466\) 0 0
\(467\) 7.78383 0.360193 0.180096 0.983649i \(-0.442359\pi\)
0.180096 + 0.983649i \(0.442359\pi\)
\(468\) 0 0
\(469\) −16.8514 −0.778127
\(470\) 0 0
\(471\) −5.47171 −0.252123
\(472\) 0 0
\(473\) −34.8817 −1.60386
\(474\) 0 0
\(475\) −1.14468 −0.0525215
\(476\) 0 0
\(477\) 24.3078 1.11298
\(478\) 0 0
\(479\) −12.5687 −0.574278 −0.287139 0.957889i \(-0.592704\pi\)
−0.287139 + 0.957889i \(0.592704\pi\)
\(480\) 0 0
\(481\) 8.18554 0.373229
\(482\) 0 0
\(483\) 0.727270 0.0330920
\(484\) 0 0
\(485\) 17.7386 0.805470
\(486\) 0 0
\(487\) 8.07349 0.365845 0.182922 0.983127i \(-0.441444\pi\)
0.182922 + 0.983127i \(0.441444\pi\)
\(488\) 0 0
\(489\) −0.0297036 −0.00134324
\(490\) 0 0
\(491\) 10.1487 0.458004 0.229002 0.973426i \(-0.426454\pi\)
0.229002 + 0.973426i \(0.426454\pi\)
\(492\) 0 0
\(493\) −32.6264 −1.46942
\(494\) 0 0
\(495\) 30.6526 1.37773
\(496\) 0 0
\(497\) −0.119664 −0.00536767
\(498\) 0 0
\(499\) −3.50745 −0.157015 −0.0785075 0.996914i \(-0.525015\pi\)
−0.0785075 + 0.996914i \(0.525015\pi\)
\(500\) 0 0
\(501\) −2.12075 −0.0947480
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) −15.6241 −0.695263
\(506\) 0 0
\(507\) −2.09044 −0.0928395
\(508\) 0 0
\(509\) 11.5909 0.513758 0.256879 0.966444i \(-0.417306\pi\)
0.256879 + 0.966444i \(0.417306\pi\)
\(510\) 0 0
\(511\) −6.89852 −0.305173
\(512\) 0 0
\(513\) 1.60822 0.0710048
\(514\) 0 0
\(515\) 6.89670 0.303905
\(516\) 0 0
\(517\) 61.7384 2.71525
\(518\) 0 0
\(519\) −4.83041 −0.212032
\(520\) 0 0
\(521\) 16.2425 0.711597 0.355799 0.934563i \(-0.384209\pi\)
0.355799 + 0.934563i \(0.384209\pi\)
\(522\) 0 0
\(523\) −7.53200 −0.329351 −0.164676 0.986348i \(-0.552658\pi\)
−0.164676 + 0.986348i \(0.552658\pi\)
\(524\) 0 0
\(525\) 0.423600 0.0184874
\(526\) 0 0
\(527\) 24.7880 1.07978
\(528\) 0 0
\(529\) −20.3329 −0.884040
\(530\) 0 0
\(531\) −33.1520 −1.43867
\(532\) 0 0
\(533\) 4.59604 0.199077
\(534\) 0 0
\(535\) −8.15844 −0.352720
\(536\) 0 0
\(537\) 2.46882 0.106537
\(538\) 0 0
\(539\) −15.8536 −0.682862
\(540\) 0 0
\(541\) −16.6480 −0.715753 −0.357876 0.933769i \(-0.616499\pi\)
−0.357876 + 0.933769i \(0.616499\pi\)
\(542\) 0 0
\(543\) −0.0337280 −0.00144741
\(544\) 0 0
\(545\) −6.41940 −0.274977
\(546\) 0 0
\(547\) −17.4585 −0.746471 −0.373235 0.927737i \(-0.621752\pi\)
−0.373235 + 0.927737i \(0.621752\pi\)
\(548\) 0 0
\(549\) −4.60803 −0.196666
\(550\) 0 0
\(551\) −10.3970 −0.442925
\(552\) 0 0
\(553\) −12.0482 −0.512340
\(554\) 0 0
\(555\) 0.783345 0.0332511
\(556\) 0 0
\(557\) −17.5491 −0.743581 −0.371790 0.928317i \(-0.621256\pi\)
−0.371790 + 0.928317i \(0.621256\pi\)
\(558\) 0 0
\(559\) −31.8979 −1.34914
\(560\) 0 0
\(561\) −4.38103 −0.184967
\(562\) 0 0
\(563\) −30.4348 −1.28267 −0.641336 0.767260i \(-0.721620\pi\)
−0.641336 + 0.767260i \(0.721620\pi\)
\(564\) 0 0
\(565\) −14.5977 −0.614130
\(566\) 0 0
\(567\) 16.9476 0.711731
\(568\) 0 0
\(569\) 10.9429 0.458749 0.229374 0.973338i \(-0.426332\pi\)
0.229374 + 0.973338i \(0.426332\pi\)
\(570\) 0 0
\(571\) −45.1429 −1.88917 −0.944586 0.328264i \(-0.893536\pi\)
−0.944586 + 0.328264i \(0.893536\pi\)
\(572\) 0 0
\(573\) 3.17598 0.132678
\(574\) 0 0
\(575\) 1.55345 0.0647832
\(576\) 0 0
\(577\) 12.4974 0.520273 0.260136 0.965572i \(-0.416232\pi\)
0.260136 + 0.965572i \(0.416232\pi\)
\(578\) 0 0
\(579\) 1.76132 0.0731978
\(580\) 0 0
\(581\) 28.2023 1.17003
\(582\) 0 0
\(583\) −42.5640 −1.76282
\(584\) 0 0
\(585\) 28.0305 1.15892
\(586\) 0 0
\(587\) 32.6269 1.34666 0.673328 0.739344i \(-0.264864\pi\)
0.673328 + 0.739344i \(0.264864\pi\)
\(588\) 0 0
\(589\) 7.89912 0.325477
\(590\) 0 0
\(591\) 1.17925 0.0485079
\(592\) 0 0
\(593\) −29.9167 −1.22853 −0.614267 0.789098i \(-0.710548\pi\)
−0.614267 + 0.789098i \(0.710548\pi\)
\(594\) 0 0
\(595\) −15.0644 −0.617579
\(596\) 0 0
\(597\) 5.54485 0.226936
\(598\) 0 0
\(599\) −7.59802 −0.310447 −0.155223 0.987879i \(-0.549610\pi\)
−0.155223 + 0.987879i \(0.549610\pi\)
\(600\) 0 0
\(601\) 5.11553 0.208667 0.104333 0.994542i \(-0.466729\pi\)
0.104333 + 0.994542i \(0.466729\pi\)
\(602\) 0 0
\(603\) 25.0710 1.02097
\(604\) 0 0
\(605\) −31.5402 −1.28229
\(606\) 0 0
\(607\) −7.85941 −0.319004 −0.159502 0.987198i \(-0.550989\pi\)
−0.159502 + 0.987198i \(0.550989\pi\)
\(608\) 0 0
\(609\) 3.84750 0.155909
\(610\) 0 0
\(611\) 56.4571 2.28401
\(612\) 0 0
\(613\) 31.1245 1.25711 0.628553 0.777767i \(-0.283648\pi\)
0.628553 + 0.777767i \(0.283648\pi\)
\(614\) 0 0
\(615\) 0.439835 0.0177359
\(616\) 0 0
\(617\) −24.9170 −1.00312 −0.501559 0.865123i \(-0.667240\pi\)
−0.501559 + 0.865123i \(0.667240\pi\)
\(618\) 0 0
\(619\) −4.68746 −0.188405 −0.0942024 0.995553i \(-0.530030\pi\)
−0.0942024 + 0.995553i \(0.530030\pi\)
\(620\) 0 0
\(621\) −2.18252 −0.0875817
\(622\) 0 0
\(623\) 0.988712 0.0396119
\(624\) 0 0
\(625\) −19.3391 −0.773565
\(626\) 0 0
\(627\) −1.39609 −0.0557544
\(628\) 0 0
\(629\) 6.54488 0.260961
\(630\) 0 0
\(631\) −23.6768 −0.942560 −0.471280 0.881984i \(-0.656208\pi\)
−0.471280 + 0.881984i \(0.656208\pi\)
\(632\) 0 0
\(633\) −1.46969 −0.0584148
\(634\) 0 0
\(635\) −21.0942 −0.837099
\(636\) 0 0
\(637\) −14.4974 −0.574409
\(638\) 0 0
\(639\) 0.178032 0.00704285
\(640\) 0 0
\(641\) −18.5880 −0.734180 −0.367090 0.930185i \(-0.619646\pi\)
−0.367090 + 0.930185i \(0.619646\pi\)
\(642\) 0 0
\(643\) −43.3788 −1.71069 −0.855346 0.518058i \(-0.826655\pi\)
−0.855346 + 0.518058i \(0.826655\pi\)
\(644\) 0 0
\(645\) −3.05258 −0.120195
\(646\) 0 0
\(647\) −9.53143 −0.374719 −0.187360 0.982291i \(-0.559993\pi\)
−0.187360 + 0.982291i \(0.559993\pi\)
\(648\) 0 0
\(649\) 58.0504 2.27868
\(650\) 0 0
\(651\) −2.92315 −0.114567
\(652\) 0 0
\(653\) 44.8203 1.75395 0.876977 0.480533i \(-0.159557\pi\)
0.876977 + 0.480533i \(0.159557\pi\)
\(654\) 0 0
\(655\) −37.4204 −1.46214
\(656\) 0 0
\(657\) 10.2634 0.400413
\(658\) 0 0
\(659\) 35.1924 1.37090 0.685450 0.728119i \(-0.259605\pi\)
0.685450 + 0.728119i \(0.259605\pi\)
\(660\) 0 0
\(661\) −5.52829 −0.215026 −0.107513 0.994204i \(-0.534289\pi\)
−0.107513 + 0.994204i \(0.534289\pi\)
\(662\) 0 0
\(663\) −4.00627 −0.155591
\(664\) 0 0
\(665\) −4.80052 −0.186156
\(666\) 0 0
\(667\) 14.1098 0.546332
\(668\) 0 0
\(669\) −3.30942 −0.127950
\(670\) 0 0
\(671\) 8.06885 0.311494
\(672\) 0 0
\(673\) −49.6266 −1.91297 −0.956483 0.291788i \(-0.905750\pi\)
−0.956483 + 0.291788i \(0.905750\pi\)
\(674\) 0 0
\(675\) −1.27122 −0.0489291
\(676\) 0 0
\(677\) −9.48396 −0.364498 −0.182249 0.983252i \(-0.558338\pi\)
−0.182249 + 0.983252i \(0.558338\pi\)
\(678\) 0 0
\(679\) −17.4775 −0.670723
\(680\) 0 0
\(681\) 4.76703 0.182673
\(682\) 0 0
\(683\) 16.4149 0.628098 0.314049 0.949407i \(-0.398314\pi\)
0.314049 + 0.949407i \(0.398314\pi\)
\(684\) 0 0
\(685\) −15.8026 −0.603787
\(686\) 0 0
\(687\) 2.69422 0.102791
\(688\) 0 0
\(689\) −38.9229 −1.48285
\(690\) 0 0
\(691\) 31.9720 1.21627 0.608135 0.793833i \(-0.291918\pi\)
0.608135 + 0.793833i \(0.291918\pi\)
\(692\) 0 0
\(693\) −30.2013 −1.14725
\(694\) 0 0
\(695\) 26.0446 0.987927
\(696\) 0 0
\(697\) 3.67484 0.139194
\(698\) 0 0
\(699\) −4.86956 −0.184184
\(700\) 0 0
\(701\) 44.4116 1.67740 0.838701 0.544593i \(-0.183316\pi\)
0.838701 + 0.544593i \(0.183316\pi\)
\(702\) 0 0
\(703\) 2.08563 0.0786612
\(704\) 0 0
\(705\) 5.40287 0.203484
\(706\) 0 0
\(707\) 15.3940 0.578953
\(708\) 0 0
\(709\) 32.6935 1.22783 0.613914 0.789373i \(-0.289594\pi\)
0.613914 + 0.789373i \(0.289594\pi\)
\(710\) 0 0
\(711\) 17.9248 0.672234
\(712\) 0 0
\(713\) −10.7199 −0.401464
\(714\) 0 0
\(715\) −49.0826 −1.83558
\(716\) 0 0
\(717\) 1.01094 0.0377541
\(718\) 0 0
\(719\) 38.9669 1.45322 0.726610 0.687050i \(-0.241095\pi\)
0.726610 + 0.687050i \(0.241095\pi\)
\(720\) 0 0
\(721\) −6.79516 −0.253065
\(722\) 0 0
\(723\) −6.31589 −0.234891
\(724\) 0 0
\(725\) 8.21826 0.305218
\(726\) 0 0
\(727\) −19.5384 −0.724639 −0.362319 0.932054i \(-0.618015\pi\)
−0.362319 + 0.932054i \(0.618015\pi\)
\(728\) 0 0
\(729\) −24.3134 −0.900497
\(730\) 0 0
\(731\) −25.5045 −0.943316
\(732\) 0 0
\(733\) 43.6080 1.61070 0.805349 0.592801i \(-0.201978\pi\)
0.805349 + 0.592801i \(0.201978\pi\)
\(734\) 0 0
\(735\) −1.38738 −0.0511744
\(736\) 0 0
\(737\) −43.9003 −1.61709
\(738\) 0 0
\(739\) 14.1061 0.518902 0.259451 0.965756i \(-0.416458\pi\)
0.259451 + 0.965756i \(0.416458\pi\)
\(740\) 0 0
\(741\) −1.27666 −0.0468994
\(742\) 0 0
\(743\) −26.9756 −0.989638 −0.494819 0.868996i \(-0.664766\pi\)
−0.494819 + 0.868996i \(0.664766\pi\)
\(744\) 0 0
\(745\) −31.1231 −1.14026
\(746\) 0 0
\(747\) −41.9584 −1.53518
\(748\) 0 0
\(749\) 8.03832 0.293714
\(750\) 0 0
\(751\) −30.5184 −1.11363 −0.556816 0.830636i \(-0.687977\pi\)
−0.556816 + 0.830636i \(0.687977\pi\)
\(752\) 0 0
\(753\) 3.41307 0.124379
\(754\) 0 0
\(755\) −13.6088 −0.495274
\(756\) 0 0
\(757\) −18.5901 −0.675669 −0.337835 0.941205i \(-0.609694\pi\)
−0.337835 + 0.941205i \(0.609694\pi\)
\(758\) 0 0
\(759\) 1.89464 0.0687710
\(760\) 0 0
\(761\) −22.5248 −0.816524 −0.408262 0.912865i \(-0.633865\pi\)
−0.408262 + 0.912865i \(0.633865\pi\)
\(762\) 0 0
\(763\) 6.32488 0.228976
\(764\) 0 0
\(765\) 22.4123 0.810317
\(766\) 0 0
\(767\) 53.0846 1.91677
\(768\) 0 0
\(769\) −14.0042 −0.505006 −0.252503 0.967596i \(-0.581254\pi\)
−0.252503 + 0.967596i \(0.581254\pi\)
\(770\) 0 0
\(771\) −3.41151 −0.122862
\(772\) 0 0
\(773\) 8.89095 0.319785 0.159893 0.987134i \(-0.448885\pi\)
0.159893 + 0.987134i \(0.448885\pi\)
\(774\) 0 0
\(775\) −6.24384 −0.224285
\(776\) 0 0
\(777\) −0.771811 −0.0276886
\(778\) 0 0
\(779\) 1.17105 0.0419572
\(780\) 0 0
\(781\) −0.311741 −0.0111550
\(782\) 0 0
\(783\) −11.5463 −0.412631
\(784\) 0 0
\(785\) −49.0149 −1.74942
\(786\) 0 0
\(787\) −42.3895 −1.51102 −0.755511 0.655136i \(-0.772611\pi\)
−0.755511 + 0.655136i \(0.772611\pi\)
\(788\) 0 0
\(789\) −0.293771 −0.0104585
\(790\) 0 0
\(791\) 14.3828 0.511393
\(792\) 0 0
\(793\) 7.37862 0.262022
\(794\) 0 0
\(795\) −3.72487 −0.132107
\(796\) 0 0
\(797\) −38.9081 −1.37820 −0.689098 0.724669i \(-0.741993\pi\)
−0.689098 + 0.724669i \(0.741993\pi\)
\(798\) 0 0
\(799\) 45.1412 1.59698
\(800\) 0 0
\(801\) −1.47097 −0.0519742
\(802\) 0 0
\(803\) −17.9716 −0.634204
\(804\) 0 0
\(805\) 6.51480 0.229616
\(806\) 0 0
\(807\) −2.18165 −0.0767977
\(808\) 0 0
\(809\) 0.525574 0.0184782 0.00923910 0.999957i \(-0.497059\pi\)
0.00923910 + 0.999957i \(0.497059\pi\)
\(810\) 0 0
\(811\) −32.8550 −1.15370 −0.576848 0.816852i \(-0.695717\pi\)
−0.576848 + 0.816852i \(0.695717\pi\)
\(812\) 0 0
\(813\) 0.837507 0.0293727
\(814\) 0 0
\(815\) −0.266081 −0.00932042
\(816\) 0 0
\(817\) −8.12742 −0.284342
\(818\) 0 0
\(819\) −27.6178 −0.965045
\(820\) 0 0
\(821\) 31.9309 1.11440 0.557198 0.830380i \(-0.311876\pi\)
0.557198 + 0.830380i \(0.311876\pi\)
\(822\) 0 0
\(823\) 15.7444 0.548815 0.274407 0.961614i \(-0.411518\pi\)
0.274407 + 0.961614i \(0.411518\pi\)
\(824\) 0 0
\(825\) 1.10354 0.0384202
\(826\) 0 0
\(827\) −6.54590 −0.227623 −0.113812 0.993502i \(-0.536306\pi\)
−0.113812 + 0.993502i \(0.536306\pi\)
\(828\) 0 0
\(829\) −24.0859 −0.836536 −0.418268 0.908324i \(-0.637363\pi\)
−0.418268 + 0.908324i \(0.637363\pi\)
\(830\) 0 0
\(831\) −1.69413 −0.0587688
\(832\) 0 0
\(833\) −11.5916 −0.401627
\(834\) 0 0
\(835\) −18.9974 −0.657432
\(836\) 0 0
\(837\) 8.77232 0.303216
\(838\) 0 0
\(839\) −25.1163 −0.867109 −0.433555 0.901127i \(-0.642741\pi\)
−0.433555 + 0.901127i \(0.642741\pi\)
\(840\) 0 0
\(841\) 45.6453 1.57398
\(842\) 0 0
\(843\) 7.21164 0.248382
\(844\) 0 0
\(845\) −18.7259 −0.644190
\(846\) 0 0
\(847\) 31.0758 1.06778
\(848\) 0 0
\(849\) 1.12448 0.0385921
\(850\) 0 0
\(851\) −2.83042 −0.0970256
\(852\) 0 0
\(853\) 42.6052 1.45878 0.729388 0.684100i \(-0.239805\pi\)
0.729388 + 0.684100i \(0.239805\pi\)
\(854\) 0 0
\(855\) 7.14204 0.244253
\(856\) 0 0
\(857\) 9.57277 0.327000 0.163500 0.986543i \(-0.447722\pi\)
0.163500 + 0.986543i \(0.447722\pi\)
\(858\) 0 0
\(859\) 22.2607 0.759525 0.379762 0.925084i \(-0.376006\pi\)
0.379762 + 0.925084i \(0.376006\pi\)
\(860\) 0 0
\(861\) −0.433359 −0.0147688
\(862\) 0 0
\(863\) 24.9858 0.850526 0.425263 0.905070i \(-0.360182\pi\)
0.425263 + 0.905070i \(0.360182\pi\)
\(864\) 0 0
\(865\) −43.2703 −1.47123
\(866\) 0 0
\(867\) 0.615343 0.0208981
\(868\) 0 0
\(869\) −31.3871 −1.06473
\(870\) 0 0
\(871\) −40.1450 −1.36026
\(872\) 0 0
\(873\) 26.0023 0.880046
\(874\) 0 0
\(875\) 23.7404 0.802573
\(876\) 0 0
\(877\) −2.99183 −0.101027 −0.0505135 0.998723i \(-0.516086\pi\)
−0.0505135 + 0.998723i \(0.516086\pi\)
\(878\) 0 0
\(879\) 3.99584 0.134776
\(880\) 0 0
\(881\) −7.79189 −0.262515 −0.131258 0.991348i \(-0.541902\pi\)
−0.131258 + 0.991348i \(0.541902\pi\)
\(882\) 0 0
\(883\) 13.2503 0.445909 0.222955 0.974829i \(-0.428430\pi\)
0.222955 + 0.974829i \(0.428430\pi\)
\(884\) 0 0
\(885\) 5.08012 0.170766
\(886\) 0 0
\(887\) 22.6732 0.761291 0.380646 0.924721i \(-0.375702\pi\)
0.380646 + 0.924721i \(0.375702\pi\)
\(888\) 0 0
\(889\) 20.7836 0.697061
\(890\) 0 0
\(891\) 44.1507 1.47910
\(892\) 0 0
\(893\) 14.3850 0.481376
\(894\) 0 0
\(895\) 22.1154 0.739235
\(896\) 0 0
\(897\) 1.73257 0.0578487
\(898\) 0 0
\(899\) −56.7120 −1.89145
\(900\) 0 0
\(901\) −31.1214 −1.03681
\(902\) 0 0
\(903\) 3.00764 0.100088
\(904\) 0 0
\(905\) −0.302131 −0.0100432
\(906\) 0 0
\(907\) 10.9764 0.364467 0.182233 0.983255i \(-0.441667\pi\)
0.182233 + 0.983255i \(0.441667\pi\)
\(908\) 0 0
\(909\) −22.9027 −0.759635
\(910\) 0 0
\(911\) 42.8706 1.42036 0.710182 0.704018i \(-0.248612\pi\)
0.710182 + 0.704018i \(0.248612\pi\)
\(912\) 0 0
\(913\) 73.4707 2.43153
\(914\) 0 0
\(915\) 0.706123 0.0233437
\(916\) 0 0
\(917\) 36.8694 1.21754
\(918\) 0 0
\(919\) −12.4054 −0.409215 −0.204608 0.978844i \(-0.565592\pi\)
−0.204608 + 0.978844i \(0.565592\pi\)
\(920\) 0 0
\(921\) −4.19842 −0.138343
\(922\) 0 0
\(923\) −0.285074 −0.00938334
\(924\) 0 0
\(925\) −1.64859 −0.0542052
\(926\) 0 0
\(927\) 10.1096 0.332043
\(928\) 0 0
\(929\) 0.805797 0.0264373 0.0132187 0.999913i \(-0.495792\pi\)
0.0132187 + 0.999913i \(0.495792\pi\)
\(930\) 0 0
\(931\) −3.69387 −0.121062
\(932\) 0 0
\(933\) 0.890450 0.0291520
\(934\) 0 0
\(935\) −39.2448 −1.28344
\(936\) 0 0
\(937\) −37.2987 −1.21850 −0.609248 0.792980i \(-0.708529\pi\)
−0.609248 + 0.792980i \(0.708529\pi\)
\(938\) 0 0
\(939\) −5.24247 −0.171081
\(940\) 0 0
\(941\) −8.20339 −0.267423 −0.133711 0.991020i \(-0.542690\pi\)
−0.133711 + 0.991020i \(0.542690\pi\)
\(942\) 0 0
\(943\) −1.58923 −0.0517526
\(944\) 0 0
\(945\) −5.33119 −0.173424
\(946\) 0 0
\(947\) −22.1004 −0.718167 −0.359084 0.933305i \(-0.616911\pi\)
−0.359084 + 0.933305i \(0.616911\pi\)
\(948\) 0 0
\(949\) −16.4343 −0.533479
\(950\) 0 0
\(951\) 0.328256 0.0106444
\(952\) 0 0
\(953\) 50.2570 1.62798 0.813992 0.580877i \(-0.197290\pi\)
0.813992 + 0.580877i \(0.197290\pi\)
\(954\) 0 0
\(955\) 28.4500 0.920621
\(956\) 0 0
\(957\) 10.0233 0.324006
\(958\) 0 0
\(959\) 15.5699 0.502780
\(960\) 0 0
\(961\) 12.0871 0.389905
\(962\) 0 0
\(963\) −11.9591 −0.385377
\(964\) 0 0
\(965\) 15.7777 0.507901
\(966\) 0 0
\(967\) −27.1579 −0.873338 −0.436669 0.899622i \(-0.643842\pi\)
−0.436669 + 0.899622i \(0.643842\pi\)
\(968\) 0 0
\(969\) −1.02078 −0.0327921
\(970\) 0 0
\(971\) 56.0456 1.79859 0.899295 0.437343i \(-0.144081\pi\)
0.899295 + 0.437343i \(0.144081\pi\)
\(972\) 0 0
\(973\) −25.6611 −0.822657
\(974\) 0 0
\(975\) 1.00914 0.0323183
\(976\) 0 0
\(977\) −32.9503 −1.05417 −0.527086 0.849812i \(-0.676715\pi\)
−0.527086 + 0.849812i \(0.676715\pi\)
\(978\) 0 0
\(979\) 2.57573 0.0823206
\(980\) 0 0
\(981\) −9.40994 −0.300436
\(982\) 0 0
\(983\) −40.2054 −1.28235 −0.641176 0.767394i \(-0.721553\pi\)
−0.641176 + 0.767394i \(0.721553\pi\)
\(984\) 0 0
\(985\) 10.5636 0.336584
\(986\) 0 0
\(987\) −5.32332 −0.169443
\(988\) 0 0
\(989\) 11.0297 0.350726
\(990\) 0 0
\(991\) 3.84136 0.122025 0.0610125 0.998137i \(-0.480567\pi\)
0.0610125 + 0.998137i \(0.480567\pi\)
\(992\) 0 0
\(993\) 2.90774 0.0922743
\(994\) 0 0
\(995\) 49.6701 1.57465
\(996\) 0 0
\(997\) −26.9615 −0.853879 −0.426939 0.904280i \(-0.640408\pi\)
−0.426939 + 0.904280i \(0.640408\pi\)
\(998\) 0 0
\(999\) 2.31619 0.0732810
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.1.16 33
4.3 odd 2 8048.2.a.y.1.18 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.f.1.16 33 1.1 even 1 trivial
8048.2.a.y.1.18 33 4.3 odd 2