Properties

Label 7623.2.a.cu.1.7
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6988960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.80692\) of defining polynomial
Character \(\chi\) \(=\) 7623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.80692 q^{2} +1.26498 q^{4} +2.14896 q^{5} -1.00000 q^{7} -1.32813 q^{8} +O(q^{10})\) \(q+1.80692 q^{2} +1.26498 q^{4} +2.14896 q^{5} -1.00000 q^{7} -1.32813 q^{8} +3.88301 q^{10} -3.66481 q^{13} -1.80692 q^{14} -4.92979 q^{16} +1.89145 q^{17} +4.44661 q^{19} +2.71839 q^{20} -5.20938 q^{23} -0.381966 q^{25} -6.62204 q^{26} -1.26498 q^{28} -3.76049 q^{29} +2.21054 q^{31} -6.25149 q^{32} +3.41770 q^{34} -2.14896 q^{35} -8.31175 q^{37} +8.03469 q^{38} -2.85410 q^{40} +0.0621247 q^{41} -8.99443 q^{43} -9.41297 q^{46} +12.3547 q^{47} +1.00000 q^{49} -0.690184 q^{50} -4.63590 q^{52} -7.01020 q^{53} +1.32813 q^{56} -6.79493 q^{58} -12.0268 q^{59} +5.13486 q^{61} +3.99428 q^{62} -1.43640 q^{64} -7.87553 q^{65} -9.38744 q^{67} +2.39264 q^{68} -3.88301 q^{70} -4.35232 q^{71} +7.65242 q^{73} -15.0187 q^{74} +5.62486 q^{76} +16.1679 q^{79} -10.5939 q^{80} +0.112255 q^{82} -7.24748 q^{83} +4.06464 q^{85} -16.2523 q^{86} -6.68300 q^{89} +3.66481 q^{91} -6.58975 q^{92} +22.3240 q^{94} +9.55559 q^{95} -13.1033 q^{97} +1.80692 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80692 1.27769 0.638844 0.769336i \(-0.279413\pi\)
0.638844 + 0.769336i \(0.279413\pi\)
\(3\) 0 0
\(4\) 1.26498 0.632489
\(5\) 2.14896 0.961045 0.480522 0.876982i \(-0.340447\pi\)
0.480522 + 0.876982i \(0.340447\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.32813 −0.469565
\(9\) 0 0
\(10\) 3.88301 1.22792
\(11\) 0 0
\(12\) 0 0
\(13\) −3.66481 −1.01644 −0.508218 0.861229i \(-0.669695\pi\)
−0.508218 + 0.861229i \(0.669695\pi\)
\(14\) −1.80692 −0.482921
\(15\) 0 0
\(16\) −4.92979 −1.23245
\(17\) 1.89145 0.458743 0.229371 0.973339i \(-0.426333\pi\)
0.229371 + 0.973339i \(0.426333\pi\)
\(18\) 0 0
\(19\) 4.44661 1.02012 0.510061 0.860138i \(-0.329623\pi\)
0.510061 + 0.860138i \(0.329623\pi\)
\(20\) 2.71839 0.607850
\(21\) 0 0
\(22\) 0 0
\(23\) −5.20938 −1.08623 −0.543116 0.839658i \(-0.682756\pi\)
−0.543116 + 0.839658i \(0.682756\pi\)
\(24\) 0 0
\(25\) −0.381966 −0.0763932
\(26\) −6.62204 −1.29869
\(27\) 0 0
\(28\) −1.26498 −0.239058
\(29\) −3.76049 −0.698306 −0.349153 0.937066i \(-0.613531\pi\)
−0.349153 + 0.937066i \(0.613531\pi\)
\(30\) 0 0
\(31\) 2.21054 0.397025 0.198512 0.980098i \(-0.436389\pi\)
0.198512 + 0.980098i \(0.436389\pi\)
\(32\) −6.25149 −1.10512
\(33\) 0 0
\(34\) 3.41770 0.586131
\(35\) −2.14896 −0.363241
\(36\) 0 0
\(37\) −8.31175 −1.36644 −0.683222 0.730211i \(-0.739422\pi\)
−0.683222 + 0.730211i \(0.739422\pi\)
\(38\) 8.03469 1.30340
\(39\) 0 0
\(40\) −2.85410 −0.451273
\(41\) 0.0621247 0.00970224 0.00485112 0.999988i \(-0.498456\pi\)
0.00485112 + 0.999988i \(0.498456\pi\)
\(42\) 0 0
\(43\) −8.99443 −1.37164 −0.685819 0.727772i \(-0.740556\pi\)
−0.685819 + 0.727772i \(0.740556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −9.41297 −1.38787
\(47\) 12.3547 1.80212 0.901061 0.433693i \(-0.142790\pi\)
0.901061 + 0.433693i \(0.142790\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.690184 −0.0976067
\(51\) 0 0
\(52\) −4.63590 −0.642884
\(53\) −7.01020 −0.962925 −0.481462 0.876467i \(-0.659894\pi\)
−0.481462 + 0.876467i \(0.659894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.32813 0.177479
\(57\) 0 0
\(58\) −6.79493 −0.892218
\(59\) −12.0268 −1.56576 −0.782879 0.622174i \(-0.786249\pi\)
−0.782879 + 0.622174i \(0.786249\pi\)
\(60\) 0 0
\(61\) 5.13486 0.657451 0.328725 0.944426i \(-0.393381\pi\)
0.328725 + 0.944426i \(0.393381\pi\)
\(62\) 3.99428 0.507274
\(63\) 0 0
\(64\) −1.43640 −0.179550
\(65\) −7.87553 −0.976840
\(66\) 0 0
\(67\) −9.38744 −1.14686 −0.573429 0.819255i \(-0.694387\pi\)
−0.573429 + 0.819255i \(0.694387\pi\)
\(68\) 2.39264 0.290150
\(69\) 0 0
\(70\) −3.88301 −0.464109
\(71\) −4.35232 −0.516525 −0.258262 0.966075i \(-0.583150\pi\)
−0.258262 + 0.966075i \(0.583150\pi\)
\(72\) 0 0
\(73\) 7.65242 0.895647 0.447824 0.894122i \(-0.352199\pi\)
0.447824 + 0.894122i \(0.352199\pi\)
\(74\) −15.0187 −1.74589
\(75\) 0 0
\(76\) 5.62486 0.645216
\(77\) 0 0
\(78\) 0 0
\(79\) 16.1679 1.81904 0.909518 0.415665i \(-0.136451\pi\)
0.909518 + 0.415665i \(0.136451\pi\)
\(80\) −10.5939 −1.18444
\(81\) 0 0
\(82\) 0.112255 0.0123964
\(83\) −7.24748 −0.795514 −0.397757 0.917491i \(-0.630211\pi\)
−0.397757 + 0.917491i \(0.630211\pi\)
\(84\) 0 0
\(85\) 4.06464 0.440872
\(86\) −16.2523 −1.75253
\(87\) 0 0
\(88\) 0 0
\(89\) −6.68300 −0.708397 −0.354198 0.935170i \(-0.615246\pi\)
−0.354198 + 0.935170i \(0.615246\pi\)
\(90\) 0 0
\(91\) 3.66481 0.384176
\(92\) −6.58975 −0.687029
\(93\) 0 0
\(94\) 22.3240 2.30255
\(95\) 9.55559 0.980383
\(96\) 0 0
\(97\) −13.1033 −1.33044 −0.665219 0.746648i \(-0.731662\pi\)
−0.665219 + 0.746648i \(0.731662\pi\)
\(98\) 1.80692 0.182527
\(99\) 0 0
\(100\) −0.483178 −0.0483178
\(101\) 1.24739 0.124120 0.0620598 0.998072i \(-0.480233\pi\)
0.0620598 + 0.998072i \(0.480233\pi\)
\(102\) 0 0
\(103\) −14.9689 −1.47493 −0.737465 0.675385i \(-0.763977\pi\)
−0.737465 + 0.675385i \(0.763977\pi\)
\(104\) 4.86735 0.477283
\(105\) 0 0
\(106\) −12.6669 −1.23032
\(107\) −10.2458 −0.990496 −0.495248 0.868752i \(-0.664923\pi\)
−0.495248 + 0.868752i \(0.664923\pi\)
\(108\) 0 0
\(109\) 3.39172 0.324868 0.162434 0.986719i \(-0.448066\pi\)
0.162434 + 0.986719i \(0.448066\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.92979 0.465821
\(113\) 8.21990 0.773264 0.386632 0.922234i \(-0.373638\pi\)
0.386632 + 0.922234i \(0.373638\pi\)
\(114\) 0 0
\(115\) −11.1948 −1.04392
\(116\) −4.75694 −0.441671
\(117\) 0 0
\(118\) −21.7315 −2.00055
\(119\) −1.89145 −0.173389
\(120\) 0 0
\(121\) 0 0
\(122\) 9.27830 0.840018
\(123\) 0 0
\(124\) 2.79628 0.251114
\(125\) −11.5656 −1.03446
\(126\) 0 0
\(127\) −3.25941 −0.289226 −0.144613 0.989488i \(-0.546194\pi\)
−0.144613 + 0.989488i \(0.546194\pi\)
\(128\) 9.90751 0.875709
\(129\) 0 0
\(130\) −14.2305 −1.24810
\(131\) 0.349304 0.0305189 0.0152594 0.999884i \(-0.495143\pi\)
0.0152594 + 0.999884i \(0.495143\pi\)
\(132\) 0 0
\(133\) −4.44661 −0.385570
\(134\) −16.9624 −1.46533
\(135\) 0 0
\(136\) −2.51209 −0.215410
\(137\) −0.776324 −0.0663258 −0.0331629 0.999450i \(-0.510558\pi\)
−0.0331629 + 0.999450i \(0.510558\pi\)
\(138\) 0 0
\(139\) 8.76940 0.743811 0.371906 0.928271i \(-0.378705\pi\)
0.371906 + 0.928271i \(0.378705\pi\)
\(140\) −2.71839 −0.229746
\(141\) 0 0
\(142\) −7.86431 −0.659958
\(143\) 0 0
\(144\) 0 0
\(145\) −8.08116 −0.671104
\(146\) 13.8273 1.14436
\(147\) 0 0
\(148\) −10.5142 −0.864260
\(149\) −2.71490 −0.222413 −0.111206 0.993797i \(-0.535471\pi\)
−0.111206 + 0.993797i \(0.535471\pi\)
\(150\) 0 0
\(151\) −3.67720 −0.299247 −0.149623 0.988743i \(-0.547806\pi\)
−0.149623 + 0.988743i \(0.547806\pi\)
\(152\) −5.90568 −0.479014
\(153\) 0 0
\(154\) 0 0
\(155\) 4.75037 0.381559
\(156\) 0 0
\(157\) 21.0688 1.68148 0.840738 0.541443i \(-0.182122\pi\)
0.840738 + 0.541443i \(0.182122\pi\)
\(158\) 29.2143 2.32416
\(159\) 0 0
\(160\) −13.4342 −1.06207
\(161\) 5.20938 0.410557
\(162\) 0 0
\(163\) −16.9490 −1.32755 −0.663774 0.747933i \(-0.731046\pi\)
−0.663774 + 0.747933i \(0.731046\pi\)
\(164\) 0.0785863 0.00613656
\(165\) 0 0
\(166\) −13.0956 −1.01642
\(167\) −15.7297 −1.21720 −0.608599 0.793478i \(-0.708268\pi\)
−0.608599 + 0.793478i \(0.708268\pi\)
\(168\) 0 0
\(169\) 0.430832 0.0331409
\(170\) 7.34450 0.563298
\(171\) 0 0
\(172\) −11.3777 −0.867545
\(173\) 9.79142 0.744428 0.372214 0.928147i \(-0.378599\pi\)
0.372214 + 0.928147i \(0.378599\pi\)
\(174\) 0 0
\(175\) 0.381966 0.0288739
\(176\) 0 0
\(177\) 0 0
\(178\) −12.0757 −0.905111
\(179\) 13.4597 1.00603 0.503013 0.864279i \(-0.332225\pi\)
0.503013 + 0.864279i \(0.332225\pi\)
\(180\) 0 0
\(181\) −15.7516 −1.17081 −0.585404 0.810742i \(-0.699064\pi\)
−0.585404 + 0.810742i \(0.699064\pi\)
\(182\) 6.62204 0.490858
\(183\) 0 0
\(184\) 6.91874 0.510057
\(185\) −17.8616 −1.31321
\(186\) 0 0
\(187\) 0 0
\(188\) 15.6284 1.13982
\(189\) 0 0
\(190\) 17.2662 1.25262
\(191\) −9.10153 −0.658563 −0.329282 0.944232i \(-0.606807\pi\)
−0.329282 + 0.944232i \(0.606807\pi\)
\(192\) 0 0
\(193\) 14.3475 1.03275 0.516377 0.856361i \(-0.327280\pi\)
0.516377 + 0.856361i \(0.327280\pi\)
\(194\) −23.6767 −1.69989
\(195\) 0 0
\(196\) 1.26498 0.0903555
\(197\) −20.6333 −1.47006 −0.735029 0.678035i \(-0.762832\pi\)
−0.735029 + 0.678035i \(0.762832\pi\)
\(198\) 0 0
\(199\) 19.5733 1.38751 0.693755 0.720211i \(-0.255955\pi\)
0.693755 + 0.720211i \(0.255955\pi\)
\(200\) 0.507301 0.0358716
\(201\) 0 0
\(202\) 2.25393 0.158586
\(203\) 3.76049 0.263935
\(204\) 0 0
\(205\) 0.133503 0.00932429
\(206\) −27.0477 −1.88450
\(207\) 0 0
\(208\) 18.0667 1.25270
\(209\) 0 0
\(210\) 0 0
\(211\) 10.1880 0.701371 0.350685 0.936493i \(-0.385949\pi\)
0.350685 + 0.936493i \(0.385949\pi\)
\(212\) −8.86774 −0.609039
\(213\) 0 0
\(214\) −18.5133 −1.26555
\(215\) −19.3287 −1.31821
\(216\) 0 0
\(217\) −2.21054 −0.150061
\(218\) 6.12858 0.415080
\(219\) 0 0
\(220\) 0 0
\(221\) −6.93179 −0.466283
\(222\) 0 0
\(223\) −24.5903 −1.64669 −0.823345 0.567542i \(-0.807895\pi\)
−0.823345 + 0.567542i \(0.807895\pi\)
\(224\) 6.25149 0.417695
\(225\) 0 0
\(226\) 14.8527 0.987990
\(227\) 23.2865 1.54558 0.772789 0.634663i \(-0.218861\pi\)
0.772789 + 0.634663i \(0.218861\pi\)
\(228\) 0 0
\(229\) −6.48746 −0.428703 −0.214352 0.976757i \(-0.568764\pi\)
−0.214352 + 0.976757i \(0.568764\pi\)
\(230\) −20.2281 −1.33380
\(231\) 0 0
\(232\) 4.99443 0.327900
\(233\) −23.0166 −1.50787 −0.753935 0.656949i \(-0.771846\pi\)
−0.753935 + 0.656949i \(0.771846\pi\)
\(234\) 0 0
\(235\) 26.5498 1.73192
\(236\) −15.2136 −0.990324
\(237\) 0 0
\(238\) −3.41770 −0.221537
\(239\) 9.08553 0.587694 0.293847 0.955852i \(-0.405064\pi\)
0.293847 + 0.955852i \(0.405064\pi\)
\(240\) 0 0
\(241\) 6.60908 0.425728 0.212864 0.977082i \(-0.431721\pi\)
0.212864 + 0.977082i \(0.431721\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.49548 0.415830
\(245\) 2.14896 0.137292
\(246\) 0 0
\(247\) −16.2960 −1.03689
\(248\) −2.93589 −0.186429
\(249\) 0 0
\(250\) −20.8982 −1.32172
\(251\) 3.59669 0.227021 0.113511 0.993537i \(-0.463790\pi\)
0.113511 + 0.993537i \(0.463790\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −5.88950 −0.369540
\(255\) 0 0
\(256\) 20.7749 1.29843
\(257\) −8.69374 −0.542301 −0.271150 0.962537i \(-0.587404\pi\)
−0.271150 + 0.962537i \(0.587404\pi\)
\(258\) 0 0
\(259\) 8.31175 0.516467
\(260\) −9.96237 −0.617840
\(261\) 0 0
\(262\) 0.631167 0.0389936
\(263\) −23.6423 −1.45785 −0.728925 0.684594i \(-0.759980\pi\)
−0.728925 + 0.684594i \(0.759980\pi\)
\(264\) 0 0
\(265\) −15.0646 −0.925414
\(266\) −8.03469 −0.492638
\(267\) 0 0
\(268\) −11.8749 −0.725375
\(269\) 11.9533 0.728808 0.364404 0.931241i \(-0.381273\pi\)
0.364404 + 0.931241i \(0.381273\pi\)
\(270\) 0 0
\(271\) −28.7131 −1.74420 −0.872099 0.489330i \(-0.837242\pi\)
−0.872099 + 0.489330i \(0.837242\pi\)
\(272\) −9.32442 −0.565376
\(273\) 0 0
\(274\) −1.40276 −0.0847438
\(275\) 0 0
\(276\) 0 0
\(277\) 15.6358 0.939464 0.469732 0.882809i \(-0.344350\pi\)
0.469732 + 0.882809i \(0.344350\pi\)
\(278\) 15.8457 0.950359
\(279\) 0 0
\(280\) 2.85410 0.170565
\(281\) 16.2703 0.970607 0.485304 0.874346i \(-0.338709\pi\)
0.485304 + 0.874346i \(0.338709\pi\)
\(282\) 0 0
\(283\) −2.18428 −0.129842 −0.0649209 0.997890i \(-0.520679\pi\)
−0.0649209 + 0.997890i \(0.520679\pi\)
\(284\) −5.50558 −0.326696
\(285\) 0 0
\(286\) 0 0
\(287\) −0.0621247 −0.00366710
\(288\) 0 0
\(289\) −13.4224 −0.789555
\(290\) −14.6020 −0.857462
\(291\) 0 0
\(292\) 9.68013 0.566487
\(293\) 12.3305 0.720353 0.360176 0.932884i \(-0.382717\pi\)
0.360176 + 0.932884i \(0.382717\pi\)
\(294\) 0 0
\(295\) −25.8451 −1.50476
\(296\) 11.0391 0.641634
\(297\) 0 0
\(298\) −4.90561 −0.284174
\(299\) 19.0914 1.10408
\(300\) 0 0
\(301\) 8.99443 0.518430
\(302\) −6.64443 −0.382344
\(303\) 0 0
\(304\) −21.9208 −1.25725
\(305\) 11.0346 0.631840
\(306\) 0 0
\(307\) −1.73840 −0.0992160 −0.0496080 0.998769i \(-0.515797\pi\)
−0.0496080 + 0.998769i \(0.515797\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.58355 0.487513
\(311\) 11.2208 0.636273 0.318137 0.948045i \(-0.396943\pi\)
0.318137 + 0.948045i \(0.396943\pi\)
\(312\) 0 0
\(313\) 29.5442 1.66994 0.834970 0.550296i \(-0.185485\pi\)
0.834970 + 0.550296i \(0.185485\pi\)
\(314\) 38.0698 2.14840
\(315\) 0 0
\(316\) 20.4521 1.15052
\(317\) 8.53297 0.479259 0.239630 0.970864i \(-0.422974\pi\)
0.239630 + 0.970864i \(0.422974\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.08677 −0.172556
\(321\) 0 0
\(322\) 9.41297 0.524564
\(323\) 8.41052 0.467974
\(324\) 0 0
\(325\) 1.39983 0.0776488
\(326\) −30.6256 −1.69619
\(327\) 0 0
\(328\) −0.0825097 −0.00455583
\(329\) −12.3547 −0.681138
\(330\) 0 0
\(331\) 3.22164 0.177078 0.0885388 0.996073i \(-0.471780\pi\)
0.0885388 + 0.996073i \(0.471780\pi\)
\(332\) −9.16789 −0.503153
\(333\) 0 0
\(334\) −28.4223 −1.55520
\(335\) −20.1732 −1.10218
\(336\) 0 0
\(337\) −3.52393 −0.191961 −0.0959803 0.995383i \(-0.530599\pi\)
−0.0959803 + 0.995383i \(0.530599\pi\)
\(338\) 0.778481 0.0423438
\(339\) 0 0
\(340\) 5.14168 0.278847
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 11.9458 0.644073
\(345\) 0 0
\(346\) 17.6924 0.951148
\(347\) −4.90171 −0.263138 −0.131569 0.991307i \(-0.542001\pi\)
−0.131569 + 0.991307i \(0.542001\pi\)
\(348\) 0 0
\(349\) −11.5598 −0.618780 −0.309390 0.950935i \(-0.600125\pi\)
−0.309390 + 0.950935i \(0.600125\pi\)
\(350\) 0.690184 0.0368919
\(351\) 0 0
\(352\) 0 0
\(353\) −29.5076 −1.57053 −0.785264 0.619161i \(-0.787473\pi\)
−0.785264 + 0.619161i \(0.787473\pi\)
\(354\) 0 0
\(355\) −9.35296 −0.496404
\(356\) −8.45385 −0.448053
\(357\) 0 0
\(358\) 24.3207 1.28539
\(359\) 7.57351 0.399715 0.199857 0.979825i \(-0.435952\pi\)
0.199857 + 0.979825i \(0.435952\pi\)
\(360\) 0 0
\(361\) 0.772330 0.0406490
\(362\) −28.4620 −1.49593
\(363\) 0 0
\(364\) 4.63590 0.242987
\(365\) 16.4447 0.860757
\(366\) 0 0
\(367\) −9.52922 −0.497421 −0.248711 0.968578i \(-0.580007\pi\)
−0.248711 + 0.968578i \(0.580007\pi\)
\(368\) 25.6812 1.33872
\(369\) 0 0
\(370\) −32.2746 −1.67788
\(371\) 7.01020 0.363951
\(372\) 0 0
\(373\) 17.3921 0.900527 0.450264 0.892896i \(-0.351330\pi\)
0.450264 + 0.892896i \(0.351330\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −16.4087 −0.846213
\(377\) 13.7815 0.709783
\(378\) 0 0
\(379\) 18.3456 0.942349 0.471175 0.882040i \(-0.343830\pi\)
0.471175 + 0.882040i \(0.343830\pi\)
\(380\) 12.0876 0.620081
\(381\) 0 0
\(382\) −16.4458 −0.841439
\(383\) 2.83652 0.144940 0.0724698 0.997371i \(-0.476912\pi\)
0.0724698 + 0.997371i \(0.476912\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.9248 1.31954
\(387\) 0 0
\(388\) −16.5754 −0.841487
\(389\) 20.6813 1.04858 0.524291 0.851539i \(-0.324330\pi\)
0.524291 + 0.851539i \(0.324330\pi\)
\(390\) 0 0
\(391\) −9.85327 −0.498301
\(392\) −1.32813 −0.0670807
\(393\) 0 0
\(394\) −37.2827 −1.87828
\(395\) 34.7443 1.74817
\(396\) 0 0
\(397\) 26.6868 1.33937 0.669685 0.742645i \(-0.266429\pi\)
0.669685 + 0.742645i \(0.266429\pi\)
\(398\) 35.3674 1.77281
\(399\) 0 0
\(400\) 1.88301 0.0941506
\(401\) 12.6012 0.629272 0.314636 0.949212i \(-0.398117\pi\)
0.314636 + 0.949212i \(0.398117\pi\)
\(402\) 0 0
\(403\) −8.10121 −0.403550
\(404\) 1.57792 0.0785043
\(405\) 0 0
\(406\) 6.79493 0.337227
\(407\) 0 0
\(408\) 0 0
\(409\) −0.0123946 −0.000612872 0 −0.000306436 1.00000i \(-0.500098\pi\)
−0.000306436 1.00000i \(0.500098\pi\)
\(410\) 0.241231 0.0119135
\(411\) 0 0
\(412\) −18.9353 −0.932876
\(413\) 12.0268 0.591801
\(414\) 0 0
\(415\) −15.5745 −0.764524
\(416\) 22.9105 1.12328
\(417\) 0 0
\(418\) 0 0
\(419\) 32.9560 1.61001 0.805003 0.593271i \(-0.202164\pi\)
0.805003 + 0.593271i \(0.202164\pi\)
\(420\) 0 0
\(421\) −27.3708 −1.33397 −0.666986 0.745070i \(-0.732416\pi\)
−0.666986 + 0.745070i \(0.732416\pi\)
\(422\) 18.4089 0.896134
\(423\) 0 0
\(424\) 9.31046 0.452156
\(425\) −0.722468 −0.0350448
\(426\) 0 0
\(427\) −5.13486 −0.248493
\(428\) −12.9607 −0.626478
\(429\) 0 0
\(430\) −34.9255 −1.68426
\(431\) −24.7125 −1.19036 −0.595180 0.803593i \(-0.702919\pi\)
−0.595180 + 0.803593i \(0.702919\pi\)
\(432\) 0 0
\(433\) −27.4299 −1.31820 −0.659099 0.752056i \(-0.729062\pi\)
−0.659099 + 0.752056i \(0.729062\pi\)
\(434\) −3.99428 −0.191732
\(435\) 0 0
\(436\) 4.29044 0.205475
\(437\) −23.1641 −1.10809
\(438\) 0 0
\(439\) 3.36929 0.160807 0.0804037 0.996762i \(-0.474379\pi\)
0.0804037 + 0.996762i \(0.474379\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.5252 −0.595764
\(443\) −7.97012 −0.378672 −0.189336 0.981912i \(-0.560634\pi\)
−0.189336 + 0.981912i \(0.560634\pi\)
\(444\) 0 0
\(445\) −14.3615 −0.680801
\(446\) −44.4329 −2.10396
\(447\) 0 0
\(448\) 1.43640 0.0678636
\(449\) −5.55142 −0.261988 −0.130994 0.991383i \(-0.541817\pi\)
−0.130994 + 0.991383i \(0.541817\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.3980 0.489080
\(453\) 0 0
\(454\) 42.0769 1.97477
\(455\) 7.87553 0.369211
\(456\) 0 0
\(457\) −16.7915 −0.785471 −0.392735 0.919651i \(-0.628471\pi\)
−0.392735 + 0.919651i \(0.628471\pi\)
\(458\) −11.7223 −0.547749
\(459\) 0 0
\(460\) −14.1611 −0.660266
\(461\) 17.3351 0.807376 0.403688 0.914897i \(-0.367728\pi\)
0.403688 + 0.914897i \(0.367728\pi\)
\(462\) 0 0
\(463\) 24.9175 1.15801 0.579006 0.815323i \(-0.303441\pi\)
0.579006 + 0.815323i \(0.303441\pi\)
\(464\) 18.5384 0.860625
\(465\) 0 0
\(466\) −41.5893 −1.92659
\(467\) 22.4093 1.03698 0.518490 0.855083i \(-0.326494\pi\)
0.518490 + 0.855083i \(0.326494\pi\)
\(468\) 0 0
\(469\) 9.38744 0.433472
\(470\) 47.9735 2.21285
\(471\) 0 0
\(472\) 15.9732 0.735225
\(473\) 0 0
\(474\) 0 0
\(475\) −1.69845 −0.0779304
\(476\) −2.39264 −0.109666
\(477\) 0 0
\(478\) 16.4169 0.750890
\(479\) 3.20429 0.146408 0.0732039 0.997317i \(-0.476678\pi\)
0.0732039 + 0.997317i \(0.476678\pi\)
\(480\) 0 0
\(481\) 30.4610 1.38890
\(482\) 11.9421 0.543948
\(483\) 0 0
\(484\) 0 0
\(485\) −28.1585 −1.27861
\(486\) 0 0
\(487\) 8.39844 0.380570 0.190285 0.981729i \(-0.439059\pi\)
0.190285 + 0.981729i \(0.439059\pi\)
\(488\) −6.81976 −0.308716
\(489\) 0 0
\(490\) 3.88301 0.175417
\(491\) 34.7888 1.57000 0.784998 0.619498i \(-0.212664\pi\)
0.784998 + 0.619498i \(0.212664\pi\)
\(492\) 0 0
\(493\) −7.11277 −0.320343
\(494\) −29.4456 −1.32482
\(495\) 0 0
\(496\) −10.8975 −0.489312
\(497\) 4.35232 0.195228
\(498\) 0 0
\(499\) 15.1305 0.677333 0.338667 0.940906i \(-0.390024\pi\)
0.338667 + 0.940906i \(0.390024\pi\)
\(500\) −14.6303 −0.654285
\(501\) 0 0
\(502\) 6.49895 0.290063
\(503\) 36.7358 1.63797 0.818985 0.573815i \(-0.194537\pi\)
0.818985 + 0.573815i \(0.194537\pi\)
\(504\) 0 0
\(505\) 2.68059 0.119285
\(506\) 0 0
\(507\) 0 0
\(508\) −4.12308 −0.182932
\(509\) −31.2822 −1.38656 −0.693279 0.720669i \(-0.743835\pi\)
−0.693279 + 0.720669i \(0.743835\pi\)
\(510\) 0 0
\(511\) −7.65242 −0.338523
\(512\) 17.7237 0.783285
\(513\) 0 0
\(514\) −15.7089 −0.692892
\(515\) −32.1676 −1.41747
\(516\) 0 0
\(517\) 0 0
\(518\) 15.0187 0.659884
\(519\) 0 0
\(520\) 10.4597 0.458690
\(521\) −41.5773 −1.82153 −0.910767 0.412921i \(-0.864509\pi\)
−0.910767 + 0.412921i \(0.864509\pi\)
\(522\) 0 0
\(523\) −17.4520 −0.763122 −0.381561 0.924344i \(-0.624613\pi\)
−0.381561 + 0.924344i \(0.624613\pi\)
\(524\) 0.441862 0.0193028
\(525\) 0 0
\(526\) −42.7199 −1.86268
\(527\) 4.18112 0.182132
\(528\) 0 0
\(529\) 4.13768 0.179899
\(530\) −27.2207 −1.18239
\(531\) 0 0
\(532\) −5.62486 −0.243869
\(533\) −0.227675 −0.00986170
\(534\) 0 0
\(535\) −22.0178 −0.951911
\(536\) 12.4677 0.538525
\(537\) 0 0
\(538\) 21.5988 0.931189
\(539\) 0 0
\(540\) 0 0
\(541\) 27.3372 1.17532 0.587659 0.809109i \(-0.300050\pi\)
0.587659 + 0.809109i \(0.300050\pi\)
\(542\) −51.8825 −2.22854
\(543\) 0 0
\(544\) −11.8244 −0.506965
\(545\) 7.28867 0.312212
\(546\) 0 0
\(547\) −31.1721 −1.33282 −0.666411 0.745585i \(-0.732170\pi\)
−0.666411 + 0.745585i \(0.732170\pi\)
\(548\) −0.982032 −0.0419503
\(549\) 0 0
\(550\) 0 0
\(551\) −16.7214 −0.712358
\(552\) 0 0
\(553\) −16.1679 −0.687531
\(554\) 28.2527 1.20034
\(555\) 0 0
\(556\) 11.0931 0.470452
\(557\) 4.18066 0.177140 0.0885700 0.996070i \(-0.471770\pi\)
0.0885700 + 0.996070i \(0.471770\pi\)
\(558\) 0 0
\(559\) 32.9629 1.39418
\(560\) 10.5939 0.447675
\(561\) 0 0
\(562\) 29.3993 1.24013
\(563\) 24.7878 1.04468 0.522340 0.852737i \(-0.325059\pi\)
0.522340 + 0.852737i \(0.325059\pi\)
\(564\) 0 0
\(565\) 17.6643 0.743141
\(566\) −3.94682 −0.165897
\(567\) 0 0
\(568\) 5.78045 0.242542
\(569\) −35.0889 −1.47100 −0.735502 0.677522i \(-0.763054\pi\)
−0.735502 + 0.677522i \(0.763054\pi\)
\(570\) 0 0
\(571\) 6.83359 0.285977 0.142988 0.989724i \(-0.454329\pi\)
0.142988 + 0.989724i \(0.454329\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.112255 −0.00468542
\(575\) 1.98981 0.0829807
\(576\) 0 0
\(577\) −45.3214 −1.88675 −0.943377 0.331721i \(-0.892371\pi\)
−0.943377 + 0.331721i \(0.892371\pi\)
\(578\) −24.2533 −1.00881
\(579\) 0 0
\(580\) −10.2225 −0.424465
\(581\) 7.24748 0.300676
\(582\) 0 0
\(583\) 0 0
\(584\) −10.1634 −0.420565
\(585\) 0 0
\(586\) 22.2802 0.920387
\(587\) 44.8902 1.85282 0.926409 0.376518i \(-0.122879\pi\)
0.926409 + 0.376518i \(0.122879\pi\)
\(588\) 0 0
\(589\) 9.82941 0.405014
\(590\) −46.7002 −1.92262
\(591\) 0 0
\(592\) 40.9752 1.68407
\(593\) 38.8436 1.59512 0.797558 0.603242i \(-0.206125\pi\)
0.797558 + 0.603242i \(0.206125\pi\)
\(594\) 0 0
\(595\) −4.06464 −0.166634
\(596\) −3.43428 −0.140674
\(597\) 0 0
\(598\) 34.4967 1.41068
\(599\) −17.2289 −0.703954 −0.351977 0.936009i \(-0.614490\pi\)
−0.351977 + 0.936009i \(0.614490\pi\)
\(600\) 0 0
\(601\) −29.0667 −1.18565 −0.592827 0.805330i \(-0.701988\pi\)
−0.592827 + 0.805330i \(0.701988\pi\)
\(602\) 16.2523 0.662393
\(603\) 0 0
\(604\) −4.65158 −0.189270
\(605\) 0 0
\(606\) 0 0
\(607\) −24.0011 −0.974174 −0.487087 0.873353i \(-0.661941\pi\)
−0.487087 + 0.873353i \(0.661941\pi\)
\(608\) −27.7979 −1.12736
\(609\) 0 0
\(610\) 19.9387 0.807294
\(611\) −45.2777 −1.83174
\(612\) 0 0
\(613\) −47.4567 −1.91676 −0.958379 0.285498i \(-0.907841\pi\)
−0.958379 + 0.285498i \(0.907841\pi\)
\(614\) −3.14117 −0.126767
\(615\) 0 0
\(616\) 0 0
\(617\) −14.0679 −0.566354 −0.283177 0.959068i \(-0.591388\pi\)
−0.283177 + 0.959068i \(0.591388\pi\)
\(618\) 0 0
\(619\) 35.1265 1.41185 0.705927 0.708284i \(-0.250530\pi\)
0.705927 + 0.708284i \(0.250530\pi\)
\(620\) 6.00910 0.241331
\(621\) 0 0
\(622\) 20.2751 0.812959
\(623\) 6.68300 0.267749
\(624\) 0 0
\(625\) −22.9443 −0.917771
\(626\) 53.3842 2.13366
\(627\) 0 0
\(628\) 26.6516 1.06351
\(629\) −15.7212 −0.626846
\(630\) 0 0
\(631\) −35.2530 −1.40340 −0.701700 0.712473i \(-0.747575\pi\)
−0.701700 + 0.712473i \(0.747575\pi\)
\(632\) −21.4731 −0.854156
\(633\) 0 0
\(634\) 15.4184 0.612344
\(635\) −7.00434 −0.277959
\(636\) 0 0
\(637\) −3.66481 −0.145205
\(638\) 0 0
\(639\) 0 0
\(640\) 21.2909 0.841595
\(641\) 36.1668 1.42850 0.714252 0.699889i \(-0.246767\pi\)
0.714252 + 0.699889i \(0.246767\pi\)
\(642\) 0 0
\(643\) −2.33953 −0.0922620 −0.0461310 0.998935i \(-0.514689\pi\)
−0.0461310 + 0.998935i \(0.514689\pi\)
\(644\) 6.58975 0.259673
\(645\) 0 0
\(646\) 15.1972 0.597925
\(647\) 28.6902 1.12793 0.563965 0.825799i \(-0.309275\pi\)
0.563965 + 0.825799i \(0.309275\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.52939 0.0992109
\(651\) 0 0
\(652\) −21.4401 −0.839659
\(653\) −35.8827 −1.40420 −0.702099 0.712079i \(-0.747754\pi\)
−0.702099 + 0.712079i \(0.747754\pi\)
\(654\) 0 0
\(655\) 0.750641 0.0293300
\(656\) −0.306261 −0.0119575
\(657\) 0 0
\(658\) −22.3240 −0.870282
\(659\) 48.1529 1.87577 0.937885 0.346945i \(-0.112781\pi\)
0.937885 + 0.346945i \(0.112781\pi\)
\(660\) 0 0
\(661\) −26.9186 −1.04701 −0.523506 0.852022i \(-0.675376\pi\)
−0.523506 + 0.852022i \(0.675376\pi\)
\(662\) 5.82127 0.226250
\(663\) 0 0
\(664\) 9.62560 0.373546
\(665\) −9.55559 −0.370550
\(666\) 0 0
\(667\) 19.5899 0.758523
\(668\) −19.8977 −0.769864
\(669\) 0 0
\(670\) −36.4515 −1.40825
\(671\) 0 0
\(672\) 0 0
\(673\) −6.53324 −0.251838 −0.125919 0.992041i \(-0.540188\pi\)
−0.125919 + 0.992041i \(0.540188\pi\)
\(674\) −6.36747 −0.245266
\(675\) 0 0
\(676\) 0.544993 0.0209613
\(677\) 41.9065 1.61060 0.805299 0.592869i \(-0.202005\pi\)
0.805299 + 0.592869i \(0.202005\pi\)
\(678\) 0 0
\(679\) 13.1033 0.502859
\(680\) −5.39838 −0.207018
\(681\) 0 0
\(682\) 0 0
\(683\) 17.5429 0.671259 0.335630 0.941994i \(-0.391051\pi\)
0.335630 + 0.941994i \(0.391051\pi\)
\(684\) 0 0
\(685\) −1.66829 −0.0637421
\(686\) −1.80692 −0.0689887
\(687\) 0 0
\(688\) 44.3406 1.69047
\(689\) 25.6910 0.978751
\(690\) 0 0
\(691\) −36.6341 −1.39363 −0.696813 0.717253i \(-0.745399\pi\)
−0.696813 + 0.717253i \(0.745399\pi\)
\(692\) 12.3859 0.470842
\(693\) 0 0
\(694\) −8.85703 −0.336208
\(695\) 18.8451 0.714836
\(696\) 0 0
\(697\) 0.117505 0.00445083
\(698\) −20.8876 −0.790608
\(699\) 0 0
\(700\) 0.483178 0.0182624
\(701\) 16.5248 0.624132 0.312066 0.950060i \(-0.398979\pi\)
0.312066 + 0.950060i \(0.398979\pi\)
\(702\) 0 0
\(703\) −36.9591 −1.39394
\(704\) 0 0
\(705\) 0 0
\(706\) −53.3179 −2.00665
\(707\) −1.24739 −0.0469128
\(708\) 0 0
\(709\) −17.8668 −0.671002 −0.335501 0.942040i \(-0.608906\pi\)
−0.335501 + 0.942040i \(0.608906\pi\)
\(710\) −16.9001 −0.634249
\(711\) 0 0
\(712\) 8.87590 0.332639
\(713\) −11.5156 −0.431261
\(714\) 0 0
\(715\) 0 0
\(716\) 17.0262 0.636300
\(717\) 0 0
\(718\) 13.6848 0.510711
\(719\) 19.7323 0.735891 0.367945 0.929847i \(-0.380061\pi\)
0.367945 + 0.929847i \(0.380061\pi\)
\(720\) 0 0
\(721\) 14.9689 0.557471
\(722\) 1.39554 0.0519367
\(723\) 0 0
\(724\) −19.9254 −0.740522
\(725\) 1.43638 0.0533459
\(726\) 0 0
\(727\) −28.6895 −1.06403 −0.532017 0.846733i \(-0.678566\pi\)
−0.532017 + 0.846733i \(0.678566\pi\)
\(728\) −4.86735 −0.180396
\(729\) 0 0
\(730\) 29.7144 1.09978
\(731\) −17.0125 −0.629229
\(732\) 0 0
\(733\) −5.36355 −0.198107 −0.0990535 0.995082i \(-0.531582\pi\)
−0.0990535 + 0.995082i \(0.531582\pi\)
\(734\) −17.2186 −0.635549
\(735\) 0 0
\(736\) 32.5664 1.20041
\(737\) 0 0
\(738\) 0 0
\(739\) 45.1733 1.66173 0.830864 0.556476i \(-0.187847\pi\)
0.830864 + 0.556476i \(0.187847\pi\)
\(740\) −22.5946 −0.830593
\(741\) 0 0
\(742\) 12.6669 0.465017
\(743\) 19.8716 0.729020 0.364510 0.931199i \(-0.381236\pi\)
0.364510 + 0.931199i \(0.381236\pi\)
\(744\) 0 0
\(745\) −5.83421 −0.213749
\(746\) 31.4262 1.15059
\(747\) 0 0
\(748\) 0 0
\(749\) 10.2458 0.374372
\(750\) 0 0
\(751\) 3.72318 0.135861 0.0679304 0.997690i \(-0.478360\pi\)
0.0679304 + 0.997690i \(0.478360\pi\)
\(752\) −60.9061 −2.22102
\(753\) 0 0
\(754\) 24.9021 0.906882
\(755\) −7.90217 −0.287589
\(756\) 0 0
\(757\) 37.2656 1.35444 0.677220 0.735780i \(-0.263184\pi\)
0.677220 + 0.735780i \(0.263184\pi\)
\(758\) 33.1491 1.20403
\(759\) 0 0
\(760\) −12.6911 −0.460354
\(761\) 21.3908 0.775416 0.387708 0.921782i \(-0.373267\pi\)
0.387708 + 0.921782i \(0.373267\pi\)
\(762\) 0 0
\(763\) −3.39172 −0.122788
\(764\) −11.5132 −0.416534
\(765\) 0 0
\(766\) 5.12539 0.185188
\(767\) 44.0760 1.59149
\(768\) 0 0
\(769\) −20.2160 −0.729006 −0.364503 0.931202i \(-0.618761\pi\)
−0.364503 + 0.931202i \(0.618761\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.1492 0.653206
\(773\) −39.5108 −1.42111 −0.710553 0.703644i \(-0.751555\pi\)
−0.710553 + 0.703644i \(0.751555\pi\)
\(774\) 0 0
\(775\) −0.844351 −0.0303300
\(776\) 17.4029 0.624728
\(777\) 0 0
\(778\) 37.3695 1.33976
\(779\) 0.276244 0.00989747
\(780\) 0 0
\(781\) 0 0
\(782\) −17.8041 −0.636674
\(783\) 0 0
\(784\) −4.92979 −0.176064
\(785\) 45.2761 1.61597
\(786\) 0 0
\(787\) −34.2457 −1.22073 −0.610364 0.792121i \(-0.708977\pi\)
−0.610364 + 0.792121i \(0.708977\pi\)
\(788\) −26.1006 −0.929795
\(789\) 0 0
\(790\) 62.7803 2.23362
\(791\) −8.21990 −0.292266
\(792\) 0 0
\(793\) −18.8183 −0.668256
\(794\) 48.2210 1.71130
\(795\) 0 0
\(796\) 24.7597 0.877585
\(797\) 16.1138 0.570779 0.285390 0.958412i \(-0.407877\pi\)
0.285390 + 0.958412i \(0.407877\pi\)
\(798\) 0 0
\(799\) 23.3683 0.826710
\(800\) 2.38786 0.0844235
\(801\) 0 0
\(802\) 22.7694 0.804014
\(803\) 0 0
\(804\) 0 0
\(805\) 11.1948 0.394564
\(806\) −14.6383 −0.515611
\(807\) 0 0
\(808\) −1.65669 −0.0582823
\(809\) −7.67619 −0.269880 −0.134940 0.990854i \(-0.543084\pi\)
−0.134940 + 0.990854i \(0.543084\pi\)
\(810\) 0 0
\(811\) −1.03802 −0.0364498 −0.0182249 0.999834i \(-0.505801\pi\)
−0.0182249 + 0.999834i \(0.505801\pi\)
\(812\) 4.75694 0.166936
\(813\) 0 0
\(814\) 0 0
\(815\) −36.4228 −1.27583
\(816\) 0 0
\(817\) −39.9947 −1.39924
\(818\) −0.0223961 −0.000783060 0
\(819\) 0 0
\(820\) 0.168879 0.00589750
\(821\) 7.44620 0.259874 0.129937 0.991522i \(-0.458522\pi\)
0.129937 + 0.991522i \(0.458522\pi\)
\(822\) 0 0
\(823\) 2.94956 0.102815 0.0514076 0.998678i \(-0.483629\pi\)
0.0514076 + 0.998678i \(0.483629\pi\)
\(824\) 19.8807 0.692576
\(825\) 0 0
\(826\) 21.7315 0.756137
\(827\) −42.6293 −1.48237 −0.741183 0.671303i \(-0.765735\pi\)
−0.741183 + 0.671303i \(0.765735\pi\)
\(828\) 0 0
\(829\) 6.02334 0.209199 0.104600 0.994514i \(-0.466644\pi\)
0.104600 + 0.994514i \(0.466644\pi\)
\(830\) −28.1420 −0.976824
\(831\) 0 0
\(832\) 5.26414 0.182501
\(833\) 1.89145 0.0655347
\(834\) 0 0
\(835\) −33.8025 −1.16978
\(836\) 0 0
\(837\) 0 0
\(838\) 59.5490 2.05709
\(839\) 2.90271 0.100213 0.0501064 0.998744i \(-0.484044\pi\)
0.0501064 + 0.998744i \(0.484044\pi\)
\(840\) 0 0
\(841\) −14.8587 −0.512368
\(842\) −49.4570 −1.70440
\(843\) 0 0
\(844\) 12.8876 0.443609
\(845\) 0.925842 0.0318499
\(846\) 0 0
\(847\) 0 0
\(848\) 34.5588 1.18675
\(849\) 0 0
\(850\) −1.30545 −0.0447764
\(851\) 43.2991 1.48427
\(852\) 0 0
\(853\) −1.92232 −0.0658191 −0.0329095 0.999458i \(-0.510477\pi\)
−0.0329095 + 0.999458i \(0.510477\pi\)
\(854\) −9.27830 −0.317497
\(855\) 0 0
\(856\) 13.6077 0.465103
\(857\) 55.9198 1.91019 0.955093 0.296308i \(-0.0957554\pi\)
0.955093 + 0.296308i \(0.0957554\pi\)
\(858\) 0 0
\(859\) 0.723243 0.0246767 0.0123384 0.999924i \(-0.496072\pi\)
0.0123384 + 0.999924i \(0.496072\pi\)
\(860\) −24.4503 −0.833750
\(861\) 0 0
\(862\) −44.6536 −1.52091
\(863\) 41.6422 1.41752 0.708759 0.705451i \(-0.249256\pi\)
0.708759 + 0.705451i \(0.249256\pi\)
\(864\) 0 0
\(865\) 21.0414 0.715429
\(866\) −49.5638 −1.68425
\(867\) 0 0
\(868\) −2.79628 −0.0949121
\(869\) 0 0
\(870\) 0 0
\(871\) 34.4032 1.16571
\(872\) −4.50464 −0.152546
\(873\) 0 0
\(874\) −41.8558 −1.41579
\(875\) 11.5656 0.390990
\(876\) 0 0
\(877\) 40.1509 1.35580 0.677900 0.735154i \(-0.262890\pi\)
0.677900 + 0.735154i \(0.262890\pi\)
\(878\) 6.08805 0.205462
\(879\) 0 0
\(880\) 0 0
\(881\) 3.28404 0.110642 0.0553210 0.998469i \(-0.482382\pi\)
0.0553210 + 0.998469i \(0.482382\pi\)
\(882\) 0 0
\(883\) −4.76427 −0.160331 −0.0801653 0.996782i \(-0.525545\pi\)
−0.0801653 + 0.996782i \(0.525545\pi\)
\(884\) −8.76855 −0.294918
\(885\) 0 0
\(886\) −14.4014 −0.483825
\(887\) 8.82252 0.296231 0.148116 0.988970i \(-0.452679\pi\)
0.148116 + 0.988970i \(0.452679\pi\)
\(888\) 0 0
\(889\) 3.25941 0.109317
\(890\) −25.9502 −0.869852
\(891\) 0 0
\(892\) −31.1062 −1.04151
\(893\) 54.9366 1.83838
\(894\) 0 0
\(895\) 28.9244 0.966835
\(896\) −9.90751 −0.330987
\(897\) 0 0
\(898\) −10.0310 −0.334739
\(899\) −8.31273 −0.277245
\(900\) 0 0
\(901\) −13.2594 −0.441735
\(902\) 0 0
\(903\) 0 0
\(904\) −10.9171 −0.363098
\(905\) −33.8496 −1.12520
\(906\) 0 0
\(907\) −18.1246 −0.601816 −0.300908 0.953653i \(-0.597290\pi\)
−0.300908 + 0.953653i \(0.597290\pi\)
\(908\) 29.4569 0.977560
\(909\) 0 0
\(910\) 14.2305 0.471736
\(911\) 40.4176 1.33910 0.669548 0.742769i \(-0.266488\pi\)
0.669548 + 0.742769i \(0.266488\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −30.3409 −1.00359
\(915\) 0 0
\(916\) −8.20648 −0.271150
\(917\) −0.349304 −0.0115350
\(918\) 0 0
\(919\) 29.0293 0.957587 0.478794 0.877928i \(-0.341074\pi\)
0.478794 + 0.877928i \(0.341074\pi\)
\(920\) 14.8681 0.490187
\(921\) 0 0
\(922\) 31.3232 1.03158
\(923\) 15.9504 0.525014
\(924\) 0 0
\(925\) 3.17481 0.104387
\(926\) 45.0240 1.47958
\(927\) 0 0
\(928\) 23.5087 0.771711
\(929\) 21.6963 0.711832 0.355916 0.934518i \(-0.384169\pi\)
0.355916 + 0.934518i \(0.384169\pi\)
\(930\) 0 0
\(931\) 4.44661 0.145732
\(932\) −29.1155 −0.953710
\(933\) 0 0
\(934\) 40.4920 1.32494
\(935\) 0 0
\(936\) 0 0
\(937\) −49.4997 −1.61709 −0.808543 0.588438i \(-0.799743\pi\)
−0.808543 + 0.588438i \(0.799743\pi\)
\(938\) 16.9624 0.553842
\(939\) 0 0
\(940\) 33.5849 1.09542
\(941\) −59.9530 −1.95441 −0.977206 0.212293i \(-0.931907\pi\)
−0.977206 + 0.212293i \(0.931907\pi\)
\(942\) 0 0
\(943\) −0.323631 −0.0105389
\(944\) 59.2896 1.92971
\(945\) 0 0
\(946\) 0 0
\(947\) −59.3880 −1.92985 −0.964925 0.262526i \(-0.915445\pi\)
−0.964925 + 0.262526i \(0.915445\pi\)
\(948\) 0 0
\(949\) −28.0446 −0.910368
\(950\) −3.06898 −0.0995708
\(951\) 0 0
\(952\) 2.51209 0.0814172
\(953\) −56.6435 −1.83486 −0.917432 0.397893i \(-0.869742\pi\)
−0.917432 + 0.397893i \(0.869742\pi\)
\(954\) 0 0
\(955\) −19.5588 −0.632909
\(956\) 11.4930 0.371710
\(957\) 0 0
\(958\) 5.78992 0.187064
\(959\) 0.776324 0.0250688
\(960\) 0 0
\(961\) −26.1135 −0.842371
\(962\) 55.0407 1.77458
\(963\) 0 0
\(964\) 8.36034 0.269268
\(965\) 30.8322 0.992523
\(966\) 0 0
\(967\) 8.51193 0.273725 0.136863 0.990590i \(-0.456298\pi\)
0.136863 + 0.990590i \(0.456298\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −50.8803 −1.63367
\(971\) −8.61967 −0.276618 −0.138309 0.990389i \(-0.544167\pi\)
−0.138309 + 0.990389i \(0.544167\pi\)
\(972\) 0 0
\(973\) −8.76940 −0.281134
\(974\) 15.1754 0.486250
\(975\) 0 0
\(976\) −25.3137 −0.810273
\(977\) 34.3030 1.09745 0.548726 0.836002i \(-0.315113\pi\)
0.548726 + 0.836002i \(0.315113\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.71839 0.0868357
\(981\) 0 0
\(982\) 62.8607 2.00597
\(983\) −55.0178 −1.75479 −0.877397 0.479765i \(-0.840722\pi\)
−0.877397 + 0.479765i \(0.840722\pi\)
\(984\) 0 0
\(985\) −44.3401 −1.41279
\(986\) −12.8522 −0.409299
\(987\) 0 0
\(988\) −20.6140 −0.655820
\(989\) 46.8554 1.48992
\(990\) 0 0
\(991\) 42.7838 1.35907 0.679536 0.733642i \(-0.262181\pi\)
0.679536 + 0.733642i \(0.262181\pi\)
\(992\) −13.8192 −0.438759
\(993\) 0 0
\(994\) 7.86431 0.249441
\(995\) 42.0622 1.33346
\(996\) 0 0
\(997\) 43.5608 1.37958 0.689792 0.724007i \(-0.257702\pi\)
0.689792 + 0.724007i \(0.257702\pi\)
\(998\) 27.3396 0.865421
\(999\) 0 0
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 7623.2.a.cu.1.7 8
3.2 odd 2 inner 7623.2.a.cu.1.2 8
11.5 even 5 693.2.m.h.190.1 16
11.9 even 5 693.2.m.h.631.1 yes 16
11.10 odd 2 7623.2.a.cv.1.2 8
33.5 odd 10 693.2.m.h.190.4 yes 16
33.20 odd 10 693.2.m.h.631.4 yes 16
33.32 even 2 7623.2.a.cv.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.h.190.1 16 11.5 even 5
693.2.m.h.190.4 yes 16 33.5 odd 10
693.2.m.h.631.1 yes 16 11.9 even 5
693.2.m.h.631.4 yes 16 33.20 odd 10
7623.2.a.cu.1.2 8 3.2 odd 2 inner
7623.2.a.cu.1.7 8 1.1 even 1 trivial
7623.2.a.cv.1.2 8 11.10 odd 2
7623.2.a.cv.1.7 8 33.32 even 2