Properties

Label 9576.2.a.bf.1.2
Level $9576$
Weight $2$
Character 9576.1
Self dual yes
Analytic conductor $76.465$
Analytic rank $1$
Dimension $2$
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] = [9576,2,Mod(1,9576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9576.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9576.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: \(76.4647449756\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9576.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.732051 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+0.732051 q^{5} -1.00000 q^{7} +3.46410 q^{11} +4.00000 q^{13} -3.26795 q^{17} +1.00000 q^{19} +0.535898 q^{23} -4.46410 q^{25} +2.73205 q^{29} -3.46410 q^{31} -0.732051 q^{35} -7.46410 q^{37} -11.4641 q^{41} -8.92820 q^{43} -5.66025 q^{47} +1.00000 q^{49} -8.19615 q^{53} +2.53590 q^{55} -8.00000 q^{59} +4.53590 q^{61} +2.92820 q^{65} -4.00000 q^{67} +6.19615 q^{71} +0.535898 q^{73} -3.46410 q^{77} -5.46410 q^{79} +0.196152 q^{83} -2.39230 q^{85} +10.3923 q^{89} -4.00000 q^{91} +0.732051 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 2 q^{71} + 8 q^{73} - 4 q^{79} - 10 q^{83} + 16 q^{85} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.732051 0.327383 0.163692 0.986512i \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.26795 −0.792594 −0.396297 0.918122i \(-0.629705\pi\)
−0.396297 + 0.918122i \(0.629705\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.535898 0.111743 0.0558713 0.998438i \(-0.482206\pi\)
0.0558713 + 0.998438i \(0.482206\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.73205 0.507329 0.253665 0.967292i \(-0.418364\pi\)
0.253665 + 0.967292i \(0.418364\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.732051 −0.123739
\(36\) 0 0
\(37\) −7.46410 −1.22709 −0.613545 0.789659i \(-0.710257\pi\)
−0.613545 + 0.789659i \(0.710257\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.4641 −1.79039 −0.895196 0.445673i \(-0.852964\pi\)
−0.895196 + 0.445673i \(0.852964\pi\)
\(42\) 0 0
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.66025 −0.825633 −0.412816 0.910814i \(-0.635455\pi\)
−0.412816 + 0.910814i \(0.635455\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.19615 −1.12583 −0.562914 0.826515i \(-0.690320\pi\)
−0.562914 + 0.826515i \(0.690320\pi\)
\(54\) 0 0
\(55\) 2.53590 0.341940
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 4.53590 0.580762 0.290381 0.956911i \(-0.406218\pi\)
0.290381 + 0.956911i \(0.406218\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.92820 0.363199
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.19615 0.735348 0.367674 0.929955i \(-0.380154\pi\)
0.367674 + 0.929955i \(0.380154\pi\)
\(72\) 0 0
\(73\) 0.535898 0.0627222 0.0313611 0.999508i \(-0.490016\pi\)
0.0313611 + 0.999508i \(0.490016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −5.46410 −0.614759 −0.307380 0.951587i \(-0.599452\pi\)
−0.307380 + 0.951587i \(0.599452\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.196152 0.0215305 0.0107653 0.999942i \(-0.496573\pi\)
0.0107653 + 0.999942i \(0.496573\pi\)
\(84\) 0 0
\(85\) −2.39230 −0.259482
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.732051 0.0751068
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.5885 −1.45161 −0.725803 0.687903i \(-0.758532\pi\)
−0.725803 + 0.687903i \(0.758532\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.19615 −0.212310 −0.106155 0.994350i \(-0.533854\pi\)
−0.106155 + 0.994350i \(0.533854\pi\)
\(108\) 0 0
\(109\) 15.8564 1.51877 0.759384 0.650643i \(-0.225500\pi\)
0.759384 + 0.650643i \(0.225500\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.73205 0.257010 0.128505 0.991709i \(-0.458982\pi\)
0.128505 + 0.991709i \(0.458982\pi\)
\(114\) 0 0
\(115\) 0.392305 0.0365826
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.26795 0.299572
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 2.92820 0.259836 0.129918 0.991525i \(-0.458529\pi\)
0.129918 + 0.991525i \(0.458529\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.19615 −0.716101 −0.358051 0.933702i \(-0.616558\pi\)
−0.358051 + 0.933702i \(0.616558\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.9282 0.933659 0.466830 0.884347i \(-0.345396\pi\)
0.466830 + 0.884347i \(0.345396\pi\)
\(138\) 0 0
\(139\) 2.53590 0.215092 0.107546 0.994200i \(-0.465701\pi\)
0.107546 + 0.994200i \(0.465701\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.46410 0.775329 0.387665 0.921800i \(-0.373282\pi\)
0.387665 + 0.921800i \(0.373282\pi\)
\(150\) 0 0
\(151\) −21.4641 −1.74672 −0.873362 0.487072i \(-0.838065\pi\)
−0.873362 + 0.487072i \(0.838065\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.53590 −0.203688
\(156\) 0 0
\(157\) 14.3923 1.14863 0.574315 0.818634i \(-0.305268\pi\)
0.574315 + 0.818634i \(0.305268\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.535898 −0.0422347
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.39230 0.339887 0.169943 0.985454i \(-0.445642\pi\)
0.169943 + 0.985454i \(0.445642\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.3923 0.790112 0.395056 0.918657i \(-0.370725\pi\)
0.395056 + 0.918657i \(0.370725\pi\)
\(174\) 0 0
\(175\) 4.46410 0.337454
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −25.5167 −1.90720 −0.953602 0.301069i \(-0.902657\pi\)
−0.953602 + 0.301069i \(0.902657\pi\)
\(180\) 0 0
\(181\) −23.8564 −1.77323 −0.886616 0.462506i \(-0.846951\pi\)
−0.886616 + 0.462506i \(0.846951\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.46410 −0.401729
\(186\) 0 0
\(187\) −11.3205 −0.827838
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.3923 1.33082 0.665410 0.746478i \(-0.268257\pi\)
0.665410 + 0.746478i \(0.268257\pi\)
\(192\) 0 0
\(193\) 17.3205 1.24676 0.623379 0.781920i \(-0.285760\pi\)
0.623379 + 0.781920i \(0.285760\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.92820 −0.493614 −0.246807 0.969065i \(-0.579381\pi\)
−0.246807 + 0.969065i \(0.579381\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.73205 −0.191752
\(204\) 0 0
\(205\) −8.39230 −0.586144
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.53590 −0.445745
\(216\) 0 0
\(217\) 3.46410 0.235159
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −13.0718 −0.879304
\(222\) 0 0
\(223\) 13.3205 0.892007 0.446004 0.895031i \(-0.352847\pi\)
0.446004 + 0.895031i \(0.352847\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.4641 −0.893644 −0.446822 0.894623i \(-0.647444\pi\)
−0.446822 + 0.894623i \(0.647444\pi\)
\(228\) 0 0
\(229\) 0.928203 0.0613374 0.0306687 0.999530i \(-0.490236\pi\)
0.0306687 + 0.999530i \(0.490236\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5359 −0.952278 −0.476139 0.879370i \(-0.657964\pi\)
−0.476139 + 0.879370i \(0.657964\pi\)
\(234\) 0 0
\(235\) −4.14359 −0.270298
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −19.8564 −1.28440 −0.642202 0.766535i \(-0.721979\pi\)
−0.642202 + 0.766535i \(0.721979\pi\)
\(240\) 0 0
\(241\) 15.0718 0.970860 0.485430 0.874276i \(-0.338663\pi\)
0.485430 + 0.874276i \(0.338663\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.732051 0.0467690
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.7321 −0.677401 −0.338701 0.940894i \(-0.609987\pi\)
−0.338701 + 0.940894i \(0.609987\pi\)
\(252\) 0 0
\(253\) 1.85641 0.116711
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.143594 0.00895712 0.00447856 0.999990i \(-0.498574\pi\)
0.00447856 + 0.999990i \(0.498574\pi\)
\(258\) 0 0
\(259\) 7.46410 0.463797
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.9282 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.9282 −0.788246 −0.394123 0.919058i \(-0.628952\pi\)
−0.394123 + 0.919058i \(0.628952\pi\)
\(270\) 0 0
\(271\) 23.3205 1.41662 0.708310 0.705902i \(-0.249458\pi\)
0.708310 + 0.705902i \(0.249458\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.4641 −0.932520
\(276\) 0 0
\(277\) 22.2487 1.33680 0.668398 0.743804i \(-0.266980\pi\)
0.668398 + 0.743804i \(0.266980\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.7321 −1.11746 −0.558730 0.829349i \(-0.688711\pi\)
−0.558730 + 0.829349i \(0.688711\pi\)
\(282\) 0 0
\(283\) 21.8564 1.29923 0.649614 0.760264i \(-0.274930\pi\)
0.649614 + 0.760264i \(0.274930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.4641 0.676705
\(288\) 0 0
\(289\) −6.32051 −0.371795
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −20.5359 −1.19972 −0.599860 0.800105i \(-0.704777\pi\)
−0.599860 + 0.800105i \(0.704777\pi\)
\(294\) 0 0
\(295\) −5.85641 −0.340973
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.14359 0.123967
\(300\) 0 0
\(301\) 8.92820 0.514613
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.32051 0.190132
\(306\) 0 0
\(307\) −21.3205 −1.21683 −0.608413 0.793621i \(-0.708193\pi\)
−0.608413 + 0.793621i \(0.708193\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −25.2679 −1.43281 −0.716407 0.697683i \(-0.754215\pi\)
−0.716407 + 0.697683i \(0.754215\pi\)
\(312\) 0 0
\(313\) 21.3205 1.20511 0.602553 0.798079i \(-0.294150\pi\)
0.602553 + 0.798079i \(0.294150\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.7321 0.827434 0.413717 0.910405i \(-0.364230\pi\)
0.413717 + 0.910405i \(0.364230\pi\)
\(318\) 0 0
\(319\) 9.46410 0.529888
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.26795 −0.181834
\(324\) 0 0
\(325\) −17.8564 −0.990495
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.66025 0.312060
\(330\) 0 0
\(331\) −16.3923 −0.901003 −0.450501 0.892776i \(-0.648755\pi\)
−0.450501 + 0.892776i \(0.648755\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.92820 −0.159985
\(336\) 0 0
\(337\) 16.9282 0.922138 0.461069 0.887364i \(-0.347466\pi\)
0.461069 + 0.887364i \(0.347466\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.14359 −0.437171 −0.218586 0.975818i \(-0.570144\pi\)
−0.218586 + 0.975818i \(0.570144\pi\)
\(348\) 0 0
\(349\) −3.85641 −0.206429 −0.103214 0.994659i \(-0.532913\pi\)
−0.103214 + 0.994659i \(0.532913\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.1962 0.968483 0.484242 0.874934i \(-0.339096\pi\)
0.484242 + 0.874934i \(0.339096\pi\)
\(354\) 0 0
\(355\) 4.53590 0.240740
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11.4641 −0.605052 −0.302526 0.953141i \(-0.597830\pi\)
−0.302526 + 0.953141i \(0.597830\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.392305 0.0205342
\(366\) 0 0
\(367\) 1.85641 0.0969036 0.0484518 0.998826i \(-0.484571\pi\)
0.0484518 + 0.998826i \(0.484571\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.19615 0.425523
\(372\) 0 0
\(373\) −29.7128 −1.53847 −0.769236 0.638965i \(-0.779363\pi\)
−0.769236 + 0.638965i \(0.779363\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.9282 0.562831
\(378\) 0 0
\(379\) 18.2487 0.937373 0.468687 0.883364i \(-0.344727\pi\)
0.468687 + 0.883364i \(0.344727\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.5359 −0.742750 −0.371375 0.928483i \(-0.621114\pi\)
−0.371375 + 0.928483i \(0.621114\pi\)
\(384\) 0 0
\(385\) −2.53590 −0.129241
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.46410 −0.479849 −0.239924 0.970792i \(-0.577123\pi\)
−0.239924 + 0.970792i \(0.577123\pi\)
\(390\) 0 0
\(391\) −1.75129 −0.0885665
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0526 −1.10125 −0.550626 0.834752i \(-0.685611\pi\)
−0.550626 + 0.834752i \(0.685611\pi\)
\(402\) 0 0
\(403\) −13.8564 −0.690237
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.8564 −1.28165
\(408\) 0 0
\(409\) −19.7128 −0.974736 −0.487368 0.873197i \(-0.662043\pi\)
−0.487368 + 0.873197i \(0.662043\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 0.143594 0.00704873
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.196152 −0.00958267 −0.00479134 0.999989i \(-0.501525\pi\)
−0.00479134 + 0.999989i \(0.501525\pi\)
\(420\) 0 0
\(421\) 9.32051 0.454254 0.227127 0.973865i \(-0.427067\pi\)
0.227127 + 0.973865i \(0.427067\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.5885 0.707644
\(426\) 0 0
\(427\) −4.53590 −0.219508
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0526 −0.773225 −0.386612 0.922242i \(-0.626355\pi\)
−0.386612 + 0.922242i \(0.626355\pi\)
\(432\) 0 0
\(433\) 8.92820 0.429062 0.214531 0.976717i \(-0.431178\pi\)
0.214531 + 0.976717i \(0.431178\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.535898 0.0256355
\(438\) 0 0
\(439\) 4.53590 0.216487 0.108243 0.994124i \(-0.465477\pi\)
0.108243 + 0.994124i \(0.465477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.85641 −0.373269 −0.186635 0.982429i \(-0.559758\pi\)
−0.186635 + 0.982429i \(0.559758\pi\)
\(444\) 0 0
\(445\) 7.60770 0.360639
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.41154 −0.161001 −0.0805003 0.996755i \(-0.525652\pi\)
−0.0805003 + 0.996755i \(0.525652\pi\)
\(450\) 0 0
\(451\) −39.7128 −1.87000
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.92820 −0.137276
\(456\) 0 0
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −41.5167 −1.93362 −0.966812 0.255490i \(-0.917763\pi\)
−0.966812 + 0.255490i \(0.917763\pi\)
\(462\) 0 0
\(463\) −13.8564 −0.643962 −0.321981 0.946746i \(-0.604349\pi\)
−0.321981 + 0.946746i \(0.604349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.73205 0.311522 0.155761 0.987795i \(-0.450217\pi\)
0.155761 + 0.987795i \(0.450217\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −30.9282 −1.42208
\(474\) 0 0
\(475\) −4.46410 −0.204827
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −29.2679 −1.33729 −0.668643 0.743583i \(-0.733125\pi\)
−0.668643 + 0.743583i \(0.733125\pi\)
\(480\) 0 0
\(481\) −29.8564 −1.36133
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.39230 0.199444
\(486\) 0 0
\(487\) −6.14359 −0.278393 −0.139196 0.990265i \(-0.544452\pi\)
−0.139196 + 0.990265i \(0.544452\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.85641 0.354555 0.177277 0.984161i \(-0.443271\pi\)
0.177277 + 0.984161i \(0.443271\pi\)
\(492\) 0 0
\(493\) −8.92820 −0.402106
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.19615 −0.277935
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.66025 0.252378 0.126189 0.992006i \(-0.459725\pi\)
0.126189 + 0.992006i \(0.459725\pi\)
\(504\) 0 0
\(505\) −10.6795 −0.475231
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.7846 0.655316 0.327658 0.944796i \(-0.393741\pi\)
0.327658 + 0.944796i \(0.393741\pi\)
\(510\) 0 0
\(511\) −0.535898 −0.0237067
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.07180 0.223490
\(516\) 0 0
\(517\) −19.6077 −0.862345
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.7128 0.776012 0.388006 0.921657i \(-0.373164\pi\)
0.388006 + 0.921657i \(0.373164\pi\)
\(522\) 0 0
\(523\) −0.535898 −0.0234332 −0.0117166 0.999931i \(-0.503730\pi\)
−0.0117166 + 0.999931i \(0.503730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.3205 0.493129
\(528\) 0 0
\(529\) −22.7128 −0.987514
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −45.8564 −1.98626
\(534\) 0 0
\(535\) −1.60770 −0.0695067
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.46410 0.149209
\(540\) 0 0
\(541\) 31.8564 1.36961 0.684807 0.728725i \(-0.259887\pi\)
0.684807 + 0.728725i \(0.259887\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.6077 0.497219
\(546\) 0 0
\(547\) 40.1051 1.71477 0.857386 0.514675i \(-0.172087\pi\)
0.857386 + 0.514675i \(0.172087\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.73205 0.116389
\(552\) 0 0
\(553\) 5.46410 0.232357
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.4641 0.570492 0.285246 0.958454i \(-0.407925\pi\)
0.285246 + 0.958454i \(0.407925\pi\)
\(558\) 0 0
\(559\) −35.7128 −1.51049
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.5359 −0.612615 −0.306308 0.951933i \(-0.599094\pi\)
−0.306308 + 0.951933i \(0.599094\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0526 0.924491 0.462246 0.886752i \(-0.347044\pi\)
0.462246 + 0.886752i \(0.347044\pi\)
\(570\) 0 0
\(571\) −17.0718 −0.714432 −0.357216 0.934022i \(-0.616274\pi\)
−0.357216 + 0.934022i \(0.616274\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.39230 −0.0997660
\(576\) 0 0
\(577\) 45.3205 1.88672 0.943359 0.331775i \(-0.107647\pi\)
0.943359 + 0.331775i \(0.107647\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.196152 −0.00813777
\(582\) 0 0
\(583\) −28.3923 −1.17589
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.0526 0.910207 0.455103 0.890439i \(-0.349602\pi\)
0.455103 + 0.890439i \(0.349602\pi\)
\(588\) 0 0
\(589\) −3.46410 −0.142736
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.9808 −1.27223 −0.636114 0.771595i \(-0.719459\pi\)
−0.636114 + 0.771595i \(0.719459\pi\)
\(594\) 0 0
\(595\) 2.39230 0.0980749
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.0526 0.819325 0.409663 0.912237i \(-0.365646\pi\)
0.409663 + 0.912237i \(0.365646\pi\)
\(600\) 0 0
\(601\) 40.6410 1.65778 0.828891 0.559410i \(-0.188972\pi\)
0.828891 + 0.559410i \(0.188972\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.732051 0.0297621
\(606\) 0 0
\(607\) −24.7846 −1.00598 −0.502988 0.864293i \(-0.667766\pi\)
−0.502988 + 0.864293i \(0.667766\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −22.6410 −0.915957
\(612\) 0 0
\(613\) 7.60770 0.307272 0.153636 0.988128i \(-0.450902\pi\)
0.153636 + 0.988128i \(0.450902\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.7128 −1.43774 −0.718872 0.695143i \(-0.755341\pi\)
−0.718872 + 0.695143i \(0.755341\pi\)
\(618\) 0 0
\(619\) −13.4641 −0.541168 −0.270584 0.962696i \(-0.587217\pi\)
−0.270584 + 0.962696i \(0.587217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.3923 0.972585
\(630\) 0 0
\(631\) −20.9282 −0.833139 −0.416569 0.909104i \(-0.636768\pi\)
−0.416569 + 0.909104i \(0.636768\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.14359 0.0850659
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.12436 0.281395 0.140698 0.990053i \(-0.455065\pi\)
0.140698 + 0.990053i \(0.455065\pi\)
\(642\) 0 0
\(643\) 41.5692 1.63933 0.819665 0.572843i \(-0.194160\pi\)
0.819665 + 0.572843i \(0.194160\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.1244 0.751856 0.375928 0.926649i \(-0.377324\pi\)
0.375928 + 0.926649i \(0.377324\pi\)
\(648\) 0 0
\(649\) −27.7128 −1.08782
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.0718 0.511539 0.255769 0.966738i \(-0.417671\pi\)
0.255769 + 0.966738i \(0.417671\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.7321 0.651788 0.325894 0.945406i \(-0.394335\pi\)
0.325894 + 0.945406i \(0.394335\pi\)
\(660\) 0 0
\(661\) 26.6410 1.03622 0.518108 0.855315i \(-0.326637\pi\)
0.518108 + 0.855315i \(0.326637\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.732051 −0.0283877
\(666\) 0 0
\(667\) 1.46410 0.0566902
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.7128 0.606586
\(672\) 0 0
\(673\) −32.2487 −1.24310 −0.621548 0.783376i \(-0.713496\pi\)
−0.621548 + 0.783376i \(0.713496\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.92820 −0.189406 −0.0947031 0.995506i \(-0.530190\pi\)
−0.0947031 + 0.995506i \(0.530190\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.2679 0.584212 0.292106 0.956386i \(-0.405644\pi\)
0.292106 + 0.956386i \(0.405644\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −32.7846 −1.24899
\(690\) 0 0
\(691\) −4.67949 −0.178016 −0.0890081 0.996031i \(-0.528370\pi\)
−0.0890081 + 0.996031i \(0.528370\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.85641 0.0704175
\(696\) 0 0
\(697\) 37.4641 1.41905
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) −7.46410 −0.281514
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.5885 0.548655
\(708\) 0 0
\(709\) 14.2487 0.535122 0.267561 0.963541i \(-0.413782\pi\)
0.267561 + 0.963541i \(0.413782\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.85641 −0.0695230
\(714\) 0 0
\(715\) 10.1436 0.379349
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.41154 0.276404 0.138202 0.990404i \(-0.455868\pi\)
0.138202 + 0.990404i \(0.455868\pi\)
\(720\) 0 0
\(721\) −6.92820 −0.258020
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.1962 −0.452954
\(726\) 0 0
\(727\) −8.39230 −0.311253 −0.155627 0.987816i \(-0.549740\pi\)
−0.155627 + 0.987816i \(0.549740\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29.1769 1.07915
\(732\) 0 0
\(733\) 7.46410 0.275693 0.137846 0.990454i \(-0.455982\pi\)
0.137846 + 0.990454i \(0.455982\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 27.0718 0.995852 0.497926 0.867219i \(-0.334095\pi\)
0.497926 + 0.867219i \(0.334095\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.3013 1.25839 0.629196 0.777247i \(-0.283384\pi\)
0.629196 + 0.777247i \(0.283384\pi\)
\(744\) 0 0
\(745\) 6.92820 0.253830
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.19615 0.0802457
\(750\) 0 0
\(751\) −16.3923 −0.598164 −0.299082 0.954227i \(-0.596680\pi\)
−0.299082 + 0.954227i \(0.596680\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15.7128 −0.571848
\(756\) 0 0
\(757\) −5.71281 −0.207636 −0.103818 0.994596i \(-0.533106\pi\)
−0.103818 + 0.994596i \(0.533106\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.26795 −0.263463 −0.131731 0.991285i \(-0.542054\pi\)
−0.131731 + 0.991285i \(0.542054\pi\)
\(762\) 0 0
\(763\) −15.8564 −0.574040
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −32.0000 −1.15545
\(768\) 0 0
\(769\) 39.4641 1.42311 0.711556 0.702629i \(-0.247991\pi\)
0.711556 + 0.702629i \(0.247991\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.60770 0.345565 0.172782 0.984960i \(-0.444724\pi\)
0.172782 + 0.984960i \(0.444724\pi\)
\(774\) 0 0
\(775\) 15.4641 0.555487
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.4641 −0.410744
\(780\) 0 0
\(781\) 21.4641 0.768046
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.5359 0.376042
\(786\) 0 0
\(787\) 16.5359 0.589441 0.294721 0.955583i \(-0.404773\pi\)
0.294721 + 0.955583i \(0.404773\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.73205 −0.0971405
\(792\) 0 0
\(793\) 18.1436 0.644298
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.2487 −1.28400 −0.641998 0.766707i \(-0.721894\pi\)
−0.641998 + 0.766707i \(0.721894\pi\)
\(798\) 0 0
\(799\) 18.4974 0.654392
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.85641 0.0655112
\(804\) 0 0
\(805\) −0.392305 −0.0138269
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.9282 1.36864 0.684321 0.729181i \(-0.260099\pi\)
0.684321 + 0.729181i \(0.260099\pi\)
\(810\) 0 0
\(811\) −27.7128 −0.973128 −0.486564 0.873645i \(-0.661750\pi\)
−0.486564 + 0.873645i \(0.661750\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.32051 −0.256426
\(816\) 0 0
\(817\) −8.92820 −0.312358
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.3205 0.674290 0.337145 0.941453i \(-0.390539\pi\)
0.337145 + 0.941453i \(0.390539\pi\)
\(822\) 0 0
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.5167 −1.58277 −0.791385 0.611318i \(-0.790639\pi\)
−0.791385 + 0.611318i \(0.790639\pi\)
\(828\) 0 0
\(829\) 39.0718 1.35702 0.678510 0.734591i \(-0.262626\pi\)
0.678510 + 0.734591i \(0.262626\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.26795 −0.113228
\(834\) 0 0
\(835\) 3.21539 0.111273
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.7846 1.13185 0.565925 0.824457i \(-0.308519\pi\)
0.565925 + 0.824457i \(0.308519\pi\)
\(840\) 0 0
\(841\) −21.5359 −0.742617
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.19615 0.0755499
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −19.0718 −0.653006 −0.326503 0.945196i \(-0.605870\pi\)
−0.326503 + 0.945196i \(0.605870\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.3923 −0.491632 −0.245816 0.969317i \(-0.579056\pi\)
−0.245816 + 0.969317i \(0.579056\pi\)
\(858\) 0 0
\(859\) −40.7846 −1.39155 −0.695776 0.718258i \(-0.744940\pi\)
−0.695776 + 0.718258i \(0.744940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.0526 0.546435 0.273218 0.961952i \(-0.411912\pi\)
0.273218 + 0.961952i \(0.411912\pi\)
\(864\) 0 0
\(865\) 7.60770 0.258669
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.9282 −0.642095
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) 13.2154 0.446252 0.223126 0.974790i \(-0.428374\pi\)
0.223126 + 0.974790i \(0.428374\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.5167 −0.590151 −0.295076 0.955474i \(-0.595345\pi\)
−0.295076 + 0.955474i \(0.595345\pi\)
\(882\) 0 0
\(883\) −39.7128 −1.33644 −0.668221 0.743963i \(-0.732944\pi\)
−0.668221 + 0.743963i \(0.732944\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.1051 1.21229 0.606146 0.795354i \(-0.292715\pi\)
0.606146 + 0.795354i \(0.292715\pi\)
\(888\) 0 0
\(889\) −2.92820 −0.0982088
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5.66025 −0.189413
\(894\) 0 0
\(895\) −18.6795 −0.624387
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9.46410 −0.315645
\(900\) 0 0
\(901\) 26.7846 0.892325
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −17.4641 −0.580526
\(906\) 0 0
\(907\) 38.6410 1.28305 0.641527 0.767101i \(-0.278301\pi\)
0.641527 + 0.767101i \(0.278301\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.0526 −1.59205 −0.796026 0.605262i \(-0.793068\pi\)
−0.796026 + 0.605262i \(0.793068\pi\)
\(912\) 0 0
\(913\) 0.679492 0.0224879
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.19615 0.270661
\(918\) 0 0
\(919\) 13.8564 0.457081 0.228540 0.973534i \(-0.426605\pi\)
0.228540 + 0.973534i \(0.426605\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.7846 0.815795
\(924\) 0 0
\(925\) 33.3205 1.09557
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.8038 0.715361 0.357681 0.933844i \(-0.383568\pi\)
0.357681 + 0.933844i \(0.383568\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.28719 −0.271020
\(936\) 0 0
\(937\) 39.8564 1.30205 0.651026 0.759055i \(-0.274339\pi\)
0.651026 + 0.759055i \(0.274339\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −40.6410 −1.32486 −0.662430 0.749124i \(-0.730475\pi\)
−0.662430 + 0.749124i \(0.730475\pi\)
\(942\) 0 0
\(943\) −6.14359 −0.200063
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −57.0333 −1.85333 −0.926667 0.375883i \(-0.877339\pi\)
−0.926667 + 0.375883i \(0.877339\pi\)
\(948\) 0 0
\(949\) 2.14359 0.0695840
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.41154 −0.110511 −0.0552554 0.998472i \(-0.517597\pi\)
−0.0552554 + 0.998472i \(0.517597\pi\)
\(954\) 0 0
\(955\) 13.4641 0.435688
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.9282 −0.352890
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.6795 0.408167
\(966\) 0 0
\(967\) −38.7846 −1.24723 −0.623614 0.781732i \(-0.714336\pi\)
−0.623614 + 0.781732i \(0.714336\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 41.0718 1.31806 0.659028 0.752118i \(-0.270968\pi\)
0.659028 + 0.752118i \(0.270968\pi\)
\(972\) 0 0
\(973\) −2.53590 −0.0812972
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.12436 −0.0999570 −0.0499785 0.998750i \(-0.515915\pi\)
−0.0499785 + 0.998750i \(0.515915\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.7846 0.407766 0.203883 0.978995i \(-0.434644\pi\)
0.203883 + 0.978995i \(0.434644\pi\)
\(984\) 0 0
\(985\) −5.07180 −0.161601
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.78461 −0.152142
\(990\) 0 0
\(991\) −52.7846 −1.67676 −0.838379 0.545087i \(-0.816496\pi\)
−0.838379 + 0.545087i \(0.816496\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.2872 −0.389530
\(996\) 0 0
\(997\) −46.1051 −1.46016 −0.730082 0.683360i \(-0.760518\pi\)
−0.730082 + 0.683360i \(0.760518\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9576.2.a.bf.1.2 2
3.2 odd 2 9576.2.a.br.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bf.1.2 2 1.1 even 1 trivial
9576.2.a.br.1.1 yes 2 3.2 odd 2