Properties

Label 6400.2.a.cm.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6400,2,Mod(1,6400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.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: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} -3.46410 q^{11} +4.89898 q^{17} +3.46410 q^{19} +3.46410 q^{21} -2.44949 q^{23} +5.65685 q^{27} -4.00000 q^{31} +4.89898 q^{33} +8.48528 q^{37} -4.24264 q^{43} +7.34847 q^{47} -1.00000 q^{49} -6.92820 q^{51} +5.65685 q^{53} -4.89898 q^{57} +10.3923 q^{59} +3.46410 q^{61} +2.44949 q^{63} +4.24264 q^{67} +3.46410 q^{69} -12.0000 q^{71} +4.89898 q^{73} +8.48528 q^{77} -4.00000 q^{79} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{89} +5.65685 q^{93} -4.89898 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} - 20 q^{81} + 24 q^{89}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) −2.44949 −0.510754 −0.255377 0.966842i \(-0.582200\pi\)
−0.255377 + 0.966842i \(0.582200\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 4.89898 0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.34847 1.07188 0.535942 0.844255i \(-0.319956\pi\)
0.535942 + 0.844255i \(0.319956\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.92820 −0.970143
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.89898 −0.648886
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 3.46410 0.443533 0.221766 0.975100i \(-0.428818\pi\)
0.221766 + 0.975100i \(0.428818\pi\)
\(62\) 0 0
\(63\) 2.44949 0.308607
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) 3.46410 0.417029
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.89898 0.573382 0.286691 0.958023i \(-0.407445\pi\)
0.286691 + 0.958023i \(0.407445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528 0.966988
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.65685 0.586588
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 −0.497416 −0.248708 0.968579i \(-0.580006\pi\)
−0.248708 + 0.968579i \(0.580006\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
\(100\) 0 0
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 0 0
\(103\) 7.34847 0.724066 0.362033 0.932165i \(-0.382083\pi\)
0.362033 + 0.932165i \(0.382083\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.41421 0.136717 0.0683586 0.997661i \(-0.478224\pi\)
0.0683586 + 0.997661i \(0.478224\pi\)
\(108\) 0 0
\(109\) 3.46410 0.331801 0.165900 0.986143i \(-0.446947\pi\)
0.165900 + 0.986143i \(0.446947\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.1464 −1.52150 −0.760750 0.649045i \(-0.775169\pi\)
−0.760750 + 0.649045i \(0.775169\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.79796 0.837096 0.418548 0.908195i \(-0.362539\pi\)
0.418548 + 0.908195i \(0.362539\pi\)
\(138\) 0 0
\(139\) −10.3923 −0.881464 −0.440732 0.897639i \(-0.645281\pi\)
−0.440732 + 0.897639i \(0.645281\pi\)
\(140\) 0 0
\(141\) −10.3923 −0.875190
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.41421 0.116642
\(148\) 0 0
\(149\) −17.3205 −1.41895 −0.709476 0.704730i \(-0.751068\pi\)
−0.709476 + 0.704730i \(0.751068\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −4.89898 −0.396059
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.48528 0.677199 0.338600 0.940931i \(-0.390047\pi\)
0.338600 + 0.940931i \(0.390047\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 21.2132 1.66155 0.830773 0.556611i \(-0.187899\pi\)
0.830773 + 0.556611i \(0.187899\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.2474 0.947736 0.473868 0.880596i \(-0.342857\pi\)
0.473868 + 0.880596i \(0.342857\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −3.46410 −0.264906
\(172\) 0 0
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.6969 −1.10469
\(178\) 0 0
\(179\) −3.46410 −0.258919 −0.129460 0.991585i \(-0.541324\pi\)
−0.129460 + 0.991585i \(0.541324\pi\)
\(180\) 0 0
\(181\) 13.8564 1.02994 0.514969 0.857209i \(-0.327803\pi\)
0.514969 + 0.857209i \(0.327803\pi\)
\(182\) 0 0
\(183\) −4.89898 −0.362143
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −16.9706 −1.24101
\(188\) 0 0
\(189\) −13.8564 −1.00791
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) −24.4949 −1.76318 −0.881591 0.472015i \(-0.843527\pi\)
−0.881591 + 0.472015i \(0.843527\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.65685 0.403034 0.201517 0.979485i \(-0.435413\pi\)
0.201517 + 0.979485i \(0.435413\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.44949 0.170251
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 24.2487 1.66935 0.834675 0.550743i \(-0.185655\pi\)
0.834675 + 0.550743i \(0.185655\pi\)
\(212\) 0 0
\(213\) 16.9706 1.16280
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.79796 0.665129
\(218\) 0 0
\(219\) −6.92820 −0.468165
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −22.0454 −1.47627 −0.738135 0.674653i \(-0.764293\pi\)
−0.738135 + 0.674653i \(0.764293\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.41421 0.0938647 0.0469323 0.998898i \(-0.485055\pi\)
0.0469323 + 0.998898i \(0.485055\pi\)
\(228\) 0 0
\(229\) 27.7128 1.83131 0.915657 0.401960i \(-0.131671\pi\)
0.915657 + 0.401960i \(0.131671\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) −14.6969 −0.962828 −0.481414 0.876493i \(-0.659877\pi\)
−0.481414 + 0.876493i \(0.659877\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) 8.48528 0.533465
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.79796 0.611180 0.305590 0.952163i \(-0.401146\pi\)
0.305590 + 0.952163i \(0.401146\pi\)
\(258\) 0 0
\(259\) −20.7846 −1.29149
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.34847 −0.453126 −0.226563 0.973997i \(-0.572749\pi\)
−0.226563 + 0.973997i \(0.572749\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8.48528 −0.519291
\(268\) 0 0
\(269\) 10.3923 0.633630 0.316815 0.948487i \(-0.397387\pi\)
0.316815 + 0.948487i \(0.397387\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −25.4558 −1.52949 −0.764747 0.644331i \(-0.777136\pi\)
−0.764747 + 0.644331i \(0.777136\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 21.2132 1.26099 0.630497 0.776192i \(-0.282851\pi\)
0.630497 + 0.776192i \(0.282851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 6.92820 0.406138
\(292\) 0 0
\(293\) −19.7990 −1.15667 −0.578335 0.815800i \(-0.696297\pi\)
−0.578335 + 0.815800i \(0.696297\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −19.5959 −1.13707
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 10.3923 0.599002
\(302\) 0 0
\(303\) 19.5959 1.12576
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −29.6985 −1.69498 −0.847491 0.530810i \(-0.821888\pi\)
−0.847491 + 0.530810i \(0.821888\pi\)
\(308\) 0 0
\(309\) −10.3923 −0.591198
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 9.79796 0.553813 0.276907 0.960897i \(-0.410691\pi\)
0.276907 + 0.960897i \(0.410691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −28.2843 −1.58860 −0.794301 0.607524i \(-0.792163\pi\)
−0.794301 + 0.607524i \(0.792163\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) 16.9706 0.944267
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4.89898 −0.270914
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) 31.1769 1.71364 0.856819 0.515617i \(-0.172437\pi\)
0.856819 + 0.515617i \(0.172437\pi\)
\(332\) 0 0
\(333\) −8.48528 −0.464991
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13.8564 0.750366
\(342\) 0 0
\(343\) 19.5959 1.05808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.5563 0.835109 0.417554 0.908652i \(-0.362887\pi\)
0.417554 + 0.908652i \(0.362887\pi\)
\(348\) 0 0
\(349\) 13.8564 0.741716 0.370858 0.928689i \(-0.379064\pi\)
0.370858 + 0.928689i \(0.379064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.3939 1.56448 0.782239 0.622978i \(-0.214078\pi\)
0.782239 + 0.622978i \(0.214078\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.9706 0.898177
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) −1.41421 −0.0742270
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 12.2474 0.639312 0.319656 0.947534i \(-0.396433\pi\)
0.319656 + 0.947534i \(0.396433\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.8564 −0.719389
\(372\) 0 0
\(373\) −8.48528 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.2487 −1.24557 −0.622786 0.782392i \(-0.713999\pi\)
−0.622786 + 0.782392i \(0.713999\pi\)
\(380\) 0 0
\(381\) 24.2487 1.24230
\(382\) 0 0
\(383\) 26.9444 1.37679 0.688397 0.725334i \(-0.258315\pi\)
0.688397 + 0.725334i \(0.258315\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.24264 0.215666
\(388\) 0 0
\(389\) −3.46410 −0.175637 −0.0878185 0.996136i \(-0.527990\pi\)
−0.0878185 + 0.996136i \(0.527990\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 4.89898 0.247121
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.9706 −0.851728 −0.425864 0.904787i \(-0.640030\pi\)
−0.425864 + 0.904787i \(0.640030\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −29.3939 −1.45700
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) −13.8564 −0.683486
\(412\) 0 0
\(413\) −25.4558 −1.25260
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.6969 0.719712
\(418\) 0 0
\(419\) 10.3923 0.507697 0.253849 0.967244i \(-0.418303\pi\)
0.253849 + 0.967244i \(0.418303\pi\)
\(420\) 0 0
\(421\) −24.2487 −1.18181 −0.590905 0.806741i \(-0.701229\pi\)
−0.590905 + 0.806741i \(0.701229\pi\)
\(422\) 0 0
\(423\) −7.34847 −0.357295
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.48528 −0.410632
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 4.89898 0.235430 0.117715 0.993047i \(-0.462443\pi\)
0.117715 + 0.993047i \(0.462443\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.48528 −0.405906
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −1.41421 −0.0671913 −0.0335957 0.999436i \(-0.510696\pi\)
−0.0335957 + 0.999436i \(0.510696\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 24.4949 1.15857
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 22.6274 1.06313
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.5959 −0.916658 −0.458329 0.888783i \(-0.651552\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) 0 0
\(459\) 27.7128 1.29352
\(460\) 0 0
\(461\) −13.8564 −0.645357 −0.322679 0.946509i \(-0.604583\pi\)
−0.322679 + 0.946509i \(0.604583\pi\)
\(462\) 0 0
\(463\) −17.1464 −0.796862 −0.398431 0.917198i \(-0.630445\pi\)
−0.398431 + 0.917198i \(0.630445\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.07107 0.327210 0.163605 0.986526i \(-0.447688\pi\)
0.163605 + 0.986526i \(0.447688\pi\)
\(468\) 0 0
\(469\) −10.3923 −0.479872
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 0 0
\(473\) 14.6969 0.675766
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.65685 −0.259010
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −8.48528 −0.386094
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.34847 0.332991 0.166495 0.986042i \(-0.446755\pi\)
0.166495 + 0.986042i \(0.446755\pi\)
\(488\) 0 0
\(489\) −30.0000 −1.35665
\(490\) 0 0
\(491\) −24.2487 −1.09433 −0.547165 0.837025i \(-0.684293\pi\)
−0.547165 + 0.837025i \(0.684293\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29.3939 1.31850
\(498\) 0 0
\(499\) −17.3205 −0.775372 −0.387686 0.921791i \(-0.626726\pi\)
−0.387686 + 0.921791i \(0.626726\pi\)
\(500\) 0 0
\(501\) −17.3205 −0.773823
\(502\) 0 0
\(503\) −12.2474 −0.546087 −0.273043 0.962002i \(-0.588030\pi\)
−0.273043 + 0.962002i \(0.588030\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 18.3848 0.816497
\(508\) 0 0
\(509\) −27.7128 −1.22835 −0.614174 0.789170i \(-0.710511\pi\)
−0.614174 + 0.789170i \(0.710511\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) 19.5959 0.865181
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −25.4558 −1.11955
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.5959 −0.853612
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) −10.3923 −0.450988
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.89898 0.211407
\(538\) 0 0
\(539\) 3.46410 0.149209
\(540\) 0 0
\(541\) −41.5692 −1.78720 −0.893600 0.448864i \(-0.851829\pi\)
−0.893600 + 0.448864i \(0.851829\pi\)
\(542\) 0 0
\(543\) −19.5959 −0.840941
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.24264 −0.181402 −0.0907011 0.995878i \(-0.528911\pi\)
−0.0907011 + 0.995878i \(0.528911\pi\)
\(548\) 0 0
\(549\) −3.46410 −0.147844
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 9.79796 0.416652
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.1421 −0.599222 −0.299611 0.954062i \(-0.596857\pi\)
−0.299611 + 0.954062i \(0.596857\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) −41.0122 −1.72846 −0.864229 0.503099i \(-0.832193\pi\)
−0.864229 + 0.503099i \(0.832193\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.2474 0.514344
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −3.46410 −0.144968 −0.0724841 0.997370i \(-0.523093\pi\)
−0.0724841 + 0.997370i \(0.523093\pi\)
\(572\) 0 0
\(573\) 33.9411 1.41791
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −29.3939 −1.22368 −0.611842 0.790980i \(-0.709571\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) 34.6410 1.43963
\(580\) 0 0
\(581\) 24.2487 1.00601
\(582\) 0 0
\(583\) −19.5959 −0.811580
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.89949 0.408596 0.204298 0.978909i \(-0.434509\pi\)
0.204298 + 0.978909i \(0.434509\pi\)
\(588\) 0 0
\(589\) −13.8564 −0.570943
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) 0 0
\(593\) −9.79796 −0.402354 −0.201177 0.979555i \(-0.564477\pi\)
−0.201177 + 0.979555i \(0.564477\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.65685 0.231520
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) −4.24264 −0.172774
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.34847 0.298265 0.149133 0.988817i \(-0.452352\pi\)
0.149133 + 0.988817i \(0.452352\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −33.9411 −1.37087 −0.685435 0.728134i \(-0.740388\pi\)
−0.685435 + 0.728134i \(0.740388\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.2929 1.38058 0.690289 0.723534i \(-0.257483\pi\)
0.690289 + 0.723534i \(0.257483\pi\)
\(618\) 0 0
\(619\) 10.3923 0.417702 0.208851 0.977947i \(-0.433028\pi\)
0.208851 + 0.977947i \(0.433028\pi\)
\(620\) 0 0
\(621\) −13.8564 −0.556038
\(622\) 0 0
\(623\) −14.6969 −0.588820
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 16.9706 0.677739
\(628\) 0 0
\(629\) 41.5692 1.65747
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −34.2929 −1.36302
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 29.6985 1.17119 0.585597 0.810602i \(-0.300860\pi\)
0.585597 + 0.810602i \(0.300860\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.2474 −0.481497 −0.240748 0.970588i \(-0.577393\pi\)
−0.240748 + 0.970588i \(0.577393\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −13.8564 −0.543075
\(652\) 0 0
\(653\) −11.3137 −0.442740 −0.221370 0.975190i \(-0.571053\pi\)
−0.221370 + 0.975190i \(0.571053\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.89898 −0.191127
\(658\) 0 0
\(659\) 24.2487 0.944596 0.472298 0.881439i \(-0.343425\pi\)
0.472298 + 0.881439i \(0.343425\pi\)
\(660\) 0 0
\(661\) 10.3923 0.404214 0.202107 0.979363i \(-0.435221\pi\)
0.202107 + 0.979363i \(0.435221\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 31.1769 1.20537
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.2929 −1.32189 −0.660946 0.750433i \(-0.729845\pi\)
−0.660946 + 0.750433i \(0.729845\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.6274 −0.869642 −0.434821 0.900517i \(-0.643188\pi\)
−0.434821 + 0.900517i \(0.643188\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 0 0
\(683\) −15.5563 −0.595247 −0.297624 0.954683i \(-0.596194\pi\)
−0.297624 + 0.954683i \(0.596194\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −39.1918 −1.49526
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −3.46410 −0.131781 −0.0658903 0.997827i \(-0.520989\pi\)
−0.0658903 + 0.997827i \(0.520989\pi\)
\(692\) 0 0
\(693\) −8.48528 −0.322329
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 20.7846 0.786146
\(700\) 0 0
\(701\) 24.2487 0.915861 0.457931 0.888988i \(-0.348591\pi\)
0.457931 + 0.888988i \(0.348591\pi\)
\(702\) 0 0
\(703\) 29.3939 1.10861
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33.9411 1.27649
\(708\) 0 0
\(709\) −41.5692 −1.56116 −0.780582 0.625053i \(-0.785077\pi\)
−0.780582 + 0.625053i \(0.785077\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 9.79796 0.366936
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.9706 0.633777
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) −5.65685 −0.210381
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 46.5403 1.72608 0.863042 0.505132i \(-0.168556\pi\)
0.863042 + 0.505132i \(0.168556\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −20.7846 −0.768747
\(732\) 0 0
\(733\) −42.4264 −1.56706 −0.783528 0.621357i \(-0.786582\pi\)
−0.783528 + 0.621357i \(0.786582\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.6969 −0.541369
\(738\) 0 0
\(739\) −3.46410 −0.127429 −0.0637145 0.997968i \(-0.520295\pi\)
−0.0637145 + 0.997968i \(0.520295\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 51.4393 1.88712 0.943562 0.331195i \(-0.107452\pi\)
0.943562 + 0.331195i \(0.107452\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9.89949 0.362204
\(748\) 0 0
\(749\) −3.46410 −0.126576
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −14.6969 −0.535586
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 25.4558 0.925208 0.462604 0.886565i \(-0.346915\pi\)
0.462604 + 0.886565i \(0.346915\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −8.48528 −0.307188
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −13.8564 −0.499026
\(772\) 0 0
\(773\) 28.2843 1.01731 0.508657 0.860969i \(-0.330142\pi\)
0.508657 + 0.860969i \(0.330142\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 29.3939 1.05450
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 41.5692 1.48746
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 21.2132 0.756169 0.378085 0.925771i \(-0.376583\pi\)
0.378085 + 0.925771i \(0.376583\pi\)
\(788\) 0 0
\(789\) 10.3923 0.369976
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.5980 1.40263 0.701316 0.712850i \(-0.252596\pi\)
0.701316 + 0.712850i \(0.252596\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −16.9706 −0.598878
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.6969 −0.517357
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 10.3923 0.364923 0.182462 0.983213i \(-0.441593\pi\)
0.182462 + 0.983213i \(0.441593\pi\)
\(812\) 0 0
\(813\) 28.2843 0.991973
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.6969 −0.514181
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.1769 −1.08808 −0.544041 0.839059i \(-0.683106\pi\)
−0.544041 + 0.839059i \(0.683106\pi\)
\(822\) 0 0
\(823\) 26.9444 0.939222 0.469611 0.882873i \(-0.344394\pi\)
0.469611 + 0.882873i \(0.344394\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.3553 −1.22943 −0.614713 0.788751i \(-0.710728\pi\)
−0.614713 + 0.788751i \(0.710728\pi\)
\(828\) 0 0
\(829\) 10.3923 0.360940 0.180470 0.983581i \(-0.442238\pi\)
0.180470 + 0.983581i \(0.442238\pi\)
\(830\) 0 0
\(831\) 36.0000 1.24883
\(832\) 0 0
\(833\) −4.89898 −0.169740
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −22.6274 −0.782118
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −16.9706 −0.584497
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.44949 −0.0841655
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) −20.7846 −0.712487
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.1918 −1.33877 −0.669384 0.742917i \(-0.733442\pi\)
−0.669384 + 0.742917i \(0.733442\pi\)
\(858\) 0 0
\(859\) −3.46410 −0.118194 −0.0590968 0.998252i \(-0.518822\pi\)
−0.0590968 + 0.998252i \(0.518822\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.44949 0.0833816 0.0416908 0.999131i \(-0.486726\pi\)
0.0416908 + 0.999131i \(0.486726\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −9.89949 −0.336204
\(868\) 0 0
\(869\) 13.8564 0.470046
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4.89898 0.165805
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.48528 −0.286528 −0.143264 0.989685i \(-0.545760\pi\)
−0.143264 + 0.989685i \(0.545760\pi\)
\(878\) 0 0
\(879\) 28.0000 0.944417
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 4.24264 0.142776 0.0713881 0.997449i \(-0.477257\pi\)
0.0713881 + 0.997449i \(0.477257\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.9444 0.904704 0.452352 0.891839i \(-0.350585\pi\)
0.452352 + 0.891839i \(0.350585\pi\)
\(888\) 0 0
\(889\) 42.0000 1.40863
\(890\) 0 0
\(891\) 17.3205 0.580259
\(892\) 0 0
\(893\) 25.4558 0.851847
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 27.7128 0.923248
\(902\) 0 0
\(903\) −14.6969 −0.489083
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.24264 −0.140875 −0.0704373 0.997516i \(-0.522439\pi\)
−0.0704373 + 0.997516i \(0.522439\pi\)
\(908\) 0 0
\(909\) 13.8564 0.459588
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 34.2929 1.13493
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.48528 0.280209
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 42.0000 1.38395
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.34847 −0.241355
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −3.46410 −0.113531
\(932\) 0 0
\(933\) 33.9411 1.11118
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.89898 0.160043 0.0800213 0.996793i \(-0.474501\pi\)
0.0800213 + 0.996793i \(0.474501\pi\)
\(938\) 0 0
\(939\) −13.8564 −0.452187
\(940\) 0 0
\(941\) 13.8564 0.451706 0.225853 0.974161i \(-0.427483\pi\)
0.225853 + 0.974161i \(0.427483\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.0122 −1.33272 −0.666359 0.745631i \(-0.732148\pi\)
−0.666359 + 0.745631i \(0.732148\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 40.0000 1.29709
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −1.41421 −0.0455724
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −17.1464 −0.551392 −0.275696 0.961245i \(-0.588908\pi\)
−0.275696 + 0.961245i \(0.588908\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 3.46410 0.111168 0.0555842 0.998454i \(-0.482298\pi\)
0.0555842 + 0.998454i \(0.482298\pi\)
\(972\) 0 0
\(973\) 25.4558 0.816077
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −44.0908 −1.41059 −0.705295 0.708914i \(-0.749185\pi\)
−0.705295 + 0.708914i \(0.749185\pi\)
\(978\) 0 0
\(979\) −20.7846 −0.664279
\(980\) 0 0
\(981\) −3.46410 −0.110600
\(982\) 0 0
\(983\) −56.3383 −1.79691 −0.898456 0.439064i \(-0.855310\pi\)
−0.898456 + 0.439064i \(0.855310\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 25.4558 0.810268
\(988\) 0 0
\(989\) 10.3923 0.330456
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −44.0908 −1.39918
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 50.9117 1.61239 0.806195 0.591650i \(-0.201523\pi\)
0.806195 + 0.591650i \(0.201523\pi\)
\(998\) 0 0
\(999\) 48.0000 1.51865
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 6400.2.a.cm.1.1 4
4.3 odd 2 6400.2.a.co.1.4 4
5.2 odd 4 1280.2.c.k.769.3 4
5.3 odd 4 1280.2.c.k.769.1 4
5.4 even 2 inner 6400.2.a.cm.1.4 4
8.3 odd 2 6400.2.a.co.1.2 4
8.5 even 2 inner 6400.2.a.cm.1.3 4
16.3 odd 4 200.2.d.e.101.4 4
16.5 even 4 800.2.d.f.401.4 4
16.11 odd 4 200.2.d.e.101.3 4
16.13 even 4 800.2.d.f.401.2 4
20.3 even 4 1280.2.c.i.769.3 4
20.7 even 4 1280.2.c.i.769.1 4
20.19 odd 2 6400.2.a.co.1.1 4
40.3 even 4 1280.2.c.i.769.2 4
40.13 odd 4 1280.2.c.k.769.4 4
40.19 odd 2 6400.2.a.co.1.3 4
40.27 even 4 1280.2.c.i.769.4 4
40.29 even 2 inner 6400.2.a.cm.1.2 4
40.37 odd 4 1280.2.c.k.769.2 4
48.5 odd 4 7200.2.k.l.3601.4 4
48.11 even 4 1800.2.k.m.901.2 4
48.29 odd 4 7200.2.k.l.3601.3 4
48.35 even 4 1800.2.k.m.901.1 4
80.3 even 4 40.2.f.a.29.3 yes 4
80.13 odd 4 160.2.f.a.49.4 4
80.19 odd 4 200.2.d.e.101.1 4
80.27 even 4 40.2.f.a.29.4 yes 4
80.29 even 4 800.2.d.f.401.3 4
80.37 odd 4 160.2.f.a.49.3 4
80.43 even 4 40.2.f.a.29.1 4
80.53 odd 4 160.2.f.a.49.1 4
80.59 odd 4 200.2.d.e.101.2 4
80.67 even 4 40.2.f.a.29.2 yes 4
80.69 even 4 800.2.d.f.401.1 4
80.77 odd 4 160.2.f.a.49.2 4
240.29 odd 4 7200.2.k.l.3601.1 4
240.53 even 4 1440.2.d.c.1009.4 4
240.59 even 4 1800.2.k.m.901.3 4
240.77 even 4 1440.2.d.c.1009.3 4
240.83 odd 4 360.2.d.b.109.2 4
240.107 odd 4 360.2.d.b.109.1 4
240.149 odd 4 7200.2.k.l.3601.2 4
240.173 even 4 1440.2.d.c.1009.1 4
240.179 even 4 1800.2.k.m.901.4 4
240.197 even 4 1440.2.d.c.1009.2 4
240.203 odd 4 360.2.d.b.109.4 4
240.227 odd 4 360.2.d.b.109.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.f.a.29.1 4 80.43 even 4
40.2.f.a.29.2 yes 4 80.67 even 4
40.2.f.a.29.3 yes 4 80.3 even 4
40.2.f.a.29.4 yes 4 80.27 even 4
160.2.f.a.49.1 4 80.53 odd 4
160.2.f.a.49.2 4 80.77 odd 4
160.2.f.a.49.3 4 80.37 odd 4
160.2.f.a.49.4 4 80.13 odd 4
200.2.d.e.101.1 4 80.19 odd 4
200.2.d.e.101.2 4 80.59 odd 4
200.2.d.e.101.3 4 16.11 odd 4
200.2.d.e.101.4 4 16.3 odd 4
360.2.d.b.109.1 4 240.107 odd 4
360.2.d.b.109.2 4 240.83 odd 4
360.2.d.b.109.3 4 240.227 odd 4
360.2.d.b.109.4 4 240.203 odd 4
800.2.d.f.401.1 4 80.69 even 4
800.2.d.f.401.2 4 16.13 even 4
800.2.d.f.401.3 4 80.29 even 4
800.2.d.f.401.4 4 16.5 even 4
1280.2.c.i.769.1 4 20.7 even 4
1280.2.c.i.769.2 4 40.3 even 4
1280.2.c.i.769.3 4 20.3 even 4
1280.2.c.i.769.4 4 40.27 even 4
1280.2.c.k.769.1 4 5.3 odd 4
1280.2.c.k.769.2 4 40.37 odd 4
1280.2.c.k.769.3 4 5.2 odd 4
1280.2.c.k.769.4 4 40.13 odd 4
1440.2.d.c.1009.1 4 240.173 even 4
1440.2.d.c.1009.2 4 240.197 even 4
1440.2.d.c.1009.3 4 240.77 even 4
1440.2.d.c.1009.4 4 240.53 even 4
1800.2.k.m.901.1 4 48.35 even 4
1800.2.k.m.901.2 4 48.11 even 4
1800.2.k.m.901.3 4 240.59 even 4
1800.2.k.m.901.4 4 240.179 even 4
6400.2.a.cm.1.1 4 1.1 even 1 trivial
6400.2.a.cm.1.2 4 40.29 even 2 inner
6400.2.a.cm.1.3 4 8.5 even 2 inner
6400.2.a.cm.1.4 4 5.4 even 2 inner
6400.2.a.co.1.1 4 20.19 odd 2
6400.2.a.co.1.2 4 8.3 odd 2
6400.2.a.co.1.3 4 40.19 odd 2
6400.2.a.co.1.4 4 4.3 odd 2
7200.2.k.l.3601.1 4 240.29 odd 4
7200.2.k.l.3601.2 4 240.149 odd 4
7200.2.k.l.3601.3 4 48.29 odd 4
7200.2.k.l.3601.4 4 48.5 odd 4