Properties

Label 3800.2.a.bb.1.5
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.22174\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.22174 q^{3} +3.08602 q^{7} -1.50735 q^{9} +O(q^{10})\) \(q+1.22174 q^{3} +3.08602 q^{7} -1.50735 q^{9} -3.83920 q^{11} +5.85633 q^{13} +6.54564 q^{17} +1.00000 q^{19} +3.77031 q^{21} +4.37163 q^{23} -5.50681 q^{27} -4.41841 q^{29} +0.451431 q^{31} -4.69051 q^{33} +2.49067 q^{37} +7.15491 q^{39} -1.03924 q^{41} +3.19209 q^{43} -3.70641 q^{47} +2.52349 q^{49} +7.99707 q^{51} +5.19667 q^{53} +1.22174 q^{57} -7.18818 q^{59} +5.02265 q^{61} -4.65171 q^{63} -13.1631 q^{67} +5.34099 q^{69} +12.2993 q^{71} -2.49886 q^{73} -11.8478 q^{77} +11.0313 q^{79} -2.20584 q^{81} +0.245834 q^{83} -5.39814 q^{87} -13.4321 q^{89} +18.0727 q^{91} +0.551531 q^{93} +11.0992 q^{97} +5.78702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) 1.22174 0.705372 0.352686 0.935742i \(-0.385268\pi\)
0.352686 + 0.935742i \(0.385268\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.08602 1.16640 0.583202 0.812327i \(-0.301800\pi\)
0.583202 + 0.812327i \(0.301800\pi\)
\(8\) 0 0
\(9\) −1.50735 −0.502450
\(10\) 0 0
\(11\) −3.83920 −1.15756 −0.578781 0.815483i \(-0.696472\pi\)
−0.578781 + 0.815483i \(0.696472\pi\)
\(12\) 0 0
\(13\) 5.85633 1.62425 0.812126 0.583482i \(-0.198310\pi\)
0.812126 + 0.583482i \(0.198310\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.54564 1.58755 0.793775 0.608211i \(-0.208113\pi\)
0.793775 + 0.608211i \(0.208113\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.77031 0.822749
\(22\) 0 0
\(23\) 4.37163 0.911547 0.455774 0.890096i \(-0.349363\pi\)
0.455774 + 0.890096i \(0.349363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.50681 −1.05979
\(28\) 0 0
\(29\) −4.41841 −0.820477 −0.410239 0.911978i \(-0.634555\pi\)
−0.410239 + 0.911978i \(0.634555\pi\)
\(30\) 0 0
\(31\) 0.451431 0.0810793 0.0405397 0.999178i \(-0.487092\pi\)
0.0405397 + 0.999178i \(0.487092\pi\)
\(32\) 0 0
\(33\) −4.69051 −0.816512
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.49067 0.409463 0.204732 0.978818i \(-0.434368\pi\)
0.204732 + 0.978818i \(0.434368\pi\)
\(38\) 0 0
\(39\) 7.15491 1.14570
\(40\) 0 0
\(41\) −1.03924 −0.162302 −0.0811508 0.996702i \(-0.525860\pi\)
−0.0811508 + 0.996702i \(0.525860\pi\)
\(42\) 0 0
\(43\) 3.19209 0.486789 0.243394 0.969927i \(-0.421739\pi\)
0.243394 + 0.969927i \(0.421739\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.70641 −0.540635 −0.270317 0.962771i \(-0.587129\pi\)
−0.270317 + 0.962771i \(0.587129\pi\)
\(48\) 0 0
\(49\) 2.52349 0.360499
\(50\) 0 0
\(51\) 7.99707 1.11981
\(52\) 0 0
\(53\) 5.19667 0.713817 0.356908 0.934139i \(-0.383831\pi\)
0.356908 + 0.934139i \(0.383831\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.22174 0.161823
\(58\) 0 0
\(59\) −7.18818 −0.935821 −0.467910 0.883776i \(-0.654993\pi\)
−0.467910 + 0.883776i \(0.654993\pi\)
\(60\) 0 0
\(61\) 5.02265 0.643085 0.321542 0.946895i \(-0.395799\pi\)
0.321542 + 0.946895i \(0.395799\pi\)
\(62\) 0 0
\(63\) −4.65171 −0.586060
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.1631 −1.60813 −0.804064 0.594542i \(-0.797333\pi\)
−0.804064 + 0.594542i \(0.797333\pi\)
\(68\) 0 0
\(69\) 5.34099 0.642980
\(70\) 0 0
\(71\) 12.2993 1.45965 0.729827 0.683632i \(-0.239601\pi\)
0.729827 + 0.683632i \(0.239601\pi\)
\(72\) 0 0
\(73\) −2.49886 −0.292470 −0.146235 0.989250i \(-0.546715\pi\)
−0.146235 + 0.989250i \(0.546715\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −11.8478 −1.35019
\(78\) 0 0
\(79\) 11.0313 1.24112 0.620558 0.784160i \(-0.286906\pi\)
0.620558 + 0.784160i \(0.286906\pi\)
\(80\) 0 0
\(81\) −2.20584 −0.245093
\(82\) 0 0
\(83\) 0.245834 0.0269838 0.0134919 0.999909i \(-0.495705\pi\)
0.0134919 + 0.999909i \(0.495705\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.39814 −0.578742
\(88\) 0 0
\(89\) −13.4321 −1.42380 −0.711898 0.702283i \(-0.752164\pi\)
−0.711898 + 0.702283i \(0.752164\pi\)
\(90\) 0 0
\(91\) 18.0727 1.89454
\(92\) 0 0
\(93\) 0.551531 0.0571911
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0992 1.12696 0.563478 0.826131i \(-0.309463\pi\)
0.563478 + 0.826131i \(0.309463\pi\)
\(98\) 0 0
\(99\) 5.78702 0.581618
\(100\) 0 0
\(101\) 16.0460 1.59664 0.798318 0.602236i \(-0.205724\pi\)
0.798318 + 0.602236i \(0.205724\pi\)
\(102\) 0 0
\(103\) 15.4569 1.52301 0.761507 0.648157i \(-0.224460\pi\)
0.761507 + 0.648157i \(0.224460\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.23615 0.892892 0.446446 0.894811i \(-0.352690\pi\)
0.446446 + 0.894811i \(0.352690\pi\)
\(108\) 0 0
\(109\) −9.87147 −0.945515 −0.472758 0.881192i \(-0.656741\pi\)
−0.472758 + 0.881192i \(0.656741\pi\)
\(110\) 0 0
\(111\) 3.04295 0.288824
\(112\) 0 0
\(113\) 18.4191 1.73272 0.866360 0.499421i \(-0.166454\pi\)
0.866360 + 0.499421i \(0.166454\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.82754 −0.816106
\(118\) 0 0
\(119\) 20.1999 1.85173
\(120\) 0 0
\(121\) 3.73946 0.339951
\(122\) 0 0
\(123\) −1.26968 −0.114483
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.1753 1.25786 0.628928 0.777464i \(-0.283494\pi\)
0.628928 + 0.777464i \(0.283494\pi\)
\(128\) 0 0
\(129\) 3.89990 0.343367
\(130\) 0 0
\(131\) 1.40424 0.122689 0.0613446 0.998117i \(-0.480461\pi\)
0.0613446 + 0.998117i \(0.480461\pi\)
\(132\) 0 0
\(133\) 3.08602 0.267592
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.4212 −1.14665 −0.573327 0.819326i \(-0.694348\pi\)
−0.573327 + 0.819326i \(0.694348\pi\)
\(138\) 0 0
\(139\) 3.42929 0.290868 0.145434 0.989368i \(-0.453542\pi\)
0.145434 + 0.989368i \(0.453542\pi\)
\(140\) 0 0
\(141\) −4.52827 −0.381349
\(142\) 0 0
\(143\) −22.4836 −1.88017
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.08305 0.254286
\(148\) 0 0
\(149\) −20.3518 −1.66729 −0.833643 0.552304i \(-0.813749\pi\)
−0.833643 + 0.552304i \(0.813749\pi\)
\(150\) 0 0
\(151\) −2.34058 −0.190474 −0.0952369 0.995455i \(-0.530361\pi\)
−0.0952369 + 0.995455i \(0.530361\pi\)
\(152\) 0 0
\(153\) −9.86658 −0.797665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.5097 1.39742 0.698711 0.715404i \(-0.253757\pi\)
0.698711 + 0.715404i \(0.253757\pi\)
\(158\) 0 0
\(159\) 6.34897 0.503506
\(160\) 0 0
\(161\) 13.4909 1.06323
\(162\) 0 0
\(163\) −6.65575 −0.521319 −0.260659 0.965431i \(-0.583940\pi\)
−0.260659 + 0.965431i \(0.583940\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.32539 −0.644238 −0.322119 0.946699i \(-0.604395\pi\)
−0.322119 + 0.946699i \(0.604395\pi\)
\(168\) 0 0
\(169\) 21.2965 1.63820
\(170\) 0 0
\(171\) −1.50735 −0.115270
\(172\) 0 0
\(173\) 20.0259 1.52254 0.761272 0.648433i \(-0.224575\pi\)
0.761272 + 0.648433i \(0.224575\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.78208 −0.660102
\(178\) 0 0
\(179\) −12.1689 −0.909544 −0.454772 0.890608i \(-0.650279\pi\)
−0.454772 + 0.890608i \(0.650279\pi\)
\(180\) 0 0
\(181\) 3.76508 0.279856 0.139928 0.990162i \(-0.455313\pi\)
0.139928 + 0.990162i \(0.455313\pi\)
\(182\) 0 0
\(183\) 6.13638 0.453614
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −25.1300 −1.83769
\(188\) 0 0
\(189\) −16.9941 −1.23614
\(190\) 0 0
\(191\) −13.0616 −0.945106 −0.472553 0.881302i \(-0.656667\pi\)
−0.472553 + 0.881302i \(0.656667\pi\)
\(192\) 0 0
\(193\) 9.44847 0.680116 0.340058 0.940405i \(-0.389553\pi\)
0.340058 + 0.940405i \(0.389553\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.3270 −1.02075 −0.510377 0.859951i \(-0.670494\pi\)
−0.510377 + 0.859951i \(0.670494\pi\)
\(198\) 0 0
\(199\) −16.1608 −1.14561 −0.572805 0.819692i \(-0.694145\pi\)
−0.572805 + 0.819692i \(0.694145\pi\)
\(200\) 0 0
\(201\) −16.0819 −1.13433
\(202\) 0 0
\(203\) −13.6353 −0.957008
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.58958 −0.458007
\(208\) 0 0
\(209\) −3.83920 −0.265563
\(210\) 0 0
\(211\) 4.61746 0.317879 0.158940 0.987288i \(-0.449193\pi\)
0.158940 + 0.987288i \(0.449193\pi\)
\(212\) 0 0
\(213\) 15.0265 1.02960
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.39312 0.0945713
\(218\) 0 0
\(219\) −3.05296 −0.206300
\(220\) 0 0
\(221\) 38.3334 2.57858
\(222\) 0 0
\(223\) −20.2785 −1.35795 −0.678975 0.734161i \(-0.737576\pi\)
−0.678975 + 0.734161i \(0.737576\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.5708 −0.767982 −0.383991 0.923337i \(-0.625451\pi\)
−0.383991 + 0.923337i \(0.625451\pi\)
\(228\) 0 0
\(229\) 14.0792 0.930378 0.465189 0.885211i \(-0.345986\pi\)
0.465189 + 0.885211i \(0.345986\pi\)
\(230\) 0 0
\(231\) −14.4750 −0.952383
\(232\) 0 0
\(233\) 3.95281 0.258957 0.129479 0.991582i \(-0.458670\pi\)
0.129479 + 0.991582i \(0.458670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.4774 0.875449
\(238\) 0 0
\(239\) 18.0661 1.16860 0.584300 0.811538i \(-0.301369\pi\)
0.584300 + 0.811538i \(0.301369\pi\)
\(240\) 0 0
\(241\) 23.0999 1.48800 0.743998 0.668182i \(-0.232927\pi\)
0.743998 + 0.668182i \(0.232927\pi\)
\(242\) 0 0
\(243\) 13.8255 0.886904
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.85633 0.372629
\(248\) 0 0
\(249\) 0.300345 0.0190336
\(250\) 0 0
\(251\) −13.0987 −0.826784 −0.413392 0.910553i \(-0.635656\pi\)
−0.413392 + 0.910553i \(0.635656\pi\)
\(252\) 0 0
\(253\) −16.7836 −1.05517
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.5191 −0.843296 −0.421648 0.906760i \(-0.638548\pi\)
−0.421648 + 0.906760i \(0.638548\pi\)
\(258\) 0 0
\(259\) 7.68624 0.477600
\(260\) 0 0
\(261\) 6.66009 0.412249
\(262\) 0 0
\(263\) −0.357938 −0.0220714 −0.0110357 0.999939i \(-0.503513\pi\)
−0.0110357 + 0.999939i \(0.503513\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −16.4105 −1.00431
\(268\) 0 0
\(269\) 15.9556 0.972829 0.486415 0.873728i \(-0.338304\pi\)
0.486415 + 0.873728i \(0.338304\pi\)
\(270\) 0 0
\(271\) 0.795153 0.0483021 0.0241511 0.999708i \(-0.492312\pi\)
0.0241511 + 0.999708i \(0.492312\pi\)
\(272\) 0 0
\(273\) 22.0802 1.33635
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.9640 −0.778934 −0.389467 0.921040i \(-0.627341\pi\)
−0.389467 + 0.921040i \(0.627341\pi\)
\(278\) 0 0
\(279\) −0.680465 −0.0407383
\(280\) 0 0
\(281\) −26.2769 −1.56755 −0.783775 0.621045i \(-0.786709\pi\)
−0.783775 + 0.621045i \(0.786709\pi\)
\(282\) 0 0
\(283\) 27.4599 1.63232 0.816160 0.577826i \(-0.196099\pi\)
0.816160 + 0.577826i \(0.196099\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.20710 −0.189309
\(288\) 0 0
\(289\) 25.8454 1.52032
\(290\) 0 0
\(291\) 13.5604 0.794923
\(292\) 0 0
\(293\) −19.7572 −1.15423 −0.577114 0.816663i \(-0.695821\pi\)
−0.577114 + 0.816663i \(0.695821\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 21.1418 1.22677
\(298\) 0 0
\(299\) 25.6017 1.48058
\(300\) 0 0
\(301\) 9.85083 0.567792
\(302\) 0 0
\(303\) 19.6040 1.12622
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −28.9367 −1.65151 −0.825753 0.564032i \(-0.809250\pi\)
−0.825753 + 0.564032i \(0.809250\pi\)
\(308\) 0 0
\(309\) 18.8843 1.07429
\(310\) 0 0
\(311\) −28.0895 −1.59281 −0.796404 0.604764i \(-0.793267\pi\)
−0.796404 + 0.604764i \(0.793267\pi\)
\(312\) 0 0
\(313\) 29.7936 1.68403 0.842016 0.539453i \(-0.181369\pi\)
0.842016 + 0.539453i \(0.181369\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.13494 −0.344572 −0.172286 0.985047i \(-0.555115\pi\)
−0.172286 + 0.985047i \(0.555115\pi\)
\(318\) 0 0
\(319\) 16.9631 0.949754
\(320\) 0 0
\(321\) 11.2842 0.629821
\(322\) 0 0
\(323\) 6.54564 0.364209
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.0604 −0.666940
\(328\) 0 0
\(329\) −11.4380 −0.630599
\(330\) 0 0
\(331\) −17.1019 −0.940003 −0.470002 0.882666i \(-0.655747\pi\)
−0.470002 + 0.882666i \(0.655747\pi\)
\(332\) 0 0
\(333\) −3.75431 −0.205735
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 17.0521 0.928885 0.464442 0.885603i \(-0.346255\pi\)
0.464442 + 0.885603i \(0.346255\pi\)
\(338\) 0 0
\(339\) 22.5033 1.22221
\(340\) 0 0
\(341\) −1.73313 −0.0938544
\(342\) 0 0
\(343\) −13.8146 −0.745917
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.1351 −1.40300 −0.701502 0.712667i \(-0.747487\pi\)
−0.701502 + 0.712667i \(0.747487\pi\)
\(348\) 0 0
\(349\) −0.577782 −0.0309279 −0.0154640 0.999880i \(-0.504923\pi\)
−0.0154640 + 0.999880i \(0.504923\pi\)
\(350\) 0 0
\(351\) −32.2497 −1.72136
\(352\) 0 0
\(353\) 30.0746 1.60071 0.800355 0.599527i \(-0.204645\pi\)
0.800355 + 0.599527i \(0.204645\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 24.6791 1.30616
\(358\) 0 0
\(359\) −25.6793 −1.35530 −0.677651 0.735383i \(-0.737002\pi\)
−0.677651 + 0.735383i \(0.737002\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 4.56865 0.239792
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.57520 0.395422 0.197711 0.980260i \(-0.436649\pi\)
0.197711 + 0.980260i \(0.436649\pi\)
\(368\) 0 0
\(369\) 1.56650 0.0815485
\(370\) 0 0
\(371\) 16.0370 0.832599
\(372\) 0 0
\(373\) −26.2562 −1.35949 −0.679747 0.733447i \(-0.737910\pi\)
−0.679747 + 0.733447i \(0.737910\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −25.8756 −1.33266
\(378\) 0 0
\(379\) −15.6277 −0.802742 −0.401371 0.915915i \(-0.631466\pi\)
−0.401371 + 0.915915i \(0.631466\pi\)
\(380\) 0 0
\(381\) 17.3185 0.887256
\(382\) 0 0
\(383\) −10.7189 −0.547711 −0.273856 0.961771i \(-0.588299\pi\)
−0.273856 + 0.961771i \(0.588299\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.81159 −0.244587
\(388\) 0 0
\(389\) 7.83988 0.397498 0.198749 0.980050i \(-0.436312\pi\)
0.198749 + 0.980050i \(0.436312\pi\)
\(390\) 0 0
\(391\) 28.6151 1.44713
\(392\) 0 0
\(393\) 1.71562 0.0865416
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.9045 1.90237 0.951187 0.308614i \(-0.0998651\pi\)
0.951187 + 0.308614i \(0.0998651\pi\)
\(398\) 0 0
\(399\) 3.77031 0.188752
\(400\) 0 0
\(401\) 18.3901 0.918355 0.459178 0.888344i \(-0.348144\pi\)
0.459178 + 0.888344i \(0.348144\pi\)
\(402\) 0 0
\(403\) 2.64372 0.131693
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.56218 −0.473979
\(408\) 0 0
\(409\) −23.2744 −1.15085 −0.575423 0.817856i \(-0.695163\pi\)
−0.575423 + 0.817856i \(0.695163\pi\)
\(410\) 0 0
\(411\) −16.3973 −0.808818
\(412\) 0 0
\(413\) −22.1828 −1.09155
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.18970 0.205170
\(418\) 0 0
\(419\) 27.5055 1.34373 0.671866 0.740673i \(-0.265493\pi\)
0.671866 + 0.740673i \(0.265493\pi\)
\(420\) 0 0
\(421\) −9.70130 −0.472812 −0.236406 0.971654i \(-0.575970\pi\)
−0.236406 + 0.971654i \(0.575970\pi\)
\(422\) 0 0
\(423\) 5.58686 0.271642
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.5000 0.750097
\(428\) 0 0
\(429\) −27.4691 −1.32622
\(430\) 0 0
\(431\) 15.6303 0.752886 0.376443 0.926440i \(-0.377147\pi\)
0.376443 + 0.926440i \(0.377147\pi\)
\(432\) 0 0
\(433\) −17.3839 −0.835416 −0.417708 0.908581i \(-0.637166\pi\)
−0.417708 + 0.908581i \(0.637166\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.37163 0.209123
\(438\) 0 0
\(439\) −25.3238 −1.20864 −0.604320 0.796742i \(-0.706555\pi\)
−0.604320 + 0.796742i \(0.706555\pi\)
\(440\) 0 0
\(441\) −3.80379 −0.181133
\(442\) 0 0
\(443\) −6.49504 −0.308589 −0.154294 0.988025i \(-0.549310\pi\)
−0.154294 + 0.988025i \(0.549310\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −24.8646 −1.17606
\(448\) 0 0
\(449\) −3.09007 −0.145829 −0.0729147 0.997338i \(-0.523230\pi\)
−0.0729147 + 0.997338i \(0.523230\pi\)
\(450\) 0 0
\(451\) 3.98984 0.187874
\(452\) 0 0
\(453\) −2.85958 −0.134355
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.7215 1.81131 0.905657 0.424012i \(-0.139378\pi\)
0.905657 + 0.424012i \(0.139378\pi\)
\(458\) 0 0
\(459\) −36.0456 −1.68246
\(460\) 0 0
\(461\) −21.9817 −1.02379 −0.511895 0.859048i \(-0.671056\pi\)
−0.511895 + 0.859048i \(0.671056\pi\)
\(462\) 0 0
\(463\) −42.4385 −1.97228 −0.986142 0.165901i \(-0.946947\pi\)
−0.986142 + 0.165901i \(0.946947\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.4303 0.667754 0.333877 0.942617i \(-0.391643\pi\)
0.333877 + 0.942617i \(0.391643\pi\)
\(468\) 0 0
\(469\) −40.6215 −1.87573
\(470\) 0 0
\(471\) 21.3922 0.985703
\(472\) 0 0
\(473\) −12.2551 −0.563488
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −7.83320 −0.358658
\(478\) 0 0
\(479\) −13.0612 −0.596781 −0.298390 0.954444i \(-0.596450\pi\)
−0.298390 + 0.954444i \(0.596450\pi\)
\(480\) 0 0
\(481\) 14.5862 0.665072
\(482\) 0 0
\(483\) 16.4824 0.749975
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.65696 0.392284 0.196142 0.980575i \(-0.437159\pi\)
0.196142 + 0.980575i \(0.437159\pi\)
\(488\) 0 0
\(489\) −8.13160 −0.367723
\(490\) 0 0
\(491\) −8.37358 −0.377894 −0.188947 0.981987i \(-0.560507\pi\)
−0.188947 + 0.981987i \(0.560507\pi\)
\(492\) 0 0
\(493\) −28.9213 −1.30255
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 37.9557 1.70255
\(498\) 0 0
\(499\) 35.0254 1.56795 0.783977 0.620790i \(-0.213188\pi\)
0.783977 + 0.620790i \(0.213188\pi\)
\(500\) 0 0
\(501\) −10.1715 −0.454427
\(502\) 0 0
\(503\) 14.7037 0.655604 0.327802 0.944746i \(-0.393692\pi\)
0.327802 + 0.944746i \(0.393692\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 26.0188 1.15554
\(508\) 0 0
\(509\) −3.73411 −0.165512 −0.0827558 0.996570i \(-0.526372\pi\)
−0.0827558 + 0.996570i \(0.526372\pi\)
\(510\) 0 0
\(511\) −7.71153 −0.341138
\(512\) 0 0
\(513\) −5.50681 −0.243132
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 14.2296 0.625819
\(518\) 0 0
\(519\) 24.4665 1.07396
\(520\) 0 0
\(521\) −3.38053 −0.148104 −0.0740518 0.997254i \(-0.523593\pi\)
−0.0740518 + 0.997254i \(0.523593\pi\)
\(522\) 0 0
\(523\) −13.7951 −0.603217 −0.301608 0.953432i \(-0.597524\pi\)
−0.301608 + 0.953432i \(0.597524\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.95490 0.128718
\(528\) 0 0
\(529\) −3.88888 −0.169082
\(530\) 0 0
\(531\) 10.8351 0.470203
\(532\) 0 0
\(533\) −6.08611 −0.263619
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14.8672 −0.641567
\(538\) 0 0
\(539\) −9.68820 −0.417300
\(540\) 0 0
\(541\) 19.4420 0.835876 0.417938 0.908476i \(-0.362753\pi\)
0.417938 + 0.908476i \(0.362753\pi\)
\(542\) 0 0
\(543\) 4.59995 0.197403
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.2715 −0.524689 −0.262345 0.964974i \(-0.584496\pi\)
−0.262345 + 0.964974i \(0.584496\pi\)
\(548\) 0 0
\(549\) −7.57090 −0.323118
\(550\) 0 0
\(551\) −4.41841 −0.188230
\(552\) 0 0
\(553\) 34.0427 1.44764
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.01766 0.254977 0.127488 0.991840i \(-0.459308\pi\)
0.127488 + 0.991840i \(0.459308\pi\)
\(558\) 0 0
\(559\) 18.6939 0.790668
\(560\) 0 0
\(561\) −30.7024 −1.29625
\(562\) 0 0
\(563\) −30.1874 −1.27225 −0.636124 0.771587i \(-0.719463\pi\)
−0.636124 + 0.771587i \(0.719463\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −6.80725 −0.285878
\(568\) 0 0
\(569\) −12.3451 −0.517535 −0.258768 0.965940i \(-0.583316\pi\)
−0.258768 + 0.965940i \(0.583316\pi\)
\(570\) 0 0
\(571\) 3.13368 0.131140 0.0655702 0.997848i \(-0.479113\pi\)
0.0655702 + 0.997848i \(0.479113\pi\)
\(572\) 0 0
\(573\) −15.9579 −0.666651
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27.2272 −1.13348 −0.566742 0.823895i \(-0.691796\pi\)
−0.566742 + 0.823895i \(0.691796\pi\)
\(578\) 0 0
\(579\) 11.5436 0.479735
\(580\) 0 0
\(581\) 0.758647 0.0314740
\(582\) 0 0
\(583\) −19.9510 −0.826288
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.7445 −1.18641 −0.593206 0.805050i \(-0.702138\pi\)
−0.593206 + 0.805050i \(0.702138\pi\)
\(588\) 0 0
\(589\) 0.451431 0.0186009
\(590\) 0 0
\(591\) −17.5038 −0.720012
\(592\) 0 0
\(593\) 27.5257 1.13034 0.565172 0.824973i \(-0.308810\pi\)
0.565172 + 0.824973i \(0.308810\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.7443 −0.808081
\(598\) 0 0
\(599\) −28.2278 −1.15335 −0.576677 0.816972i \(-0.695651\pi\)
−0.576677 + 0.816972i \(0.695651\pi\)
\(600\) 0 0
\(601\) 8.32545 0.339602 0.169801 0.985478i \(-0.445687\pi\)
0.169801 + 0.985478i \(0.445687\pi\)
\(602\) 0 0
\(603\) 19.8414 0.808005
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11.7364 0.476364 0.238182 0.971220i \(-0.423448\pi\)
0.238182 + 0.971220i \(0.423448\pi\)
\(608\) 0 0
\(609\) −16.6588 −0.675047
\(610\) 0 0
\(611\) −21.7059 −0.878128
\(612\) 0 0
\(613\) −31.5934 −1.27604 −0.638022 0.770018i \(-0.720247\pi\)
−0.638022 + 0.770018i \(0.720247\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.3247 1.18057 0.590284 0.807196i \(-0.299016\pi\)
0.590284 + 0.807196i \(0.299016\pi\)
\(618\) 0 0
\(619\) −37.8615 −1.52178 −0.760890 0.648881i \(-0.775237\pi\)
−0.760890 + 0.648881i \(0.775237\pi\)
\(620\) 0 0
\(621\) −24.0737 −0.966045
\(622\) 0 0
\(623\) −41.4516 −1.66072
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.69051 −0.187321
\(628\) 0 0
\(629\) 16.3030 0.650044
\(630\) 0 0
\(631\) −24.0801 −0.958615 −0.479308 0.877647i \(-0.659112\pi\)
−0.479308 + 0.877647i \(0.659112\pi\)
\(632\) 0 0
\(633\) 5.64134 0.224223
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.7784 0.585542
\(638\) 0 0
\(639\) −18.5393 −0.733404
\(640\) 0 0
\(641\) −48.2114 −1.90424 −0.952118 0.305732i \(-0.901099\pi\)
−0.952118 + 0.305732i \(0.901099\pi\)
\(642\) 0 0
\(643\) 18.6087 0.733856 0.366928 0.930249i \(-0.380410\pi\)
0.366928 + 0.930249i \(0.380410\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.7599 0.658900 0.329450 0.944173i \(-0.393137\pi\)
0.329450 + 0.944173i \(0.393137\pi\)
\(648\) 0 0
\(649\) 27.5968 1.08327
\(650\) 0 0
\(651\) 1.70203 0.0667079
\(652\) 0 0
\(653\) −2.12145 −0.0830186 −0.0415093 0.999138i \(-0.513217\pi\)
−0.0415093 + 0.999138i \(0.513217\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.76666 0.146951
\(658\) 0 0
\(659\) 0.175317 0.00682937 0.00341468 0.999994i \(-0.498913\pi\)
0.00341468 + 0.999994i \(0.498913\pi\)
\(660\) 0 0
\(661\) 16.9306 0.658526 0.329263 0.944238i \(-0.393200\pi\)
0.329263 + 0.944238i \(0.393200\pi\)
\(662\) 0 0
\(663\) 46.8334 1.81886
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −19.3156 −0.747904
\(668\) 0 0
\(669\) −24.7751 −0.957860
\(670\) 0 0
\(671\) −19.2830 −0.744411
\(672\) 0 0
\(673\) −21.8162 −0.840952 −0.420476 0.907304i \(-0.638137\pi\)
−0.420476 + 0.907304i \(0.638137\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.0305 −0.500803 −0.250402 0.968142i \(-0.580563\pi\)
−0.250402 + 0.968142i \(0.580563\pi\)
\(678\) 0 0
\(679\) 34.2524 1.31449
\(680\) 0 0
\(681\) −14.1365 −0.541713
\(682\) 0 0
\(683\) 8.41103 0.321839 0.160920 0.986968i \(-0.448554\pi\)
0.160920 + 0.986968i \(0.448554\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 17.2011 0.656263
\(688\) 0 0
\(689\) 30.4334 1.15942
\(690\) 0 0
\(691\) −8.74129 −0.332534 −0.166267 0.986081i \(-0.553171\pi\)
−0.166267 + 0.986081i \(0.553171\pi\)
\(692\) 0 0
\(693\) 17.8588 0.678402
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.80247 −0.257662
\(698\) 0 0
\(699\) 4.82931 0.182661
\(700\) 0 0
\(701\) −45.5898 −1.72190 −0.860952 0.508686i \(-0.830131\pi\)
−0.860952 + 0.508686i \(0.830131\pi\)
\(702\) 0 0
\(703\) 2.49067 0.0939373
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 49.5182 1.86232
\(708\) 0 0
\(709\) −34.2985 −1.28811 −0.644053 0.764981i \(-0.722748\pi\)
−0.644053 + 0.764981i \(0.722748\pi\)
\(710\) 0 0
\(711\) −16.6280 −0.623600
\(712\) 0 0
\(713\) 1.97349 0.0739077
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.0721 0.824297
\(718\) 0 0
\(719\) −15.0753 −0.562215 −0.281107 0.959676i \(-0.590702\pi\)
−0.281107 + 0.959676i \(0.590702\pi\)
\(720\) 0 0
\(721\) 47.7003 1.77645
\(722\) 0 0
\(723\) 28.2221 1.04959
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.0189 0.557021 0.278510 0.960433i \(-0.410159\pi\)
0.278510 + 0.960433i \(0.410159\pi\)
\(728\) 0 0
\(729\) 23.5087 0.870691
\(730\) 0 0
\(731\) 20.8942 0.772802
\(732\) 0 0
\(733\) −32.0731 −1.18465 −0.592324 0.805700i \(-0.701789\pi\)
−0.592324 + 0.805700i \(0.701789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 50.5358 1.86151
\(738\) 0 0
\(739\) −32.0663 −1.17958 −0.589788 0.807558i \(-0.700789\pi\)
−0.589788 + 0.807558i \(0.700789\pi\)
\(740\) 0 0
\(741\) 7.15491 0.262842
\(742\) 0 0
\(743\) −14.3624 −0.526904 −0.263452 0.964673i \(-0.584861\pi\)
−0.263452 + 0.964673i \(0.584861\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.370558 −0.0135580
\(748\) 0 0
\(749\) 28.5029 1.04147
\(750\) 0 0
\(751\) 21.2055 0.773800 0.386900 0.922122i \(-0.373546\pi\)
0.386900 + 0.922122i \(0.373546\pi\)
\(752\) 0 0
\(753\) −16.0032 −0.583190
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −41.8045 −1.51941 −0.759705 0.650267i \(-0.774657\pi\)
−0.759705 + 0.650267i \(0.774657\pi\)
\(758\) 0 0
\(759\) −20.5051 −0.744289
\(760\) 0 0
\(761\) −1.64223 −0.0595307 −0.0297654 0.999557i \(-0.509476\pi\)
−0.0297654 + 0.999557i \(0.509476\pi\)
\(762\) 0 0
\(763\) −30.4635 −1.10285
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −42.0963 −1.52001
\(768\) 0 0
\(769\) 30.3955 1.09609 0.548044 0.836449i \(-0.315373\pi\)
0.548044 + 0.836449i \(0.315373\pi\)
\(770\) 0 0
\(771\) −16.5168 −0.594837
\(772\) 0 0
\(773\) 47.6753 1.71476 0.857380 0.514684i \(-0.172091\pi\)
0.857380 + 0.514684i \(0.172091\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.39059 0.336886
\(778\) 0 0
\(779\) −1.03924 −0.0372346
\(780\) 0 0
\(781\) −47.2194 −1.68964
\(782\) 0 0
\(783\) 24.3313 0.869531
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.3173 1.22328 0.611639 0.791137i \(-0.290510\pi\)
0.611639 + 0.791137i \(0.290510\pi\)
\(788\) 0 0
\(789\) −0.437307 −0.0155685
\(790\) 0 0
\(791\) 56.8415 2.02105
\(792\) 0 0
\(793\) 29.4143 1.04453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.4643 1.39790 0.698949 0.715172i \(-0.253651\pi\)
0.698949 + 0.715172i \(0.253651\pi\)
\(798\) 0 0
\(799\) −24.2608 −0.858285
\(800\) 0 0
\(801\) 20.2468 0.715387
\(802\) 0 0
\(803\) 9.59363 0.338552
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.4936 0.686207
\(808\) 0 0
\(809\) −36.3760 −1.27891 −0.639456 0.768827i \(-0.720841\pi\)
−0.639456 + 0.768827i \(0.720841\pi\)
\(810\) 0 0
\(811\) −45.6700 −1.60369 −0.801845 0.597532i \(-0.796148\pi\)
−0.801845 + 0.597532i \(0.796148\pi\)
\(812\) 0 0
\(813\) 0.971471 0.0340710
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.19209 0.111677
\(818\) 0 0
\(819\) −27.2419 −0.951910
\(820\) 0 0
\(821\) 18.4543 0.644061 0.322030 0.946729i \(-0.395635\pi\)
0.322030 + 0.946729i \(0.395635\pi\)
\(822\) 0 0
\(823\) −13.0227 −0.453942 −0.226971 0.973902i \(-0.572882\pi\)
−0.226971 + 0.973902i \(0.572882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.0455 1.32297 0.661485 0.749958i \(-0.269926\pi\)
0.661485 + 0.749958i \(0.269926\pi\)
\(828\) 0 0
\(829\) 35.9506 1.24862 0.624308 0.781178i \(-0.285381\pi\)
0.624308 + 0.781178i \(0.285381\pi\)
\(830\) 0 0
\(831\) −15.8387 −0.549438
\(832\) 0 0
\(833\) 16.5179 0.572311
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.48594 −0.0859268
\(838\) 0 0
\(839\) −3.66477 −0.126522 −0.0632610 0.997997i \(-0.520150\pi\)
−0.0632610 + 0.997997i \(0.520150\pi\)
\(840\) 0 0
\(841\) −9.47769 −0.326817
\(842\) 0 0
\(843\) −32.1036 −1.10571
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11.5400 0.396521
\(848\) 0 0
\(849\) 33.5488 1.15139
\(850\) 0 0
\(851\) 10.8883 0.373245
\(852\) 0 0
\(853\) −31.5802 −1.08129 −0.540643 0.841252i \(-0.681819\pi\)
−0.540643 + 0.841252i \(0.681819\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.40752 −0.0822392 −0.0411196 0.999154i \(-0.513092\pi\)
−0.0411196 + 0.999154i \(0.513092\pi\)
\(858\) 0 0
\(859\) 41.4550 1.41443 0.707214 0.707000i \(-0.249952\pi\)
0.707214 + 0.707000i \(0.249952\pi\)
\(860\) 0 0
\(861\) −3.91825 −0.133534
\(862\) 0 0
\(863\) −6.99173 −0.238001 −0.119001 0.992894i \(-0.537969\pi\)
−0.119001 + 0.992894i \(0.537969\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 31.5764 1.07239
\(868\) 0 0
\(869\) −42.3513 −1.43667
\(870\) 0 0
\(871\) −77.0874 −2.61201
\(872\) 0 0
\(873\) −16.7304 −0.566239
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −39.8619 −1.34604 −0.673020 0.739624i \(-0.735003\pi\)
−0.673020 + 0.739624i \(0.735003\pi\)
\(878\) 0 0
\(879\) −24.1382 −0.814161
\(880\) 0 0
\(881\) 24.7409 0.833541 0.416771 0.909012i \(-0.363162\pi\)
0.416771 + 0.909012i \(0.363162\pi\)
\(882\) 0 0
\(883\) −31.5648 −1.06224 −0.531120 0.847297i \(-0.678229\pi\)
−0.531120 + 0.847297i \(0.678229\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11.7866 −0.395755 −0.197877 0.980227i \(-0.563405\pi\)
−0.197877 + 0.980227i \(0.563405\pi\)
\(888\) 0 0
\(889\) 43.7452 1.46717
\(890\) 0 0
\(891\) 8.46866 0.283711
\(892\) 0 0
\(893\) −3.70641 −0.124030
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 31.2786 1.04436
\(898\) 0 0
\(899\) −1.99460 −0.0665238
\(900\) 0 0
\(901\) 34.0155 1.13322
\(902\) 0 0
\(903\) 12.0352 0.400505
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.08545 −0.135655 −0.0678276 0.997697i \(-0.521607\pi\)
−0.0678276 + 0.997697i \(0.521607\pi\)
\(908\) 0 0
\(909\) −24.1869 −0.802230
\(910\) 0 0
\(911\) 4.88428 0.161823 0.0809116 0.996721i \(-0.474217\pi\)
0.0809116 + 0.996721i \(0.474217\pi\)
\(912\) 0 0
\(913\) −0.943806 −0.0312354
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.33351 0.143105
\(918\) 0 0
\(919\) −53.8155 −1.77521 −0.887605 0.460606i \(-0.847632\pi\)
−0.887605 + 0.460606i \(0.847632\pi\)
\(920\) 0 0
\(921\) −35.3532 −1.16493
\(922\) 0 0
\(923\) 72.0285 2.37085
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −23.2990 −0.765239
\(928\) 0 0
\(929\) −43.3501 −1.42227 −0.711135 0.703055i \(-0.751819\pi\)
−0.711135 + 0.703055i \(0.751819\pi\)
\(930\) 0 0
\(931\) 2.52349 0.0827042
\(932\) 0 0
\(933\) −34.3181 −1.12352
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 41.5106 1.35609 0.678046 0.735019i \(-0.262827\pi\)
0.678046 + 0.735019i \(0.262827\pi\)
\(938\) 0 0
\(939\) 36.4000 1.18787
\(940\) 0 0
\(941\) 10.2485 0.334091 0.167045 0.985949i \(-0.446577\pi\)
0.167045 + 0.985949i \(0.446577\pi\)
\(942\) 0 0
\(943\) −4.54316 −0.147946
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.3985 0.597871 0.298936 0.954273i \(-0.403368\pi\)
0.298936 + 0.954273i \(0.403368\pi\)
\(948\) 0 0
\(949\) −14.6341 −0.475044
\(950\) 0 0
\(951\) −7.49530 −0.243052
\(952\) 0 0
\(953\) 26.4170 0.855732 0.427866 0.903842i \(-0.359266\pi\)
0.427866 + 0.903842i \(0.359266\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.7246 0.669930
\(958\) 0 0
\(959\) −41.4182 −1.33746
\(960\) 0 0
\(961\) −30.7962 −0.993426
\(962\) 0 0
\(963\) −13.9221 −0.448634
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −56.2597 −1.80919 −0.904596 0.426271i \(-0.859827\pi\)
−0.904596 + 0.426271i \(0.859827\pi\)
\(968\) 0 0
\(969\) 7.99707 0.256903
\(970\) 0 0
\(971\) −38.5451 −1.23697 −0.618485 0.785796i \(-0.712253\pi\)
−0.618485 + 0.785796i \(0.712253\pi\)
\(972\) 0 0
\(973\) 10.5828 0.339270
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.9699 0.926831 0.463415 0.886141i \(-0.346624\pi\)
0.463415 + 0.886141i \(0.346624\pi\)
\(978\) 0 0
\(979\) 51.5684 1.64813
\(980\) 0 0
\(981\) 14.8798 0.475075
\(982\) 0 0
\(983\) 1.75595 0.0560061 0.0280030 0.999608i \(-0.491085\pi\)
0.0280030 + 0.999608i \(0.491085\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −13.9743 −0.444807
\(988\) 0 0
\(989\) 13.9546 0.443731
\(990\) 0 0
\(991\) 1.13587 0.0360822 0.0180411 0.999837i \(-0.494257\pi\)
0.0180411 + 0.999837i \(0.494257\pi\)
\(992\) 0 0
\(993\) −20.8940 −0.663052
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −16.2745 −0.515420 −0.257710 0.966222i \(-0.582968\pi\)
−0.257710 + 0.966222i \(0.582968\pi\)
\(998\) 0 0
\(999\) −13.7156 −0.433944
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 3800.2.a.bb.1.5 6
4.3 odd 2 7600.2.a.cm.1.2 6
5.2 odd 4 3800.2.d.p.3649.4 12
5.3 odd 4 3800.2.d.p.3649.9 12
5.4 even 2 3800.2.a.bd.1.2 yes 6
20.19 odd 2 7600.2.a.ci.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.bb.1.5 6 1.1 even 1 trivial
3800.2.a.bd.1.2 yes 6 5.4 even 2
3800.2.d.p.3649.4 12 5.2 odd 4
3800.2.d.p.3649.9 12 5.3 odd 4
7600.2.a.ci.1.5 6 20.19 odd 2
7600.2.a.cm.1.2 6 4.3 odd 2