Properties

Label 4840.2.a.bd.1.6
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.27997\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.19828 q^{3} +1.00000 q^{5} +2.58485 q^{7} +1.83244 q^{9} +O(q^{10})\) \(q+2.19828 q^{3} +1.00000 q^{5} +2.58485 q^{7} +1.83244 q^{9} +1.66906 q^{13} +2.19828 q^{15} +6.39656 q^{17} +0.330943 q^{19} +5.68224 q^{21} +1.33727 q^{23} +1.00000 q^{25} -2.56662 q^{27} +0.941551 q^{29} +5.78263 q^{31} +2.58485 q^{35} -7.41429 q^{37} +3.66906 q^{39} +5.49063 q^{41} -4.54413 q^{43} +1.83244 q^{45} -7.21601 q^{47} -0.318527 q^{49} +14.0614 q^{51} +6.55577 q^{53} +0.727506 q^{57} -7.74191 q^{59} +3.88821 q^{61} +4.73660 q^{63} +1.66906 q^{65} +9.68736 q^{67} +2.93969 q^{69} -11.4882 q^{71} -12.1229 q^{73} +2.19828 q^{75} +7.61507 q^{79} -11.1395 q^{81} +9.29886 q^{83} +6.39656 q^{85} +2.06979 q^{87} -17.7651 q^{89} +4.31427 q^{91} +12.7118 q^{93} +0.330943 q^{95} +4.03731 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} - 16 q^{57} + 4 q^{59} + 16 q^{61} + 28 q^{63} - 2 q^{67} + 8 q^{69} - 28 q^{71} + 16 q^{73} - 2 q^{75} - 10 q^{81} + 12 q^{83} + 8 q^{85} - 24 q^{87} + 18 q^{89} - 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100}) \) 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
\(3\) 2.19828 1.26918 0.634589 0.772850i \(-0.281169\pi\)
0.634589 + 0.772850i \(0.281169\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.58485 0.976983 0.488492 0.872569i \(-0.337547\pi\)
0.488492 + 0.872569i \(0.337547\pi\)
\(8\) 0 0
\(9\) 1.83244 0.610814
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1.66906 0.462913 0.231457 0.972845i \(-0.425651\pi\)
0.231457 + 0.972845i \(0.425651\pi\)
\(14\) 0 0
\(15\) 2.19828 0.567594
\(16\) 0 0
\(17\) 6.39656 1.55139 0.775697 0.631105i \(-0.217398\pi\)
0.775697 + 0.631105i \(0.217398\pi\)
\(18\) 0 0
\(19\) 0.330943 0.0759236 0.0379618 0.999279i \(-0.487913\pi\)
0.0379618 + 0.999279i \(0.487913\pi\)
\(20\) 0 0
\(21\) 5.68224 1.23997
\(22\) 0 0
\(23\) 1.33727 0.278839 0.139420 0.990233i \(-0.455476\pi\)
0.139420 + 0.990233i \(0.455476\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.56662 −0.493947
\(28\) 0 0
\(29\) 0.941551 0.174842 0.0874208 0.996171i \(-0.472138\pi\)
0.0874208 + 0.996171i \(0.472138\pi\)
\(30\) 0 0
\(31\) 5.78263 1.03859 0.519295 0.854595i \(-0.326194\pi\)
0.519295 + 0.854595i \(0.326194\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.58485 0.436920
\(36\) 0 0
\(37\) −7.41429 −1.21890 −0.609451 0.792824i \(-0.708610\pi\)
−0.609451 + 0.792824i \(0.708610\pi\)
\(38\) 0 0
\(39\) 3.66906 0.587519
\(40\) 0 0
\(41\) 5.49063 0.857492 0.428746 0.903425i \(-0.358955\pi\)
0.428746 + 0.903425i \(0.358955\pi\)
\(42\) 0 0
\(43\) −4.54413 −0.692973 −0.346487 0.938055i \(-0.612625\pi\)
−0.346487 + 0.938055i \(0.612625\pi\)
\(44\) 0 0
\(45\) 1.83244 0.273164
\(46\) 0 0
\(47\) −7.21601 −1.05256 −0.526281 0.850310i \(-0.676414\pi\)
−0.526281 + 0.850310i \(0.676414\pi\)
\(48\) 0 0
\(49\) −0.318527 −0.0455039
\(50\) 0 0
\(51\) 14.0614 1.96900
\(52\) 0 0
\(53\) 6.55577 0.900505 0.450252 0.892901i \(-0.351334\pi\)
0.450252 + 0.892901i \(0.351334\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.727506 0.0963606
\(58\) 0 0
\(59\) −7.74191 −1.00791 −0.503955 0.863730i \(-0.668122\pi\)
−0.503955 + 0.863730i \(0.668122\pi\)
\(60\) 0 0
\(61\) 3.88821 0.497834 0.248917 0.968525i \(-0.419925\pi\)
0.248917 + 0.968525i \(0.419925\pi\)
\(62\) 0 0
\(63\) 4.73660 0.596755
\(64\) 0 0
\(65\) 1.66906 0.207021
\(66\) 0 0
\(67\) 9.68736 1.18350 0.591750 0.806122i \(-0.298437\pi\)
0.591750 + 0.806122i \(0.298437\pi\)
\(68\) 0 0
\(69\) 2.93969 0.353897
\(70\) 0 0
\(71\) −11.4882 −1.36340 −0.681701 0.731631i \(-0.738760\pi\)
−0.681701 + 0.731631i \(0.738760\pi\)
\(72\) 0 0
\(73\) −12.1229 −1.41888 −0.709439 0.704767i \(-0.751051\pi\)
−0.709439 + 0.704767i \(0.751051\pi\)
\(74\) 0 0
\(75\) 2.19828 0.253836
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.61507 0.856762 0.428381 0.903598i \(-0.359084\pi\)
0.428381 + 0.903598i \(0.359084\pi\)
\(80\) 0 0
\(81\) −11.1395 −1.23772
\(82\) 0 0
\(83\) 9.29886 1.02068 0.510341 0.859972i \(-0.329519\pi\)
0.510341 + 0.859972i \(0.329519\pi\)
\(84\) 0 0
\(85\) 6.39656 0.693805
\(86\) 0 0
\(87\) 2.06979 0.221905
\(88\) 0 0
\(89\) −17.7651 −1.88309 −0.941546 0.336885i \(-0.890627\pi\)
−0.941546 + 0.336885i \(0.890627\pi\)
\(90\) 0 0
\(91\) 4.31427 0.452258
\(92\) 0 0
\(93\) 12.7118 1.31816
\(94\) 0 0
\(95\) 0.330943 0.0339541
\(96\) 0 0
\(97\) 4.03731 0.409927 0.204963 0.978770i \(-0.434292\pi\)
0.204963 + 0.978770i \(0.434292\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.90476 −0.985560 −0.492780 0.870154i \(-0.664019\pi\)
−0.492780 + 0.870154i \(0.664019\pi\)
\(102\) 0 0
\(103\) −11.9304 −1.17553 −0.587766 0.809031i \(-0.699993\pi\)
−0.587766 + 0.809031i \(0.699993\pi\)
\(104\) 0 0
\(105\) 5.68224 0.554530
\(106\) 0 0
\(107\) 11.4320 1.10517 0.552585 0.833457i \(-0.313641\pi\)
0.552585 + 0.833457i \(0.313641\pi\)
\(108\) 0 0
\(109\) 13.2534 1.26945 0.634724 0.772739i \(-0.281114\pi\)
0.634724 + 0.772739i \(0.281114\pi\)
\(110\) 0 0
\(111\) −16.2987 −1.54700
\(112\) 0 0
\(113\) 14.4580 1.36010 0.680048 0.733168i \(-0.261959\pi\)
0.680048 + 0.733168i \(0.261959\pi\)
\(114\) 0 0
\(115\) 1.33727 0.124701
\(116\) 0 0
\(117\) 3.05845 0.282754
\(118\) 0 0
\(119\) 16.5342 1.51569
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 12.0700 1.08831
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.9795 1.59542 0.797711 0.603040i \(-0.206044\pi\)
0.797711 + 0.603040i \(0.206044\pi\)
\(128\) 0 0
\(129\) −9.98928 −0.879507
\(130\) 0 0
\(131\) −9.70451 −0.847887 −0.423943 0.905689i \(-0.639354\pi\)
−0.423943 + 0.905689i \(0.639354\pi\)
\(132\) 0 0
\(133\) 0.855440 0.0741761
\(134\) 0 0
\(135\) −2.56662 −0.220900
\(136\) 0 0
\(137\) 19.4516 1.66186 0.830931 0.556375i \(-0.187808\pi\)
0.830931 + 0.556375i \(0.187808\pi\)
\(138\) 0 0
\(139\) 1.88457 0.159847 0.0799235 0.996801i \(-0.474532\pi\)
0.0799235 + 0.996801i \(0.474532\pi\)
\(140\) 0 0
\(141\) −15.8628 −1.33589
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.941551 0.0781915
\(146\) 0 0
\(147\) −0.700213 −0.0577526
\(148\) 0 0
\(149\) −8.19423 −0.671298 −0.335649 0.941987i \(-0.608956\pi\)
−0.335649 + 0.941987i \(0.608956\pi\)
\(150\) 0 0
\(151\) 3.64408 0.296551 0.148275 0.988946i \(-0.452628\pi\)
0.148275 + 0.988946i \(0.452628\pi\)
\(152\) 0 0
\(153\) 11.7213 0.947613
\(154\) 0 0
\(155\) 5.78263 0.464472
\(156\) 0 0
\(157\) −8.12289 −0.648277 −0.324139 0.946010i \(-0.605074\pi\)
−0.324139 + 0.946010i \(0.605074\pi\)
\(158\) 0 0
\(159\) 14.4114 1.14290
\(160\) 0 0
\(161\) 3.45664 0.272421
\(162\) 0 0
\(163\) 12.1858 0.954464 0.477232 0.878777i \(-0.341640\pi\)
0.477232 + 0.878777i \(0.341640\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.9941 −1.77933 −0.889667 0.456610i \(-0.849064\pi\)
−0.889667 + 0.456610i \(0.849064\pi\)
\(168\) 0 0
\(169\) −10.2142 −0.785711
\(170\) 0 0
\(171\) 0.606434 0.0463752
\(172\) 0 0
\(173\) 13.0803 0.994475 0.497237 0.867615i \(-0.334348\pi\)
0.497237 + 0.867615i \(0.334348\pi\)
\(174\) 0 0
\(175\) 2.58485 0.195397
\(176\) 0 0
\(177\) −17.0189 −1.27922
\(178\) 0 0
\(179\) 2.75740 0.206098 0.103049 0.994676i \(-0.467140\pi\)
0.103049 + 0.994676i \(0.467140\pi\)
\(180\) 0 0
\(181\) 3.38392 0.251524 0.125762 0.992060i \(-0.459862\pi\)
0.125762 + 0.992060i \(0.459862\pi\)
\(182\) 0 0
\(183\) 8.54737 0.631840
\(184\) 0 0
\(185\) −7.41429 −0.545109
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.63434 −0.482577
\(190\) 0 0
\(191\) −0.0917500 −0.00663880 −0.00331940 0.999994i \(-0.501057\pi\)
−0.00331940 + 0.999994i \(0.501057\pi\)
\(192\) 0 0
\(193\) −24.9932 −1.79905 −0.899525 0.436869i \(-0.856087\pi\)
−0.899525 + 0.436869i \(0.856087\pi\)
\(194\) 0 0
\(195\) 3.66906 0.262747
\(196\) 0 0
\(197\) 18.6454 1.32843 0.664216 0.747540i \(-0.268765\pi\)
0.664216 + 0.747540i \(0.268765\pi\)
\(198\) 0 0
\(199\) −21.9376 −1.55511 −0.777557 0.628812i \(-0.783541\pi\)
−0.777557 + 0.628812i \(0.783541\pi\)
\(200\) 0 0
\(201\) 21.2956 1.50207
\(202\) 0 0
\(203\) 2.43377 0.170817
\(204\) 0 0
\(205\) 5.49063 0.383482
\(206\) 0 0
\(207\) 2.45046 0.170319
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.8128 1.15744 0.578719 0.815527i \(-0.303553\pi\)
0.578719 + 0.815527i \(0.303553\pi\)
\(212\) 0 0
\(213\) −25.2544 −1.73040
\(214\) 0 0
\(215\) −4.54413 −0.309907
\(216\) 0 0
\(217\) 14.9473 1.01469
\(218\) 0 0
\(219\) −26.6495 −1.80081
\(220\) 0 0
\(221\) 10.6762 0.718161
\(222\) 0 0
\(223\) 0.156711 0.0104942 0.00524708 0.999986i \(-0.498330\pi\)
0.00524708 + 0.999986i \(0.498330\pi\)
\(224\) 0 0
\(225\) 1.83244 0.122163
\(226\) 0 0
\(227\) 4.26616 0.283155 0.141577 0.989927i \(-0.454783\pi\)
0.141577 + 0.989927i \(0.454783\pi\)
\(228\) 0 0
\(229\) −3.55280 −0.234776 −0.117388 0.993086i \(-0.537452\pi\)
−0.117388 + 0.993086i \(0.537452\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.20904 −0.144719 −0.0723596 0.997379i \(-0.523053\pi\)
−0.0723596 + 0.997379i \(0.523053\pi\)
\(234\) 0 0
\(235\) −7.21601 −0.470720
\(236\) 0 0
\(237\) 16.7401 1.08738
\(238\) 0 0
\(239\) 12.7323 0.823586 0.411793 0.911277i \(-0.364903\pi\)
0.411793 + 0.911277i \(0.364903\pi\)
\(240\) 0 0
\(241\) −13.6950 −0.882171 −0.441085 0.897465i \(-0.645406\pi\)
−0.441085 + 0.897465i \(0.645406\pi\)
\(242\) 0 0
\(243\) −16.7879 −1.07694
\(244\) 0 0
\(245\) −0.318527 −0.0203500
\(246\) 0 0
\(247\) 0.552363 0.0351460
\(248\) 0 0
\(249\) 20.4415 1.29543
\(250\) 0 0
\(251\) −9.12212 −0.575783 −0.287892 0.957663i \(-0.592954\pi\)
−0.287892 + 0.957663i \(0.592954\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 14.0614 0.880562
\(256\) 0 0
\(257\) 23.2138 1.44804 0.724019 0.689780i \(-0.242293\pi\)
0.724019 + 0.689780i \(0.242293\pi\)
\(258\) 0 0
\(259\) −19.1649 −1.19085
\(260\) 0 0
\(261\) 1.72534 0.106796
\(262\) 0 0
\(263\) −30.5012 −1.88078 −0.940392 0.340092i \(-0.889542\pi\)
−0.940392 + 0.340092i \(0.889542\pi\)
\(264\) 0 0
\(265\) 6.55577 0.402718
\(266\) 0 0
\(267\) −39.0526 −2.38998
\(268\) 0 0
\(269\) −5.13465 −0.313065 −0.156533 0.987673i \(-0.550032\pi\)
−0.156533 + 0.987673i \(0.550032\pi\)
\(270\) 0 0
\(271\) 12.6879 0.770734 0.385367 0.922763i \(-0.374075\pi\)
0.385367 + 0.922763i \(0.374075\pi\)
\(272\) 0 0
\(273\) 9.48398 0.573996
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.3144 −0.739899 −0.369949 0.929052i \(-0.620625\pi\)
−0.369949 + 0.929052i \(0.620625\pi\)
\(278\) 0 0
\(279\) 10.5963 0.634386
\(280\) 0 0
\(281\) 13.3736 0.797800 0.398900 0.916994i \(-0.369392\pi\)
0.398900 + 0.916994i \(0.369392\pi\)
\(282\) 0 0
\(283\) −11.0470 −0.656678 −0.328339 0.944560i \(-0.606489\pi\)
−0.328339 + 0.944560i \(0.606489\pi\)
\(284\) 0 0
\(285\) 0.727506 0.0430938
\(286\) 0 0
\(287\) 14.1925 0.837756
\(288\) 0 0
\(289\) 23.9160 1.40682
\(290\) 0 0
\(291\) 8.87515 0.520270
\(292\) 0 0
\(293\) 25.4592 1.48734 0.743672 0.668545i \(-0.233082\pi\)
0.743672 + 0.668545i \(0.233082\pi\)
\(294\) 0 0
\(295\) −7.74191 −0.450751
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.23197 0.129078
\(300\) 0 0
\(301\) −11.7459 −0.677023
\(302\) 0 0
\(303\) −21.7734 −1.25085
\(304\) 0 0
\(305\) 3.88821 0.222638
\(306\) 0 0
\(307\) −30.1355 −1.71992 −0.859961 0.510360i \(-0.829512\pi\)
−0.859961 + 0.510360i \(0.829512\pi\)
\(308\) 0 0
\(309\) −26.2263 −1.49196
\(310\) 0 0
\(311\) 12.3904 0.702598 0.351299 0.936263i \(-0.385740\pi\)
0.351299 + 0.936263i \(0.385740\pi\)
\(312\) 0 0
\(313\) 26.2056 1.48123 0.740613 0.671932i \(-0.234535\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(314\) 0 0
\(315\) 4.73660 0.266877
\(316\) 0 0
\(317\) −16.9782 −0.953589 −0.476794 0.879015i \(-0.658201\pi\)
−0.476794 + 0.879015i \(0.658201\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 25.1307 1.40266
\(322\) 0 0
\(323\) 2.11690 0.117787
\(324\) 0 0
\(325\) 1.66906 0.0925826
\(326\) 0 0
\(327\) 29.1348 1.61116
\(328\) 0 0
\(329\) −18.6523 −1.02834
\(330\) 0 0
\(331\) −13.7482 −0.755668 −0.377834 0.925873i \(-0.623331\pi\)
−0.377834 + 0.925873i \(0.623331\pi\)
\(332\) 0 0
\(333\) −13.5863 −0.744522
\(334\) 0 0
\(335\) 9.68736 0.529277
\(336\) 0 0
\(337\) −16.9287 −0.922163 −0.461081 0.887358i \(-0.652538\pi\)
−0.461081 + 0.887358i \(0.652538\pi\)
\(338\) 0 0
\(339\) 31.7828 1.72620
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −18.9173 −1.02144
\(344\) 0 0
\(345\) 2.93969 0.158268
\(346\) 0 0
\(347\) 9.03447 0.484996 0.242498 0.970152i \(-0.422033\pi\)
0.242498 + 0.970152i \(0.422033\pi\)
\(348\) 0 0
\(349\) −27.5646 −1.47550 −0.737748 0.675076i \(-0.764111\pi\)
−0.737748 + 0.675076i \(0.764111\pi\)
\(350\) 0 0
\(351\) −4.28384 −0.228654
\(352\) 0 0
\(353\) 9.53305 0.507393 0.253696 0.967284i \(-0.418354\pi\)
0.253696 + 0.967284i \(0.418354\pi\)
\(354\) 0 0
\(355\) −11.4882 −0.609732
\(356\) 0 0
\(357\) 36.3468 1.92368
\(358\) 0 0
\(359\) −26.2842 −1.38723 −0.693615 0.720346i \(-0.743983\pi\)
−0.693615 + 0.720346i \(0.743983\pi\)
\(360\) 0 0
\(361\) −18.8905 −0.994236
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.1229 −0.634541
\(366\) 0 0
\(367\) −21.0794 −1.10033 −0.550167 0.835054i \(-0.685436\pi\)
−0.550167 + 0.835054i \(0.685436\pi\)
\(368\) 0 0
\(369\) 10.0613 0.523768
\(370\) 0 0
\(371\) 16.9457 0.879778
\(372\) 0 0
\(373\) 4.86333 0.251813 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(374\) 0 0
\(375\) 2.19828 0.113519
\(376\) 0 0
\(377\) 1.57150 0.0809364
\(378\) 0 0
\(379\) 17.6659 0.907435 0.453718 0.891146i \(-0.350097\pi\)
0.453718 + 0.891146i \(0.350097\pi\)
\(380\) 0 0
\(381\) 39.5240 2.02488
\(382\) 0 0
\(383\) −20.9804 −1.07205 −0.536024 0.844203i \(-0.680074\pi\)
−0.536024 + 0.844203i \(0.680074\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.32685 −0.423278
\(388\) 0 0
\(389\) −9.45185 −0.479228 −0.239614 0.970868i \(-0.577021\pi\)
−0.239614 + 0.970868i \(0.577021\pi\)
\(390\) 0 0
\(391\) 8.55391 0.432590
\(392\) 0 0
\(393\) −21.3332 −1.07612
\(394\) 0 0
\(395\) 7.61507 0.383156
\(396\) 0 0
\(397\) 26.5417 1.33209 0.666046 0.745911i \(-0.267986\pi\)
0.666046 + 0.745911i \(0.267986\pi\)
\(398\) 0 0
\(399\) 1.88050 0.0941426
\(400\) 0 0
\(401\) 9.91003 0.494883 0.247442 0.968903i \(-0.420410\pi\)
0.247442 + 0.968903i \(0.420410\pi\)
\(402\) 0 0
\(403\) 9.65154 0.480777
\(404\) 0 0
\(405\) −11.1395 −0.553525
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −24.9962 −1.23598 −0.617991 0.786185i \(-0.712053\pi\)
−0.617991 + 0.786185i \(0.712053\pi\)
\(410\) 0 0
\(411\) 42.7601 2.10920
\(412\) 0 0
\(413\) −20.0117 −0.984711
\(414\) 0 0
\(415\) 9.29886 0.456463
\(416\) 0 0
\(417\) 4.14281 0.202874
\(418\) 0 0
\(419\) 31.6860 1.54796 0.773982 0.633208i \(-0.218262\pi\)
0.773982 + 0.633208i \(0.218262\pi\)
\(420\) 0 0
\(421\) 27.6659 1.34835 0.674176 0.738571i \(-0.264499\pi\)
0.674176 + 0.738571i \(0.264499\pi\)
\(422\) 0 0
\(423\) −13.2229 −0.642920
\(424\) 0 0
\(425\) 6.39656 0.310279
\(426\) 0 0
\(427\) 10.0505 0.486375
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.37658 0.403486 0.201743 0.979439i \(-0.435339\pi\)
0.201743 + 0.979439i \(0.435339\pi\)
\(432\) 0 0
\(433\) 16.9179 0.813021 0.406511 0.913646i \(-0.366745\pi\)
0.406511 + 0.913646i \(0.366745\pi\)
\(434\) 0 0
\(435\) 2.06979 0.0992390
\(436\) 0 0
\(437\) 0.442559 0.0211705
\(438\) 0 0
\(439\) −15.9993 −0.763605 −0.381802 0.924244i \(-0.624697\pi\)
−0.381802 + 0.924244i \(0.624697\pi\)
\(440\) 0 0
\(441\) −0.583683 −0.0277944
\(442\) 0 0
\(443\) −21.6509 −1.02867 −0.514333 0.857591i \(-0.671960\pi\)
−0.514333 + 0.857591i \(0.671960\pi\)
\(444\) 0 0
\(445\) −17.7651 −0.842144
\(446\) 0 0
\(447\) −18.0132 −0.851997
\(448\) 0 0
\(449\) 10.9199 0.515341 0.257671 0.966233i \(-0.417045\pi\)
0.257671 + 0.966233i \(0.417045\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.01070 0.376376
\(454\) 0 0
\(455\) 4.31427 0.202256
\(456\) 0 0
\(457\) 32.1926 1.50591 0.752953 0.658074i \(-0.228629\pi\)
0.752953 + 0.658074i \(0.228629\pi\)
\(458\) 0 0
\(459\) −16.4176 −0.766306
\(460\) 0 0
\(461\) 1.21757 0.0567081 0.0283540 0.999598i \(-0.490973\pi\)
0.0283540 + 0.999598i \(0.490973\pi\)
\(462\) 0 0
\(463\) −25.9855 −1.20765 −0.603824 0.797118i \(-0.706357\pi\)
−0.603824 + 0.797118i \(0.706357\pi\)
\(464\) 0 0
\(465\) 12.7118 0.589498
\(466\) 0 0
\(467\) 20.3664 0.942442 0.471221 0.882015i \(-0.343813\pi\)
0.471221 + 0.882015i \(0.343813\pi\)
\(468\) 0 0
\(469\) 25.0404 1.15626
\(470\) 0 0
\(471\) −17.8564 −0.822780
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.330943 0.0151847
\(476\) 0 0
\(477\) 12.0131 0.550041
\(478\) 0 0
\(479\) −29.9861 −1.37010 −0.685050 0.728496i \(-0.740220\pi\)
−0.685050 + 0.728496i \(0.740220\pi\)
\(480\) 0 0
\(481\) −12.3749 −0.564245
\(482\) 0 0
\(483\) 7.59867 0.345751
\(484\) 0 0
\(485\) 4.03731 0.183325
\(486\) 0 0
\(487\) −28.7529 −1.30292 −0.651460 0.758683i \(-0.725843\pi\)
−0.651460 + 0.758683i \(0.725843\pi\)
\(488\) 0 0
\(489\) 26.7878 1.21139
\(490\) 0 0
\(491\) 16.9472 0.764816 0.382408 0.923994i \(-0.375095\pi\)
0.382408 + 0.923994i \(0.375095\pi\)
\(492\) 0 0
\(493\) 6.02269 0.271248
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −29.6954 −1.33202
\(498\) 0 0
\(499\) −35.5491 −1.59140 −0.795699 0.605692i \(-0.792896\pi\)
−0.795699 + 0.605692i \(0.792896\pi\)
\(500\) 0 0
\(501\) −50.5474 −2.25829
\(502\) 0 0
\(503\) 15.3213 0.683142 0.341571 0.939856i \(-0.389041\pi\)
0.341571 + 0.939856i \(0.389041\pi\)
\(504\) 0 0
\(505\) −9.90476 −0.440756
\(506\) 0 0
\(507\) −22.4538 −0.997208
\(508\) 0 0
\(509\) −34.5306 −1.53054 −0.765272 0.643707i \(-0.777395\pi\)
−0.765272 + 0.643707i \(0.777395\pi\)
\(510\) 0 0
\(511\) −31.3359 −1.38622
\(512\) 0 0
\(513\) −0.849406 −0.0375022
\(514\) 0 0
\(515\) −11.9304 −0.525714
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 28.7541 1.26217
\(520\) 0 0
\(521\) 7.89583 0.345922 0.172961 0.984929i \(-0.444666\pi\)
0.172961 + 0.984929i \(0.444666\pi\)
\(522\) 0 0
\(523\) 39.2227 1.71509 0.857545 0.514410i \(-0.171989\pi\)
0.857545 + 0.514410i \(0.171989\pi\)
\(524\) 0 0
\(525\) 5.68224 0.247993
\(526\) 0 0
\(527\) 36.9890 1.61126
\(528\) 0 0
\(529\) −21.2117 −0.922249
\(530\) 0 0
\(531\) −14.1866 −0.615646
\(532\) 0 0
\(533\) 9.16417 0.396944
\(534\) 0 0
\(535\) 11.4320 0.494247
\(536\) 0 0
\(537\) 6.06155 0.261575
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.28640 0.184287 0.0921433 0.995746i \(-0.470628\pi\)
0.0921433 + 0.995746i \(0.470628\pi\)
\(542\) 0 0
\(543\) 7.43880 0.319229
\(544\) 0 0
\(545\) 13.2534 0.567714
\(546\) 0 0
\(547\) −0.942165 −0.0402841 −0.0201420 0.999797i \(-0.506412\pi\)
−0.0201420 + 0.999797i \(0.506412\pi\)
\(548\) 0 0
\(549\) 7.12491 0.304084
\(550\) 0 0
\(551\) 0.311600 0.0132746
\(552\) 0 0
\(553\) 19.6838 0.837042
\(554\) 0 0
\(555\) −16.2987 −0.691841
\(556\) 0 0
\(557\) −20.7709 −0.880089 −0.440045 0.897976i \(-0.645037\pi\)
−0.440045 + 0.897976i \(0.645037\pi\)
\(558\) 0 0
\(559\) −7.58441 −0.320786
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.09795 −0.130563 −0.0652815 0.997867i \(-0.520795\pi\)
−0.0652815 + 0.997867i \(0.520795\pi\)
\(564\) 0 0
\(565\) 14.4580 0.608253
\(566\) 0 0
\(567\) −28.7939 −1.20923
\(568\) 0 0
\(569\) −4.83283 −0.202603 −0.101301 0.994856i \(-0.532301\pi\)
−0.101301 + 0.994856i \(0.532301\pi\)
\(570\) 0 0
\(571\) 9.40074 0.393409 0.196704 0.980463i \(-0.436976\pi\)
0.196704 + 0.980463i \(0.436976\pi\)
\(572\) 0 0
\(573\) −0.201692 −0.00842582
\(574\) 0 0
\(575\) 1.33727 0.0557679
\(576\) 0 0
\(577\) 6.62302 0.275720 0.137860 0.990452i \(-0.455978\pi\)
0.137860 + 0.990452i \(0.455978\pi\)
\(578\) 0 0
\(579\) −54.9421 −2.28332
\(580\) 0 0
\(581\) 24.0362 0.997189
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.05845 0.126451
\(586\) 0 0
\(587\) −28.2488 −1.16595 −0.582976 0.812489i \(-0.698112\pi\)
−0.582976 + 0.812489i \(0.698112\pi\)
\(588\) 0 0
\(589\) 1.91372 0.0788535
\(590\) 0 0
\(591\) 40.9879 1.68602
\(592\) 0 0
\(593\) −1.97934 −0.0812818 −0.0406409 0.999174i \(-0.512940\pi\)
−0.0406409 + 0.999174i \(0.512940\pi\)
\(594\) 0 0
\(595\) 16.5342 0.677835
\(596\) 0 0
\(597\) −48.2250 −1.97372
\(598\) 0 0
\(599\) −20.3610 −0.831927 −0.415964 0.909381i \(-0.636556\pi\)
−0.415964 + 0.909381i \(0.636556\pi\)
\(600\) 0 0
\(601\) −25.0897 −1.02343 −0.511716 0.859155i \(-0.670990\pi\)
−0.511716 + 0.859155i \(0.670990\pi\)
\(602\) 0 0
\(603\) 17.7515 0.722898
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −9.61170 −0.390127 −0.195063 0.980791i \(-0.562491\pi\)
−0.195063 + 0.980791i \(0.562491\pi\)
\(608\) 0 0
\(609\) 5.35011 0.216798
\(610\) 0 0
\(611\) −12.0439 −0.487245
\(612\) 0 0
\(613\) −16.4057 −0.662618 −0.331309 0.943522i \(-0.607490\pi\)
−0.331309 + 0.943522i \(0.607490\pi\)
\(614\) 0 0
\(615\) 12.0700 0.486707
\(616\) 0 0
\(617\) 36.5821 1.47274 0.736370 0.676579i \(-0.236538\pi\)
0.736370 + 0.676579i \(0.236538\pi\)
\(618\) 0 0
\(619\) 31.6865 1.27359 0.636793 0.771035i \(-0.280261\pi\)
0.636793 + 0.771035i \(0.280261\pi\)
\(620\) 0 0
\(621\) −3.43226 −0.137732
\(622\) 0 0
\(623\) −45.9201 −1.83975
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −47.4260 −1.89100
\(630\) 0 0
\(631\) −22.3894 −0.891307 −0.445654 0.895206i \(-0.647029\pi\)
−0.445654 + 0.895206i \(0.647029\pi\)
\(632\) 0 0
\(633\) 36.9592 1.46900
\(634\) 0 0
\(635\) 17.9795 0.713495
\(636\) 0 0
\(637\) −0.531640 −0.0210643
\(638\) 0 0
\(639\) −21.0515 −0.832785
\(640\) 0 0
\(641\) 25.8587 1.02136 0.510678 0.859772i \(-0.329394\pi\)
0.510678 + 0.859772i \(0.329394\pi\)
\(642\) 0 0
\(643\) −46.6628 −1.84020 −0.920101 0.391681i \(-0.871894\pi\)
−0.920101 + 0.391681i \(0.871894\pi\)
\(644\) 0 0
\(645\) −9.98928 −0.393327
\(646\) 0 0
\(647\) −25.9855 −1.02159 −0.510797 0.859701i \(-0.670650\pi\)
−0.510797 + 0.859701i \(0.670650\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 32.8583 1.28782
\(652\) 0 0
\(653\) −3.44698 −0.134891 −0.0674453 0.997723i \(-0.521485\pi\)
−0.0674453 + 0.997723i \(0.521485\pi\)
\(654\) 0 0
\(655\) −9.70451 −0.379187
\(656\) 0 0
\(657\) −22.2145 −0.866670
\(658\) 0 0
\(659\) −15.6066 −0.607945 −0.303973 0.952681i \(-0.598313\pi\)
−0.303973 + 0.952681i \(0.598313\pi\)
\(660\) 0 0
\(661\) 39.5126 1.53686 0.768431 0.639933i \(-0.221038\pi\)
0.768431 + 0.639933i \(0.221038\pi\)
\(662\) 0 0
\(663\) 23.4694 0.911474
\(664\) 0 0
\(665\) 0.855440 0.0331725
\(666\) 0 0
\(667\) 1.25910 0.0487527
\(668\) 0 0
\(669\) 0.344496 0.0133190
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 4.33800 0.167218 0.0836089 0.996499i \(-0.473355\pi\)
0.0836089 + 0.996499i \(0.473355\pi\)
\(674\) 0 0
\(675\) −2.56662 −0.0987893
\(676\) 0 0
\(677\) 41.6655 1.60134 0.800668 0.599109i \(-0.204478\pi\)
0.800668 + 0.599109i \(0.204478\pi\)
\(678\) 0 0
\(679\) 10.4359 0.400492
\(680\) 0 0
\(681\) 9.37822 0.359374
\(682\) 0 0
\(683\) −43.4283 −1.66174 −0.830869 0.556468i \(-0.812156\pi\)
−0.830869 + 0.556468i \(0.812156\pi\)
\(684\) 0 0
\(685\) 19.4516 0.743207
\(686\) 0 0
\(687\) −7.81007 −0.297973
\(688\) 0 0
\(689\) 10.9420 0.416856
\(690\) 0 0
\(691\) −19.6046 −0.745793 −0.372897 0.927873i \(-0.621635\pi\)
−0.372897 + 0.927873i \(0.621635\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.88457 0.0714857
\(696\) 0 0
\(697\) 35.1212 1.33031
\(698\) 0 0
\(699\) −4.85610 −0.183675
\(700\) 0 0
\(701\) 36.9648 1.39614 0.698071 0.716029i \(-0.254042\pi\)
0.698071 + 0.716029i \(0.254042\pi\)
\(702\) 0 0
\(703\) −2.45371 −0.0925434
\(704\) 0 0
\(705\) −15.8628 −0.597428
\(706\) 0 0
\(707\) −25.6024 −0.962876
\(708\) 0 0
\(709\) −41.3706 −1.55371 −0.776854 0.629681i \(-0.783185\pi\)
−0.776854 + 0.629681i \(0.783185\pi\)
\(710\) 0 0
\(711\) 13.9542 0.523322
\(712\) 0 0
\(713\) 7.73292 0.289600
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.9893 1.04528
\(718\) 0 0
\(719\) 2.88409 0.107558 0.0537792 0.998553i \(-0.482873\pi\)
0.0537792 + 0.998553i \(0.482873\pi\)
\(720\) 0 0
\(721\) −30.8382 −1.14848
\(722\) 0 0
\(723\) −30.1054 −1.11963
\(724\) 0 0
\(725\) 0.941551 0.0349683
\(726\) 0 0
\(727\) −17.9716 −0.666530 −0.333265 0.942833i \(-0.608150\pi\)
−0.333265 + 0.942833i \(0.608150\pi\)
\(728\) 0 0
\(729\) −3.48598 −0.129110
\(730\) 0 0
\(731\) −29.0668 −1.07508
\(732\) 0 0
\(733\) −15.3188 −0.565812 −0.282906 0.959148i \(-0.591298\pi\)
−0.282906 + 0.959148i \(0.591298\pi\)
\(734\) 0 0
\(735\) −0.700213 −0.0258277
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 6.38055 0.234712 0.117356 0.993090i \(-0.462558\pi\)
0.117356 + 0.993090i \(0.462558\pi\)
\(740\) 0 0
\(741\) 1.21425 0.0446066
\(742\) 0 0
\(743\) 32.3860 1.18813 0.594064 0.804418i \(-0.297522\pi\)
0.594064 + 0.804418i \(0.297522\pi\)
\(744\) 0 0
\(745\) −8.19423 −0.300213
\(746\) 0 0
\(747\) 17.0396 0.623447
\(748\) 0 0
\(749\) 29.5500 1.07973
\(750\) 0 0
\(751\) 32.5871 1.18912 0.594560 0.804051i \(-0.297326\pi\)
0.594560 + 0.804051i \(0.297326\pi\)
\(752\) 0 0
\(753\) −20.0530 −0.730772
\(754\) 0 0
\(755\) 3.64408 0.132621
\(756\) 0 0
\(757\) 32.7857 1.19162 0.595808 0.803127i \(-0.296832\pi\)
0.595808 + 0.803127i \(0.296832\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.3176 1.49776 0.748881 0.662705i \(-0.230592\pi\)
0.748881 + 0.662705i \(0.230592\pi\)
\(762\) 0 0
\(763\) 34.2582 1.24023
\(764\) 0 0
\(765\) 11.7213 0.423786
\(766\) 0 0
\(767\) −12.9217 −0.466575
\(768\) 0 0
\(769\) 20.4922 0.738967 0.369483 0.929237i \(-0.379535\pi\)
0.369483 + 0.929237i \(0.379535\pi\)
\(770\) 0 0
\(771\) 51.0305 1.83782
\(772\) 0 0
\(773\) −1.88634 −0.0678470 −0.0339235 0.999424i \(-0.510800\pi\)
−0.0339235 + 0.999424i \(0.510800\pi\)
\(774\) 0 0
\(775\) 5.78263 0.207718
\(776\) 0 0
\(777\) −42.1298 −1.51140
\(778\) 0 0
\(779\) 1.81709 0.0651039
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −2.41660 −0.0863624
\(784\) 0 0
\(785\) −8.12289 −0.289918
\(786\) 0 0
\(787\) −20.6178 −0.734946 −0.367473 0.930034i \(-0.619777\pi\)
−0.367473 + 0.930034i \(0.619777\pi\)
\(788\) 0 0
\(789\) −67.0502 −2.38705
\(790\) 0 0
\(791\) 37.3719 1.32879
\(792\) 0 0
\(793\) 6.48964 0.230454
\(794\) 0 0
\(795\) 14.4114 0.511121
\(796\) 0 0
\(797\) 5.96779 0.211390 0.105695 0.994399i \(-0.466293\pi\)
0.105695 + 0.994399i \(0.466293\pi\)
\(798\) 0 0
\(799\) −46.1576 −1.63294
\(800\) 0 0
\(801\) −32.5534 −1.15022
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 3.45664 0.121831
\(806\) 0 0
\(807\) −11.2874 −0.397336
\(808\) 0 0
\(809\) 27.3857 0.962829 0.481415 0.876493i \(-0.340123\pi\)
0.481415 + 0.876493i \(0.340123\pi\)
\(810\) 0 0
\(811\) −10.5465 −0.370336 −0.185168 0.982707i \(-0.559283\pi\)
−0.185168 + 0.982707i \(0.559283\pi\)
\(812\) 0 0
\(813\) 27.8915 0.978199
\(814\) 0 0
\(815\) 12.1858 0.426849
\(816\) 0 0
\(817\) −1.50385 −0.0526130
\(818\) 0 0
\(819\) 7.90565 0.276246
\(820\) 0 0
\(821\) 41.0152 1.43144 0.715720 0.698387i \(-0.246099\pi\)
0.715720 + 0.698387i \(0.246099\pi\)
\(822\) 0 0
\(823\) −26.8483 −0.935871 −0.467936 0.883763i \(-0.655002\pi\)
−0.467936 + 0.883763i \(0.655002\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.1904 −0.423903 −0.211951 0.977280i \(-0.567982\pi\)
−0.211951 + 0.977280i \(0.567982\pi\)
\(828\) 0 0
\(829\) −34.5013 −1.19828 −0.599139 0.800645i \(-0.704490\pi\)
−0.599139 + 0.800645i \(0.704490\pi\)
\(830\) 0 0
\(831\) −27.0705 −0.939064
\(832\) 0 0
\(833\) −2.03748 −0.0705945
\(834\) 0 0
\(835\) −22.9941 −0.795742
\(836\) 0 0
\(837\) −14.8418 −0.513008
\(838\) 0 0
\(839\) −13.3549 −0.461064 −0.230532 0.973065i \(-0.574047\pi\)
−0.230532 + 0.973065i \(0.574047\pi\)
\(840\) 0 0
\(841\) −28.1135 −0.969430
\(842\) 0 0
\(843\) 29.3989 1.01255
\(844\) 0 0
\(845\) −10.2142 −0.351381
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −24.2845 −0.833442
\(850\) 0 0
\(851\) −9.91489 −0.339878
\(852\) 0 0
\(853\) −39.5614 −1.35456 −0.677278 0.735727i \(-0.736840\pi\)
−0.677278 + 0.735727i \(0.736840\pi\)
\(854\) 0 0
\(855\) 0.606434 0.0207396
\(856\) 0 0
\(857\) 10.6805 0.364838 0.182419 0.983221i \(-0.441607\pi\)
0.182419 + 0.983221i \(0.441607\pi\)
\(858\) 0 0
\(859\) 2.20217 0.0751369 0.0375685 0.999294i \(-0.488039\pi\)
0.0375685 + 0.999294i \(0.488039\pi\)
\(860\) 0 0
\(861\) 31.1991 1.06326
\(862\) 0 0
\(863\) 4.30423 0.146518 0.0732588 0.997313i \(-0.476660\pi\)
0.0732588 + 0.997313i \(0.476660\pi\)
\(864\) 0 0
\(865\) 13.0803 0.444743
\(866\) 0 0
\(867\) 52.5741 1.78551
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.1688 0.547858
\(872\) 0 0
\(873\) 7.39814 0.250389
\(874\) 0 0
\(875\) 2.58485 0.0873840
\(876\) 0 0
\(877\) −41.8693 −1.41383 −0.706913 0.707300i \(-0.749913\pi\)
−0.706913 + 0.707300i \(0.749913\pi\)
\(878\) 0 0
\(879\) 55.9666 1.88770
\(880\) 0 0
\(881\) 55.2076 1.85999 0.929996 0.367570i \(-0.119810\pi\)
0.929996 + 0.367570i \(0.119810\pi\)
\(882\) 0 0
\(883\) 27.4993 0.925426 0.462713 0.886508i \(-0.346876\pi\)
0.462713 + 0.886508i \(0.346876\pi\)
\(884\) 0 0
\(885\) −17.0189 −0.572084
\(886\) 0 0
\(887\) −28.9410 −0.971743 −0.485871 0.874030i \(-0.661498\pi\)
−0.485871 + 0.874030i \(0.661498\pi\)
\(888\) 0 0
\(889\) 46.4744 1.55870
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.38809 −0.0799143
\(894\) 0 0
\(895\) 2.75740 0.0921698
\(896\) 0 0
\(897\) 4.90651 0.163824
\(898\) 0 0
\(899\) 5.44464 0.181589
\(900\) 0 0
\(901\) 41.9344 1.39704
\(902\) 0 0
\(903\) −25.8208 −0.859263
\(904\) 0 0
\(905\) 3.38392 0.112485
\(906\) 0 0
\(907\) −55.6297 −1.84715 −0.923577 0.383412i \(-0.874749\pi\)
−0.923577 + 0.383412i \(0.874749\pi\)
\(908\) 0 0
\(909\) −18.1499 −0.601994
\(910\) 0 0
\(911\) −49.2558 −1.63192 −0.815959 0.578110i \(-0.803790\pi\)
−0.815959 + 0.578110i \(0.803790\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 8.54737 0.282567
\(916\) 0 0
\(917\) −25.0847 −0.828371
\(918\) 0 0
\(919\) 2.39222 0.0789121 0.0394560 0.999221i \(-0.487437\pi\)
0.0394560 + 0.999221i \(0.487437\pi\)
\(920\) 0 0
\(921\) −66.2462 −2.18289
\(922\) 0 0
\(923\) −19.1745 −0.631137
\(924\) 0 0
\(925\) −7.41429 −0.243780
\(926\) 0 0
\(927\) −21.8617 −0.718032
\(928\) 0 0
\(929\) −36.1713 −1.18674 −0.593371 0.804929i \(-0.702203\pi\)
−0.593371 + 0.804929i \(0.702203\pi\)
\(930\) 0 0
\(931\) −0.105414 −0.00345482
\(932\) 0 0
\(933\) 27.2377 0.891722
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.30586 0.140666 0.0703331 0.997524i \(-0.477594\pi\)
0.0703331 + 0.997524i \(0.477594\pi\)
\(938\) 0 0
\(939\) 57.6072 1.87994
\(940\) 0 0
\(941\) 0.649571 0.0211754 0.0105877 0.999944i \(-0.496630\pi\)
0.0105877 + 0.999944i \(0.496630\pi\)
\(942\) 0 0
\(943\) 7.34244 0.239103
\(944\) 0 0
\(945\) −6.63434 −0.215815
\(946\) 0 0
\(947\) 39.0346 1.26845 0.634226 0.773147i \(-0.281319\pi\)
0.634226 + 0.773147i \(0.281319\pi\)
\(948\) 0 0
\(949\) −20.2338 −0.656817
\(950\) 0 0
\(951\) −37.3228 −1.21027
\(952\) 0 0
\(953\) −11.1295 −0.360521 −0.180260 0.983619i \(-0.557694\pi\)
−0.180260 + 0.983619i \(0.557694\pi\)
\(954\) 0 0
\(955\) −0.0917500 −0.00296896
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 50.2796 1.62361
\(960\) 0 0
\(961\) 2.43880 0.0786709
\(962\) 0 0
\(963\) 20.9484 0.675053
\(964\) 0 0
\(965\) −24.9932 −0.804560
\(966\) 0 0
\(967\) 5.30703 0.170663 0.0853313 0.996353i \(-0.472805\pi\)
0.0853313 + 0.996353i \(0.472805\pi\)
\(968\) 0 0
\(969\) 4.65354 0.149493
\(970\) 0 0
\(971\) −4.61394 −0.148068 −0.0740342 0.997256i \(-0.523587\pi\)
−0.0740342 + 0.997256i \(0.523587\pi\)
\(972\) 0 0
\(973\) 4.87133 0.156168
\(974\) 0 0
\(975\) 3.66906 0.117504
\(976\) 0 0
\(977\) 55.8973 1.78831 0.894156 0.447756i \(-0.147777\pi\)
0.894156 + 0.447756i \(0.147777\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 24.2861 0.775396
\(982\) 0 0
\(983\) 7.55085 0.240835 0.120417 0.992723i \(-0.461577\pi\)
0.120417 + 0.992723i \(0.461577\pi\)
\(984\) 0 0
\(985\) 18.6454 0.594093
\(986\) 0 0
\(987\) −41.0031 −1.30514
\(988\) 0 0
\(989\) −6.07672 −0.193228
\(990\) 0 0
\(991\) −23.2874 −0.739749 −0.369874 0.929082i \(-0.620599\pi\)
−0.369874 + 0.929082i \(0.620599\pi\)
\(992\) 0 0
\(993\) −30.2224 −0.959077
\(994\) 0 0
\(995\) −21.9376 −0.695468
\(996\) 0 0
\(997\) −21.1584 −0.670092 −0.335046 0.942202i \(-0.608752\pi\)
−0.335046 + 0.942202i \(0.608752\pi\)
\(998\) 0 0
\(999\) 19.0297 0.602072
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 4840.2.a.bd.1.6 yes 6
4.3 odd 2 9680.2.a.da.1.1 6
11.10 odd 2 4840.2.a.bc.1.6 6
44.43 even 2 9680.2.a.db.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.6 6 11.10 odd 2
4840.2.a.bd.1.6 yes 6 1.1 even 1 trivial
9680.2.a.da.1.1 6 4.3 odd 2
9680.2.a.db.1.1 6 44.43 even 2