Properties

Label 9072.2.a.y.1.1
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
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] = [9072,2,Mod(1,9072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, 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("9072.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.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: \(72.4402847137\)
Analytic rank: \(0\)
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: no (minimal twist has level 1134)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.73205 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-3.73205 q^{5} +1.00000 q^{7} -4.19615 q^{11} +0.464102 q^{13} -7.00000 q^{17} +2.73205 q^{19} +6.19615 q^{23} +8.92820 q^{25} -8.46410 q^{29} +2.19615 q^{31} -3.73205 q^{35} -6.66025 q^{37} -9.46410 q^{41} -5.46410 q^{43} +1.26795 q^{47} +1.00000 q^{49} +2.53590 q^{53} +15.6603 q^{55} -6.19615 q^{59} -9.92820 q^{61} -1.73205 q^{65} +3.26795 q^{67} +13.4641 q^{71} +11.7321 q^{73} -4.19615 q^{77} -15.1244 q^{79} -14.5885 q^{83} +26.1244 q^{85} +3.92820 q^{89} +0.464102 q^{91} -10.1962 q^{95} -2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} - 14 q^{17} + 2 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} - 12 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} + 12 q^{53} + 14 q^{55} - 2 q^{59} - 6 q^{61} + 10 q^{67} + 20 q^{71} + 20 q^{73} + 2 q^{77} - 6 q^{79} + 2 q^{83} + 28 q^{85} - 6 q^{89} - 6 q^{91} - 10 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(5\) −3.73205 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.19615 −1.26519 −0.632594 0.774484i \(-0.718010\pi\)
−0.632594 + 0.774484i \(0.718010\pi\)
\(12\) 0 0
\(13\) 0.464102 0.128719 0.0643593 0.997927i \(-0.479500\pi\)
0.0643593 + 0.997927i \(0.479500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.73205 0.626775 0.313388 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.19615 1.29199 0.645994 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.46410 −1.57174 −0.785872 0.618389i \(-0.787786\pi\)
−0.785872 + 0.618389i \(0.787786\pi\)
\(30\) 0 0
\(31\) 2.19615 0.394441 0.197220 0.980359i \(-0.436809\pi\)
0.197220 + 0.980359i \(0.436809\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −6.66025 −1.09494 −0.547470 0.836826i \(-0.684409\pi\)
−0.547470 + 0.836826i \(0.684409\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.46410 −1.47804 −0.739022 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) −5.46410 −0.833268 −0.416634 0.909074i \(-0.636790\pi\)
−0.416634 + 0.909074i \(0.636790\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.26795 0.184949 0.0924747 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.53590 0.348332 0.174166 0.984716i \(-0.444277\pi\)
0.174166 + 0.984716i \(0.444277\pi\)
\(54\) 0 0
\(55\) 15.6603 2.11163
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.19615 −0.806670 −0.403335 0.915052i \(-0.632149\pi\)
−0.403335 + 0.915052i \(0.632149\pi\)
\(60\) 0 0
\(61\) −9.92820 −1.27118 −0.635588 0.772028i \(-0.719242\pi\)
−0.635588 + 0.772028i \(0.719242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) 3.26795 0.399244 0.199622 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4641 1.59789 0.798947 0.601401i \(-0.205391\pi\)
0.798947 + 0.601401i \(0.205391\pi\)
\(72\) 0 0
\(73\) 11.7321 1.37313 0.686566 0.727067i \(-0.259117\pi\)
0.686566 + 0.727067i \(0.259117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.19615 −0.478196
\(78\) 0 0
\(79\) −15.1244 −1.70162 −0.850811 0.525471i \(-0.823889\pi\)
−0.850811 + 0.525471i \(0.823889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.5885 −1.60129 −0.800646 0.599138i \(-0.795510\pi\)
−0.800646 + 0.599138i \(0.795510\pi\)
\(84\) 0 0
\(85\) 26.1244 2.83358
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.92820 0.416389 0.208194 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 0.464102 0.0486511
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.1962 −1.04610
\(96\) 0 0
\(97\) −2.92820 −0.297314 −0.148657 0.988889i \(-0.547495\pi\)
−0.148657 + 0.988889i \(0.547495\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.92820 0.490375 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(102\) 0 0
\(103\) −12.3923 −1.22105 −0.610525 0.791997i \(-0.709042\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −7.19615 −0.689266 −0.344633 0.938737i \(-0.611997\pi\)
−0.344633 + 0.938737i \(0.611997\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.26795 0.213351 0.106675 0.994294i \(-0.465979\pi\)
0.106675 + 0.994294i \(0.465979\pi\)
\(114\) 0 0
\(115\) −23.1244 −2.15636
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 6.60770 0.600700
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −14.6603 −1.31125
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.4641 1.52585 0.762923 0.646490i \(-0.223764\pi\)
0.762923 + 0.646490i \(0.223764\pi\)
\(132\) 0 0
\(133\) 2.73205 0.236899
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.7321 1.00234 0.501168 0.865350i \(-0.332904\pi\)
0.501168 + 0.865350i \(0.332904\pi\)
\(138\) 0 0
\(139\) 6.73205 0.571005 0.285503 0.958378i \(-0.407839\pi\)
0.285503 + 0.958378i \(0.407839\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.94744 −0.162853
\(144\) 0 0
\(145\) 31.5885 2.62328
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 16.1962 1.31802 0.659012 0.752132i \(-0.270975\pi\)
0.659012 + 0.752132i \(0.270975\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.19615 −0.658331
\(156\) 0 0
\(157\) −1.00000 −0.0798087 −0.0399043 0.999204i \(-0.512705\pi\)
−0.0399043 + 0.999204i \(0.512705\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.19615 0.488325
\(162\) 0 0
\(163\) 6.53590 0.511931 0.255966 0.966686i \(-0.417607\pi\)
0.255966 + 0.966686i \(0.417607\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.1962 −0.943767 −0.471883 0.881661i \(-0.656426\pi\)
−0.471883 + 0.881661i \(0.656426\pi\)
\(168\) 0 0
\(169\) −12.7846 −0.983432
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.73205 −0.739914 −0.369957 0.929049i \(-0.620628\pi\)
−0.369957 + 0.929049i \(0.620628\pi\)
\(174\) 0 0
\(175\) 8.92820 0.674909
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.19615 0.612609 0.306305 0.951934i \(-0.400907\pi\)
0.306305 + 0.951934i \(0.400907\pi\)
\(180\) 0 0
\(181\) 4.39230 0.326477 0.163239 0.986587i \(-0.447806\pi\)
0.163239 + 0.986587i \(0.447806\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 24.8564 1.82748
\(186\) 0 0
\(187\) 29.3731 2.14797
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.6603 0.843706 0.421853 0.906664i \(-0.361380\pi\)
0.421853 + 0.906664i \(0.361380\pi\)
\(192\) 0 0
\(193\) −8.85641 −0.637498 −0.318749 0.947839i \(-0.603263\pi\)
−0.318749 + 0.947839i \(0.603263\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.7846 1.83708 0.918539 0.395331i \(-0.129370\pi\)
0.918539 + 0.395331i \(0.129370\pi\)
\(198\) 0 0
\(199\) −5.12436 −0.363256 −0.181628 0.983367i \(-0.558137\pi\)
−0.181628 + 0.983367i \(0.558137\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.46410 −0.594063
\(204\) 0 0
\(205\) 35.3205 2.46689
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.4641 −0.792988
\(210\) 0 0
\(211\) 20.7321 1.42725 0.713627 0.700526i \(-0.247051\pi\)
0.713627 + 0.700526i \(0.247051\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.3923 1.39074
\(216\) 0 0
\(217\) 2.19615 0.149085
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.24871 −0.218532
\(222\) 0 0
\(223\) 18.5359 1.24126 0.620628 0.784105i \(-0.286878\pi\)
0.620628 + 0.784105i \(0.286878\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.07180 −0.336627 −0.168313 0.985734i \(-0.553832\pi\)
−0.168313 + 0.985734i \(0.553832\pi\)
\(228\) 0 0
\(229\) 4.46410 0.294996 0.147498 0.989062i \(-0.452878\pi\)
0.147498 + 0.989062i \(0.452878\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.1962 0.864509 0.432254 0.901752i \(-0.357718\pi\)
0.432254 + 0.901752i \(0.357718\pi\)
\(234\) 0 0
\(235\) −4.73205 −0.308685
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.0526 1.81457 0.907285 0.420517i \(-0.138151\pi\)
0.907285 + 0.420517i \(0.138151\pi\)
\(240\) 0 0
\(241\) −17.7321 −1.14222 −0.571111 0.820873i \(-0.693487\pi\)
−0.571111 + 0.820873i \(0.693487\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.73205 −0.238432
\(246\) 0 0
\(247\) 1.26795 0.0806777
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0526 1.01323 0.506614 0.862173i \(-0.330897\pi\)
0.506614 + 0.862173i \(0.330897\pi\)
\(252\) 0 0
\(253\) −26.0000 −1.63461
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.464102 0.0289499 0.0144749 0.999895i \(-0.495392\pi\)
0.0144749 + 0.999895i \(0.495392\pi\)
\(258\) 0 0
\(259\) −6.66025 −0.413848
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −23.6603 −1.45895 −0.729477 0.684005i \(-0.760236\pi\)
−0.729477 + 0.684005i \(0.760236\pi\)
\(264\) 0 0
\(265\) −9.46410 −0.581375
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.5885 1.56016 0.780078 0.625682i \(-0.215179\pi\)
0.780078 + 0.625682i \(0.215179\pi\)
\(270\) 0 0
\(271\) −25.5167 −1.55003 −0.775013 0.631945i \(-0.782257\pi\)
−0.775013 + 0.631945i \(0.782257\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −37.4641 −2.25917
\(276\) 0 0
\(277\) 22.7846 1.36899 0.684497 0.729015i \(-0.260022\pi\)
0.684497 + 0.729015i \(0.260022\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.80385 0.167264 0.0836318 0.996497i \(-0.473348\pi\)
0.0836318 + 0.996497i \(0.473348\pi\)
\(282\) 0 0
\(283\) 19.3205 1.14848 0.574242 0.818685i \(-0.305297\pi\)
0.574242 + 0.818685i \(0.305297\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.46410 −0.558648
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −20.6603 −1.20698 −0.603492 0.797369i \(-0.706225\pi\)
−0.603492 + 0.797369i \(0.706225\pi\)
\(294\) 0 0
\(295\) 23.1244 1.34635
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.87564 0.166303
\(300\) 0 0
\(301\) −5.46410 −0.314946
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 37.0526 2.12162
\(306\) 0 0
\(307\) −5.85641 −0.334243 −0.167121 0.985936i \(-0.553447\pi\)
−0.167121 + 0.985936i \(0.553447\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.196152 −0.0111228 −0.00556139 0.999985i \(-0.501770\pi\)
−0.00556139 + 0.999985i \(0.501770\pi\)
\(312\) 0 0
\(313\) −5.58846 −0.315878 −0.157939 0.987449i \(-0.550485\pi\)
−0.157939 + 0.987449i \(0.550485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.6077 0.595788 0.297894 0.954599i \(-0.403716\pi\)
0.297894 + 0.954599i \(0.403716\pi\)
\(318\) 0 0
\(319\) 35.5167 1.98855
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −19.1244 −1.06411
\(324\) 0 0
\(325\) 4.14359 0.229845
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.26795 0.0699043
\(330\) 0 0
\(331\) 8.39230 0.461283 0.230641 0.973039i \(-0.425918\pi\)
0.230641 + 0.973039i \(0.425918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.1962 −0.666347
\(336\) 0 0
\(337\) 4.39230 0.239264 0.119632 0.992818i \(-0.461829\pi\)
0.119632 + 0.992818i \(0.461829\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −9.21539 −0.499041
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.5359 0.780328 0.390164 0.920745i \(-0.372418\pi\)
0.390164 + 0.920745i \(0.372418\pi\)
\(348\) 0 0
\(349\) −5.46410 −0.292487 −0.146243 0.989249i \(-0.546718\pi\)
−0.146243 + 0.989249i \(0.546718\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −50.2487 −2.66692
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.92820 0.154545 0.0772723 0.997010i \(-0.475379\pi\)
0.0772723 + 0.997010i \(0.475379\pi\)
\(360\) 0 0
\(361\) −11.5359 −0.607153
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −43.7846 −2.29179
\(366\) 0 0
\(367\) 13.1244 0.685086 0.342543 0.939502i \(-0.388712\pi\)
0.342543 + 0.939502i \(0.388712\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.53590 0.131657
\(372\) 0 0
\(373\) −33.8564 −1.75302 −0.876509 0.481385i \(-0.840134\pi\)
−0.876509 + 0.481385i \(0.840134\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.92820 −0.202313
\(378\) 0 0
\(379\) −17.5167 −0.899770 −0.449885 0.893086i \(-0.648535\pi\)
−0.449885 + 0.893086i \(0.648535\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −35.7128 −1.82484 −0.912420 0.409256i \(-0.865788\pi\)
−0.912420 + 0.409256i \(0.865788\pi\)
\(384\) 0 0
\(385\) 15.6603 0.798120
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.53590 0.331383 0.165692 0.986178i \(-0.447014\pi\)
0.165692 + 0.986178i \(0.447014\pi\)
\(390\) 0 0
\(391\) −43.3731 −2.19347
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 56.4449 2.84005
\(396\) 0 0
\(397\) −21.0000 −1.05396 −0.526980 0.849878i \(-0.676676\pi\)
−0.526980 + 0.849878i \(0.676676\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.5167 1.72368 0.861840 0.507180i \(-0.169312\pi\)
0.861840 + 0.507180i \(0.169312\pi\)
\(402\) 0 0
\(403\) 1.01924 0.0507719
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 27.9474 1.38530
\(408\) 0 0
\(409\) 34.6603 1.71384 0.856920 0.515450i \(-0.172375\pi\)
0.856920 + 0.515450i \(0.172375\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.19615 −0.304893
\(414\) 0 0
\(415\) 54.4449 2.67259
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.53590 −0.123887 −0.0619434 0.998080i \(-0.519730\pi\)
−0.0619434 + 0.998080i \(0.519730\pi\)
\(420\) 0 0
\(421\) −24.1244 −1.17575 −0.587875 0.808952i \(-0.700035\pi\)
−0.587875 + 0.808952i \(0.700035\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −62.4974 −3.03157
\(426\) 0 0
\(427\) −9.92820 −0.480459
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.4641 −1.03389 −0.516945 0.856019i \(-0.672931\pi\)
−0.516945 + 0.856019i \(0.672931\pi\)
\(432\) 0 0
\(433\) 12.2679 0.589560 0.294780 0.955565i \(-0.404754\pi\)
0.294780 + 0.955565i \(0.404754\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.9282 0.809786
\(438\) 0 0
\(439\) 11.3205 0.540298 0.270149 0.962818i \(-0.412927\pi\)
0.270149 + 0.962818i \(0.412927\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.7321 −0.889987 −0.444993 0.895534i \(-0.646794\pi\)
−0.444993 + 0.895534i \(0.646794\pi\)
\(444\) 0 0
\(445\) −14.6603 −0.694963
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.8564 0.559538 0.279769 0.960067i \(-0.409742\pi\)
0.279769 + 0.960067i \(0.409742\pi\)
\(450\) 0 0
\(451\) 39.7128 1.87000
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.73205 −0.0811998
\(456\) 0 0
\(457\) 20.8564 0.975622 0.487811 0.872949i \(-0.337796\pi\)
0.487811 + 0.872949i \(0.337796\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −34.7846 −1.62008 −0.810040 0.586374i \(-0.800555\pi\)
−0.810040 + 0.586374i \(0.800555\pi\)
\(462\) 0 0
\(463\) 32.5885 1.51451 0.757257 0.653117i \(-0.226539\pi\)
0.757257 + 0.653117i \(0.226539\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.5885 0.675073 0.337537 0.941312i \(-0.390406\pi\)
0.337537 + 0.941312i \(0.390406\pi\)
\(468\) 0 0
\(469\) 3.26795 0.150900
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.9282 1.05424
\(474\) 0 0
\(475\) 24.3923 1.11920
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.5167 1.07450 0.537252 0.843422i \(-0.319462\pi\)
0.537252 + 0.843422i \(0.319462\pi\)
\(480\) 0 0
\(481\) −3.09103 −0.140939
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.9282 0.496224
\(486\) 0 0
\(487\) 28.5885 1.29547 0.647733 0.761867i \(-0.275717\pi\)
0.647733 + 0.761867i \(0.275717\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.4641 1.51021 0.755107 0.655602i \(-0.227585\pi\)
0.755107 + 0.655602i \(0.227585\pi\)
\(492\) 0 0
\(493\) 59.2487 2.66843
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.4641 0.603947
\(498\) 0 0
\(499\) −30.1962 −1.35177 −0.675883 0.737009i \(-0.736237\pi\)
−0.675883 + 0.737009i \(0.736237\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.94744 0.0868321 0.0434161 0.999057i \(-0.486176\pi\)
0.0434161 + 0.999057i \(0.486176\pi\)
\(504\) 0 0
\(505\) −18.3923 −0.818447
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.14359 −0.183662 −0.0918308 0.995775i \(-0.529272\pi\)
−0.0918308 + 0.995775i \(0.529272\pi\)
\(510\) 0 0
\(511\) 11.7321 0.518995
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 46.2487 2.03796
\(516\) 0 0
\(517\) −5.32051 −0.233996
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.3731 −0.669661
\(528\) 0 0
\(529\) 15.3923 0.669231
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.39230 −0.190252
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.19615 −0.180741
\(540\) 0 0
\(541\) 20.6603 0.888254 0.444127 0.895964i \(-0.353514\pi\)
0.444127 + 0.895964i \(0.353514\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.8564 1.15040
\(546\) 0 0
\(547\) 19.2679 0.823838 0.411919 0.911220i \(-0.364859\pi\)
0.411919 + 0.911220i \(0.364859\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23.1244 −0.985131
\(552\) 0 0
\(553\) −15.1244 −0.643153
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.0718 −0.426756 −0.213378 0.976970i \(-0.568447\pi\)
−0.213378 + 0.976970i \(0.568447\pi\)
\(558\) 0 0
\(559\) −2.53590 −0.107257
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −35.7128 −1.50512 −0.752558 0.658526i \(-0.771180\pi\)
−0.752558 + 0.658526i \(0.771180\pi\)
\(564\) 0 0
\(565\) −8.46410 −0.356087
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.8038 −0.536765 −0.268383 0.963312i \(-0.586489\pi\)
−0.268383 + 0.963312i \(0.586489\pi\)
\(570\) 0 0
\(571\) 19.2679 0.806339 0.403169 0.915125i \(-0.367909\pi\)
0.403169 + 0.915125i \(0.367909\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 55.3205 2.30702
\(576\) 0 0
\(577\) 7.33975 0.305558 0.152779 0.988260i \(-0.451178\pi\)
0.152779 + 0.988260i \(0.451178\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.5885 −0.605231
\(582\) 0 0
\(583\) −10.6410 −0.440706
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.2679 0.465078 0.232539 0.972587i \(-0.425297\pi\)
0.232539 + 0.972587i \(0.425297\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −40.1769 −1.64987 −0.824934 0.565229i \(-0.808788\pi\)
−0.824934 + 0.565229i \(0.808788\pi\)
\(594\) 0 0
\(595\) 26.1244 1.07099
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.12436 0.372811 0.186406 0.982473i \(-0.440316\pi\)
0.186406 + 0.982473i \(0.440316\pi\)
\(600\) 0 0
\(601\) 8.80385 0.359116 0.179558 0.983747i \(-0.442533\pi\)
0.179558 + 0.983747i \(0.442533\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.6603 −1.00258
\(606\) 0 0
\(607\) −6.58846 −0.267417 −0.133709 0.991021i \(-0.542689\pi\)
−0.133709 + 0.991021i \(0.542689\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.588457 0.0238064
\(612\) 0 0
\(613\) −14.7846 −0.597145 −0.298572 0.954387i \(-0.596510\pi\)
−0.298572 + 0.954387i \(0.596510\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.9808 1.60956 0.804782 0.593570i \(-0.202282\pi\)
0.804782 + 0.593570i \(0.202282\pi\)
\(618\) 0 0
\(619\) −31.7128 −1.27465 −0.637323 0.770597i \(-0.719958\pi\)
−0.637323 + 0.770597i \(0.719958\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.92820 0.157380
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 46.6218 1.85893
\(630\) 0 0
\(631\) −13.6603 −0.543806 −0.271903 0.962325i \(-0.587653\pi\)
−0.271903 + 0.962325i \(0.587653\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 44.7846 1.77722
\(636\) 0 0
\(637\) 0.464102 0.0183884
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19.4449 0.768026 0.384013 0.923328i \(-0.374542\pi\)
0.384013 + 0.923328i \(0.374542\pi\)
\(642\) 0 0
\(643\) 40.5885 1.60065 0.800326 0.599565i \(-0.204660\pi\)
0.800326 + 0.599565i \(0.204660\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.3923 0.644448 0.322224 0.946663i \(-0.395570\pi\)
0.322224 + 0.946663i \(0.395570\pi\)
\(648\) 0 0
\(649\) 26.0000 1.02059
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.2487 −0.714127 −0.357064 0.934080i \(-0.616222\pi\)
−0.357064 + 0.934080i \(0.616222\pi\)
\(654\) 0 0
\(655\) −65.1769 −2.54667
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.6077 −0.607989 −0.303995 0.952674i \(-0.598321\pi\)
−0.303995 + 0.952674i \(0.598321\pi\)
\(660\) 0 0
\(661\) 14.8564 0.577847 0.288924 0.957352i \(-0.406703\pi\)
0.288924 + 0.957352i \(0.406703\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.1962 −0.395390
\(666\) 0 0
\(667\) −52.4449 −2.03067
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 41.6603 1.60828
\(672\) 0 0
\(673\) 16.3205 0.629109 0.314555 0.949239i \(-0.398145\pi\)
0.314555 + 0.949239i \(0.398145\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 0 0
\(679\) −2.92820 −0.112374
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.8564 0.989368 0.494684 0.869073i \(-0.335284\pi\)
0.494684 + 0.869073i \(0.335284\pi\)
\(684\) 0 0
\(685\) −43.7846 −1.67292
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.17691 0.0448369
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −25.1244 −0.953021
\(696\) 0 0
\(697\) 66.2487 2.50935
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.60770 −0.249569 −0.124785 0.992184i \(-0.539824\pi\)
−0.124785 + 0.992184i \(0.539824\pi\)
\(702\) 0 0
\(703\) −18.1962 −0.686281
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.92820 0.185344
\(708\) 0 0
\(709\) −28.1244 −1.05623 −0.528116 0.849172i \(-0.677101\pi\)
−0.528116 + 0.849172i \(0.677101\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.6077 0.509612
\(714\) 0 0
\(715\) 7.26795 0.271806
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.53590 −0.0945731 −0.0472865 0.998881i \(-0.515057\pi\)
−0.0472865 + 0.998881i \(0.515057\pi\)
\(720\) 0 0
\(721\) −12.3923 −0.461514
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −75.5692 −2.80657
\(726\) 0 0
\(727\) −16.6795 −0.618608 −0.309304 0.950963i \(-0.600096\pi\)
−0.309304 + 0.950963i \(0.600096\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.2487 1.41468
\(732\) 0 0
\(733\) 12.6795 0.468328 0.234164 0.972197i \(-0.424765\pi\)
0.234164 + 0.972197i \(0.424765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.7128 −0.505118
\(738\) 0 0
\(739\) 16.7321 0.615498 0.307749 0.951468i \(-0.400424\pi\)
0.307749 + 0.951468i \(0.400424\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.6077 −0.719337 −0.359668 0.933080i \(-0.617110\pi\)
−0.359668 + 0.933080i \(0.617110\pi\)
\(744\) 0 0
\(745\) 33.5885 1.23059
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 49.8564 1.81929 0.909643 0.415391i \(-0.136355\pi\)
0.909643 + 0.415391i \(0.136355\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −60.4449 −2.19981
\(756\) 0 0
\(757\) −20.7846 −0.755429 −0.377715 0.925922i \(-0.623290\pi\)
−0.377715 + 0.925922i \(0.623290\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −37.0000 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(762\) 0 0
\(763\) −7.19615 −0.260518
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.87564 −0.103833
\(768\) 0 0
\(769\) −35.5885 −1.28335 −0.641676 0.766976i \(-0.721761\pi\)
−0.641676 + 0.766976i \(0.721761\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.1244 −0.723823 −0.361911 0.932213i \(-0.617876\pi\)
−0.361911 + 0.932213i \(0.617876\pi\)
\(774\) 0 0
\(775\) 19.6077 0.704329
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −25.8564 −0.926402
\(780\) 0 0
\(781\) −56.4974 −2.02164
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.73205 0.133203
\(786\) 0 0
\(787\) 7.60770 0.271185 0.135593 0.990765i \(-0.456706\pi\)
0.135593 + 0.990765i \(0.456706\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.26795 0.0806390
\(792\) 0 0
\(793\) −4.60770 −0.163624
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.4449 −1.04299 −0.521495 0.853254i \(-0.674626\pi\)
−0.521495 + 0.853254i \(0.674626\pi\)
\(798\) 0 0
\(799\) −8.87564 −0.313998
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −49.2295 −1.73727
\(804\) 0 0
\(805\) −23.1244 −0.815026
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −7.87564 −0.276893 −0.138446 0.990370i \(-0.544211\pi\)
−0.138446 + 0.990370i \(0.544211\pi\)
\(810\) 0 0
\(811\) −7.80385 −0.274030 −0.137015 0.990569i \(-0.543751\pi\)
−0.137015 + 0.990569i \(0.543751\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.3923 −0.854425
\(816\) 0 0
\(817\) −14.9282 −0.522272
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0718 −0.421309 −0.210654 0.977561i \(-0.567559\pi\)
−0.210654 + 0.977561i \(0.567559\pi\)
\(822\) 0 0
\(823\) 40.7846 1.42166 0.710831 0.703363i \(-0.248319\pi\)
0.710831 + 0.703363i \(0.248319\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.3205 0.393653 0.196826 0.980438i \(-0.436936\pi\)
0.196826 + 0.980438i \(0.436936\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.00000 −0.242536
\(834\) 0 0
\(835\) 45.5167 1.57517
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.46410 −0.188642 −0.0943209 0.995542i \(-0.530068\pi\)
−0.0943209 + 0.995542i \(0.530068\pi\)
\(840\) 0 0
\(841\) 42.6410 1.47038
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 47.7128 1.64137
\(846\) 0 0
\(847\) 6.60770 0.227043
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −41.2679 −1.41465
\(852\) 0 0
\(853\) 5.71281 0.195603 0.0978015 0.995206i \(-0.468819\pi\)
0.0978015 + 0.995206i \(0.468819\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.8564 −0.370848 −0.185424 0.982659i \(-0.559366\pi\)
−0.185424 + 0.982659i \(0.559366\pi\)
\(858\) 0 0
\(859\) −3.60770 −0.123093 −0.0615465 0.998104i \(-0.519603\pi\)
−0.0615465 + 0.998104i \(0.519603\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.1244 0.582920 0.291460 0.956583i \(-0.405859\pi\)
0.291460 + 0.956583i \(0.405859\pi\)
\(864\) 0 0
\(865\) 36.3205 1.23493
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 63.4641 2.15287
\(870\) 0 0
\(871\) 1.51666 0.0513901
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −14.6603 −0.495607
\(876\) 0 0
\(877\) 8.51666 0.287587 0.143794 0.989608i \(-0.454070\pi\)
0.143794 + 0.989608i \(0.454070\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.2487 0.614815 0.307407 0.951578i \(-0.400539\pi\)
0.307407 + 0.951578i \(0.400539\pi\)
\(882\) 0 0
\(883\) −7.66025 −0.257788 −0.128894 0.991658i \(-0.541143\pi\)
−0.128894 + 0.991658i \(0.541143\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.44486 0.0820905 0.0410452 0.999157i \(-0.486931\pi\)
0.0410452 + 0.999157i \(0.486931\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) −30.5885 −1.02246
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.5885 −0.619960
\(900\) 0 0
\(901\) −17.7513 −0.591381
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.3923 −0.544899
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.24871 −0.207029 −0.103515 0.994628i \(-0.533009\pi\)
−0.103515 + 0.994628i \(0.533009\pi\)
\(912\) 0 0
\(913\) 61.2154 2.02593
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.4641 0.576715
\(918\) 0 0
\(919\) 24.9808 0.824039 0.412020 0.911175i \(-0.364824\pi\)
0.412020 + 0.911175i \(0.364824\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.24871 0.205679
\(924\) 0 0
\(925\) −59.4641 −1.95517
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −45.4974 −1.49272 −0.746361 0.665541i \(-0.768201\pi\)
−0.746361 + 0.665541i \(0.768201\pi\)
\(930\) 0 0
\(931\) 2.73205 0.0895393
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −109.622 −3.58502
\(936\) 0 0
\(937\) 53.8372 1.75878 0.879392 0.476099i \(-0.157950\pi\)
0.879392 + 0.476099i \(0.157950\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.87564 0.321937 0.160968 0.986960i \(-0.448538\pi\)
0.160968 + 0.986960i \(0.448538\pi\)
\(942\) 0 0
\(943\) −58.6410 −1.90961
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.24871 0.0730733 0.0365366 0.999332i \(-0.488367\pi\)
0.0365366 + 0.999332i \(0.488367\pi\)
\(948\) 0 0
\(949\) 5.44486 0.176748
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.4115 0.337263 0.168631 0.985679i \(-0.446065\pi\)
0.168631 + 0.985679i \(0.446065\pi\)
\(954\) 0 0
\(955\) −43.5167 −1.40817
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.7321 0.378848
\(960\) 0 0
\(961\) −26.1769 −0.844417
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 33.0526 1.06400
\(966\) 0 0
\(967\) −13.6603 −0.439284 −0.219642 0.975581i \(-0.570489\pi\)
−0.219642 + 0.975581i \(0.570489\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.1244 1.06301 0.531506 0.847055i \(-0.321626\pi\)
0.531506 + 0.847055i \(0.321626\pi\)
\(972\) 0 0
\(973\) 6.73205 0.215820
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.28719 −0.0731736 −0.0365868 0.999330i \(-0.511649\pi\)
−0.0365868 + 0.999330i \(0.511649\pi\)
\(978\) 0 0
\(979\) −16.4833 −0.526810
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.6410 −0.339396 −0.169698 0.985496i \(-0.554279\pi\)
−0.169698 + 0.985496i \(0.554279\pi\)
\(984\) 0 0
\(985\) −96.2295 −3.06613
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −33.8564 −1.07657
\(990\) 0 0
\(991\) 10.3397 0.328453 0.164226 0.986423i \(-0.447487\pi\)
0.164226 + 0.986423i \(0.447487\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.1244 0.606283
\(996\) 0 0
\(997\) −7.24871 −0.229569 −0.114784 0.993390i \(-0.536618\pi\)
−0.114784 + 0.993390i \(0.536618\pi\)
\(998\) 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 9072.2.a.y.1.1 2
3.2 odd 2 9072.2.a.bp.1.2 2
4.3 odd 2 1134.2.a.m.1.1 yes 2
12.11 even 2 1134.2.a.l.1.2 2
28.27 even 2 7938.2.a.bt.1.2 2
36.7 odd 6 1134.2.f.r.757.2 4
36.11 even 6 1134.2.f.s.757.1 4
36.23 even 6 1134.2.f.s.379.1 4
36.31 odd 6 1134.2.f.r.379.2 4
84.83 odd 2 7938.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 2 12.11 even 2
1134.2.a.m.1.1 yes 2 4.3 odd 2
1134.2.f.r.379.2 4 36.31 odd 6
1134.2.f.r.757.2 4 36.7 odd 6
1134.2.f.s.379.1 4 36.23 even 6
1134.2.f.s.757.1 4 36.11 even 6
7938.2.a.bg.1.1 2 84.83 odd 2
7938.2.a.bt.1.2 2 28.27 even 2
9072.2.a.y.1.1 2 1.1 even 1 trivial
9072.2.a.bp.1.2 2 3.2 odd 2