Properties

Label 3744.2.m.h.1585.14
Level $3744$
Weight $2$
Character 3744.1585
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1585,3744)]
 
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, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1585.14
Root \(-0.752864 - 0.902863i\) of defining polynomial
Character \(\chi\) \(=\) 3744.1585
Dual form 3744.2.m.h.1585.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.68999 q^{5} -4.15163i q^{7} +O(q^{10})\) \(q+2.68999 q^{5} -4.15163i q^{7} +4.35250 q^{11} +(3.53159 - 0.726543i) q^{13} +5.87130 q^{17} +5.71423 q^{19} -3.62866 q^{23} +2.23607 q^{25} +3.08672i q^{29} -9.28334i q^{31} -11.1679i q^{35} -2.69790 q^{37} +11.1074i q^{41} +3.80423i q^{43} +4.91034i q^{47} -10.2361 q^{49} -1.17902i q^{53} +11.7082 q^{55} +2.29753 q^{59} +7.05342i q^{61} +(9.49996 - 1.95440i) q^{65} -10.0795 q^{67} +2.08191i q^{71} +13.4350i q^{73} -18.0700i q^{77} -10.9443 q^{79} -9.73249 q^{83} +15.7938 q^{85} -12.1877i q^{89} +(-3.01634 - 14.6619i) q^{91} +15.3713 q^{95} +5.13170i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) 2.68999 1.20300 0.601501 0.798872i \(-0.294570\pi\)
0.601501 + 0.798872i \(0.294570\pi\)
\(6\) 0 0
\(7\) 4.15163i 1.56917i −0.620021 0.784585i \(-0.712876\pi\)
0.620021 0.784585i \(-0.287124\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.35250 1.31233 0.656164 0.754618i \(-0.272178\pi\)
0.656164 + 0.754618i \(0.272178\pi\)
\(12\) 0 0
\(13\) 3.53159 0.726543i 0.979487 0.201507i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.87130 1.42400 0.711999 0.702180i \(-0.247790\pi\)
0.711999 + 0.702180i \(0.247790\pi\)
\(18\) 0 0
\(19\) 5.71423 1.31094 0.655468 0.755223i \(-0.272472\pi\)
0.655468 + 0.755223i \(0.272472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.62866 −0.756628 −0.378314 0.925677i \(-0.623496\pi\)
−0.378314 + 0.925677i \(0.623496\pi\)
\(24\) 0 0
\(25\) 2.23607 0.447214
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.08672i 0.573190i 0.958052 + 0.286595i \(0.0925234\pi\)
−0.958052 + 0.286595i \(0.907477\pi\)
\(30\) 0 0
\(31\) 9.28334i 1.66734i −0.552266 0.833668i \(-0.686237\pi\)
0.552266 0.833668i \(-0.313763\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.1679i 1.88771i
\(36\) 0 0
\(37\) −2.69790 −0.443531 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.1074i 1.73468i 0.497715 + 0.867340i \(0.334172\pi\)
−0.497715 + 0.867340i \(0.665828\pi\)
\(42\) 0 0
\(43\) 3.80423i 0.580139i 0.957006 + 0.290070i \(0.0936784\pi\)
−0.957006 + 0.290070i \(0.906322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.91034i 0.716247i 0.933674 + 0.358123i \(0.116583\pi\)
−0.933674 + 0.358123i \(0.883417\pi\)
\(48\) 0 0
\(49\) −10.2361 −1.46230
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.17902i 0.161951i −0.996716 0.0809757i \(-0.974196\pi\)
0.996716 0.0809757i \(-0.0258036\pi\)
\(54\) 0 0
\(55\) 11.7082 1.57873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.29753 0.299113 0.149556 0.988753i \(-0.452215\pi\)
0.149556 + 0.988753i \(0.452215\pi\)
\(60\) 0 0
\(61\) 7.05342i 0.903098i 0.892246 + 0.451549i \(0.149128\pi\)
−0.892246 + 0.451549i \(0.850872\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.49996 1.95440i 1.17832 0.242413i
\(66\) 0 0
\(67\) −10.0795 −1.23141 −0.615705 0.787977i \(-0.711129\pi\)
−0.615705 + 0.787977i \(0.711129\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.08191i 0.247078i 0.992340 + 0.123539i \(0.0394244\pi\)
−0.992340 + 0.123539i \(0.960576\pi\)
\(72\) 0 0
\(73\) 13.4350i 1.57244i 0.617944 + 0.786222i \(0.287966\pi\)
−0.617944 + 0.786222i \(0.712034\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.0700i 2.05927i
\(78\) 0 0
\(79\) −10.9443 −1.23133 −0.615663 0.788009i \(-0.711112\pi\)
−0.615663 + 0.788009i \(0.711112\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.73249 −1.06828 −0.534140 0.845396i \(-0.679364\pi\)
−0.534140 + 0.845396i \(0.679364\pi\)
\(84\) 0 0
\(85\) 15.7938 1.71307
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.1877i 1.29190i −0.763381 0.645949i \(-0.776462\pi\)
0.763381 0.645949i \(-0.223538\pi\)
\(90\) 0 0
\(91\) −3.01634 14.6619i −0.316198 1.53698i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.3713 1.57706
\(96\) 0 0
\(97\) 5.13170i 0.521045i 0.965468 + 0.260523i \(0.0838949\pi\)
−0.965468 + 0.260523i \(0.916105\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.9833i 1.49089i −0.666566 0.745446i \(-0.732237\pi\)
0.666566 0.745446i \(-0.267763\pi\)
\(102\) 0 0
\(103\) −1.70820 −0.168314 −0.0841572 0.996452i \(-0.526820\pi\)
−0.0841572 + 0.996452i \(0.526820\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.99442i 0.482829i −0.970422 0.241415i \(-0.922389\pi\)
0.970422 0.241415i \(-0.0776114\pi\)
\(108\) 0 0
\(109\) −8.73057 −0.836237 −0.418119 0.908392i \(-0.637310\pi\)
−0.418119 + 0.908392i \(0.637310\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.87130 0.552325 0.276163 0.961111i \(-0.410937\pi\)
0.276163 + 0.961111i \(0.410937\pi\)
\(114\) 0 0
\(115\) −9.76108 −0.910225
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.3755i 2.23450i
\(120\) 0 0
\(121\) 7.94427 0.722207
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.43496 −0.665003
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.80982i 0.769718i 0.922975 + 0.384859i \(0.125750\pi\)
−0.922975 + 0.384859i \(0.874250\pi\)
\(132\) 0 0
\(133\) 23.7234i 2.05708i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.4397i 0.891922i −0.895052 0.445961i \(-0.852862\pi\)
0.895052 0.445961i \(-0.147138\pi\)
\(138\) 0 0
\(139\) 14.3188i 1.21451i 0.794507 + 0.607254i \(0.207729\pi\)
−0.794507 + 0.607254i \(0.792271\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 15.3713 3.16228i 1.28541 0.264443i
\(144\) 0 0
\(145\) 8.30327i 0.689549i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.635021 −0.0520230 −0.0260115 0.999662i \(-0.508281\pi\)
−0.0260115 + 0.999662i \(0.508281\pi\)
\(150\) 0 0
\(151\) 4.15163i 0.337855i 0.985628 + 0.168928i \(0.0540304\pi\)
−0.985628 + 0.168928i \(0.945970\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 24.9721i 2.00581i
\(156\) 0 0
\(157\) 16.1150i 1.28611i 0.765818 + 0.643057i \(0.222334\pi\)
−0.765818 + 0.643057i \(0.777666\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0649i 1.18728i
\(162\) 0 0
\(163\) −10.0795 −0.789489 −0.394745 0.918791i \(-0.629167\pi\)
−0.394745 + 0.918791i \(0.629167\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.2967i 1.88013i 0.340992 + 0.940066i \(0.389237\pi\)
−0.340992 + 0.940066i \(0.610763\pi\)
\(168\) 0 0
\(169\) 11.9443 5.13170i 0.918790 0.394746i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.17902i 0.0896395i −0.998995 0.0448198i \(-0.985729\pi\)
0.998995 0.0448198i \(-0.0142714\pi\)
\(174\) 0 0
\(175\) 9.28334i 0.701754i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.7987i 1.40508i −0.711645 0.702539i \(-0.752049\pi\)
0.711645 0.702539i \(-0.247951\pi\)
\(180\) 0 0
\(181\) 13.2088i 0.981802i −0.871215 0.490901i \(-0.836668\pi\)
0.871215 0.490901i \(-0.163332\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.25732 −0.533569
\(186\) 0 0
\(187\) 25.5548 1.86875
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.3713 1.11223 0.556113 0.831107i \(-0.312292\pi\)
0.556113 + 0.831107i \(0.312292\pi\)
\(192\) 0 0
\(193\) 13.4350i 0.967070i 0.875325 + 0.483535i \(0.160647\pi\)
−0.875325 + 0.483535i \(0.839353\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.6101 0.755936 0.377968 0.925819i \(-0.376623\pi\)
0.377968 + 0.925819i \(0.376623\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.8149 0.899433
\(204\) 0 0
\(205\) 29.8788i 2.08682i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.8712 1.72038
\(210\) 0 0
\(211\) 17.0130i 1.17122i −0.810591 0.585612i \(-0.800854\pi\)
0.810591 0.585612i \(-0.199146\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.2333i 0.697908i
\(216\) 0 0
\(217\) −38.5410 −2.61633
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.7350 4.26575i 1.39479 0.286945i
\(222\) 0 0
\(223\) 9.28334i 0.621658i 0.950466 + 0.310829i \(0.100607\pi\)
−0.950466 + 0.310829i \(0.899393\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.02749 0.0681967 0.0340983 0.999418i \(-0.489144\pi\)
0.0340983 + 0.999418i \(0.489144\pi\)
\(228\) 0 0
\(229\) −18.4917 −1.22196 −0.610981 0.791645i \(-0.709225\pi\)
−0.610981 + 0.791645i \(0.709225\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.25732 −0.475443 −0.237722 0.971333i \(-0.576401\pi\)
−0.237722 + 0.971333i \(0.576401\pi\)
\(234\) 0 0
\(235\) 13.2088i 0.861646i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.07107i 0.457389i 0.973498 + 0.228695i \(0.0734457\pi\)
−0.973498 + 0.228695i \(0.926554\pi\)
\(240\) 0 0
\(241\) 18.5667i 1.19598i −0.801502 0.597992i \(-0.795965\pi\)
0.801502 0.597992i \(-0.204035\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −27.5350 −1.75914
\(246\) 0 0
\(247\) 20.1803 4.15163i 1.28404 0.264162i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.35805i 0.148839i −0.997227 0.0744193i \(-0.976290\pi\)
0.997227 0.0744193i \(-0.0237103\pi\)
\(252\) 0 0
\(253\) −15.7938 −0.992945
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −26.2572 −1.63788 −0.818941 0.573878i \(-0.805438\pi\)
−0.818941 + 0.573878i \(0.805438\pi\)
\(258\) 0 0
\(259\) 11.2007i 0.695976i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.25732 −0.447506 −0.223753 0.974646i \(-0.571831\pi\)
−0.223753 + 0.974646i \(0.571831\pi\)
\(264\) 0 0
\(265\) 3.17157i 0.194828i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.2490i 1.17363i −0.809720 0.586817i \(-0.800381\pi\)
0.809720 0.586817i \(-0.199619\pi\)
\(270\) 0 0
\(271\) 9.28334i 0.563923i 0.959426 + 0.281961i \(0.0909850\pi\)
−0.959426 + 0.281961i \(0.909015\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.73249 0.586891
\(276\) 0 0
\(277\) 21.7153i 1.30475i −0.757898 0.652373i \(-0.773774\pi\)
0.757898 0.652373i \(-0.226226\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.35931i 0.558330i 0.960243 + 0.279165i \(0.0900576\pi\)
−0.960243 + 0.279165i \(0.909942\pi\)
\(282\) 0 0
\(283\) 12.3107i 0.731797i −0.930655 0.365899i \(-0.880762\pi\)
0.930655 0.365899i \(-0.119238\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 46.1138 2.72201
\(288\) 0 0
\(289\) 17.4721 1.02777
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.4800 1.48856 0.744278 0.667869i \(-0.232794\pi\)
0.744278 + 0.667869i \(0.232794\pi\)
\(294\) 0 0
\(295\) 6.18034 0.359833
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.8149 + 2.63638i −0.741108 + 0.152466i
\(300\) 0 0
\(301\) 15.7938 0.910337
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.9737i 1.08643i
\(306\) 0 0
\(307\) −4.04684 −0.230966 −0.115483 0.993309i \(-0.536842\pi\)
−0.115483 + 0.993309i \(0.536842\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.8860 −0.617288 −0.308644 0.951178i \(-0.599875\pi\)
−0.308644 + 0.951178i \(0.599875\pi\)
\(312\) 0 0
\(313\) −2.47214 −0.139733 −0.0698667 0.997556i \(-0.522257\pi\)
−0.0698667 + 0.997556i \(0.522257\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.2349 0.799512 0.399756 0.916622i \(-0.369095\pi\)
0.399756 + 0.916622i \(0.369095\pi\)
\(318\) 0 0
\(319\) 13.4350i 0.752214i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 33.5500 1.86677
\(324\) 0 0
\(325\) 7.89688 1.62460i 0.438040 0.0901165i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 20.3859 1.12391
\(330\) 0 0
\(331\) 3.01634 0.165793 0.0828965 0.996558i \(-0.473583\pi\)
0.0828965 + 0.996558i \(0.473583\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.1139 −1.48139
\(336\) 0 0
\(337\) 14.4721 0.788347 0.394174 0.919036i \(-0.371031\pi\)
0.394174 + 0.919036i \(0.371031\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 40.4057i 2.18809i
\(342\) 0 0
\(343\) 13.4350i 0.725420i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.5092i 1.53045i 0.643761 + 0.765227i \(0.277373\pi\)
−0.643761 + 0.765227i \(0.722627\pi\)
\(348\) 0 0
\(349\) 9.76108 0.522499 0.261249 0.965271i \(-0.415866\pi\)
0.261249 + 0.965271i \(0.415866\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.86629i 0.418680i −0.977843 0.209340i \(-0.932868\pi\)
0.977843 0.209340i \(-0.0671316\pi\)
\(354\) 0 0
\(355\) 5.60034i 0.297235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.1406i 0.693533i −0.937951 0.346767i \(-0.887280\pi\)
0.937951 0.346767i \(-0.112720\pi\)
\(360\) 0 0
\(361\) 13.6525 0.718551
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.1400i 1.89165i
\(366\) 0 0
\(367\) 9.12461 0.476301 0.238150 0.971228i \(-0.423459\pi\)
0.238150 + 0.971228i \(0.423459\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.89487 −0.254129
\(372\) 0 0
\(373\) 0.343027i 0.0177613i 0.999961 + 0.00888063i \(0.00282683\pi\)
−0.999961 + 0.00888063i \(0.997173\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.24264 + 10.9010i 0.115502 + 0.561432i
\(378\) 0 0
\(379\) 18.8101 0.966210 0.483105 0.875563i \(-0.339509\pi\)
0.483105 + 0.875563i \(0.339509\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.2241i 0.829010i 0.910047 + 0.414505i \(0.136045\pi\)
−0.910047 + 0.414505i \(0.863955\pi\)
\(384\) 0 0
\(385\) 48.6082i 2.47730i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.6182i 0.589067i 0.955641 + 0.294534i \(0.0951643\pi\)
−0.955641 + 0.294534i \(0.904836\pi\)
\(390\) 0 0
\(391\) −21.3050 −1.07744
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −29.4400 −1.48129
\(396\) 0 0
\(397\) −24.5243 −1.23084 −0.615420 0.788199i \(-0.711014\pi\)
−0.615420 + 0.788199i \(0.711014\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.3438i 1.16574i −0.812567 0.582868i \(-0.801931\pi\)
0.812567 0.582868i \(-0.198069\pi\)
\(402\) 0 0
\(403\) −6.74474 32.7849i −0.335979 1.63313i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.7426 −0.582059
\(408\) 0 0
\(409\) 15.3951i 0.761239i −0.924732 0.380620i \(-0.875711\pi\)
0.924732 0.380620i \(-0.124289\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.53850i 0.469359i
\(414\) 0 0
\(415\) −26.1803 −1.28514
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.9833i 0.731981i −0.930619 0.365990i \(-0.880730\pi\)
0.930619 0.365990i \(-0.119270\pi\)
\(420\) 0 0
\(421\) 15.7938 0.769741 0.384870 0.922971i \(-0.374246\pi\)
0.384870 + 0.922971i \(0.374246\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.1286 0.636832
\(426\) 0 0
\(427\) 29.2832 1.41711
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.8005i 1.00193i 0.865468 + 0.500963i \(0.167021\pi\)
−0.865468 + 0.500963i \(0.832979\pi\)
\(432\) 0 0
\(433\) 14.1803 0.681464 0.340732 0.940161i \(-0.389325\pi\)
0.340732 + 0.940161i \(0.389325\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.7350 −0.991891
\(438\) 0 0
\(439\) −24.1803 −1.15406 −0.577032 0.816721i \(-0.695789\pi\)
−0.577032 + 0.816721i \(0.695789\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.17902i 0.0560171i −0.999608 0.0280086i \(-0.991083\pi\)
0.999608 0.0280086i \(-0.00891656\pi\)
\(444\) 0 0
\(445\) 32.7849i 1.55416i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.3221i 1.57257i −0.617865 0.786284i \(-0.712002\pi\)
0.617865 0.786284i \(-0.287998\pi\)
\(450\) 0 0
\(451\) 48.3449i 2.27647i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.11393 39.4404i −0.380387 1.84899i
\(456\) 0 0
\(457\) 38.3448i 1.79369i −0.442342 0.896847i \(-0.645852\pi\)
0.442342 0.896847i \(-0.354148\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.6951 −1.15016 −0.575082 0.818096i \(-0.695030\pi\)
−0.575082 + 0.818096i \(0.695030\pi\)
\(462\) 0 0
\(463\) 4.15163i 0.192943i −0.995336 0.0964714i \(-0.969244\pi\)
0.995336 0.0964714i \(-0.0307556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.17345i 0.285673i 0.989746 + 0.142837i \(0.0456223\pi\)
−0.989746 + 0.142837i \(0.954378\pi\)
\(468\) 0 0
\(469\) 41.8465i 1.93229i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.5579i 0.761333i
\(474\) 0 0
\(475\) 12.7774 0.586268
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3122i 0.471174i −0.971853 0.235587i \(-0.924299\pi\)
0.971853 0.235587i \(-0.0757013\pi\)
\(480\) 0 0
\(481\) −9.52786 + 1.96014i −0.434433 + 0.0893745i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.8042i 0.626819i
\(486\) 0 0
\(487\) 17.5866i 0.796925i 0.917185 + 0.398463i \(0.130456\pi\)
−0.917185 + 0.398463i \(0.869544\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.1511i 1.18018i −0.807336 0.590092i \(-0.799091\pi\)
0.807336 0.590092i \(-0.200909\pi\)
\(492\) 0 0
\(493\) 18.1231i 0.816222i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.64335 0.387707
\(498\) 0 0
\(499\) −39.3628 −1.76212 −0.881059 0.473006i \(-0.843169\pi\)
−0.881059 + 0.473006i \(0.843169\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.9999 −0.847164 −0.423582 0.905858i \(-0.639228\pi\)
−0.423582 + 0.905858i \(0.639228\pi\)
\(504\) 0 0
\(505\) 40.3049i 1.79355i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −27.5350 −1.22047 −0.610233 0.792222i \(-0.708924\pi\)
−0.610233 + 0.792222i \(0.708924\pi\)
\(510\) 0 0
\(511\) 55.7771 2.46743
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.59506 −0.202482
\(516\) 0 0
\(517\) 21.3723i 0.939951i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.8712 1.08963 0.544814 0.838557i \(-0.316600\pi\)
0.544814 + 0.838557i \(0.316600\pi\)
\(522\) 0 0
\(523\) 21.9273i 0.958814i 0.877593 + 0.479407i \(0.159148\pi\)
−0.877593 + 0.479407i \(0.840852\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 54.5052i 2.37429i
\(528\) 0 0
\(529\) −9.83282 −0.427514
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.06998 + 39.2267i 0.349550 + 1.69910i
\(534\) 0 0
\(535\) 13.4350i 0.580844i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −44.5525 −1.91901
\(540\) 0 0
\(541\) 41.3486 1.77771 0.888857 0.458184i \(-0.151500\pi\)
0.888857 + 0.458184i \(0.151500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −23.4852 −1.00600
\(546\) 0 0
\(547\) 0.212002i 0.00906456i −0.999990 0.00453228i \(-0.998557\pi\)
0.999990 0.00453228i \(-0.00144267\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.6383i 0.751415i
\(552\) 0 0
\(553\) 45.4366i 1.93216i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.47492 −0.147237 −0.0736186 0.997286i \(-0.523455\pi\)
−0.0736186 + 0.997286i \(0.523455\pi\)
\(558\) 0 0
\(559\) 2.76393 + 13.4350i 0.116902 + 0.568239i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 21.1567i 0.891649i 0.895120 + 0.445825i \(0.147090\pi\)
−0.895120 + 0.445825i \(0.852910\pi\)
\(564\) 0 0
\(565\) 15.7938 0.664448
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.48527 −0.188032 −0.0940162 0.995571i \(-0.529971\pi\)
−0.0940162 + 0.995571i \(0.529971\pi\)
\(570\) 0 0
\(571\) 5.81234i 0.243239i 0.992577 + 0.121619i \(0.0388087\pi\)
−0.992577 + 0.121619i \(0.961191\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.11393 −0.338374
\(576\) 0 0
\(577\) 14.6464i 0.609738i 0.952394 + 0.304869i \(0.0986126\pi\)
−0.952394 + 0.304869i \(0.901387\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.4057i 1.67631i
\(582\) 0 0
\(583\) 5.13170i 0.212533i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.0875 1.03547 0.517736 0.855540i \(-0.326775\pi\)
0.517736 + 0.855540i \(0.326775\pi\)
\(588\) 0 0
\(589\) 53.0472i 2.18577i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.1769i 0.705370i 0.935742 + 0.352685i \(0.114731\pi\)
−0.935742 + 0.352685i \(0.885269\pi\)
\(594\) 0 0
\(595\) 65.5699i 2.68810i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27.9705 −1.14284 −0.571421 0.820657i \(-0.693608\pi\)
−0.571421 + 0.820657i \(0.693608\pi\)
\(600\) 0 0
\(601\) −7.41641 −0.302522 −0.151261 0.988494i \(-0.548333\pi\)
−0.151261 + 0.988494i \(0.548333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.3700 0.868816
\(606\) 0 0
\(607\) −2.29180 −0.0930211 −0.0465106 0.998918i \(-0.514810\pi\)
−0.0465106 + 0.998918i \(0.514810\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.56757 + 17.3413i 0.144329 + 0.701555i
\(612\) 0 0
\(613\) −34.2854 −1.38477 −0.692387 0.721526i \(-0.743441\pi\)
−0.692387 + 0.721526i \(0.743441\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.53089i 0.262924i 0.991321 + 0.131462i \(0.0419671\pi\)
−0.991321 + 0.131462i \(0.958033\pi\)
\(618\) 0 0
\(619\) 32.2996 1.29823 0.649115 0.760691i \(-0.275140\pi\)
0.649115 + 0.760691i \(0.275140\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −50.5990 −2.02721
\(624\) 0 0
\(625\) −31.1803 −1.24721
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15.8401 −0.631588
\(630\) 0 0
\(631\) 7.32320i 0.291532i 0.989319 + 0.145766i \(0.0465647\pi\)
−0.989319 + 0.145766i \(0.953435\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.1400 0.640495
\(636\) 0 0
\(637\) −36.1496 + 7.43694i −1.43230 + 0.294662i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −27.6433 −1.09184 −0.545922 0.837836i \(-0.683820\pi\)
−0.545922 + 0.837836i \(0.683820\pi\)
\(642\) 0 0
\(643\) −38.9691 −1.53679 −0.768396 0.639974i \(-0.778945\pi\)
−0.768396 + 0.639974i \(0.778945\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.2279 0.637983 0.318992 0.947758i \(-0.396656\pi\)
0.318992 + 0.947758i \(0.396656\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.7694i 1.47803i 0.673689 + 0.739015i \(0.264709\pi\)
−0.673689 + 0.739015i \(0.735291\pi\)
\(654\) 0 0
\(655\) 23.6984i 0.925972i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.45735i 0.0567704i 0.999597 + 0.0283852i \(0.00903651\pi\)
−0.999597 + 0.0283852i \(0.990963\pi\)
\(660\) 0 0
\(661\) −23.4938 −0.913804 −0.456902 0.889517i \(-0.651041\pi\)
−0.456902 + 0.889517i \(0.651041\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 63.8158i 2.47467i
\(666\) 0 0
\(667\) 11.2007i 0.433692i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 30.7000i 1.18516i
\(672\) 0 0
\(673\) 18.6525 0.719000 0.359500 0.933145i \(-0.382947\pi\)
0.359500 + 0.933145i \(0.382947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.5036i 1.28765i −0.765173 0.643824i \(-0.777347\pi\)
0.765173 0.643824i \(-0.222653\pi\)
\(678\) 0 0
\(679\) 21.3050 0.817609
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.0875 0.959947 0.479974 0.877283i \(-0.340646\pi\)
0.479974 + 0.877283i \(0.340646\pi\)
\(684\) 0 0
\(685\) 28.0827i 1.07298i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.856611 4.16383i −0.0326343 0.158629i
\(690\) 0 0
\(691\) −1.34895 −0.0513164 −0.0256582 0.999671i \(-0.508168\pi\)
−0.0256582 + 0.999671i \(0.508168\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.5176i 1.46106i
\(696\) 0 0
\(697\) 65.2147i 2.47018i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.44477i 0.205646i −0.994700 0.102823i \(-0.967212\pi\)
0.994700 0.102823i \(-0.0327876\pi\)
\(702\) 0 0
\(703\) −15.4164 −0.581441
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −62.2051 −2.33946
\(708\) 0 0
\(709\) −15.7938 −0.593147 −0.296573 0.955010i \(-0.595844\pi\)
−0.296573 + 0.955010i \(0.595844\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33.6861i 1.26155i
\(714\) 0 0
\(715\) 41.3486 8.50651i 1.54635 0.318125i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27.1139 1.01118 0.505588 0.862775i \(-0.331276\pi\)
0.505588 + 0.862775i \(0.331276\pi\)
\(720\) 0 0
\(721\) 7.09184i 0.264114i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.90212i 0.256338i
\(726\) 0 0
\(727\) −22.3607 −0.829312 −0.414656 0.909978i \(-0.636098\pi\)
−0.414656 + 0.909978i \(0.636098\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.3357i 0.826117i
\(732\) 0 0
\(733\) −24.9179 −0.920365 −0.460183 0.887824i \(-0.652216\pi\)
−0.460183 + 0.887824i \(0.652216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −43.8711 −1.61601
\(738\) 0 0
\(739\) −29.6017 −1.08892 −0.544458 0.838788i \(-0.683264\pi\)
−0.544458 + 0.838788i \(0.683264\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.31990i 0.121795i 0.998144 + 0.0608977i \(0.0193963\pi\)
−0.998144 + 0.0608977i \(0.980604\pi\)
\(744\) 0 0
\(745\) −1.70820 −0.0625837
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20.7350 −0.757641
\(750\) 0 0
\(751\) −12.5410 −0.457628 −0.228814 0.973470i \(-0.573485\pi\)
−0.228814 + 0.973470i \(0.573485\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.1679i 0.406440i
\(756\) 0 0
\(757\) 5.25731i 0.191080i −0.995426 0.0955401i \(-0.969542\pi\)
0.995426 0.0955401i \(-0.0304578\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.2681i 0.480968i 0.970653 + 0.240484i \(0.0773062\pi\)
−0.970653 + 0.240484i \(0.922694\pi\)
\(762\) 0 0
\(763\) 36.2461i 1.31220i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.11393 1.66925i 0.292977 0.0602732i
\(768\) 0 0
\(769\) 46.6480i 1.68217i 0.540902 + 0.841086i \(0.318083\pi\)
−0.540902 + 0.841086i \(0.681917\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 37.0249 1.33169 0.665846 0.746089i \(-0.268071\pi\)
0.665846 + 0.746089i \(0.268071\pi\)
\(774\) 0 0
\(775\) 20.7582i 0.745656i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63.4702i 2.27405i
\(780\) 0 0
\(781\) 9.06154i 0.324247i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 43.3491i 1.54720i
\(786\) 0 0
\(787\) −5.07735 −0.180988 −0.0904940 0.995897i \(-0.528845\pi\)
−0.0904940 + 0.995897i \(0.528845\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.3755i 0.866692i
\(792\) 0 0
\(793\) 5.12461 + 24.9098i 0.181980 + 0.884573i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.4336i 0.546687i 0.961917 + 0.273343i \(0.0881295\pi\)
−0.961917 + 0.273343i \(0.911870\pi\)
\(798\) 0 0
\(799\) 28.8301i 1.01993i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.4757i 2.06356i
\(804\) 0 0
\(805\) 40.5244i 1.42830i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.3859 −0.716732 −0.358366 0.933581i \(-0.616666\pi\)
−0.358366 + 0.933581i \(0.616666\pi\)
\(810\) 0 0
\(811\) −40.6365 −1.42694 −0.713471 0.700685i \(-0.752878\pi\)
−0.713471 + 0.700685i \(0.752878\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −27.1139 −0.949757
\(816\) 0 0
\(817\) 21.7382i 0.760525i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0750 1.04963 0.524813 0.851217i \(-0.324135\pi\)
0.524813 + 0.851217i \(0.324135\pi\)
\(822\) 0 0
\(823\) 8.18034 0.285149 0.142574 0.989784i \(-0.454462\pi\)
0.142574 + 0.989784i \(0.454462\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 40.7424 1.41675 0.708376 0.705836i \(-0.249428\pi\)
0.708376 + 0.705836i \(0.249428\pi\)
\(828\) 0 0
\(829\) 8.84953i 0.307357i −0.988121 0.153679i \(-0.950888\pi\)
0.988121 0.153679i \(-0.0491120\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −60.0990 −2.08231
\(834\) 0 0
\(835\) 65.3579i 2.26180i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.8763i 1.06597i −0.846126 0.532984i \(-0.821071\pi\)
0.846126 0.532984i \(-0.178929\pi\)
\(840\) 0 0
\(841\) 19.4721 0.671453
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 32.1300 13.8042i 1.10531 0.474881i
\(846\) 0 0
\(847\) 32.9817i 1.13327i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.78975 0.335588
\(852\) 0 0
\(853\) 8.09369 0.277123 0.138561 0.990354i \(-0.455752\pi\)
0.138561 + 0.990354i \(0.455752\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37.9998 −1.29805 −0.649025 0.760767i \(-0.724823\pi\)
−0.649025 + 0.760767i \(0.724823\pi\)
\(858\) 0 0
\(859\) 6.49839i 0.221722i −0.993836 0.110861i \(-0.964639\pi\)
0.993836 0.110861i \(-0.0353609\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.7093i 0.841115i 0.907266 + 0.420558i \(0.138166\pi\)
−0.907266 + 0.420558i \(0.861834\pi\)
\(864\) 0 0
\(865\) 3.17157i 0.107836i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −47.6350 −1.61591
\(870\) 0 0
\(871\) −35.5967 + 7.32320i −1.20615 + 0.248137i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 30.8672i 1.04350i
\(876\) 0 0
\(877\) 54.4444 1.83846 0.919229 0.393723i \(-0.128813\pi\)
0.919229 + 0.393723i \(0.128813\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 11.7426 0.395618 0.197809 0.980241i \(-0.436617\pi\)
0.197809 + 0.980241i \(0.436617\pi\)
\(882\) 0 0
\(883\) 17.0130i 0.572534i −0.958150 0.286267i \(-0.907586\pi\)
0.958150 0.286267i \(-0.0924144\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.8859 1.00347 0.501735 0.865021i \(-0.332695\pi\)
0.501735 + 0.865021i \(0.332695\pi\)
\(888\) 0 0
\(889\) 24.9098i 0.835448i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 28.0588i 0.938953i
\(894\) 0 0
\(895\) 50.5683i 1.69031i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 28.6551 0.955701
\(900\) 0 0
\(901\) 6.92240i 0.230619i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 35.5316i 1.18111i
\(906\) 0 0
\(907\) 24.6215i 0.817542i −0.912637 0.408771i \(-0.865957\pi\)
0.912637 0.408771i \(-0.134043\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.1433 −0.601115 −0.300557 0.953764i \(-0.597173\pi\)
−0.300557 + 0.953764i \(0.597173\pi\)
\(912\) 0 0
\(913\) −42.3607 −1.40193
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.5752 1.20782
\(918\) 0 0
\(919\) 21.4164 0.706462 0.353231 0.935536i \(-0.385083\pi\)
0.353231 + 0.935536i \(0.385083\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.51260 + 7.35247i 0.0497878 + 0.242010i
\(924\) 0 0
\(925\) −6.03268 −0.198353
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.62210i 0.0860282i 0.999074 + 0.0430141i \(0.0136960\pi\)
−0.999074 + 0.0430141i \(0.986304\pi\)
\(930\) 0 0
\(931\) −58.4913 −1.91697
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 68.7424 2.24812
\(936\) 0 0
\(937\) 0.652476 0.0213155 0.0106577 0.999943i \(-0.496607\pi\)
0.0106577 + 0.999943i \(0.496607\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.9099 0.355652 0.177826 0.984062i \(-0.443094\pi\)
0.177826 + 0.984062i \(0.443094\pi\)
\(942\) 0 0
\(943\) 40.3049i 1.31251i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.59735 0.0844024 0.0422012 0.999109i \(-0.486563\pi\)
0.0422012 + 0.999109i \(0.486563\pi\)
\(948\) 0 0
\(949\) 9.76108 + 47.4468i 0.316858 + 1.54019i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14.5146 0.470176 0.235088 0.971974i \(-0.424462\pi\)
0.235088 + 0.971974i \(0.424462\pi\)
\(954\) 0 0
\(955\) 41.3486 1.33801
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −43.3417 −1.39958
\(960\) 0 0
\(961\) −55.1803 −1.78001
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 36.1400i 1.16339i
\(966\) 0 0
\(967\) 31.0216i 0.997587i −0.866721 0.498793i \(-0.833777\pi\)
0.866721 0.498793i \(-0.166223\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 52.3023i 1.67846i −0.543776 0.839230i \(-0.683006\pi\)
0.543776 0.839230i \(-0.316994\pi\)
\(972\) 0 0
\(973\) 59.4466 1.90577
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.8909i 1.43619i −0.695947 0.718093i \(-0.745015\pi\)
0.695947 0.718093i \(-0.254985\pi\)
\(978\) 0 0
\(979\) 53.0472i 1.69539i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 43.4280i 1.38514i 0.721352 + 0.692568i \(0.243521\pi\)
−0.721352 + 0.692568i \(0.756479\pi\)
\(984\) 0 0
\(985\) 28.5410 0.909393
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.8042i 0.438950i
\(990\) 0 0
\(991\) −12.1803 −0.386921 −0.193461 0.981108i \(-0.561971\pi\)
−0.193461 + 0.981108i \(0.561971\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −48.4199 −1.53501
\(996\) 0 0
\(997\) 55.0553i 1.74362i −0.489846 0.871809i \(-0.662947\pi\)
0.489846 0.871809i \(-0.337053\pi\)
\(998\) 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.m.h.1585.14 16
3.2 odd 2 inner 3744.2.m.h.1585.2 16
4.3 odd 2 936.2.m.h.181.6 yes 16
8.3 odd 2 936.2.m.h.181.10 yes 16
8.5 even 2 inner 3744.2.m.h.1585.1 16
12.11 even 2 936.2.m.h.181.12 yes 16
13.12 even 2 inner 3744.2.m.h.1585.3 16
24.5 odd 2 inner 3744.2.m.h.1585.13 16
24.11 even 2 936.2.m.h.181.8 yes 16
39.38 odd 2 inner 3744.2.m.h.1585.15 16
52.51 odd 2 936.2.m.h.181.11 yes 16
104.51 odd 2 936.2.m.h.181.7 yes 16
104.77 even 2 inner 3744.2.m.h.1585.16 16
156.155 even 2 936.2.m.h.181.5 16
312.77 odd 2 inner 3744.2.m.h.1585.4 16
312.155 even 2 936.2.m.h.181.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
936.2.m.h.181.5 16 156.155 even 2
936.2.m.h.181.6 yes 16 4.3 odd 2
936.2.m.h.181.7 yes 16 104.51 odd 2
936.2.m.h.181.8 yes 16 24.11 even 2
936.2.m.h.181.9 yes 16 312.155 even 2
936.2.m.h.181.10 yes 16 8.3 odd 2
936.2.m.h.181.11 yes 16 52.51 odd 2
936.2.m.h.181.12 yes 16 12.11 even 2
3744.2.m.h.1585.1 16 8.5 even 2 inner
3744.2.m.h.1585.2 16 3.2 odd 2 inner
3744.2.m.h.1585.3 16 13.12 even 2 inner
3744.2.m.h.1585.4 16 312.77 odd 2 inner
3744.2.m.h.1585.13 16 24.5 odd 2 inner
3744.2.m.h.1585.14 16 1.1 even 1 trivial
3744.2.m.h.1585.15 16 39.38 odd 2 inner
3744.2.m.h.1585.16 16 104.77 even 2 inner