Properties

Label 644.2.a.c.1.2
Level $644$
Weight $2$
Character 644.1
Self dual yes
Analytic conductor $5.142$
Analytic rank $0$
Dimension $5$
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] = [644,2,Mod(1,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, 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("644.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8580816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.604308\) of defining polynomial
Character \(\chi\) \(=\) 644.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.604308 q^{3} -3.83889 q^{5} +1.00000 q^{7} -2.63481 q^{9} +O(q^{10})\) \(q-0.604308 q^{3} -3.83889 q^{5} +1.00000 q^{7} -2.63481 q^{9} +5.52849 q^{11} +0.106323 q^{13} +2.31987 q^{15} -0.497985 q^{17} +1.82942 q^{19} -0.604308 q^{21} +1.00000 q^{23} +9.73710 q^{25} +3.40516 q^{27} +7.25159 q^{29} +3.94522 q^{31} -3.34091 q^{33} -3.83889 q^{35} +5.37919 q^{37} -0.0642519 q^{39} +5.10229 q^{41} +2.40816 q^{43} +10.1148 q^{45} -8.35338 q^{47} +1.00000 q^{49} +0.300936 q^{51} -8.25653 q^{53} -21.2233 q^{55} -1.10554 q^{57} +0.668560 q^{59} +3.21808 q^{61} -2.63481 q^{63} -0.408163 q^{65} +11.1453 q^{67} -0.604308 q^{69} -0.897708 q^{71} +10.5755 q^{73} -5.88421 q^{75} +5.52849 q^{77} -2.77011 q^{79} +5.84667 q^{81} -14.4362 q^{83} +1.91171 q^{85} -4.38219 q^{87} -16.9762 q^{89} +0.106323 q^{91} -2.38413 q^{93} -7.02297 q^{95} +12.2471 q^{97} -14.5665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9} + 2 q^{11} + 3 q^{13} - 6 q^{15} + 4 q^{17} + 6 q^{19} + q^{21} + 5 q^{23} + 15 q^{25} + q^{27} + 5 q^{29} - q^{31} + 4 q^{35} + 22 q^{37} - q^{39} + 15 q^{41} + 12 q^{43} + 36 q^{45} - 21 q^{47} + 5 q^{49} + 24 q^{51} + 2 q^{53} - 22 q^{55} - 34 q^{57} - 12 q^{61} + 10 q^{63} - 2 q^{65} + 22 q^{67} + q^{69} - 15 q^{71} + 17 q^{73} - 47 q^{75} + 2 q^{77} + 2 q^{79} + 37 q^{81} - 16 q^{83} - 8 q^{85} - 33 q^{87} - 24 q^{89} + 3 q^{91} + 5 q^{93} - 8 q^{95} + 38 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) −0.604308 −0.348897 −0.174449 0.984666i \(-0.555814\pi\)
−0.174449 + 0.984666i \(0.555814\pi\)
\(4\) 0 0
\(5\) −3.83889 −1.71681 −0.858403 0.512976i \(-0.828543\pi\)
−0.858403 + 0.512976i \(0.828543\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.63481 −0.878271
\(10\) 0 0
\(11\) 5.52849 1.66690 0.833451 0.552593i \(-0.186362\pi\)
0.833451 + 0.552593i \(0.186362\pi\)
\(12\) 0 0
\(13\) 0.106323 0.0294887 0.0147444 0.999891i \(-0.495307\pi\)
0.0147444 + 0.999891i \(0.495307\pi\)
\(14\) 0 0
\(15\) 2.31987 0.598989
\(16\) 0 0
\(17\) −0.497985 −0.120779 −0.0603895 0.998175i \(-0.519234\pi\)
−0.0603895 + 0.998175i \(0.519234\pi\)
\(18\) 0 0
\(19\) 1.82942 0.419699 0.209849 0.977734i \(-0.432703\pi\)
0.209849 + 0.977734i \(0.432703\pi\)
\(20\) 0 0
\(21\) −0.604308 −0.131871
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 9.73710 1.94742
\(26\) 0 0
\(27\) 3.40516 0.655324
\(28\) 0 0
\(29\) 7.25159 1.34659 0.673293 0.739375i \(-0.264879\pi\)
0.673293 + 0.739375i \(0.264879\pi\)
\(30\) 0 0
\(31\) 3.94522 0.708582 0.354291 0.935135i \(-0.384722\pi\)
0.354291 + 0.935135i \(0.384722\pi\)
\(32\) 0 0
\(33\) −3.34091 −0.581578
\(34\) 0 0
\(35\) −3.83889 −0.648891
\(36\) 0 0
\(37\) 5.37919 0.884333 0.442167 0.896933i \(-0.354210\pi\)
0.442167 + 0.896933i \(0.354210\pi\)
\(38\) 0 0
\(39\) −0.0642519 −0.0102885
\(40\) 0 0
\(41\) 5.10229 0.796844 0.398422 0.917202i \(-0.369558\pi\)
0.398422 + 0.917202i \(0.369558\pi\)
\(42\) 0 0
\(43\) 2.40816 0.367241 0.183621 0.982997i \(-0.441218\pi\)
0.183621 + 0.982997i \(0.441218\pi\)
\(44\) 0 0
\(45\) 10.1148 1.50782
\(46\) 0 0
\(47\) −8.35338 −1.21847 −0.609233 0.792991i \(-0.708523\pi\)
−0.609233 + 0.792991i \(0.708523\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.300936 0.0421395
\(52\) 0 0
\(53\) −8.25653 −1.13412 −0.567060 0.823676i \(-0.691919\pi\)
−0.567060 + 0.823676i \(0.691919\pi\)
\(54\) 0 0
\(55\) −21.2233 −2.86175
\(56\) 0 0
\(57\) −1.10554 −0.146432
\(58\) 0 0
\(59\) 0.668560 0.0870391 0.0435195 0.999053i \(-0.486143\pi\)
0.0435195 + 0.999053i \(0.486143\pi\)
\(60\) 0 0
\(61\) 3.21808 0.412033 0.206017 0.978548i \(-0.433950\pi\)
0.206017 + 0.978548i \(0.433950\pi\)
\(62\) 0 0
\(63\) −2.63481 −0.331955
\(64\) 0 0
\(65\) −0.408163 −0.0506264
\(66\) 0 0
\(67\) 11.1453 1.36161 0.680806 0.732464i \(-0.261630\pi\)
0.680806 + 0.732464i \(0.261630\pi\)
\(68\) 0 0
\(69\) −0.604308 −0.0727501
\(70\) 0 0
\(71\) −0.897708 −0.106538 −0.0532691 0.998580i \(-0.516964\pi\)
−0.0532691 + 0.998580i \(0.516964\pi\)
\(72\) 0 0
\(73\) 10.5755 1.23777 0.618884 0.785482i \(-0.287585\pi\)
0.618884 + 0.785482i \(0.287585\pi\)
\(74\) 0 0
\(75\) −5.88421 −0.679450
\(76\) 0 0
\(77\) 5.52849 0.630030
\(78\) 0 0
\(79\) −2.77011 −0.311661 −0.155831 0.987784i \(-0.549805\pi\)
−0.155831 + 0.987784i \(0.549805\pi\)
\(80\) 0 0
\(81\) 5.84667 0.649630
\(82\) 0 0
\(83\) −14.4362 −1.58458 −0.792288 0.610148i \(-0.791110\pi\)
−0.792288 + 0.610148i \(0.791110\pi\)
\(84\) 0 0
\(85\) 1.91171 0.207354
\(86\) 0 0
\(87\) −4.38219 −0.469820
\(88\) 0 0
\(89\) −16.9762 −1.79948 −0.899738 0.436430i \(-0.856243\pi\)
−0.899738 + 0.436430i \(0.856243\pi\)
\(90\) 0 0
\(91\) 0.106323 0.0111457
\(92\) 0 0
\(93\) −2.38413 −0.247222
\(94\) 0 0
\(95\) −7.02297 −0.720541
\(96\) 0 0
\(97\) 12.2471 1.24350 0.621750 0.783216i \(-0.286422\pi\)
0.621750 + 0.783216i \(0.286422\pi\)
\(98\) 0 0
\(99\) −14.5665 −1.46399
\(100\) 0 0
\(101\) −10.4782 −1.04262 −0.521312 0.853366i \(-0.674557\pi\)
−0.521312 + 0.853366i \(0.674557\pi\)
\(102\) 0 0
\(103\) 4.67376 0.460519 0.230259 0.973129i \(-0.426043\pi\)
0.230259 + 0.973129i \(0.426043\pi\)
\(104\) 0 0
\(105\) 2.31987 0.226396
\(106\) 0 0
\(107\) 17.9764 1.73784 0.868921 0.494950i \(-0.164814\pi\)
0.868921 + 0.494950i \(0.164814\pi\)
\(108\) 0 0
\(109\) 4.79138 0.458931 0.229466 0.973317i \(-0.426302\pi\)
0.229466 + 0.973317i \(0.426302\pi\)
\(110\) 0 0
\(111\) −3.25069 −0.308542
\(112\) 0 0
\(113\) −8.62988 −0.811831 −0.405915 0.913911i \(-0.633047\pi\)
−0.405915 + 0.913911i \(0.633047\pi\)
\(114\) 0 0
\(115\) −3.83889 −0.357979
\(116\) 0 0
\(117\) −0.280141 −0.0258991
\(118\) 0 0
\(119\) −0.497985 −0.0456502
\(120\) 0 0
\(121\) 19.5642 1.77856
\(122\) 0 0
\(123\) −3.08336 −0.278017
\(124\) 0 0
\(125\) −18.1852 −1.62654
\(126\) 0 0
\(127\) −15.4198 −1.36829 −0.684144 0.729347i \(-0.739824\pi\)
−0.684144 + 0.729347i \(0.739824\pi\)
\(128\) 0 0
\(129\) −1.45527 −0.128130
\(130\) 0 0
\(131\) 13.4194 1.17246 0.586230 0.810144i \(-0.300611\pi\)
0.586230 + 0.810144i \(0.300611\pi\)
\(132\) 0 0
\(133\) 1.82942 0.158631
\(134\) 0 0
\(135\) −13.0721 −1.12506
\(136\) 0 0
\(137\) −0.825394 −0.0705182 −0.0352591 0.999378i \(-0.511226\pi\)
−0.0352591 + 0.999378i \(0.511226\pi\)
\(138\) 0 0
\(139\) 12.6573 1.07357 0.536787 0.843718i \(-0.319638\pi\)
0.536787 + 0.843718i \(0.319638\pi\)
\(140\) 0 0
\(141\) 5.04801 0.425119
\(142\) 0 0
\(143\) 0.587806 0.0491548
\(144\) 0 0
\(145\) −27.8381 −2.31183
\(146\) 0 0
\(147\) −0.604308 −0.0498425
\(148\) 0 0
\(149\) 14.6738 1.20212 0.601060 0.799204i \(-0.294745\pi\)
0.601060 + 0.799204i \(0.294745\pi\)
\(150\) 0 0
\(151\) −10.5334 −0.857198 −0.428599 0.903495i \(-0.640993\pi\)
−0.428599 + 0.903495i \(0.640993\pi\)
\(152\) 0 0
\(153\) 1.31210 0.106077
\(154\) 0 0
\(155\) −15.1453 −1.21650
\(156\) 0 0
\(157\) 23.8536 1.90372 0.951861 0.306531i \(-0.0991683\pi\)
0.951861 + 0.306531i \(0.0991683\pi\)
\(158\) 0 0
\(159\) 4.98948 0.395692
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −7.40089 −0.579682 −0.289841 0.957075i \(-0.593602\pi\)
−0.289841 + 0.957075i \(0.593602\pi\)
\(164\) 0 0
\(165\) 12.8254 0.998456
\(166\) 0 0
\(167\) −16.5973 −1.28434 −0.642168 0.766564i \(-0.721965\pi\)
−0.642168 + 0.766564i \(0.721965\pi\)
\(168\) 0 0
\(169\) −12.9887 −0.999130
\(170\) 0 0
\(171\) −4.82019 −0.368609
\(172\) 0 0
\(173\) −9.19955 −0.699429 −0.349714 0.936856i \(-0.613721\pi\)
−0.349714 + 0.936856i \(0.613721\pi\)
\(174\) 0 0
\(175\) 9.73710 0.736056
\(176\) 0 0
\(177\) −0.404016 −0.0303677
\(178\) 0 0
\(179\) 12.9507 0.967977 0.483989 0.875074i \(-0.339188\pi\)
0.483989 + 0.875074i \(0.339188\pi\)
\(180\) 0 0
\(181\) 15.7676 1.17200 0.585999 0.810312i \(-0.300702\pi\)
0.585999 + 0.810312i \(0.300702\pi\)
\(182\) 0 0
\(183\) −1.94471 −0.143757
\(184\) 0 0
\(185\) −20.6501 −1.51823
\(186\) 0 0
\(187\) −2.75310 −0.201327
\(188\) 0 0
\(189\) 3.40516 0.247689
\(190\) 0 0
\(191\) 12.7348 0.921455 0.460728 0.887542i \(-0.347588\pi\)
0.460728 + 0.887542i \(0.347588\pi\)
\(192\) 0 0
\(193\) −13.4100 −0.965270 −0.482635 0.875822i \(-0.660320\pi\)
−0.482635 + 0.875822i \(0.660320\pi\)
\(194\) 0 0
\(195\) 0.246656 0.0176634
\(196\) 0 0
\(197\) −11.7841 −0.839583 −0.419792 0.907621i \(-0.637897\pi\)
−0.419792 + 0.907621i \(0.637897\pi\)
\(198\) 0 0
\(199\) −22.3266 −1.58269 −0.791345 0.611369i \(-0.790619\pi\)
−0.791345 + 0.611369i \(0.790619\pi\)
\(200\) 0 0
\(201\) −6.73517 −0.475062
\(202\) 0 0
\(203\) 7.25159 0.508962
\(204\) 0 0
\(205\) −19.5872 −1.36803
\(206\) 0 0
\(207\) −2.63481 −0.183132
\(208\) 0 0
\(209\) 10.1140 0.699597
\(210\) 0 0
\(211\) 2.36428 0.162764 0.0813820 0.996683i \(-0.474067\pi\)
0.0813820 + 0.996683i \(0.474067\pi\)
\(212\) 0 0
\(213\) 0.542492 0.0371709
\(214\) 0 0
\(215\) −9.24468 −0.630482
\(216\) 0 0
\(217\) 3.94522 0.267819
\(218\) 0 0
\(219\) −6.39085 −0.431854
\(220\) 0 0
\(221\) −0.0529473 −0.00356162
\(222\) 0 0
\(223\) −7.97119 −0.533790 −0.266895 0.963726i \(-0.585998\pi\)
−0.266895 + 0.963726i \(0.585998\pi\)
\(224\) 0 0
\(225\) −25.6554 −1.71036
\(226\) 0 0
\(227\) 6.68182 0.443488 0.221744 0.975105i \(-0.428825\pi\)
0.221744 + 0.975105i \(0.428825\pi\)
\(228\) 0 0
\(229\) 1.08478 0.0716846 0.0358423 0.999357i \(-0.488589\pi\)
0.0358423 + 0.999357i \(0.488589\pi\)
\(230\) 0 0
\(231\) −3.34091 −0.219816
\(232\) 0 0
\(233\) 25.2370 1.65333 0.826667 0.562692i \(-0.190234\pi\)
0.826667 + 0.562692i \(0.190234\pi\)
\(234\) 0 0
\(235\) 32.0677 2.09187
\(236\) 0 0
\(237\) 1.67400 0.108738
\(238\) 0 0
\(239\) −2.83670 −0.183491 −0.0917454 0.995782i \(-0.529245\pi\)
−0.0917454 + 0.995782i \(0.529245\pi\)
\(240\) 0 0
\(241\) 3.71466 0.239282 0.119641 0.992817i \(-0.461826\pi\)
0.119641 + 0.992817i \(0.461826\pi\)
\(242\) 0 0
\(243\) −13.7487 −0.881978
\(244\) 0 0
\(245\) −3.83889 −0.245258
\(246\) 0 0
\(247\) 0.194510 0.0123764
\(248\) 0 0
\(249\) 8.72389 0.552854
\(250\) 0 0
\(251\) 25.5962 1.61562 0.807810 0.589443i \(-0.200653\pi\)
0.807810 + 0.589443i \(0.200653\pi\)
\(252\) 0 0
\(253\) 5.52849 0.347573
\(254\) 0 0
\(255\) −1.15526 −0.0723453
\(256\) 0 0
\(257\) −9.17237 −0.572157 −0.286078 0.958206i \(-0.592352\pi\)
−0.286078 + 0.958206i \(0.592352\pi\)
\(258\) 0 0
\(259\) 5.37919 0.334247
\(260\) 0 0
\(261\) −19.1066 −1.18267
\(262\) 0 0
\(263\) 3.48642 0.214982 0.107491 0.994206i \(-0.465718\pi\)
0.107491 + 0.994206i \(0.465718\pi\)
\(264\) 0 0
\(265\) 31.6959 1.94707
\(266\) 0 0
\(267\) 10.2589 0.627832
\(268\) 0 0
\(269\) −2.24389 −0.136813 −0.0684063 0.997658i \(-0.521791\pi\)
−0.0684063 + 0.997658i \(0.521791\pi\)
\(270\) 0 0
\(271\) −0.0288129 −0.00175026 −0.000875131 1.00000i \(-0.500279\pi\)
−0.000875131 1.00000i \(0.500279\pi\)
\(272\) 0 0
\(273\) −0.0642519 −0.00388870
\(274\) 0 0
\(275\) 53.8315 3.24616
\(276\) 0 0
\(277\) 6.91898 0.415721 0.207861 0.978158i \(-0.433350\pi\)
0.207861 + 0.978158i \(0.433350\pi\)
\(278\) 0 0
\(279\) −10.3949 −0.622327
\(280\) 0 0
\(281\) 32.5561 1.94214 0.971068 0.238803i \(-0.0767552\pi\)
0.971068 + 0.238803i \(0.0767552\pi\)
\(282\) 0 0
\(283\) 1.22755 0.0729704 0.0364852 0.999334i \(-0.488384\pi\)
0.0364852 + 0.999334i \(0.488384\pi\)
\(284\) 0 0
\(285\) 4.24403 0.251395
\(286\) 0 0
\(287\) 5.10229 0.301179
\(288\) 0 0
\(289\) −16.7520 −0.985412
\(290\) 0 0
\(291\) −7.40099 −0.433854
\(292\) 0 0
\(293\) 34.0388 1.98857 0.994284 0.106768i \(-0.0340502\pi\)
0.994284 + 0.106768i \(0.0340502\pi\)
\(294\) 0 0
\(295\) −2.56653 −0.149429
\(296\) 0 0
\(297\) 18.8254 1.09236
\(298\) 0 0
\(299\) 0.106323 0.00614882
\(300\) 0 0
\(301\) 2.40816 0.138804
\(302\) 0 0
\(303\) 6.33208 0.363769
\(304\) 0 0
\(305\) −12.3539 −0.707381
\(306\) 0 0
\(307\) −8.25612 −0.471202 −0.235601 0.971850i \(-0.575706\pi\)
−0.235601 + 0.971850i \(0.575706\pi\)
\(308\) 0 0
\(309\) −2.82439 −0.160674
\(310\) 0 0
\(311\) 25.0844 1.42240 0.711202 0.702988i \(-0.248151\pi\)
0.711202 + 0.702988i \(0.248151\pi\)
\(312\) 0 0
\(313\) 13.1377 0.742588 0.371294 0.928515i \(-0.378914\pi\)
0.371294 + 0.928515i \(0.378914\pi\)
\(314\) 0 0
\(315\) 10.1148 0.569902
\(316\) 0 0
\(317\) −29.0697 −1.63272 −0.816359 0.577545i \(-0.804011\pi\)
−0.816359 + 0.577545i \(0.804011\pi\)
\(318\) 0 0
\(319\) 40.0903 2.24463
\(320\) 0 0
\(321\) −10.8633 −0.606329
\(322\) 0 0
\(323\) −0.911026 −0.0506908
\(324\) 0 0
\(325\) 1.03528 0.0574270
\(326\) 0 0
\(327\) −2.89547 −0.160120
\(328\) 0 0
\(329\) −8.35338 −0.460537
\(330\) 0 0
\(331\) −26.1357 −1.43655 −0.718273 0.695762i \(-0.755067\pi\)
−0.718273 + 0.695762i \(0.755067\pi\)
\(332\) 0 0
\(333\) −14.1732 −0.776684
\(334\) 0 0
\(335\) −42.7855 −2.33762
\(336\) 0 0
\(337\) −8.04308 −0.438134 −0.219067 0.975710i \(-0.570301\pi\)
−0.219067 + 0.975710i \(0.570301\pi\)
\(338\) 0 0
\(339\) 5.21510 0.283245
\(340\) 0 0
\(341\) 21.8111 1.18114
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.31987 0.124898
\(346\) 0 0
\(347\) −12.3941 −0.665350 −0.332675 0.943042i \(-0.607951\pi\)
−0.332675 + 0.943042i \(0.607951\pi\)
\(348\) 0 0
\(349\) 14.8977 0.797456 0.398728 0.917069i \(-0.369452\pi\)
0.398728 + 0.917069i \(0.369452\pi\)
\(350\) 0 0
\(351\) 0.362047 0.0193247
\(352\) 0 0
\(353\) 4.35298 0.231686 0.115843 0.993268i \(-0.463043\pi\)
0.115843 + 0.993268i \(0.463043\pi\)
\(354\) 0 0
\(355\) 3.44620 0.182906
\(356\) 0 0
\(357\) 0.300936 0.0159272
\(358\) 0 0
\(359\) 35.2673 1.86134 0.930668 0.365865i \(-0.119227\pi\)
0.930668 + 0.365865i \(0.119227\pi\)
\(360\) 0 0
\(361\) −15.6532 −0.823853
\(362\) 0 0
\(363\) −11.8228 −0.620536
\(364\) 0 0
\(365\) −40.5982 −2.12501
\(366\) 0 0
\(367\) −10.4520 −0.545592 −0.272796 0.962072i \(-0.587948\pi\)
−0.272796 + 0.962072i \(0.587948\pi\)
\(368\) 0 0
\(369\) −13.4436 −0.699845
\(370\) 0 0
\(371\) −8.25653 −0.428657
\(372\) 0 0
\(373\) 14.2827 0.739531 0.369766 0.929125i \(-0.379438\pi\)
0.369766 + 0.929125i \(0.379438\pi\)
\(374\) 0 0
\(375\) 10.9895 0.567494
\(376\) 0 0
\(377\) 0.771012 0.0397091
\(378\) 0 0
\(379\) 21.6910 1.11419 0.557096 0.830448i \(-0.311916\pi\)
0.557096 + 0.830448i \(0.311916\pi\)
\(380\) 0 0
\(381\) 9.31832 0.477392
\(382\) 0 0
\(383\) −32.9913 −1.68578 −0.842888 0.538089i \(-0.819146\pi\)
−0.842888 + 0.538089i \(0.819146\pi\)
\(384\) 0 0
\(385\) −21.2233 −1.08164
\(386\) 0 0
\(387\) −6.34506 −0.322537
\(388\) 0 0
\(389\) −31.3987 −1.59197 −0.795987 0.605313i \(-0.793048\pi\)
−0.795987 + 0.605313i \(0.793048\pi\)
\(390\) 0 0
\(391\) −0.497985 −0.0251842
\(392\) 0 0
\(393\) −8.10946 −0.409068
\(394\) 0 0
\(395\) 10.6341 0.535062
\(396\) 0 0
\(397\) −5.75917 −0.289044 −0.144522 0.989502i \(-0.546165\pi\)
−0.144522 + 0.989502i \(0.546165\pi\)
\(398\) 0 0
\(399\) −1.10554 −0.0553460
\(400\) 0 0
\(401\) 23.7917 1.18810 0.594051 0.804427i \(-0.297528\pi\)
0.594051 + 0.804427i \(0.297528\pi\)
\(402\) 0 0
\(403\) 0.419468 0.0208952
\(404\) 0 0
\(405\) −22.4447 −1.11529
\(406\) 0 0
\(407\) 29.7388 1.47410
\(408\) 0 0
\(409\) 25.8809 1.27973 0.639865 0.768487i \(-0.278990\pi\)
0.639865 + 0.768487i \(0.278990\pi\)
\(410\) 0 0
\(411\) 0.498792 0.0246036
\(412\) 0 0
\(413\) 0.668560 0.0328977
\(414\) 0 0
\(415\) 55.4189 2.72041
\(416\) 0 0
\(417\) −7.64888 −0.374567
\(418\) 0 0
\(419\) −11.5322 −0.563383 −0.281691 0.959505i \(-0.590895\pi\)
−0.281691 + 0.959505i \(0.590895\pi\)
\(420\) 0 0
\(421\) −2.41123 −0.117516 −0.0587580 0.998272i \(-0.518714\pi\)
−0.0587580 + 0.998272i \(0.518714\pi\)
\(422\) 0 0
\(423\) 22.0096 1.07014
\(424\) 0 0
\(425\) −4.84893 −0.235208
\(426\) 0 0
\(427\) 3.21808 0.155734
\(428\) 0 0
\(429\) −0.355216 −0.0171500
\(430\) 0 0
\(431\) −27.0144 −1.30124 −0.650620 0.759404i \(-0.725491\pi\)
−0.650620 + 0.759404i \(0.725491\pi\)
\(432\) 0 0
\(433\) −24.4550 −1.17523 −0.587617 0.809139i \(-0.699934\pi\)
−0.587617 + 0.809139i \(0.699934\pi\)
\(434\) 0 0
\(435\) 16.8228 0.806590
\(436\) 0 0
\(437\) 1.82942 0.0875133
\(438\) 0 0
\(439\) −17.3451 −0.827835 −0.413918 0.910314i \(-0.635840\pi\)
−0.413918 + 0.910314i \(0.635840\pi\)
\(440\) 0 0
\(441\) −2.63481 −0.125467
\(442\) 0 0
\(443\) −23.9547 −1.13812 −0.569061 0.822296i \(-0.692693\pi\)
−0.569061 + 0.822296i \(0.692693\pi\)
\(444\) 0 0
\(445\) 65.1699 3.08935
\(446\) 0 0
\(447\) −8.86747 −0.419417
\(448\) 0 0
\(449\) 28.5180 1.34585 0.672923 0.739712i \(-0.265038\pi\)
0.672923 + 0.739712i \(0.265038\pi\)
\(450\) 0 0
\(451\) 28.2080 1.32826
\(452\) 0 0
\(453\) 6.36543 0.299074
\(454\) 0 0
\(455\) −0.408163 −0.0191350
\(456\) 0 0
\(457\) 3.68686 0.172464 0.0862319 0.996275i \(-0.472517\pi\)
0.0862319 + 0.996275i \(0.472517\pi\)
\(458\) 0 0
\(459\) −1.69572 −0.0791493
\(460\) 0 0
\(461\) 32.3634 1.50731 0.753656 0.657269i \(-0.228288\pi\)
0.753656 + 0.657269i \(0.228288\pi\)
\(462\) 0 0
\(463\) 3.33664 0.155067 0.0775333 0.996990i \(-0.475296\pi\)
0.0775333 + 0.996990i \(0.475296\pi\)
\(464\) 0 0
\(465\) 9.15240 0.424433
\(466\) 0 0
\(467\) −24.8817 −1.15139 −0.575694 0.817665i \(-0.695268\pi\)
−0.575694 + 0.817665i \(0.695268\pi\)
\(468\) 0 0
\(469\) 11.1453 0.514641
\(470\) 0 0
\(471\) −14.4149 −0.664203
\(472\) 0 0
\(473\) 13.3135 0.612156
\(474\) 0 0
\(475\) 17.8133 0.817330
\(476\) 0 0
\(477\) 21.7544 0.996065
\(478\) 0 0
\(479\) 14.8584 0.678898 0.339449 0.940625i \(-0.389759\pi\)
0.339449 + 0.940625i \(0.389759\pi\)
\(480\) 0 0
\(481\) 0.571932 0.0260779
\(482\) 0 0
\(483\) −0.604308 −0.0274970
\(484\) 0 0
\(485\) −47.0151 −2.13485
\(486\) 0 0
\(487\) −2.76114 −0.125119 −0.0625597 0.998041i \(-0.519926\pi\)
−0.0625597 + 0.998041i \(0.519926\pi\)
\(488\) 0 0
\(489\) 4.47242 0.202250
\(490\) 0 0
\(491\) −26.4760 −1.19485 −0.597423 0.801927i \(-0.703809\pi\)
−0.597423 + 0.801927i \(0.703809\pi\)
\(492\) 0 0
\(493\) −3.61118 −0.162639
\(494\) 0 0
\(495\) 55.9194 2.51339
\(496\) 0 0
\(497\) −0.897708 −0.0402677
\(498\) 0 0
\(499\) −23.5058 −1.05226 −0.526132 0.850403i \(-0.676358\pi\)
−0.526132 + 0.850403i \(0.676358\pi\)
\(500\) 0 0
\(501\) 10.0299 0.448101
\(502\) 0 0
\(503\) −18.9285 −0.843979 −0.421989 0.906601i \(-0.638668\pi\)
−0.421989 + 0.906601i \(0.638668\pi\)
\(504\) 0 0
\(505\) 40.2248 1.78998
\(506\) 0 0
\(507\) 7.84917 0.348594
\(508\) 0 0
\(509\) −34.7910 −1.54208 −0.771041 0.636785i \(-0.780264\pi\)
−0.771041 + 0.636785i \(0.780264\pi\)
\(510\) 0 0
\(511\) 10.5755 0.467832
\(512\) 0 0
\(513\) 6.22949 0.275039
\(514\) 0 0
\(515\) −17.9421 −0.790621
\(516\) 0 0
\(517\) −46.1816 −2.03106
\(518\) 0 0
\(519\) 5.55936 0.244029
\(520\) 0 0
\(521\) −27.2804 −1.19518 −0.597588 0.801803i \(-0.703874\pi\)
−0.597588 + 0.801803i \(0.703874\pi\)
\(522\) 0 0
\(523\) −44.6189 −1.95105 −0.975525 0.219888i \(-0.929431\pi\)
−0.975525 + 0.219888i \(0.929431\pi\)
\(524\) 0 0
\(525\) −5.88421 −0.256808
\(526\) 0 0
\(527\) −1.96466 −0.0855818
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.76153 −0.0764439
\(532\) 0 0
\(533\) 0.542492 0.0234979
\(534\) 0 0
\(535\) −69.0094 −2.98354
\(536\) 0 0
\(537\) −7.82618 −0.337725
\(538\) 0 0
\(539\) 5.52849 0.238129
\(540\) 0 0
\(541\) −44.9839 −1.93401 −0.967004 0.254759i \(-0.918004\pi\)
−0.967004 + 0.254759i \(0.918004\pi\)
\(542\) 0 0
\(543\) −9.52849 −0.408907
\(544\) 0 0
\(545\) −18.3936 −0.787896
\(546\) 0 0
\(547\) 27.9736 1.19607 0.598033 0.801472i \(-0.295949\pi\)
0.598033 + 0.801472i \(0.295949\pi\)
\(548\) 0 0
\(549\) −8.47905 −0.361877
\(550\) 0 0
\(551\) 13.2662 0.565161
\(552\) 0 0
\(553\) −2.77011 −0.117797
\(554\) 0 0
\(555\) 12.4790 0.529706
\(556\) 0 0
\(557\) 20.8945 0.885327 0.442663 0.896688i \(-0.354034\pi\)
0.442663 + 0.896688i \(0.354034\pi\)
\(558\) 0 0
\(559\) 0.256043 0.0108295
\(560\) 0 0
\(561\) 1.66372 0.0702424
\(562\) 0 0
\(563\) −22.1140 −0.931992 −0.465996 0.884787i \(-0.654304\pi\)
−0.465996 + 0.884787i \(0.654304\pi\)
\(564\) 0 0
\(565\) 33.1292 1.39376
\(566\) 0 0
\(567\) 5.84667 0.245537
\(568\) 0 0
\(569\) 26.3090 1.10293 0.551465 0.834198i \(-0.314069\pi\)
0.551465 + 0.834198i \(0.314069\pi\)
\(570\) 0 0
\(571\) 27.7899 1.16297 0.581486 0.813556i \(-0.302471\pi\)
0.581486 + 0.813556i \(0.302471\pi\)
\(572\) 0 0
\(573\) −7.69572 −0.321493
\(574\) 0 0
\(575\) 9.73710 0.406065
\(576\) 0 0
\(577\) 10.4669 0.435744 0.217872 0.975977i \(-0.430088\pi\)
0.217872 + 0.975977i \(0.430088\pi\)
\(578\) 0 0
\(579\) 8.10374 0.336780
\(580\) 0 0
\(581\) −14.4362 −0.598913
\(582\) 0 0
\(583\) −45.6461 −1.89047
\(584\) 0 0
\(585\) 1.07543 0.0444637
\(586\) 0 0
\(587\) 21.8082 0.900123 0.450061 0.892998i \(-0.351402\pi\)
0.450061 + 0.892998i \(0.351402\pi\)
\(588\) 0 0
\(589\) 7.21748 0.297391
\(590\) 0 0
\(591\) 7.12123 0.292928
\(592\) 0 0
\(593\) −21.4823 −0.882171 −0.441086 0.897465i \(-0.645406\pi\)
−0.441086 + 0.897465i \(0.645406\pi\)
\(594\) 0 0
\(595\) 1.91171 0.0783725
\(596\) 0 0
\(597\) 13.4921 0.552197
\(598\) 0 0
\(599\) −22.6127 −0.923932 −0.461966 0.886898i \(-0.652856\pi\)
−0.461966 + 0.886898i \(0.652856\pi\)
\(600\) 0 0
\(601\) 35.2833 1.43923 0.719617 0.694371i \(-0.244317\pi\)
0.719617 + 0.694371i \(0.244317\pi\)
\(602\) 0 0
\(603\) −29.3657 −1.19586
\(604\) 0 0
\(605\) −75.1048 −3.05345
\(606\) 0 0
\(607\) 13.4081 0.544219 0.272109 0.962266i \(-0.412279\pi\)
0.272109 + 0.962266i \(0.412279\pi\)
\(608\) 0 0
\(609\) −4.38219 −0.177575
\(610\) 0 0
\(611\) −0.888157 −0.0359310
\(612\) 0 0
\(613\) −23.9863 −0.968796 −0.484398 0.874848i \(-0.660961\pi\)
−0.484398 + 0.874848i \(0.660961\pi\)
\(614\) 0 0
\(615\) 11.8367 0.477301
\(616\) 0 0
\(617\) −37.8338 −1.52313 −0.761566 0.648087i \(-0.775569\pi\)
−0.761566 + 0.648087i \(0.775569\pi\)
\(618\) 0 0
\(619\) −40.4917 −1.62750 −0.813749 0.581216i \(-0.802577\pi\)
−0.813749 + 0.581216i \(0.802577\pi\)
\(620\) 0 0
\(621\) 3.40516 0.136644
\(622\) 0 0
\(623\) −16.9762 −0.680138
\(624\) 0 0
\(625\) 21.1257 0.845027
\(626\) 0 0
\(627\) −6.11194 −0.244087
\(628\) 0 0
\(629\) −2.67875 −0.106809
\(630\) 0 0
\(631\) 27.3532 1.08892 0.544458 0.838788i \(-0.316736\pi\)
0.544458 + 0.838788i \(0.316736\pi\)
\(632\) 0 0
\(633\) −1.42876 −0.0567879
\(634\) 0 0
\(635\) 59.1951 2.34908
\(636\) 0 0
\(637\) 0.106323 0.00421268
\(638\) 0 0
\(639\) 2.36529 0.0935695
\(640\) 0 0
\(641\) −40.4456 −1.59750 −0.798752 0.601661i \(-0.794506\pi\)
−0.798752 + 0.601661i \(0.794506\pi\)
\(642\) 0 0
\(643\) −20.9045 −0.824395 −0.412197 0.911095i \(-0.635239\pi\)
−0.412197 + 0.911095i \(0.635239\pi\)
\(644\) 0 0
\(645\) 5.58663 0.219973
\(646\) 0 0
\(647\) −4.03117 −0.158482 −0.0792408 0.996856i \(-0.525250\pi\)
−0.0792408 + 0.996856i \(0.525250\pi\)
\(648\) 0 0
\(649\) 3.69612 0.145086
\(650\) 0 0
\(651\) −2.38413 −0.0934412
\(652\) 0 0
\(653\) −42.1107 −1.64792 −0.823960 0.566648i \(-0.808240\pi\)
−0.823960 + 0.566648i \(0.808240\pi\)
\(654\) 0 0
\(655\) −51.5157 −2.01289
\(656\) 0 0
\(657\) −27.8644 −1.08710
\(658\) 0 0
\(659\) 35.1364 1.36872 0.684361 0.729144i \(-0.260081\pi\)
0.684361 + 0.729144i \(0.260081\pi\)
\(660\) 0 0
\(661\) 9.65893 0.375689 0.187844 0.982199i \(-0.439850\pi\)
0.187844 + 0.982199i \(0.439850\pi\)
\(662\) 0 0
\(663\) 0.0319965 0.00124264
\(664\) 0 0
\(665\) −7.02297 −0.272339
\(666\) 0 0
\(667\) 7.25159 0.280783
\(668\) 0 0
\(669\) 4.81705 0.186238
\(670\) 0 0
\(671\) 17.7911 0.686819
\(672\) 0 0
\(673\) −46.9398 −1.80940 −0.904698 0.426054i \(-0.859903\pi\)
−0.904698 + 0.426054i \(0.859903\pi\)
\(674\) 0 0
\(675\) 33.1564 1.27619
\(676\) 0 0
\(677\) −20.4968 −0.787755 −0.393878 0.919163i \(-0.628867\pi\)
−0.393878 + 0.919163i \(0.628867\pi\)
\(678\) 0 0
\(679\) 12.2471 0.469999
\(680\) 0 0
\(681\) −4.03787 −0.154732
\(682\) 0 0
\(683\) 19.2648 0.737147 0.368574 0.929599i \(-0.379846\pi\)
0.368574 + 0.929599i \(0.379846\pi\)
\(684\) 0 0
\(685\) 3.16860 0.121066
\(686\) 0 0
\(687\) −0.655544 −0.0250105
\(688\) 0 0
\(689\) −0.877860 −0.0334438
\(690\) 0 0
\(691\) 19.1176 0.727267 0.363634 0.931542i \(-0.381536\pi\)
0.363634 + 0.931542i \(0.381536\pi\)
\(692\) 0 0
\(693\) −14.5665 −0.553337
\(694\) 0 0
\(695\) −48.5899 −1.84312
\(696\) 0 0
\(697\) −2.54086 −0.0962421
\(698\) 0 0
\(699\) −15.2509 −0.576844
\(700\) 0 0
\(701\) −15.9164 −0.601153 −0.300577 0.953758i \(-0.597179\pi\)
−0.300577 + 0.953758i \(0.597179\pi\)
\(702\) 0 0
\(703\) 9.84083 0.371154
\(704\) 0 0
\(705\) −19.3788 −0.729847
\(706\) 0 0
\(707\) −10.4782 −0.394075
\(708\) 0 0
\(709\) −45.7591 −1.71852 −0.859259 0.511541i \(-0.829075\pi\)
−0.859259 + 0.511541i \(0.829075\pi\)
\(710\) 0 0
\(711\) 7.29871 0.273723
\(712\) 0 0
\(713\) 3.94522 0.147750
\(714\) 0 0
\(715\) −2.25653 −0.0843893
\(716\) 0 0
\(717\) 1.71424 0.0640194
\(718\) 0 0
\(719\) −49.6642 −1.85216 −0.926080 0.377326i \(-0.876844\pi\)
−0.926080 + 0.377326i \(0.876844\pi\)
\(720\) 0 0
\(721\) 4.67376 0.174060
\(722\) 0 0
\(723\) −2.24480 −0.0834850
\(724\) 0 0
\(725\) 70.6095 2.62237
\(726\) 0 0
\(727\) 35.8348 1.32904 0.664520 0.747271i \(-0.268636\pi\)
0.664520 + 0.747271i \(0.268636\pi\)
\(728\) 0 0
\(729\) −9.23158 −0.341911
\(730\) 0 0
\(731\) −1.19923 −0.0443551
\(732\) 0 0
\(733\) 12.9548 0.478497 0.239249 0.970958i \(-0.423099\pi\)
0.239249 + 0.970958i \(0.423099\pi\)
\(734\) 0 0
\(735\) 2.31987 0.0855698
\(736\) 0 0
\(737\) 61.6165 2.26967
\(738\) 0 0
\(739\) −44.6560 −1.64270 −0.821348 0.570427i \(-0.806778\pi\)
−0.821348 + 0.570427i \(0.806778\pi\)
\(740\) 0 0
\(741\) −0.117544 −0.00431809
\(742\) 0 0
\(743\) 19.1471 0.702438 0.351219 0.936293i \(-0.385767\pi\)
0.351219 + 0.936293i \(0.385767\pi\)
\(744\) 0 0
\(745\) −56.3310 −2.06381
\(746\) 0 0
\(747\) 38.0366 1.39169
\(748\) 0 0
\(749\) 17.9764 0.656843
\(750\) 0 0
\(751\) 5.13072 0.187223 0.0936113 0.995609i \(-0.470159\pi\)
0.0936113 + 0.995609i \(0.470159\pi\)
\(752\) 0 0
\(753\) −15.4680 −0.563685
\(754\) 0 0
\(755\) 40.4367 1.47164
\(756\) 0 0
\(757\) −41.5986 −1.51193 −0.755963 0.654615i \(-0.772831\pi\)
−0.755963 + 0.654615i \(0.772831\pi\)
\(758\) 0 0
\(759\) −3.34091 −0.121267
\(760\) 0 0
\(761\) 45.0221 1.63205 0.816024 0.578018i \(-0.196174\pi\)
0.816024 + 0.578018i \(0.196174\pi\)
\(762\) 0 0
\(763\) 4.79138 0.173460
\(764\) 0 0
\(765\) −5.03700 −0.182113
\(766\) 0 0
\(767\) 0.0710834 0.00256667
\(768\) 0 0
\(769\) 24.3467 0.877963 0.438982 0.898496i \(-0.355339\pi\)
0.438982 + 0.898496i \(0.355339\pi\)
\(770\) 0 0
\(771\) 5.54293 0.199624
\(772\) 0 0
\(773\) 28.9581 1.04155 0.520775 0.853694i \(-0.325643\pi\)
0.520775 + 0.853694i \(0.325643\pi\)
\(774\) 0 0
\(775\) 38.4150 1.37991
\(776\) 0 0
\(777\) −3.25069 −0.116618
\(778\) 0 0
\(779\) 9.33426 0.334435
\(780\) 0 0
\(781\) −4.96297 −0.177589
\(782\) 0 0
\(783\) 24.6928 0.882450
\(784\) 0 0
\(785\) −91.5713 −3.26832
\(786\) 0 0
\(787\) −10.3643 −0.369447 −0.184723 0.982791i \(-0.559139\pi\)
−0.184723 + 0.982791i \(0.559139\pi\)
\(788\) 0 0
\(789\) −2.10687 −0.0750065
\(790\) 0 0
\(791\) −8.62988 −0.306843
\(792\) 0 0
\(793\) 0.342157 0.0121503
\(794\) 0 0
\(795\) −19.1541 −0.679326
\(796\) 0 0
\(797\) 29.0652 1.02954 0.514772 0.857327i \(-0.327877\pi\)
0.514772 + 0.857327i \(0.327877\pi\)
\(798\) 0 0
\(799\) 4.15986 0.147165
\(800\) 0 0
\(801\) 44.7292 1.58043
\(802\) 0 0
\(803\) 58.4665 2.06324
\(804\) 0 0
\(805\) −3.83889 −0.135303
\(806\) 0 0
\(807\) 1.35600 0.0477336
\(808\) 0 0
\(809\) 32.1570 1.13058 0.565291 0.824892i \(-0.308764\pi\)
0.565291 + 0.824892i \(0.308764\pi\)
\(810\) 0 0
\(811\) 14.5270 0.510113 0.255056 0.966926i \(-0.417906\pi\)
0.255056 + 0.966926i \(0.417906\pi\)
\(812\) 0 0
\(813\) 0.0174119 0.000610662 0
\(814\) 0 0
\(815\) 28.4112 0.995202
\(816\) 0 0
\(817\) 4.40555 0.154131
\(818\) 0 0
\(819\) −0.280141 −0.00978893
\(820\) 0 0
\(821\) 0.201239 0.00702329 0.00351164 0.999994i \(-0.498882\pi\)
0.00351164 + 0.999994i \(0.498882\pi\)
\(822\) 0 0
\(823\) −22.3719 −0.779836 −0.389918 0.920850i \(-0.627497\pi\)
−0.389918 + 0.920850i \(0.627497\pi\)
\(824\) 0 0
\(825\) −32.5308 −1.13258
\(826\) 0 0
\(827\) 9.19173 0.319628 0.159814 0.987147i \(-0.448911\pi\)
0.159814 + 0.987147i \(0.448911\pi\)
\(828\) 0 0
\(829\) −10.8345 −0.376296 −0.188148 0.982141i \(-0.560249\pi\)
−0.188148 + 0.982141i \(0.560249\pi\)
\(830\) 0 0
\(831\) −4.18120 −0.145044
\(832\) 0 0
\(833\) −0.497985 −0.0172541
\(834\) 0 0
\(835\) 63.7152 2.20495
\(836\) 0 0
\(837\) 13.4341 0.464350
\(838\) 0 0
\(839\) −9.77829 −0.337584 −0.168792 0.985652i \(-0.553987\pi\)
−0.168792 + 0.985652i \(0.553987\pi\)
\(840\) 0 0
\(841\) 23.5856 0.813296
\(842\) 0 0
\(843\) −19.6739 −0.677606
\(844\) 0 0
\(845\) 49.8622 1.71531
\(846\) 0 0
\(847\) 19.5642 0.672234
\(848\) 0 0
\(849\) −0.741820 −0.0254592
\(850\) 0 0
\(851\) 5.37919 0.184396
\(852\) 0 0
\(853\) 0.927467 0.0317559 0.0158779 0.999874i \(-0.494946\pi\)
0.0158779 + 0.999874i \(0.494946\pi\)
\(854\) 0 0
\(855\) 18.5042 0.632830
\(856\) 0 0
\(857\) −2.66044 −0.0908789 −0.0454394 0.998967i \(-0.514469\pi\)
−0.0454394 + 0.998967i \(0.514469\pi\)
\(858\) 0 0
\(859\) 37.4822 1.27888 0.639438 0.768843i \(-0.279167\pi\)
0.639438 + 0.768843i \(0.279167\pi\)
\(860\) 0 0
\(861\) −3.08336 −0.105080
\(862\) 0 0
\(863\) −14.1751 −0.482528 −0.241264 0.970460i \(-0.577562\pi\)
−0.241264 + 0.970460i \(0.577562\pi\)
\(864\) 0 0
\(865\) 35.3161 1.20078
\(866\) 0 0
\(867\) 10.1234 0.343808
\(868\) 0 0
\(869\) −15.3145 −0.519509
\(870\) 0 0
\(871\) 1.18500 0.0401522
\(872\) 0 0
\(873\) −32.2687 −1.09213
\(874\) 0 0
\(875\) −18.1852 −0.614773
\(876\) 0 0
\(877\) −4.13991 −0.139795 −0.0698974 0.997554i \(-0.522267\pi\)
−0.0698974 + 0.997554i \(0.522267\pi\)
\(878\) 0 0
\(879\) −20.5699 −0.693806
\(880\) 0 0
\(881\) 31.9906 1.07779 0.538896 0.842372i \(-0.318842\pi\)
0.538896 + 0.842372i \(0.318842\pi\)
\(882\) 0 0
\(883\) −30.9333 −1.04099 −0.520495 0.853865i \(-0.674253\pi\)
−0.520495 + 0.853865i \(0.674253\pi\)
\(884\) 0 0
\(885\) 1.55097 0.0521354
\(886\) 0 0
\(887\) 27.9581 0.938740 0.469370 0.883002i \(-0.344481\pi\)
0.469370 + 0.883002i \(0.344481\pi\)
\(888\) 0 0
\(889\) −15.4198 −0.517164
\(890\) 0 0
\(891\) 32.3233 1.08287
\(892\) 0 0
\(893\) −15.2819 −0.511389
\(894\) 0 0
\(895\) −49.7162 −1.66183
\(896\) 0 0
\(897\) −0.0642519 −0.00214531
\(898\) 0 0
\(899\) 28.6091 0.954167
\(900\) 0 0
\(901\) 4.11162 0.136978
\(902\) 0 0
\(903\) −1.45527 −0.0484284
\(904\) 0 0
\(905\) −60.5302 −2.01209
\(906\) 0 0
\(907\) −32.4224 −1.07657 −0.538284 0.842763i \(-0.680927\pi\)
−0.538284 + 0.842763i \(0.680927\pi\)
\(908\) 0 0
\(909\) 27.6082 0.915706
\(910\) 0 0
\(911\) −22.5349 −0.746613 −0.373306 0.927708i \(-0.621776\pi\)
−0.373306 + 0.927708i \(0.621776\pi\)
\(912\) 0 0
\(913\) −79.8102 −2.64133
\(914\) 0 0
\(915\) 7.46555 0.246803
\(916\) 0 0
\(917\) 13.4194 0.443148
\(918\) 0 0
\(919\) 37.8810 1.24958 0.624790 0.780793i \(-0.285185\pi\)
0.624790 + 0.780793i \(0.285185\pi\)
\(920\) 0 0
\(921\) 4.98924 0.164401
\(922\) 0 0
\(923\) −0.0954471 −0.00314168
\(924\) 0 0
\(925\) 52.3777 1.72217
\(926\) 0 0
\(927\) −12.3145 −0.404460
\(928\) 0 0
\(929\) 55.2532 1.81280 0.906399 0.422423i \(-0.138820\pi\)
0.906399 + 0.422423i \(0.138820\pi\)
\(930\) 0 0
\(931\) 1.82942 0.0599570
\(932\) 0 0
\(933\) −15.1587 −0.496273
\(934\) 0 0
\(935\) 10.5689 0.345639
\(936\) 0 0
\(937\) −15.0702 −0.492323 −0.246162 0.969229i \(-0.579169\pi\)
−0.246162 + 0.969229i \(0.579169\pi\)
\(938\) 0 0
\(939\) −7.93923 −0.259087
\(940\) 0 0
\(941\) 13.2068 0.430529 0.215265 0.976556i \(-0.430939\pi\)
0.215265 + 0.976556i \(0.430939\pi\)
\(942\) 0 0
\(943\) 5.10229 0.166154
\(944\) 0 0
\(945\) −13.0721 −0.425234
\(946\) 0 0
\(947\) 22.9948 0.747230 0.373615 0.927584i \(-0.378118\pi\)
0.373615 + 0.927584i \(0.378118\pi\)
\(948\) 0 0
\(949\) 1.12442 0.0365002
\(950\) 0 0
\(951\) 17.5671 0.569651
\(952\) 0 0
\(953\) −20.8303 −0.674758 −0.337379 0.941369i \(-0.609540\pi\)
−0.337379 + 0.941369i \(0.609540\pi\)
\(954\) 0 0
\(955\) −48.8874 −1.58196
\(956\) 0 0
\(957\) −24.2269 −0.783145
\(958\) 0 0
\(959\) −0.825394 −0.0266534
\(960\) 0 0
\(961\) −15.4353 −0.497912
\(962\) 0 0
\(963\) −47.3644 −1.52630
\(964\) 0 0
\(965\) 51.4794 1.65718
\(966\) 0 0
\(967\) 7.93655 0.255222 0.127611 0.991824i \(-0.459269\pi\)
0.127611 + 0.991824i \(0.459269\pi\)
\(968\) 0 0
\(969\) 0.550540 0.0176859
\(970\) 0 0
\(971\) 11.2326 0.360470 0.180235 0.983624i \(-0.442314\pi\)
0.180235 + 0.983624i \(0.442314\pi\)
\(972\) 0 0
\(973\) 12.6573 0.405773
\(974\) 0 0
\(975\) −0.625627 −0.0200361
\(976\) 0 0
\(977\) 12.9955 0.415762 0.207881 0.978154i \(-0.433343\pi\)
0.207881 + 0.978154i \(0.433343\pi\)
\(978\) 0 0
\(979\) −93.8529 −2.99955
\(980\) 0 0
\(981\) −12.6244 −0.403066
\(982\) 0 0
\(983\) −39.1117 −1.24747 −0.623735 0.781636i \(-0.714386\pi\)
−0.623735 + 0.781636i \(0.714386\pi\)
\(984\) 0 0
\(985\) 45.2379 1.44140
\(986\) 0 0
\(987\) 5.04801 0.160680
\(988\) 0 0
\(989\) 2.40816 0.0765751
\(990\) 0 0
\(991\) −25.1445 −0.798741 −0.399371 0.916790i \(-0.630771\pi\)
−0.399371 + 0.916790i \(0.630771\pi\)
\(992\) 0 0
\(993\) 15.7940 0.501207
\(994\) 0 0
\(995\) 85.7095 2.71717
\(996\) 0 0
\(997\) 40.1007 1.27000 0.635001 0.772511i \(-0.280999\pi\)
0.635001 + 0.772511i \(0.280999\pi\)
\(998\) 0 0
\(999\) 18.3170 0.579525
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 644.2.a.c.1.2 5
3.2 odd 2 5796.2.a.s.1.5 5
4.3 odd 2 2576.2.a.bc.1.4 5
7.6 odd 2 4508.2.a.g.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.c.1.2 5 1.1 even 1 trivial
2576.2.a.bc.1.4 5 4.3 odd 2
4508.2.a.g.1.4 5 7.6 odd 2
5796.2.a.s.1.5 5 3.2 odd 2