Properties

Label 3344.2.a.t.1.1
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.15351\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.26452 q^{3} +0.637602 q^{5} -2.66942 q^{7} +2.12805 q^{9} +O(q^{10})\) \(q-2.26452 q^{3} +0.637602 q^{5} -2.66942 q^{7} +2.12805 q^{9} -1.00000 q^{11} -1.20725 q^{13} -1.44386 q^{15} +0.222022 q^{17} +1.00000 q^{19} +6.04495 q^{21} +9.48717 q^{23} -4.59346 q^{25} +1.97454 q^{27} +4.94518 q^{29} +3.05563 q^{31} +2.26452 q^{33} -1.70203 q^{35} +7.18015 q^{37} +2.73384 q^{39} -6.92650 q^{41} -1.53538 q^{43} +1.35685 q^{45} -5.94581 q^{47} +0.125785 q^{49} -0.502774 q^{51} -9.63879 q^{53} -0.637602 q^{55} -2.26452 q^{57} -2.65817 q^{59} -0.809792 q^{61} -5.68066 q^{63} -0.769744 q^{65} +2.22447 q^{67} -21.4839 q^{69} -2.58912 q^{71} +16.6108 q^{73} +10.4020 q^{75} +2.66942 q^{77} -5.69298 q^{79} -10.8556 q^{81} +2.93784 q^{83} +0.141562 q^{85} -11.1985 q^{87} -5.54136 q^{89} +3.22265 q^{91} -6.91954 q^{93} +0.637602 q^{95} +13.7728 q^{97} -2.12805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.26452 −1.30742 −0.653711 0.756745i \(-0.726789\pi\)
−0.653711 + 0.756745i \(0.726789\pi\)
\(4\) 0 0
\(5\) 0.637602 0.285144 0.142572 0.989784i \(-0.454463\pi\)
0.142572 + 0.989784i \(0.454463\pi\)
\(6\) 0 0
\(7\) −2.66942 −1.00894 −0.504472 0.863428i \(-0.668313\pi\)
−0.504472 + 0.863428i \(0.668313\pi\)
\(8\) 0 0
\(9\) 2.12805 0.709351
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.20725 −0.334831 −0.167415 0.985886i \(-0.553542\pi\)
−0.167415 + 0.985886i \(0.553542\pi\)
\(14\) 0 0
\(15\) −1.44386 −0.372804
\(16\) 0 0
\(17\) 0.222022 0.0538483 0.0269242 0.999637i \(-0.491429\pi\)
0.0269242 + 0.999637i \(0.491429\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 6.04495 1.31912
\(22\) 0 0
\(23\) 9.48717 1.97821 0.989106 0.147206i \(-0.0470279\pi\)
0.989106 + 0.147206i \(0.0470279\pi\)
\(24\) 0 0
\(25\) −4.59346 −0.918693
\(26\) 0 0
\(27\) 1.97454 0.380001
\(28\) 0 0
\(29\) 4.94518 0.918297 0.459148 0.888360i \(-0.348155\pi\)
0.459148 + 0.888360i \(0.348155\pi\)
\(30\) 0 0
\(31\) 3.05563 0.548808 0.274404 0.961615i \(-0.411520\pi\)
0.274404 + 0.961615i \(0.411520\pi\)
\(32\) 0 0
\(33\) 2.26452 0.394202
\(34\) 0 0
\(35\) −1.70203 −0.287695
\(36\) 0 0
\(37\) 7.18015 1.18041 0.590205 0.807254i \(-0.299047\pi\)
0.590205 + 0.807254i \(0.299047\pi\)
\(38\) 0 0
\(39\) 2.73384 0.437765
\(40\) 0 0
\(41\) −6.92650 −1.08174 −0.540869 0.841107i \(-0.681904\pi\)
−0.540869 + 0.841107i \(0.681904\pi\)
\(42\) 0 0
\(43\) −1.53538 −0.234144 −0.117072 0.993123i \(-0.537351\pi\)
−0.117072 + 0.993123i \(0.537351\pi\)
\(44\) 0 0
\(45\) 1.35685 0.202267
\(46\) 0 0
\(47\) −5.94581 −0.867285 −0.433642 0.901085i \(-0.642772\pi\)
−0.433642 + 0.901085i \(0.642772\pi\)
\(48\) 0 0
\(49\) 0.125785 0.0179692
\(50\) 0 0
\(51\) −0.502774 −0.0704024
\(52\) 0 0
\(53\) −9.63879 −1.32399 −0.661995 0.749509i \(-0.730290\pi\)
−0.661995 + 0.749509i \(0.730290\pi\)
\(54\) 0 0
\(55\) −0.637602 −0.0859742
\(56\) 0 0
\(57\) −2.26452 −0.299943
\(58\) 0 0
\(59\) −2.65817 −0.346065 −0.173032 0.984916i \(-0.555357\pi\)
−0.173032 + 0.984916i \(0.555357\pi\)
\(60\) 0 0
\(61\) −0.809792 −0.103683 −0.0518416 0.998655i \(-0.516509\pi\)
−0.0518416 + 0.998655i \(0.516509\pi\)
\(62\) 0 0
\(63\) −5.68066 −0.715696
\(64\) 0 0
\(65\) −0.769744 −0.0954750
\(66\) 0 0
\(67\) 2.22447 0.271763 0.135881 0.990725i \(-0.456613\pi\)
0.135881 + 0.990725i \(0.456613\pi\)
\(68\) 0 0
\(69\) −21.4839 −2.58636
\(70\) 0 0
\(71\) −2.58912 −0.307272 −0.153636 0.988127i \(-0.549098\pi\)
−0.153636 + 0.988127i \(0.549098\pi\)
\(72\) 0 0
\(73\) 16.6108 1.94414 0.972071 0.234687i \(-0.0754065\pi\)
0.972071 + 0.234687i \(0.0754065\pi\)
\(74\) 0 0
\(75\) 10.4020 1.20112
\(76\) 0 0
\(77\) 2.66942 0.304208
\(78\) 0 0
\(79\) −5.69298 −0.640510 −0.320255 0.947331i \(-0.603769\pi\)
−0.320255 + 0.947331i \(0.603769\pi\)
\(80\) 0 0
\(81\) −10.8556 −1.20617
\(82\) 0 0
\(83\) 2.93784 0.322470 0.161235 0.986916i \(-0.448452\pi\)
0.161235 + 0.986916i \(0.448452\pi\)
\(84\) 0 0
\(85\) 0.141562 0.0153545
\(86\) 0 0
\(87\) −11.1985 −1.20060
\(88\) 0 0
\(89\) −5.54136 −0.587383 −0.293692 0.955900i \(-0.594884\pi\)
−0.293692 + 0.955900i \(0.594884\pi\)
\(90\) 0 0
\(91\) 3.22265 0.337826
\(92\) 0 0
\(93\) −6.91954 −0.717523
\(94\) 0 0
\(95\) 0.637602 0.0654166
\(96\) 0 0
\(97\) 13.7728 1.39842 0.699209 0.714917i \(-0.253536\pi\)
0.699209 + 0.714917i \(0.253536\pi\)
\(98\) 0 0
\(99\) −2.12805 −0.213877
\(100\) 0 0
\(101\) 12.0308 1.19711 0.598555 0.801082i \(-0.295742\pi\)
0.598555 + 0.801082i \(0.295742\pi\)
\(102\) 0 0
\(103\) −8.53068 −0.840553 −0.420276 0.907396i \(-0.638067\pi\)
−0.420276 + 0.907396i \(0.638067\pi\)
\(104\) 0 0
\(105\) 3.85427 0.376138
\(106\) 0 0
\(107\) −1.64585 −0.159110 −0.0795552 0.996830i \(-0.525350\pi\)
−0.0795552 + 0.996830i \(0.525350\pi\)
\(108\) 0 0
\(109\) 16.8065 1.60977 0.804886 0.593430i \(-0.202226\pi\)
0.804886 + 0.593430i \(0.202226\pi\)
\(110\) 0 0
\(111\) −16.2596 −1.54329
\(112\) 0 0
\(113\) −11.2947 −1.06252 −0.531258 0.847210i \(-0.678280\pi\)
−0.531258 + 0.847210i \(0.678280\pi\)
\(114\) 0 0
\(115\) 6.04904 0.564076
\(116\) 0 0
\(117\) −2.56909 −0.237512
\(118\) 0 0
\(119\) −0.592670 −0.0543300
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 15.6852 1.41429
\(124\) 0 0
\(125\) −6.11681 −0.547104
\(126\) 0 0
\(127\) −18.5894 −1.64954 −0.824771 0.565467i \(-0.808696\pi\)
−0.824771 + 0.565467i \(0.808696\pi\)
\(128\) 0 0
\(129\) 3.47690 0.306124
\(130\) 0 0
\(131\) 3.09098 0.270060 0.135030 0.990842i \(-0.456887\pi\)
0.135030 + 0.990842i \(0.456887\pi\)
\(132\) 0 0
\(133\) −2.66942 −0.231468
\(134\) 0 0
\(135\) 1.25897 0.108355
\(136\) 0 0
\(137\) −8.67619 −0.741257 −0.370629 0.928781i \(-0.620858\pi\)
−0.370629 + 0.928781i \(0.620858\pi\)
\(138\) 0 0
\(139\) −14.5426 −1.23348 −0.616742 0.787166i \(-0.711548\pi\)
−0.616742 + 0.787166i \(0.711548\pi\)
\(140\) 0 0
\(141\) 13.4644 1.13391
\(142\) 0 0
\(143\) 1.20725 0.100955
\(144\) 0 0
\(145\) 3.15306 0.261847
\(146\) 0 0
\(147\) −0.284842 −0.0234933
\(148\) 0 0
\(149\) −18.4716 −1.51325 −0.756625 0.653849i \(-0.773153\pi\)
−0.756625 + 0.653849i \(0.773153\pi\)
\(150\) 0 0
\(151\) 0.913992 0.0743796 0.0371898 0.999308i \(-0.488159\pi\)
0.0371898 + 0.999308i \(0.488159\pi\)
\(152\) 0 0
\(153\) 0.472475 0.0381973
\(154\) 0 0
\(155\) 1.94828 0.156489
\(156\) 0 0
\(157\) −2.39421 −0.191079 −0.0955395 0.995426i \(-0.530458\pi\)
−0.0955395 + 0.995426i \(0.530458\pi\)
\(158\) 0 0
\(159\) 21.8272 1.73101
\(160\) 0 0
\(161\) −25.3252 −1.99591
\(162\) 0 0
\(163\) −14.1986 −1.11212 −0.556061 0.831141i \(-0.687688\pi\)
−0.556061 + 0.831141i \(0.687688\pi\)
\(164\) 0 0
\(165\) 1.44386 0.112405
\(166\) 0 0
\(167\) −9.25056 −0.715830 −0.357915 0.933754i \(-0.616512\pi\)
−0.357915 + 0.933754i \(0.616512\pi\)
\(168\) 0 0
\(169\) −11.5426 −0.887888
\(170\) 0 0
\(171\) 2.12805 0.162736
\(172\) 0 0
\(173\) 5.39376 0.410080 0.205040 0.978754i \(-0.434268\pi\)
0.205040 + 0.978754i \(0.434268\pi\)
\(174\) 0 0
\(175\) 12.2619 0.926910
\(176\) 0 0
\(177\) 6.01949 0.452453
\(178\) 0 0
\(179\) −1.30720 −0.0977048 −0.0488524 0.998806i \(-0.515556\pi\)
−0.0488524 + 0.998806i \(0.515556\pi\)
\(180\) 0 0
\(181\) −10.7624 −0.799961 −0.399980 0.916524i \(-0.630983\pi\)
−0.399980 + 0.916524i \(0.630983\pi\)
\(182\) 0 0
\(183\) 1.83379 0.135558
\(184\) 0 0
\(185\) 4.57808 0.336587
\(186\) 0 0
\(187\) −0.222022 −0.0162359
\(188\) 0 0
\(189\) −5.27088 −0.383400
\(190\) 0 0
\(191\) −15.3041 −1.10737 −0.553684 0.832727i \(-0.686778\pi\)
−0.553684 + 0.832727i \(0.686778\pi\)
\(192\) 0 0
\(193\) −18.3032 −1.31749 −0.658747 0.752364i \(-0.728913\pi\)
−0.658747 + 0.752364i \(0.728913\pi\)
\(194\) 0 0
\(195\) 1.74310 0.124826
\(196\) 0 0
\(197\) −0.576171 −0.0410505 −0.0205252 0.999789i \(-0.506534\pi\)
−0.0205252 + 0.999789i \(0.506534\pi\)
\(198\) 0 0
\(199\) 10.1678 0.720778 0.360389 0.932802i \(-0.382644\pi\)
0.360389 + 0.932802i \(0.382644\pi\)
\(200\) 0 0
\(201\) −5.03736 −0.355308
\(202\) 0 0
\(203\) −13.2007 −0.926510
\(204\) 0 0
\(205\) −4.41635 −0.308451
\(206\) 0 0
\(207\) 20.1892 1.40325
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 7.25837 0.499687 0.249843 0.968286i \(-0.419621\pi\)
0.249843 + 0.968286i \(0.419621\pi\)
\(212\) 0 0
\(213\) 5.86312 0.401734
\(214\) 0 0
\(215\) −0.978963 −0.0667647
\(216\) 0 0
\(217\) −8.15675 −0.553716
\(218\) 0 0
\(219\) −37.6154 −2.54181
\(220\) 0 0
\(221\) −0.268036 −0.0180301
\(222\) 0 0
\(223\) −16.2772 −1.09000 −0.545000 0.838436i \(-0.683471\pi\)
−0.545000 + 0.838436i \(0.683471\pi\)
\(224\) 0 0
\(225\) −9.77513 −0.651675
\(226\) 0 0
\(227\) 20.9260 1.38890 0.694452 0.719539i \(-0.255647\pi\)
0.694452 + 0.719539i \(0.255647\pi\)
\(228\) 0 0
\(229\) −10.8083 −0.714234 −0.357117 0.934060i \(-0.616240\pi\)
−0.357117 + 0.934060i \(0.616240\pi\)
\(230\) 0 0
\(231\) −6.04495 −0.397728
\(232\) 0 0
\(233\) −25.4173 −1.66514 −0.832570 0.553919i \(-0.813132\pi\)
−0.832570 + 0.553919i \(0.813132\pi\)
\(234\) 0 0
\(235\) −3.79106 −0.247301
\(236\) 0 0
\(237\) 12.8919 0.837417
\(238\) 0 0
\(239\) −0.352238 −0.0227844 −0.0113922 0.999935i \(-0.503626\pi\)
−0.0113922 + 0.999935i \(0.503626\pi\)
\(240\) 0 0
\(241\) −15.6832 −1.01024 −0.505122 0.863048i \(-0.668552\pi\)
−0.505122 + 0.863048i \(0.668552\pi\)
\(242\) 0 0
\(243\) 18.6590 1.19697
\(244\) 0 0
\(245\) 0.0802004 0.00512382
\(246\) 0 0
\(247\) −1.20725 −0.0768154
\(248\) 0 0
\(249\) −6.65281 −0.421605
\(250\) 0 0
\(251\) −1.16332 −0.0734282 −0.0367141 0.999326i \(-0.511689\pi\)
−0.0367141 + 0.999326i \(0.511689\pi\)
\(252\) 0 0
\(253\) −9.48717 −0.596453
\(254\) 0 0
\(255\) −0.320570 −0.0200748
\(256\) 0 0
\(257\) −15.0510 −0.938857 −0.469428 0.882971i \(-0.655540\pi\)
−0.469428 + 0.882971i \(0.655540\pi\)
\(258\) 0 0
\(259\) −19.1668 −1.19097
\(260\) 0 0
\(261\) 10.5236 0.651394
\(262\) 0 0
\(263\) 20.6609 1.27400 0.637002 0.770862i \(-0.280174\pi\)
0.637002 + 0.770862i \(0.280174\pi\)
\(264\) 0 0
\(265\) −6.14571 −0.377528
\(266\) 0 0
\(267\) 12.5485 0.767958
\(268\) 0 0
\(269\) −2.82662 −0.172342 −0.0861711 0.996280i \(-0.527463\pi\)
−0.0861711 + 0.996280i \(0.527463\pi\)
\(270\) 0 0
\(271\) −23.5970 −1.43341 −0.716707 0.697375i \(-0.754351\pi\)
−0.716707 + 0.697375i \(0.754351\pi\)
\(272\) 0 0
\(273\) −7.29776 −0.441680
\(274\) 0 0
\(275\) 4.59346 0.276996
\(276\) 0 0
\(277\) −18.7773 −1.12822 −0.564110 0.825700i \(-0.690781\pi\)
−0.564110 + 0.825700i \(0.690781\pi\)
\(278\) 0 0
\(279\) 6.50254 0.389297
\(280\) 0 0
\(281\) 27.2632 1.62639 0.813194 0.581993i \(-0.197727\pi\)
0.813194 + 0.581993i \(0.197727\pi\)
\(282\) 0 0
\(283\) 27.2601 1.62044 0.810221 0.586124i \(-0.199347\pi\)
0.810221 + 0.586124i \(0.199347\pi\)
\(284\) 0 0
\(285\) −1.44386 −0.0855270
\(286\) 0 0
\(287\) 18.4897 1.09141
\(288\) 0 0
\(289\) −16.9507 −0.997100
\(290\) 0 0
\(291\) −31.1888 −1.82832
\(292\) 0 0
\(293\) 30.9996 1.81101 0.905507 0.424332i \(-0.139491\pi\)
0.905507 + 0.424332i \(0.139491\pi\)
\(294\) 0 0
\(295\) −1.69486 −0.0986784
\(296\) 0 0
\(297\) −1.97454 −0.114575
\(298\) 0 0
\(299\) −11.4534 −0.662366
\(300\) 0 0
\(301\) 4.09858 0.236238
\(302\) 0 0
\(303\) −27.2440 −1.56513
\(304\) 0 0
\(305\) −0.516325 −0.0295647
\(306\) 0 0
\(307\) 18.2495 1.04156 0.520778 0.853692i \(-0.325642\pi\)
0.520778 + 0.853692i \(0.325642\pi\)
\(308\) 0 0
\(309\) 19.3179 1.09896
\(310\) 0 0
\(311\) −30.8931 −1.75179 −0.875894 0.482503i \(-0.839728\pi\)
−0.875894 + 0.482503i \(0.839728\pi\)
\(312\) 0 0
\(313\) 28.7973 1.62772 0.813859 0.581062i \(-0.197363\pi\)
0.813859 + 0.581062i \(0.197363\pi\)
\(314\) 0 0
\(315\) −3.62200 −0.204076
\(316\) 0 0
\(317\) 5.96040 0.334769 0.167385 0.985892i \(-0.446468\pi\)
0.167385 + 0.985892i \(0.446468\pi\)
\(318\) 0 0
\(319\) −4.94518 −0.276877
\(320\) 0 0
\(321\) 3.72706 0.208024
\(322\) 0 0
\(323\) 0.222022 0.0123536
\(324\) 0 0
\(325\) 5.54545 0.307606
\(326\) 0 0
\(327\) −38.0587 −2.10465
\(328\) 0 0
\(329\) 15.8718 0.875043
\(330\) 0 0
\(331\) −33.4010 −1.83589 −0.917944 0.396711i \(-0.870151\pi\)
−0.917944 + 0.396711i \(0.870151\pi\)
\(332\) 0 0
\(333\) 15.2797 0.837325
\(334\) 0 0
\(335\) 1.41833 0.0774915
\(336\) 0 0
\(337\) 24.9988 1.36177 0.680885 0.732390i \(-0.261595\pi\)
0.680885 + 0.732390i \(0.261595\pi\)
\(338\) 0 0
\(339\) 25.5771 1.38916
\(340\) 0 0
\(341\) −3.05563 −0.165472
\(342\) 0 0
\(343\) 18.3501 0.990815
\(344\) 0 0
\(345\) −13.6982 −0.737485
\(346\) 0 0
\(347\) 8.35577 0.448561 0.224281 0.974525i \(-0.427997\pi\)
0.224281 + 0.974525i \(0.427997\pi\)
\(348\) 0 0
\(349\) −28.3911 −1.51974 −0.759871 0.650074i \(-0.774738\pi\)
−0.759871 + 0.650074i \(0.774738\pi\)
\(350\) 0 0
\(351\) −2.38377 −0.127236
\(352\) 0 0
\(353\) −12.4082 −0.660423 −0.330212 0.943907i \(-0.607120\pi\)
−0.330212 + 0.943907i \(0.607120\pi\)
\(354\) 0 0
\(355\) −1.65083 −0.0876169
\(356\) 0 0
\(357\) 1.34211 0.0710322
\(358\) 0 0
\(359\) −12.1134 −0.639319 −0.319659 0.947533i \(-0.603568\pi\)
−0.319659 + 0.947533i \(0.603568\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.26452 −0.118856
\(364\) 0 0
\(365\) 10.5911 0.554361
\(366\) 0 0
\(367\) 4.06064 0.211964 0.105982 0.994368i \(-0.466201\pi\)
0.105982 + 0.994368i \(0.466201\pi\)
\(368\) 0 0
\(369\) −14.7399 −0.767331
\(370\) 0 0
\(371\) 25.7299 1.33583
\(372\) 0 0
\(373\) 35.4898 1.83759 0.918796 0.394732i \(-0.129163\pi\)
0.918796 + 0.394732i \(0.129163\pi\)
\(374\) 0 0
\(375\) 13.8516 0.715296
\(376\) 0 0
\(377\) −5.97006 −0.307474
\(378\) 0 0
\(379\) −13.5584 −0.696449 −0.348224 0.937411i \(-0.613215\pi\)
−0.348224 + 0.937411i \(0.613215\pi\)
\(380\) 0 0
\(381\) 42.0961 2.15665
\(382\) 0 0
\(383\) 23.8326 1.21779 0.608894 0.793251i \(-0.291613\pi\)
0.608894 + 0.793251i \(0.291613\pi\)
\(384\) 0 0
\(385\) 1.70203 0.0867432
\(386\) 0 0
\(387\) −3.26737 −0.166090
\(388\) 0 0
\(389\) −12.7567 −0.646789 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(390\) 0 0
\(391\) 2.10636 0.106523
\(392\) 0 0
\(393\) −6.99958 −0.353082
\(394\) 0 0
\(395\) −3.62986 −0.182638
\(396\) 0 0
\(397\) −30.3169 −1.52156 −0.760780 0.649009i \(-0.775184\pi\)
−0.760780 + 0.649009i \(0.775184\pi\)
\(398\) 0 0
\(399\) 6.04495 0.302626
\(400\) 0 0
\(401\) 32.6370 1.62982 0.814908 0.579591i \(-0.196788\pi\)
0.814908 + 0.579591i \(0.196788\pi\)
\(402\) 0 0
\(403\) −3.68891 −0.183758
\(404\) 0 0
\(405\) −6.92152 −0.343933
\(406\) 0 0
\(407\) −7.18015 −0.355907
\(408\) 0 0
\(409\) −23.1587 −1.14512 −0.572562 0.819862i \(-0.694050\pi\)
−0.572562 + 0.819862i \(0.694050\pi\)
\(410\) 0 0
\(411\) 19.6474 0.969135
\(412\) 0 0
\(413\) 7.09578 0.349160
\(414\) 0 0
\(415\) 1.87318 0.0919506
\(416\) 0 0
\(417\) 32.9319 1.61268
\(418\) 0 0
\(419\) 14.2213 0.694755 0.347377 0.937725i \(-0.387072\pi\)
0.347377 + 0.937725i \(0.387072\pi\)
\(420\) 0 0
\(421\) 15.9605 0.777866 0.388933 0.921266i \(-0.372844\pi\)
0.388933 + 0.921266i \(0.372844\pi\)
\(422\) 0 0
\(423\) −12.6530 −0.615209
\(424\) 0 0
\(425\) −1.01985 −0.0494701
\(426\) 0 0
\(427\) 2.16167 0.104611
\(428\) 0 0
\(429\) −2.73384 −0.131991
\(430\) 0 0
\(431\) −24.5100 −1.18061 −0.590304 0.807181i \(-0.700992\pi\)
−0.590304 + 0.807181i \(0.700992\pi\)
\(432\) 0 0
\(433\) −19.1839 −0.921921 −0.460961 0.887421i \(-0.652495\pi\)
−0.460961 + 0.887421i \(0.652495\pi\)
\(434\) 0 0
\(435\) −7.14016 −0.342344
\(436\) 0 0
\(437\) 9.48717 0.453833
\(438\) 0 0
\(439\) −13.7961 −0.658450 −0.329225 0.944251i \(-0.606788\pi\)
−0.329225 + 0.944251i \(0.606788\pi\)
\(440\) 0 0
\(441\) 0.267676 0.0127465
\(442\) 0 0
\(443\) −14.1259 −0.671140 −0.335570 0.942015i \(-0.608929\pi\)
−0.335570 + 0.942015i \(0.608929\pi\)
\(444\) 0 0
\(445\) −3.53318 −0.167489
\(446\) 0 0
\(447\) 41.8292 1.97846
\(448\) 0 0
\(449\) −21.0016 −0.991129 −0.495565 0.868571i \(-0.665039\pi\)
−0.495565 + 0.868571i \(0.665039\pi\)
\(450\) 0 0
\(451\) 6.92650 0.326156
\(452\) 0 0
\(453\) −2.06975 −0.0972455
\(454\) 0 0
\(455\) 2.05477 0.0963290
\(456\) 0 0
\(457\) −22.4621 −1.05073 −0.525367 0.850876i \(-0.676072\pi\)
−0.525367 + 0.850876i \(0.676072\pi\)
\(458\) 0 0
\(459\) 0.438393 0.0204624
\(460\) 0 0
\(461\) −31.1213 −1.44946 −0.724732 0.689031i \(-0.758036\pi\)
−0.724732 + 0.689031i \(0.758036\pi\)
\(462\) 0 0
\(463\) −31.6932 −1.47291 −0.736455 0.676487i \(-0.763502\pi\)
−0.736455 + 0.676487i \(0.763502\pi\)
\(464\) 0 0
\(465\) −4.41191 −0.204597
\(466\) 0 0
\(467\) 1.75474 0.0811995 0.0405997 0.999175i \(-0.487073\pi\)
0.0405997 + 0.999175i \(0.487073\pi\)
\(468\) 0 0
\(469\) −5.93804 −0.274193
\(470\) 0 0
\(471\) 5.42174 0.249821
\(472\) 0 0
\(473\) 1.53538 0.0705970
\(474\) 0 0
\(475\) −4.59346 −0.210763
\(476\) 0 0
\(477\) −20.5118 −0.939173
\(478\) 0 0
\(479\) 11.1294 0.508515 0.254257 0.967137i \(-0.418169\pi\)
0.254257 + 0.967137i \(0.418169\pi\)
\(480\) 0 0
\(481\) −8.66823 −0.395237
\(482\) 0 0
\(483\) 57.3495 2.60949
\(484\) 0 0
\(485\) 8.78158 0.398751
\(486\) 0 0
\(487\) −3.27007 −0.148181 −0.0740905 0.997252i \(-0.523605\pi\)
−0.0740905 + 0.997252i \(0.523605\pi\)
\(488\) 0 0
\(489\) 32.1531 1.45401
\(490\) 0 0
\(491\) −29.2686 −1.32087 −0.660437 0.750882i \(-0.729629\pi\)
−0.660437 + 0.750882i \(0.729629\pi\)
\(492\) 0 0
\(493\) 1.09794 0.0494487
\(494\) 0 0
\(495\) −1.35685 −0.0609859
\(496\) 0 0
\(497\) 6.91145 0.310021
\(498\) 0 0
\(499\) 1.04828 0.0469275 0.0234638 0.999725i \(-0.492531\pi\)
0.0234638 + 0.999725i \(0.492531\pi\)
\(500\) 0 0
\(501\) 20.9481 0.935891
\(502\) 0 0
\(503\) −28.4724 −1.26952 −0.634761 0.772709i \(-0.718901\pi\)
−0.634761 + 0.772709i \(0.718901\pi\)
\(504\) 0 0
\(505\) 7.67086 0.341349
\(506\) 0 0
\(507\) 26.1383 1.16084
\(508\) 0 0
\(509\) 39.6801 1.75879 0.879396 0.476091i \(-0.157947\pi\)
0.879396 + 0.476091i \(0.157947\pi\)
\(510\) 0 0
\(511\) −44.3410 −1.96153
\(512\) 0 0
\(513\) 1.97454 0.0871782
\(514\) 0 0
\(515\) −5.43918 −0.239679
\(516\) 0 0
\(517\) 5.94581 0.261496
\(518\) 0 0
\(519\) −12.2143 −0.536147
\(520\) 0 0
\(521\) −15.3168 −0.671043 −0.335522 0.942033i \(-0.608913\pi\)
−0.335522 + 0.942033i \(0.608913\pi\)
\(522\) 0 0
\(523\) 4.17850 0.182713 0.0913566 0.995818i \(-0.470880\pi\)
0.0913566 + 0.995818i \(0.470880\pi\)
\(524\) 0 0
\(525\) −27.7672 −1.21186
\(526\) 0 0
\(527\) 0.678418 0.0295524
\(528\) 0 0
\(529\) 67.0064 2.91332
\(530\) 0 0
\(531\) −5.65674 −0.245481
\(532\) 0 0
\(533\) 8.36201 0.362199
\(534\) 0 0
\(535\) −1.04940 −0.0453694
\(536\) 0 0
\(537\) 2.96018 0.127741
\(538\) 0 0
\(539\) −0.125785 −0.00541792
\(540\) 0 0
\(541\) −14.6201 −0.628565 −0.314283 0.949329i \(-0.601764\pi\)
−0.314283 + 0.949329i \(0.601764\pi\)
\(542\) 0 0
\(543\) 24.3716 1.04589
\(544\) 0 0
\(545\) 10.7159 0.459017
\(546\) 0 0
\(547\) 34.7220 1.48461 0.742303 0.670064i \(-0.233733\pi\)
0.742303 + 0.670064i \(0.233733\pi\)
\(548\) 0 0
\(549\) −1.72328 −0.0735478
\(550\) 0 0
\(551\) 4.94518 0.210672
\(552\) 0 0
\(553\) 15.1969 0.646240
\(554\) 0 0
\(555\) −10.3672 −0.440061
\(556\) 0 0
\(557\) −7.07603 −0.299821 −0.149910 0.988700i \(-0.547899\pi\)
−0.149910 + 0.988700i \(0.547899\pi\)
\(558\) 0 0
\(559\) 1.85359 0.0783985
\(560\) 0 0
\(561\) 0.502774 0.0212271
\(562\) 0 0
\(563\) 32.5712 1.37271 0.686357 0.727265i \(-0.259209\pi\)
0.686357 + 0.727265i \(0.259209\pi\)
\(564\) 0 0
\(565\) −7.20152 −0.302970
\(566\) 0 0
\(567\) 28.9780 1.21696
\(568\) 0 0
\(569\) 32.6457 1.36858 0.684290 0.729210i \(-0.260112\pi\)
0.684290 + 0.729210i \(0.260112\pi\)
\(570\) 0 0
\(571\) 35.8059 1.49843 0.749216 0.662326i \(-0.230431\pi\)
0.749216 + 0.662326i \(0.230431\pi\)
\(572\) 0 0
\(573\) 34.6565 1.44780
\(574\) 0 0
\(575\) −43.5790 −1.81737
\(576\) 0 0
\(577\) 15.7218 0.654509 0.327254 0.944936i \(-0.393877\pi\)
0.327254 + 0.944936i \(0.393877\pi\)
\(578\) 0 0
\(579\) 41.4480 1.72252
\(580\) 0 0
\(581\) −7.84233 −0.325355
\(582\) 0 0
\(583\) 9.63879 0.399198
\(584\) 0 0
\(585\) −1.63806 −0.0677253
\(586\) 0 0
\(587\) −24.4810 −1.01044 −0.505220 0.862991i \(-0.668588\pi\)
−0.505220 + 0.862991i \(0.668588\pi\)
\(588\) 0 0
\(589\) 3.05563 0.125905
\(590\) 0 0
\(591\) 1.30475 0.0536703
\(592\) 0 0
\(593\) −21.2995 −0.874666 −0.437333 0.899300i \(-0.644077\pi\)
−0.437333 + 0.899300i \(0.644077\pi\)
\(594\) 0 0
\(595\) −0.377887 −0.0154919
\(596\) 0 0
\(597\) −23.0253 −0.942361
\(598\) 0 0
\(599\) 8.90839 0.363987 0.181994 0.983300i \(-0.441745\pi\)
0.181994 + 0.983300i \(0.441745\pi\)
\(600\) 0 0
\(601\) −8.36794 −0.341335 −0.170668 0.985329i \(-0.554592\pi\)
−0.170668 + 0.985329i \(0.554592\pi\)
\(602\) 0 0
\(603\) 4.73379 0.192775
\(604\) 0 0
\(605\) 0.637602 0.0259222
\(606\) 0 0
\(607\) −43.2321 −1.75474 −0.877368 0.479819i \(-0.840702\pi\)
−0.877368 + 0.479819i \(0.840702\pi\)
\(608\) 0 0
\(609\) 29.8933 1.21134
\(610\) 0 0
\(611\) 7.17807 0.290394
\(612\) 0 0
\(613\) −12.4893 −0.504438 −0.252219 0.967670i \(-0.581160\pi\)
−0.252219 + 0.967670i \(0.581160\pi\)
\(614\) 0 0
\(615\) 10.0009 0.403276
\(616\) 0 0
\(617\) −18.4079 −0.741073 −0.370536 0.928818i \(-0.620826\pi\)
−0.370536 + 0.928818i \(0.620826\pi\)
\(618\) 0 0
\(619\) 7.30900 0.293773 0.146887 0.989153i \(-0.453075\pi\)
0.146887 + 0.989153i \(0.453075\pi\)
\(620\) 0 0
\(621\) 18.7328 0.751722
\(622\) 0 0
\(623\) 14.7922 0.592637
\(624\) 0 0
\(625\) 19.0672 0.762689
\(626\) 0 0
\(627\) 2.26452 0.0904362
\(628\) 0 0
\(629\) 1.59415 0.0635631
\(630\) 0 0
\(631\) 11.8397 0.471329 0.235665 0.971834i \(-0.424273\pi\)
0.235665 + 0.971834i \(0.424273\pi\)
\(632\) 0 0
\(633\) −16.4367 −0.653301
\(634\) 0 0
\(635\) −11.8526 −0.470357
\(636\) 0 0
\(637\) −0.151853 −0.00601664
\(638\) 0 0
\(639\) −5.50979 −0.217964
\(640\) 0 0
\(641\) −36.5974 −1.44551 −0.722756 0.691104i \(-0.757125\pi\)
−0.722756 + 0.691104i \(0.757125\pi\)
\(642\) 0 0
\(643\) 14.0452 0.553888 0.276944 0.960886i \(-0.410678\pi\)
0.276944 + 0.960886i \(0.410678\pi\)
\(644\) 0 0
\(645\) 2.21688 0.0872896
\(646\) 0 0
\(647\) 43.5266 1.71121 0.855604 0.517631i \(-0.173186\pi\)
0.855604 + 0.517631i \(0.173186\pi\)
\(648\) 0 0
\(649\) 2.65817 0.104342
\(650\) 0 0
\(651\) 18.4711 0.723941
\(652\) 0 0
\(653\) 44.0328 1.72314 0.861568 0.507642i \(-0.169483\pi\)
0.861568 + 0.507642i \(0.169483\pi\)
\(654\) 0 0
\(655\) 1.97081 0.0770060
\(656\) 0 0
\(657\) 35.3486 1.37908
\(658\) 0 0
\(659\) 3.86784 0.150670 0.0753349 0.997158i \(-0.475997\pi\)
0.0753349 + 0.997158i \(0.475997\pi\)
\(660\) 0 0
\(661\) 7.96999 0.309997 0.154998 0.987915i \(-0.450463\pi\)
0.154998 + 0.987915i \(0.450463\pi\)
\(662\) 0 0
\(663\) 0.606973 0.0235729
\(664\) 0 0
\(665\) −1.70203 −0.0660017
\(666\) 0 0
\(667\) 46.9158 1.81659
\(668\) 0 0
\(669\) 36.8600 1.42509
\(670\) 0 0
\(671\) 0.809792 0.0312617
\(672\) 0 0
\(673\) 21.7872 0.839835 0.419917 0.907562i \(-0.362059\pi\)
0.419917 + 0.907562i \(0.362059\pi\)
\(674\) 0 0
\(675\) −9.06999 −0.349104
\(676\) 0 0
\(677\) 33.2004 1.27599 0.637997 0.770039i \(-0.279763\pi\)
0.637997 + 0.770039i \(0.279763\pi\)
\(678\) 0 0
\(679\) −36.7654 −1.41093
\(680\) 0 0
\(681\) −47.3873 −1.81588
\(682\) 0 0
\(683\) 34.9595 1.33769 0.668843 0.743403i \(-0.266790\pi\)
0.668843 + 0.743403i \(0.266790\pi\)
\(684\) 0 0
\(685\) −5.53196 −0.211365
\(686\) 0 0
\(687\) 24.4757 0.933805
\(688\) 0 0
\(689\) 11.6364 0.443312
\(690\) 0 0
\(691\) 30.4626 1.15885 0.579426 0.815025i \(-0.303277\pi\)
0.579426 + 0.815025i \(0.303277\pi\)
\(692\) 0 0
\(693\) 5.68066 0.215790
\(694\) 0 0
\(695\) −9.27236 −0.351721
\(696\) 0 0
\(697\) −1.53784 −0.0582497
\(698\) 0 0
\(699\) 57.5579 2.17704
\(700\) 0 0
\(701\) 10.3329 0.390269 0.195135 0.980776i \(-0.437486\pi\)
0.195135 + 0.980776i \(0.437486\pi\)
\(702\) 0 0
\(703\) 7.18015 0.270805
\(704\) 0 0
\(705\) 8.58493 0.323327
\(706\) 0 0
\(707\) −32.1152 −1.20782
\(708\) 0 0
\(709\) −45.4499 −1.70691 −0.853454 0.521169i \(-0.825496\pi\)
−0.853454 + 0.521169i \(0.825496\pi\)
\(710\) 0 0
\(711\) −12.1150 −0.454347
\(712\) 0 0
\(713\) 28.9893 1.08566
\(714\) 0 0
\(715\) 0.769744 0.0287868
\(716\) 0 0
\(717\) 0.797651 0.0297888
\(718\) 0 0
\(719\) 7.51239 0.280165 0.140082 0.990140i \(-0.455263\pi\)
0.140082 + 0.990140i \(0.455263\pi\)
\(720\) 0 0
\(721\) 22.7719 0.848071
\(722\) 0 0
\(723\) 35.5149 1.32081
\(724\) 0 0
\(725\) −22.7155 −0.843632
\(726\) 0 0
\(727\) −4.79898 −0.177984 −0.0889922 0.996032i \(-0.528365\pi\)
−0.0889922 + 0.996032i \(0.528365\pi\)
\(728\) 0 0
\(729\) −9.68700 −0.358778
\(730\) 0 0
\(731\) −0.340889 −0.0126082
\(732\) 0 0
\(733\) −8.55077 −0.315830 −0.157915 0.987453i \(-0.550477\pi\)
−0.157915 + 0.987453i \(0.550477\pi\)
\(734\) 0 0
\(735\) −0.181616 −0.00669899
\(736\) 0 0
\(737\) −2.22447 −0.0819395
\(738\) 0 0
\(739\) −38.8820 −1.43030 −0.715149 0.698972i \(-0.753641\pi\)
−0.715149 + 0.698972i \(0.753641\pi\)
\(740\) 0 0
\(741\) 2.73384 0.100430
\(742\) 0 0
\(743\) −28.4350 −1.04318 −0.521589 0.853197i \(-0.674660\pi\)
−0.521589 + 0.853197i \(0.674660\pi\)
\(744\) 0 0
\(745\) −11.7775 −0.431494
\(746\) 0 0
\(747\) 6.25189 0.228745
\(748\) 0 0
\(749\) 4.39346 0.160534
\(750\) 0 0
\(751\) −18.2732 −0.666799 −0.333400 0.942786i \(-0.608196\pi\)
−0.333400 + 0.942786i \(0.608196\pi\)
\(752\) 0 0
\(753\) 2.63436 0.0960015
\(754\) 0 0
\(755\) 0.582763 0.0212089
\(756\) 0 0
\(757\) 1.10637 0.0402117 0.0201058 0.999798i \(-0.493600\pi\)
0.0201058 + 0.999798i \(0.493600\pi\)
\(758\) 0 0
\(759\) 21.4839 0.779816
\(760\) 0 0
\(761\) 1.10110 0.0399147 0.0199573 0.999801i \(-0.493647\pi\)
0.0199573 + 0.999801i \(0.493647\pi\)
\(762\) 0 0
\(763\) −44.8636 −1.62417
\(764\) 0 0
\(765\) 0.301251 0.0108918
\(766\) 0 0
\(767\) 3.20908 0.115873
\(768\) 0 0
\(769\) 30.2914 1.09234 0.546169 0.837675i \(-0.316086\pi\)
0.546169 + 0.837675i \(0.316086\pi\)
\(770\) 0 0
\(771\) 34.0833 1.22748
\(772\) 0 0
\(773\) −43.9677 −1.58141 −0.790704 0.612199i \(-0.790285\pi\)
−0.790704 + 0.612199i \(0.790285\pi\)
\(774\) 0 0
\(775\) −14.0359 −0.504186
\(776\) 0 0
\(777\) 43.4036 1.55710
\(778\) 0 0
\(779\) −6.92650 −0.248168
\(780\) 0 0
\(781\) 2.58912 0.0926461
\(782\) 0 0
\(783\) 9.76447 0.348954
\(784\) 0 0
\(785\) −1.52655 −0.0544851
\(786\) 0 0
\(787\) 37.8350 1.34867 0.674337 0.738424i \(-0.264430\pi\)
0.674337 + 0.738424i \(0.264430\pi\)
\(788\) 0 0
\(789\) −46.7870 −1.66566
\(790\) 0 0
\(791\) 30.1502 1.07202
\(792\) 0 0
\(793\) 0.977621 0.0347163
\(794\) 0 0
\(795\) 13.9171 0.493588
\(796\) 0 0
\(797\) 11.2902 0.399918 0.199959 0.979804i \(-0.435919\pi\)
0.199959 + 0.979804i \(0.435919\pi\)
\(798\) 0 0
\(799\) −1.32010 −0.0467018
\(800\) 0 0
\(801\) −11.7923 −0.416661
\(802\) 0 0
\(803\) −16.6108 −0.586181
\(804\) 0 0
\(805\) −16.1474 −0.569121
\(806\) 0 0
\(807\) 6.40095 0.225324
\(808\) 0 0
\(809\) 17.8104 0.626180 0.313090 0.949723i \(-0.398636\pi\)
0.313090 + 0.949723i \(0.398636\pi\)
\(810\) 0 0
\(811\) −28.1638 −0.988965 −0.494482 0.869188i \(-0.664642\pi\)
−0.494482 + 0.869188i \(0.664642\pi\)
\(812\) 0 0
\(813\) 53.4358 1.87408
\(814\) 0 0
\(815\) −9.05307 −0.317115
\(816\) 0 0
\(817\) −1.53538 −0.0537162
\(818\) 0 0
\(819\) 6.85797 0.239637
\(820\) 0 0
\(821\) −44.2644 −1.54484 −0.772418 0.635114i \(-0.780953\pi\)
−0.772418 + 0.635114i \(0.780953\pi\)
\(822\) 0 0
\(823\) 22.4258 0.781716 0.390858 0.920451i \(-0.372178\pi\)
0.390858 + 0.920451i \(0.372178\pi\)
\(824\) 0 0
\(825\) −10.4020 −0.362151
\(826\) 0 0
\(827\) −19.7303 −0.686090 −0.343045 0.939319i \(-0.611458\pi\)
−0.343045 + 0.939319i \(0.611458\pi\)
\(828\) 0 0
\(829\) 5.70836 0.198259 0.0991297 0.995075i \(-0.468394\pi\)
0.0991297 + 0.995075i \(0.468394\pi\)
\(830\) 0 0
\(831\) 42.5216 1.47506
\(832\) 0 0
\(833\) 0.0279270 0.000967612 0
\(834\) 0 0
\(835\) −5.89817 −0.204115
\(836\) 0 0
\(837\) 6.03348 0.208547
\(838\) 0 0
\(839\) 0.0666011 0.00229932 0.00114966 0.999999i \(-0.499634\pi\)
0.00114966 + 0.999999i \(0.499634\pi\)
\(840\) 0 0
\(841\) −4.54521 −0.156731
\(842\) 0 0
\(843\) −61.7381 −2.12637
\(844\) 0 0
\(845\) −7.35955 −0.253176
\(846\) 0 0
\(847\) −2.66942 −0.0917222
\(848\) 0 0
\(849\) −61.7310 −2.11860
\(850\) 0 0
\(851\) 68.1193 2.33510
\(852\) 0 0
\(853\) −14.8615 −0.508848 −0.254424 0.967093i \(-0.581886\pi\)
−0.254424 + 0.967093i \(0.581886\pi\)
\(854\) 0 0
\(855\) 1.35685 0.0464033
\(856\) 0 0
\(857\) 12.5169 0.427569 0.213785 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(858\) 0 0
\(859\) 21.2738 0.725854 0.362927 0.931818i \(-0.381777\pi\)
0.362927 + 0.931818i \(0.381777\pi\)
\(860\) 0 0
\(861\) −41.8703 −1.42694
\(862\) 0 0
\(863\) 34.4402 1.17236 0.586179 0.810181i \(-0.300631\pi\)
0.586179 + 0.810181i \(0.300631\pi\)
\(864\) 0 0
\(865\) 3.43907 0.116932
\(866\) 0 0
\(867\) 38.3852 1.30363
\(868\) 0 0
\(869\) 5.69298 0.193121
\(870\) 0 0
\(871\) −2.68549 −0.0909944
\(872\) 0 0
\(873\) 29.3093 0.991969
\(874\) 0 0
\(875\) 16.3283 0.551998
\(876\) 0 0
\(877\) −0.910260 −0.0307373 −0.0153687 0.999882i \(-0.504892\pi\)
−0.0153687 + 0.999882i \(0.504892\pi\)
\(878\) 0 0
\(879\) −70.1991 −2.36776
\(880\) 0 0
\(881\) 33.7502 1.13707 0.568537 0.822658i \(-0.307510\pi\)
0.568537 + 0.822658i \(0.307510\pi\)
\(882\) 0 0
\(883\) −39.8800 −1.34207 −0.671035 0.741426i \(-0.734150\pi\)
−0.671035 + 0.741426i \(0.734150\pi\)
\(884\) 0 0
\(885\) 3.83804 0.129014
\(886\) 0 0
\(887\) 16.1868 0.543500 0.271750 0.962368i \(-0.412398\pi\)
0.271750 + 0.962368i \(0.412398\pi\)
\(888\) 0 0
\(889\) 49.6228 1.66430
\(890\) 0 0
\(891\) 10.8556 0.363675
\(892\) 0 0
\(893\) −5.94581 −0.198969
\(894\) 0 0
\(895\) −0.833474 −0.0278599
\(896\) 0 0
\(897\) 25.9364 0.865991
\(898\) 0 0
\(899\) 15.1106 0.503968
\(900\) 0 0
\(901\) −2.14003 −0.0712946
\(902\) 0 0
\(903\) −9.28131 −0.308863
\(904\) 0 0
\(905\) −6.86211 −0.228104
\(906\) 0 0
\(907\) 14.3005 0.474839 0.237419 0.971407i \(-0.423698\pi\)
0.237419 + 0.971407i \(0.423698\pi\)
\(908\) 0 0
\(909\) 25.6022 0.849171
\(910\) 0 0
\(911\) 8.50108 0.281653 0.140827 0.990034i \(-0.455024\pi\)
0.140827 + 0.990034i \(0.455024\pi\)
\(912\) 0 0
\(913\) −2.93784 −0.0972285
\(914\) 0 0
\(915\) 1.16923 0.0386535
\(916\) 0 0
\(917\) −8.25111 −0.272476
\(918\) 0 0
\(919\) 26.0156 0.858177 0.429088 0.903263i \(-0.358835\pi\)
0.429088 + 0.903263i \(0.358835\pi\)
\(920\) 0 0
\(921\) −41.3265 −1.36175
\(922\) 0 0
\(923\) 3.12571 0.102884
\(924\) 0 0
\(925\) −32.9818 −1.08443
\(926\) 0 0
\(927\) −18.1537 −0.596247
\(928\) 0 0
\(929\) 5.99773 0.196779 0.0983896 0.995148i \(-0.468631\pi\)
0.0983896 + 0.995148i \(0.468631\pi\)
\(930\) 0 0
\(931\) 0.125785 0.00412242
\(932\) 0 0
\(933\) 69.9581 2.29033
\(934\) 0 0
\(935\) −0.141562 −0.00462957
\(936\) 0 0
\(937\) −50.5847 −1.65253 −0.826265 0.563282i \(-0.809539\pi\)
−0.826265 + 0.563282i \(0.809539\pi\)
\(938\) 0 0
\(939\) −65.2120 −2.12811
\(940\) 0 0
\(941\) −21.8640 −0.712747 −0.356373 0.934344i \(-0.615987\pi\)
−0.356373 + 0.934344i \(0.615987\pi\)
\(942\) 0 0
\(943\) −65.7129 −2.13991
\(944\) 0 0
\(945\) −3.36072 −0.109324
\(946\) 0 0
\(947\) 33.1722 1.07795 0.538976 0.842321i \(-0.318812\pi\)
0.538976 + 0.842321i \(0.318812\pi\)
\(948\) 0 0
\(949\) −20.0533 −0.650958
\(950\) 0 0
\(951\) −13.4974 −0.437685
\(952\) 0 0
\(953\) 22.4494 0.727209 0.363604 0.931553i \(-0.381546\pi\)
0.363604 + 0.931553i \(0.381546\pi\)
\(954\) 0 0
\(955\) −9.75794 −0.315760
\(956\) 0 0
\(957\) 11.1985 0.361995
\(958\) 0 0
\(959\) 23.1604 0.747887
\(960\) 0 0
\(961\) −21.6631 −0.698810
\(962\) 0 0
\(963\) −3.50246 −0.112865
\(964\) 0 0
\(965\) −11.6702 −0.375676
\(966\) 0 0
\(967\) 0.790013 0.0254051 0.0127026 0.999919i \(-0.495957\pi\)
0.0127026 + 0.999919i \(0.495957\pi\)
\(968\) 0 0
\(969\) −0.502774 −0.0161514
\(970\) 0 0
\(971\) −51.6970 −1.65904 −0.829518 0.558479i \(-0.811385\pi\)
−0.829518 + 0.558479i \(0.811385\pi\)
\(972\) 0 0
\(973\) 38.8201 1.24452
\(974\) 0 0
\(975\) −12.5578 −0.402171
\(976\) 0 0
\(977\) 21.4536 0.686360 0.343180 0.939270i \(-0.388496\pi\)
0.343180 + 0.939270i \(0.388496\pi\)
\(978\) 0 0
\(979\) 5.54136 0.177103
\(980\) 0 0
\(981\) 35.7651 1.14189
\(982\) 0 0
\(983\) 42.9916 1.37122 0.685610 0.727969i \(-0.259536\pi\)
0.685610 + 0.727969i \(0.259536\pi\)
\(984\) 0 0
\(985\) −0.367368 −0.0117053
\(986\) 0 0
\(987\) −35.9421 −1.14405
\(988\) 0 0
\(989\) −14.5664 −0.463186
\(990\) 0 0
\(991\) −57.7319 −1.83392 −0.916958 0.398985i \(-0.869363\pi\)
−0.916958 + 0.398985i \(0.869363\pi\)
\(992\) 0 0
\(993\) 75.6373 2.40028
\(994\) 0 0
\(995\) 6.48303 0.205526
\(996\) 0 0
\(997\) −30.3487 −0.961152 −0.480576 0.876953i \(-0.659572\pi\)
−0.480576 + 0.876953i \(0.659572\pi\)
\(998\) 0 0
\(999\) 14.1775 0.448557
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 3344.2.a.t.1.1 5
4.3 odd 2 209.2.a.c.1.3 5
12.11 even 2 1881.2.a.k.1.3 5
20.19 odd 2 5225.2.a.h.1.3 5
44.43 even 2 2299.2.a.n.1.3 5
76.75 even 2 3971.2.a.h.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.3 5 4.3 odd 2
1881.2.a.k.1.3 5 12.11 even 2
2299.2.a.n.1.3 5 44.43 even 2
3344.2.a.t.1.1 5 1.1 even 1 trivial
3971.2.a.h.1.3 5 76.75 even 2
5225.2.a.h.1.3 5 20.19 odd 2