Properties

Label 3744.2.a.y.1.1
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
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] = [3744,2,Mod(1,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, 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("3744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.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: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.561553 q^{5} -0.561553 q^{7} +O(q^{10})\) \(q-0.561553 q^{5} -0.561553 q^{7} -2.00000 q^{11} -1.00000 q^{13} +0.561553 q^{17} +6.00000 q^{19} -4.68466 q^{25} +8.24621 q^{29} +7.12311 q^{31} +0.315342 q^{35} -9.68466 q^{37} -7.12311 q^{41} -8.80776 q^{43} -1.68466 q^{47} -6.68466 q^{49} +4.87689 q^{53} +1.12311 q^{55} +6.00000 q^{59} +13.3693 q^{61} +0.561553 q^{65} +6.00000 q^{67} +1.68466 q^{71} +10.0000 q^{73} +1.12311 q^{77} +12.0000 q^{79} +17.3693 q^{83} -0.315342 q^{85} +8.24621 q^{89} +0.561553 q^{91} -3.36932 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} + 3 q^{7} - 4 q^{11} - 2 q^{13} - 3 q^{17} + 12 q^{19} + 3 q^{25} + 6 q^{31} + 13 q^{35} - 7 q^{37} - 6 q^{41} + 3 q^{43} + 9 q^{47} - q^{49} + 18 q^{53} - 6 q^{55} + 12 q^{59} + 2 q^{61} - 3 q^{65} + 12 q^{67} - 9 q^{71} + 20 q^{73} - 6 q^{77} + 24 q^{79} + 10 q^{83} - 13 q^{85} - 3 q^{91} + 18 q^{95} - 12 q^{97}+O(q^{100}) \) 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
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.561553 0.136197 0.0680983 0.997679i \(-0.478307\pi\)
0.0680983 + 0.997679i \(0.478307\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) 7.12311 1.27935 0.639674 0.768647i \(-0.279069\pi\)
0.639674 + 0.768647i \(0.279069\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.315342 0.0533025
\(36\) 0 0
\(37\) −9.68466 −1.59215 −0.796074 0.605199i \(-0.793093\pi\)
−0.796074 + 0.605199i \(0.793093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) −8.80776 −1.34317 −0.671586 0.740927i \(-0.734386\pi\)
−0.671586 + 0.740927i \(0.734386\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.68466 −0.245733 −0.122866 0.992423i \(-0.539209\pi\)
−0.122866 + 0.992423i \(0.539209\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.87689 0.669893 0.334946 0.942237i \(-0.391282\pi\)
0.334946 + 0.942237i \(0.391282\pi\)
\(54\) 0 0
\(55\) 1.12311 0.151440
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 13.3693 1.71177 0.855883 0.517170i \(-0.173014\pi\)
0.855883 + 0.517170i \(0.173014\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.561553 0.0696521
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.68466 0.199932 0.0999661 0.994991i \(-0.468127\pi\)
0.0999661 + 0.994991i \(0.468127\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.12311 0.127990
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.3693 1.90653 0.953265 0.302135i \(-0.0976994\pi\)
0.953265 + 0.302135i \(0.0976994\pi\)
\(84\) 0 0
\(85\) −0.315342 −0.0342036
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.24621 0.874097 0.437048 0.899438i \(-0.356024\pi\)
0.437048 + 0.899438i \(0.356024\pi\)
\(90\) 0 0
\(91\) 0.561553 0.0588667
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.36932 −0.345685
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.12311 0.708776 0.354388 0.935099i \(-0.384689\pi\)
0.354388 + 0.935099i \(0.384689\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −1.68466 −0.161361 −0.0806805 0.996740i \(-0.525709\pi\)
−0.0806805 + 0.996740i \(0.525709\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.24621 −0.775738 −0.387869 0.921714i \(-0.626789\pi\)
−0.387869 + 0.921714i \(0.626789\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.315342 −0.0289073
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) 2.24621 0.199319 0.0996595 0.995022i \(-0.468225\pi\)
0.0996595 + 0.995022i \(0.468225\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.6847 −1.37037 −0.685187 0.728367i \(-0.740280\pi\)
−0.685187 + 0.728367i \(0.740280\pi\)
\(132\) 0 0
\(133\) −3.36932 −0.292157
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.87689 −0.416661 −0.208331 0.978058i \(-0.566803\pi\)
−0.208331 + 0.978058i \(0.566803\pi\)
\(138\) 0 0
\(139\) 7.68466 0.651804 0.325902 0.945404i \(-0.394332\pi\)
0.325902 + 0.945404i \(0.394332\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −4.63068 −0.384557
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.24621 0.675556 0.337778 0.941226i \(-0.390325\pi\)
0.337778 + 0.941226i \(0.390325\pi\)
\(150\) 0 0
\(151\) 10.3153 0.839451 0.419725 0.907651i \(-0.362126\pi\)
0.419725 + 0.907651i \(0.362126\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.36932 0.733862 0.366931 0.930248i \(-0.380409\pi\)
0.366931 + 0.930248i \(0.380409\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.36932 0.725020 0.362510 0.931980i \(-0.381920\pi\)
0.362510 + 0.931980i \(0.381920\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.3693 −1.62468 −0.812340 0.583185i \(-0.801806\pi\)
−0.812340 + 0.583185i \(0.801806\pi\)
\(174\) 0 0
\(175\) 2.63068 0.198861
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.0540 1.42416 0.712080 0.702098i \(-0.247753\pi\)
0.712080 + 0.702098i \(0.247753\pi\)
\(180\) 0 0
\(181\) 17.3693 1.29105 0.645526 0.763739i \(-0.276638\pi\)
0.645526 + 0.763739i \(0.276638\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.43845 0.399843
\(186\) 0 0
\(187\) −1.12311 −0.0821296
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3693 1.40151 0.700757 0.713400i \(-0.252846\pi\)
0.700757 + 0.713400i \(0.252846\pi\)
\(192\) 0 0
\(193\) 1.36932 0.0985656 0.0492828 0.998785i \(-0.484306\pi\)
0.0492828 + 0.998785i \(0.484306\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.6847 −0.974992 −0.487496 0.873125i \(-0.662090\pi\)
−0.487496 + 0.873125i \(0.662090\pi\)
\(198\) 0 0
\(199\) 15.3693 1.08950 0.544751 0.838598i \(-0.316624\pi\)
0.544751 + 0.838598i \(0.316624\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.63068 −0.325010
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 3.19224 0.219763 0.109881 0.993945i \(-0.464953\pi\)
0.109881 + 0.993945i \(0.464953\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.94602 0.337316
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.561553 −0.0377741
\(222\) 0 0
\(223\) 26.8078 1.79518 0.897590 0.440831i \(-0.145316\pi\)
0.897590 + 0.440831i \(0.145316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.3693 −1.41833 −0.709166 0.705042i \(-0.750928\pi\)
−0.709166 + 0.705042i \(0.750928\pi\)
\(228\) 0 0
\(229\) 13.6847 0.904308 0.452154 0.891940i \(-0.350656\pi\)
0.452154 + 0.891940i \(0.350656\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.6847 −0.896512 −0.448256 0.893905i \(-0.647955\pi\)
−0.448256 + 0.893905i \(0.647955\pi\)
\(234\) 0 0
\(235\) 0.946025 0.0617118
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −21.6847 −1.40266 −0.701332 0.712835i \(-0.747411\pi\)
−0.701332 + 0.712835i \(0.747411\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.75379 0.239821
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.9309 −0.993740 −0.496870 0.867825i \(-0.665517\pi\)
−0.496870 + 0.867825i \(0.665517\pi\)
\(258\) 0 0
\(259\) 5.43845 0.337929
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −26.7386 −1.64877 −0.824387 0.566026i \(-0.808480\pi\)
−0.824387 + 0.566026i \(0.808480\pi\)
\(264\) 0 0
\(265\) −2.73863 −0.168233
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.3693 1.30291 0.651455 0.758687i \(-0.274159\pi\)
0.651455 + 0.758687i \(0.274159\pi\)
\(270\) 0 0
\(271\) 1.68466 0.102336 0.0511679 0.998690i \(-0.483706\pi\)
0.0511679 + 0.998690i \(0.483706\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.36932 0.564991
\(276\) 0 0
\(277\) −1.36932 −0.0822743 −0.0411371 0.999154i \(-0.513098\pi\)
−0.0411371 + 0.999154i \(0.513098\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.2462 −1.20779 −0.603894 0.797065i \(-0.706385\pi\)
−0.603894 + 0.797065i \(0.706385\pi\)
\(282\) 0 0
\(283\) 2.24621 0.133523 0.0667617 0.997769i \(-0.478733\pi\)
0.0667617 + 0.997769i \(0.478733\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 23.4384 1.36929 0.684644 0.728877i \(-0.259958\pi\)
0.684644 + 0.728877i \(0.259958\pi\)
\(294\) 0 0
\(295\) −3.36932 −0.196169
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.94602 0.285084
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.50758 −0.429883
\(306\) 0 0
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.3693 0.871514 0.435757 0.900064i \(-0.356481\pi\)
0.435757 + 0.900064i \(0.356481\pi\)
\(312\) 0 0
\(313\) 17.0540 0.963948 0.481974 0.876186i \(-0.339920\pi\)
0.481974 + 0.876186i \(0.339920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.2462 1.81113 0.905564 0.424210i \(-0.139448\pi\)
0.905564 + 0.424210i \(0.139448\pi\)
\(318\) 0 0
\(319\) −16.4924 −0.923398
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.36932 0.187474
\(324\) 0 0
\(325\) 4.68466 0.259858
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.946025 0.0521560
\(330\) 0 0
\(331\) 21.3693 1.17456 0.587282 0.809382i \(-0.300198\pi\)
0.587282 + 0.809382i \(0.300198\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.36932 −0.184085
\(336\) 0 0
\(337\) 1.05398 0.0574137 0.0287068 0.999588i \(-0.490861\pi\)
0.0287068 + 0.999588i \(0.490861\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.2462 −0.771476
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.3153 0.875853 0.437927 0.899011i \(-0.355713\pi\)
0.437927 + 0.899011i \(0.355713\pi\)
\(348\) 0 0
\(349\) 29.0540 1.55522 0.777612 0.628745i \(-0.216431\pi\)
0.777612 + 0.628745i \(0.216431\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.8617 1.37648 0.688241 0.725482i \(-0.258383\pi\)
0.688241 + 0.725482i \(0.258383\pi\)
\(354\) 0 0
\(355\) −0.946025 −0.0502098
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.3693 0.916717 0.458359 0.888767i \(-0.348437\pi\)
0.458359 + 0.888767i \(0.348437\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.61553 −0.293930
\(366\) 0 0
\(367\) −27.3693 −1.42867 −0.714333 0.699806i \(-0.753270\pi\)
−0.714333 + 0.699806i \(0.753270\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.73863 −0.142183
\(372\) 0 0
\(373\) −13.3693 −0.692237 −0.346118 0.938191i \(-0.612500\pi\)
−0.346118 + 0.938191i \(0.612500\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.24621 −0.424701
\(378\) 0 0
\(379\) 16.8769 0.866908 0.433454 0.901176i \(-0.357295\pi\)
0.433454 + 0.901176i \(0.357295\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.68466 0.0860820 0.0430410 0.999073i \(-0.486295\pi\)
0.0430410 + 0.999073i \(0.486295\pi\)
\(384\) 0 0
\(385\) −0.630683 −0.0321426
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.2462 −1.02652 −0.513262 0.858232i \(-0.671563\pi\)
−0.513262 + 0.858232i \(0.671563\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.73863 −0.339057
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −31.1231 −1.55421 −0.777107 0.629369i \(-0.783314\pi\)
−0.777107 + 0.629369i \(0.783314\pi\)
\(402\) 0 0
\(403\) −7.12311 −0.354827
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.3693 0.960101
\(408\) 0 0
\(409\) −6.63068 −0.327866 −0.163933 0.986471i \(-0.552418\pi\)
−0.163933 + 0.986471i \(0.552418\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.36932 −0.165793
\(414\) 0 0
\(415\) −9.75379 −0.478795
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.0540 −0.735435 −0.367717 0.929938i \(-0.619861\pi\)
−0.367717 + 0.929938i \(0.619861\pi\)
\(420\) 0 0
\(421\) −33.6847 −1.64169 −0.820845 0.571151i \(-0.806497\pi\)
−0.820845 + 0.571151i \(0.806497\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.63068 −0.127607
\(426\) 0 0
\(427\) −7.50758 −0.363317
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.68466 −0.273820 −0.136910 0.990583i \(-0.543717\pi\)
−0.136910 + 0.990583i \(0.543717\pi\)
\(432\) 0 0
\(433\) 14.3153 0.687951 0.343976 0.938979i \(-0.388226\pi\)
0.343976 + 0.938979i \(0.388226\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −41.6155 −1.98620 −0.993100 0.117267i \(-0.962587\pi\)
−0.993100 + 0.117267i \(0.962587\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.68466 0.365109 0.182555 0.983196i \(-0.441563\pi\)
0.182555 + 0.983196i \(0.441563\pi\)
\(444\) 0 0
\(445\) −4.63068 −0.219515
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 14.2462 0.670828
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.315342 −0.0147834
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.1922 0.987021 0.493510 0.869740i \(-0.335713\pi\)
0.493510 + 0.869740i \(0.335713\pi\)
\(462\) 0 0
\(463\) 23.6155 1.09751 0.548753 0.835984i \(-0.315103\pi\)
0.548753 + 0.835984i \(0.315103\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.7386 −0.682023 −0.341011 0.940059i \(-0.610769\pi\)
−0.341011 + 0.940059i \(0.610769\pi\)
\(468\) 0 0
\(469\) −3.36932 −0.155581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.6155 0.809963
\(474\) 0 0
\(475\) −28.1080 −1.28968
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −33.6847 −1.53909 −0.769546 0.638592i \(-0.779517\pi\)
−0.769546 + 0.638592i \(0.779517\pi\)
\(480\) 0 0
\(481\) 9.68466 0.441582
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.36932 0.152993
\(486\) 0 0
\(487\) −19.1231 −0.866551 −0.433275 0.901262i \(-0.642642\pi\)
−0.433275 + 0.901262i \(0.642642\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.6847 −1.61043 −0.805213 0.592986i \(-0.797949\pi\)
−0.805213 + 0.592986i \(0.797949\pi\)
\(492\) 0 0
\(493\) 4.63068 0.208555
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.946025 −0.0424350
\(498\) 0 0
\(499\) 16.8769 0.755514 0.377757 0.925905i \(-0.376696\pi\)
0.377757 + 0.925905i \(0.376696\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.63068 0.206472 0.103236 0.994657i \(-0.467080\pi\)
0.103236 + 0.994657i \(0.467080\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.4924 0.465068 0.232534 0.972588i \(-0.425298\pi\)
0.232534 + 0.972588i \(0.425298\pi\)
\(510\) 0 0
\(511\) −5.61553 −0.248416
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.36932 0.148182
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.5616 −0.550332 −0.275166 0.961397i \(-0.588733\pi\)
−0.275166 + 0.961397i \(0.588733\pi\)
\(522\) 0 0
\(523\) −40.4924 −1.77061 −0.885305 0.465011i \(-0.846050\pi\)
−0.885305 + 0.465011i \(0.846050\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.12311 0.308536
\(534\) 0 0
\(535\) −6.73863 −0.291337
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.3693 0.575857
\(540\) 0 0
\(541\) −18.3153 −0.787438 −0.393719 0.919231i \(-0.628812\pi\)
−0.393719 + 0.919231i \(0.628812\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.946025 0.0405232
\(546\) 0 0
\(547\) 23.0540 0.985717 0.492858 0.870110i \(-0.335952\pi\)
0.492858 + 0.870110i \(0.335952\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 49.4773 2.10780
\(552\) 0 0
\(553\) −6.73863 −0.286556
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.3002 1.83469 0.917344 0.398096i \(-0.130329\pi\)
0.917344 + 0.398096i \(0.130329\pi\)
\(558\) 0 0
\(559\) 8.80776 0.372529
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −43.6847 −1.84109 −0.920544 0.390638i \(-0.872254\pi\)
−0.920544 + 0.390638i \(0.872254\pi\)
\(564\) 0 0
\(565\) 4.63068 0.194814
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.6847 1.07676 0.538378 0.842703i \(-0.319037\pi\)
0.538378 + 0.842703i \(0.319037\pi\)
\(570\) 0 0
\(571\) −17.4384 −0.729776 −0.364888 0.931051i \(-0.618893\pi\)
−0.364888 + 0.931051i \(0.618893\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.73863 −0.197272 −0.0986360 0.995124i \(-0.531448\pi\)
−0.0986360 + 0.995124i \(0.531448\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.75379 −0.404655
\(582\) 0 0
\(583\) −9.75379 −0.403961
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.36932 0.386713 0.193357 0.981129i \(-0.438063\pi\)
0.193357 + 0.981129i \(0.438063\pi\)
\(588\) 0 0
\(589\) 42.7386 1.76101
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20.2462 −0.831412 −0.415706 0.909499i \(-0.636466\pi\)
−0.415706 + 0.909499i \(0.636466\pi\)
\(594\) 0 0
\(595\) 0.177081 0.00725961
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −35.3693 −1.44515 −0.722576 0.691292i \(-0.757042\pi\)
−0.722576 + 0.691292i \(0.757042\pi\)
\(600\) 0 0
\(601\) −21.0540 −0.858810 −0.429405 0.903112i \(-0.641277\pi\)
−0.429405 + 0.903112i \(0.641277\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.93087 0.159813
\(606\) 0 0
\(607\) 31.8617 1.29323 0.646614 0.762817i \(-0.276184\pi\)
0.646614 + 0.762817i \(0.276184\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.68466 0.0681540
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.6155 0.950725 0.475363 0.879790i \(-0.342317\pi\)
0.475363 + 0.879790i \(0.342317\pi\)
\(618\) 0 0
\(619\) 9.36932 0.376585 0.188292 0.982113i \(-0.439705\pi\)
0.188292 + 0.982113i \(0.439705\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.63068 −0.185524
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.43845 −0.216845
\(630\) 0 0
\(631\) 41.0540 1.63433 0.817166 0.576402i \(-0.195544\pi\)
0.817166 + 0.576402i \(0.195544\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.26137 −0.0500558
\(636\) 0 0
\(637\) 6.68466 0.264856
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.4924 −0.414426 −0.207213 0.978296i \(-0.566439\pi\)
−0.207213 + 0.978296i \(0.566439\pi\)
\(642\) 0 0
\(643\) −18.0000 −0.709851 −0.354925 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.1080 1.18367 0.591833 0.806061i \(-0.298405\pi\)
0.591833 + 0.806061i \(0.298405\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 47.6155 1.86334 0.931670 0.363306i \(-0.118352\pi\)
0.931670 + 0.363306i \(0.118352\pi\)
\(654\) 0 0
\(655\) 8.80776 0.344148
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −20.7386 −0.806639 −0.403320 0.915059i \(-0.632144\pi\)
−0.403320 + 0.915059i \(0.632144\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.89205 0.0733705
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −26.7386 −1.03223
\(672\) 0 0
\(673\) −21.0540 −0.811571 −0.405786 0.913968i \(-0.633002\pi\)
−0.405786 + 0.913968i \(0.633002\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.3693 −0.821290 −0.410645 0.911795i \(-0.634696\pi\)
−0.410645 + 0.911795i \(0.634696\pi\)
\(678\) 0 0
\(679\) 3.36932 0.129303
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.36932 0.205451 0.102726 0.994710i \(-0.467244\pi\)
0.102726 + 0.994710i \(0.467244\pi\)
\(684\) 0 0
\(685\) 2.73863 0.104638
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.87689 −0.185795
\(690\) 0 0
\(691\) −4.87689 −0.185526 −0.0927629 0.995688i \(-0.529570\pi\)
−0.0927629 + 0.995688i \(0.529570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.31534 −0.163690
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.36932 −0.353874 −0.176937 0.984222i \(-0.556619\pi\)
−0.176937 + 0.984222i \(0.556619\pi\)
\(702\) 0 0
\(703\) −58.1080 −2.19158
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 16.7386 0.628633 0.314316 0.949318i \(-0.398225\pi\)
0.314316 + 0.949318i \(0.398225\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.12311 −0.0420018
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.3693 0.573179 0.286589 0.958054i \(-0.407478\pi\)
0.286589 + 0.958054i \(0.407478\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −38.6307 −1.43471
\(726\) 0 0
\(727\) −41.6155 −1.54343 −0.771717 0.635966i \(-0.780602\pi\)
−0.771717 + 0.635966i \(0.780602\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.94602 −0.182935
\(732\) 0 0
\(733\) −25.0540 −0.925390 −0.462695 0.886518i \(-0.653117\pi\)
−0.462695 + 0.886518i \(0.653117\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −52.1080 −1.91682 −0.958411 0.285392i \(-0.907876\pi\)
−0.958411 + 0.285392i \(0.907876\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.6847 −0.502041 −0.251021 0.967982i \(-0.580766\pi\)
−0.251021 + 0.967982i \(0.580766\pi\)
\(744\) 0 0
\(745\) −4.63068 −0.169655
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.73863 −0.246224
\(750\) 0 0
\(751\) −26.2462 −0.957738 −0.478869 0.877886i \(-0.658953\pi\)
−0.478869 + 0.877886i \(0.658953\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.79261 −0.210815
\(756\) 0 0
\(757\) −9.36932 −0.340534 −0.170267 0.985398i \(-0.554463\pi\)
−0.170267 + 0.985398i \(0.554463\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0.946025 0.0342484
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 9.36932 0.337866 0.168933 0.985628i \(-0.445968\pi\)
0.168933 + 0.985628i \(0.445968\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.6847 0.923813 0.461906 0.886929i \(-0.347166\pi\)
0.461906 + 0.886929i \(0.347166\pi\)
\(774\) 0 0
\(775\) −33.3693 −1.19866
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −42.7386 −1.53127
\(780\) 0 0
\(781\) −3.36932 −0.120564
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.12311 −0.0400854
\(786\) 0 0
\(787\) 35.6155 1.26956 0.634778 0.772694i \(-0.281091\pi\)
0.634778 + 0.772694i \(0.281091\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.63068 0.164648
\(792\) 0 0
\(793\) −13.3693 −0.474758
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.7386 0.451226 0.225613 0.974217i \(-0.427562\pi\)
0.225613 + 0.974217i \(0.427562\pi\)
\(798\) 0 0
\(799\) −0.946025 −0.0334679
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39.9309 1.40389 0.701947 0.712229i \(-0.252314\pi\)
0.701947 + 0.712229i \(0.252314\pi\)
\(810\) 0 0
\(811\) 25.8617 0.908128 0.454064 0.890969i \(-0.349974\pi\)
0.454064 + 0.890969i \(0.349974\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.26137 −0.184298
\(816\) 0 0
\(817\) −52.8466 −1.84887
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.05398 −0.176385 −0.0881925 0.996103i \(-0.528109\pi\)
−0.0881925 + 0.996103i \(0.528109\pi\)
\(822\) 0 0
\(823\) −39.3693 −1.37233 −0.686164 0.727447i \(-0.740707\pi\)
−0.686164 + 0.727447i \(0.740707\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.7386 1.27753 0.638764 0.769403i \(-0.279446\pi\)
0.638764 + 0.769403i \(0.279446\pi\)
\(828\) 0 0
\(829\) −32.7386 −1.13706 −0.568530 0.822663i \(-0.692488\pi\)
−0.568530 + 0.822663i \(0.692488\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.75379 −0.130061
\(834\) 0 0
\(835\) −5.26137 −0.182077
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36.1080 −1.24658 −0.623292 0.781989i \(-0.714205\pi\)
−0.623292 + 0.781989i \(0.714205\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.561553 −0.0193180
\(846\) 0 0
\(847\) 3.93087 0.135066
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 21.6847 0.742469 0.371234 0.928539i \(-0.378935\pi\)
0.371234 + 0.928539i \(0.378935\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.0540 0.852847 0.426424 0.904524i \(-0.359773\pi\)
0.426424 + 0.904524i \(0.359773\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −6.00000 −0.203302
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.05398 −0.103243
\(876\) 0 0
\(877\) −31.7926 −1.07356 −0.536780 0.843722i \(-0.680360\pi\)
−0.536780 + 0.843722i \(0.680360\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.3002 0.650240 0.325120 0.945673i \(-0.394595\pi\)
0.325120 + 0.945673i \(0.394595\pi\)
\(882\) 0 0
\(883\) −17.4384 −0.586850 −0.293425 0.955982i \(-0.594795\pi\)
−0.293425 + 0.955982i \(0.594795\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) −1.26137 −0.0423049
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.1080 −0.338250
\(894\) 0 0
\(895\) −10.6998 −0.357655
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 58.7386 1.95904
\(900\) 0 0
\(901\) 2.73863 0.0912371
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.75379 −0.324227
\(906\) 0 0
\(907\) 26.4233 0.877371 0.438686 0.898641i \(-0.355444\pi\)
0.438686 + 0.898641i \(0.355444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 27.3693 0.906786 0.453393 0.891311i \(-0.350213\pi\)
0.453393 + 0.891311i \(0.350213\pi\)
\(912\) 0 0
\(913\) −34.7386 −1.14968
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.80776 0.290858
\(918\) 0 0
\(919\) 33.7538 1.11343 0.556717 0.830702i \(-0.312061\pi\)
0.556717 + 0.830702i \(0.312061\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.68466 −0.0554512
\(924\) 0 0
\(925\) 45.3693 1.49173
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35.6155 −1.16851 −0.584254 0.811571i \(-0.698613\pi\)
−0.584254 + 0.811571i \(0.698613\pi\)
\(930\) 0 0
\(931\) −40.1080 −1.31448
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.630683 0.0206255
\(936\) 0 0
\(937\) 40.7386 1.33087 0.665437 0.746454i \(-0.268245\pi\)
0.665437 + 0.746454i \(0.268245\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −37.6847 −1.22848 −0.614242 0.789117i \(-0.710538\pi\)
−0.614242 + 0.789117i \(0.710538\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.1922 0.686484 0.343242 0.939247i \(-0.388475\pi\)
0.343242 + 0.939247i \(0.388475\pi\)
\(954\) 0 0
\(955\) −10.8769 −0.351968
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.73863 0.0884351
\(960\) 0 0
\(961\) 19.7386 0.636730
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.768944 −0.0247532
\(966\) 0 0
\(967\) −8.42329 −0.270875 −0.135437 0.990786i \(-0.543244\pi\)
−0.135437 + 0.990786i \(0.543244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.6847 0.503345 0.251672 0.967813i \(-0.419019\pi\)
0.251672 + 0.967813i \(0.419019\pi\)
\(972\) 0 0
\(973\) −4.31534 −0.138343
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.9848 −1.63115 −0.815575 0.578652i \(-0.803579\pi\)
−0.815575 + 0.578652i \(0.803579\pi\)
\(978\) 0 0
\(979\) −16.4924 −0.527100
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −40.4233 −1.28930 −0.644651 0.764477i \(-0.722997\pi\)
−0.644651 + 0.764477i \(0.722997\pi\)
\(984\) 0 0
\(985\) 7.68466 0.244854
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 3.36932 0.107030 0.0535149 0.998567i \(-0.482958\pi\)
0.0535149 + 0.998567i \(0.482958\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.63068 −0.273611
\(996\) 0 0
\(997\) 44.7386 1.41689 0.708443 0.705768i \(-0.249398\pi\)
0.708443 + 0.705768i \(0.249398\pi\)
\(998\) 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 3744.2.a.y.1.1 2
3.2 odd 2 416.2.a.e.1.2 yes 2
4.3 odd 2 3744.2.a.x.1.1 2
8.3 odd 2 7488.2.a.ce.1.2 2
8.5 even 2 7488.2.a.cf.1.2 2
12.11 even 2 416.2.a.c.1.1 2
24.5 odd 2 832.2.a.l.1.1 2
24.11 even 2 832.2.a.o.1.2 2
39.38 odd 2 5408.2.a.bd.1.2 2
48.5 odd 4 3328.2.b.u.1665.1 4
48.11 even 4 3328.2.b.ba.1665.4 4
48.29 odd 4 3328.2.b.u.1665.4 4
48.35 even 4 3328.2.b.ba.1665.1 4
156.155 even 2 5408.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.c.1.1 2 12.11 even 2
416.2.a.e.1.2 yes 2 3.2 odd 2
832.2.a.l.1.1 2 24.5 odd 2
832.2.a.o.1.2 2 24.11 even 2
3328.2.b.u.1665.1 4 48.5 odd 4
3328.2.b.u.1665.4 4 48.29 odd 4
3328.2.b.ba.1665.1 4 48.35 even 4
3328.2.b.ba.1665.4 4 48.11 even 4
3744.2.a.x.1.1 2 4.3 odd 2
3744.2.a.y.1.1 2 1.1 even 1 trivial
5408.2.a.p.1.1 2 156.155 even 2
5408.2.a.bd.1.2 2 39.38 odd 2
7488.2.a.ce.1.2 2 8.3 odd 2
7488.2.a.cf.1.2 2 8.5 even 2