Properties

Label 3744.2.g.e.1873.3
Level $3744$
Weight $2$
Character 3744.1873
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1873,3744)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.1873");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.g (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: \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} - 16 x^{7} + 8 x^{6} - 80 x^{5} + 144 x^{4} - 128 x^{3} + 128 x^{2} - 256 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1873.3
Root \(1.27276 + 0.616518i\) of defining polynomial
Character \(\chi\) \(=\) 3744.1873
Dual form 3744.2.g.e.1873.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.29521i q^{5} -2.97802 q^{7} +O(q^{10})\) \(q-3.29521i q^{5} -2.97802 q^{7} +4.64206i q^{11} -1.00000i q^{13} -2.52410 q^{17} +2.18396i q^{19} +7.18826 q^{23} -5.85840 q^{25} +8.39420i q^{29} +5.60297 q^{31} +9.81321i q^{35} -5.55709i q^{37} +7.09899 q^{41} -6.08120i q^{43} -4.27323 q^{47} +1.86863 q^{49} +2.30359i q^{53} +15.2966 q^{55} +13.8574i q^{59} -13.0161i q^{61} -3.29521 q^{65} +2.39703i q^{67} +10.0246 q^{71} +7.23131 q^{73} -13.8242i q^{77} -10.1262 q^{79} -11.4615i q^{83} +8.31745i q^{85} +7.07469 q^{89} +2.97802i q^{91} +7.19662 q^{95} +3.60104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) \) 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
\(5\) − 3.29521i − 1.47366i −0.676077 0.736831i \(-0.736321\pi\)
0.676077 0.736831i \(-0.263679\pi\)
\(6\) 0 0
\(7\) −2.97802 −1.12559 −0.562794 0.826597i \(-0.690273\pi\)
−0.562794 + 0.826597i \(0.690273\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.64206i 1.39963i 0.714322 + 0.699817i \(0.246735\pi\)
−0.714322 + 0.699817i \(0.753265\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.52410 −0.612185 −0.306093 0.952002i \(-0.599022\pi\)
−0.306093 + 0.952002i \(0.599022\pi\)
\(18\) 0 0
\(19\) 2.18396i 0.501036i 0.968112 + 0.250518i \(0.0806009\pi\)
−0.968112 + 0.250518i \(0.919399\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.18826 1.49886 0.749428 0.662086i \(-0.230328\pi\)
0.749428 + 0.662086i \(0.230328\pi\)
\(24\) 0 0
\(25\) −5.85840 −1.17168
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.39420i 1.55876i 0.626549 + 0.779382i \(0.284467\pi\)
−0.626549 + 0.779382i \(0.715533\pi\)
\(30\) 0 0
\(31\) 5.60297 1.00632 0.503162 0.864192i \(-0.332170\pi\)
0.503162 + 0.864192i \(0.332170\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.81321i 1.65874i
\(36\) 0 0
\(37\) − 5.55709i − 0.913580i −0.889574 0.456790i \(-0.848999\pi\)
0.889574 0.456790i \(-0.151001\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.09899 1.10868 0.554338 0.832291i \(-0.312971\pi\)
0.554338 + 0.832291i \(0.312971\pi\)
\(42\) 0 0
\(43\) − 6.08120i − 0.927374i −0.885999 0.463687i \(-0.846526\pi\)
0.885999 0.463687i \(-0.153474\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.27323 −0.623315 −0.311658 0.950194i \(-0.600884\pi\)
−0.311658 + 0.950194i \(0.600884\pi\)
\(48\) 0 0
\(49\) 1.86863 0.266947
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.30359i 0.316422i 0.987405 + 0.158211i \(0.0505727\pi\)
−0.987405 + 0.158211i \(0.949427\pi\)
\(54\) 0 0
\(55\) 15.2966 2.06259
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.8574i 1.80408i 0.431649 + 0.902042i \(0.357932\pi\)
−0.431649 + 0.902042i \(0.642068\pi\)
\(60\) 0 0
\(61\) − 13.0161i − 1.66655i −0.552862 0.833273i \(-0.686464\pi\)
0.552862 0.833273i \(-0.313536\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.29521 −0.408720
\(66\) 0 0
\(67\) 2.39703i 0.292844i 0.989222 + 0.146422i \(0.0467758\pi\)
−0.989222 + 0.146422i \(0.953224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0246 1.18970 0.594849 0.803838i \(-0.297212\pi\)
0.594849 + 0.803838i \(0.297212\pi\)
\(72\) 0 0
\(73\) 7.23131 0.846361 0.423181 0.906045i \(-0.360914\pi\)
0.423181 + 0.906045i \(0.360914\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 13.8242i − 1.57541i
\(78\) 0 0
\(79\) −10.1262 −1.13929 −0.569644 0.821891i \(-0.692919\pi\)
−0.569644 + 0.821891i \(0.692919\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 11.4615i − 1.25806i −0.777380 0.629031i \(-0.783452\pi\)
0.777380 0.629031i \(-0.216548\pi\)
\(84\) 0 0
\(85\) 8.31745i 0.902154i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.07469 0.749916 0.374958 0.927042i \(-0.377657\pi\)
0.374958 + 0.927042i \(0.377657\pi\)
\(90\) 0 0
\(91\) 2.97802i 0.312182i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.19662 0.738357
\(96\) 0 0
\(97\) 3.60104 0.365630 0.182815 0.983147i \(-0.441479\pi\)
0.182815 + 0.983147i \(0.441479\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.26799i 0.822696i 0.911478 + 0.411348i \(0.134942\pi\)
−0.911478 + 0.411348i \(0.865058\pi\)
\(102\) 0 0
\(103\) 9.79636 0.965264 0.482632 0.875823i \(-0.339681\pi\)
0.482632 + 0.875823i \(0.339681\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.98315i − 0.771760i −0.922549 0.385880i \(-0.873898\pi\)
0.922549 0.385880i \(-0.126102\pi\)
\(108\) 0 0
\(109\) − 5.98977i − 0.573716i −0.957973 0.286858i \(-0.907389\pi\)
0.957973 0.286858i \(-0.0926108\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.2287 1.52667 0.763334 0.646004i \(-0.223561\pi\)
0.763334 + 0.646004i \(0.223561\pi\)
\(114\) 0 0
\(115\) − 23.6868i − 2.20881i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.51684 0.689068
\(120\) 0 0
\(121\) −10.5487 −0.958976
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.82861i 0.252998i
\(126\) 0 0
\(127\) −6.37434 −0.565631 −0.282815 0.959174i \(-0.591268\pi\)
−0.282815 + 0.959174i \(0.591268\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.98315i 0.348010i 0.984745 + 0.174005i \(0.0556708\pi\)
−0.984745 + 0.174005i \(0.944329\pi\)
\(132\) 0 0
\(133\) − 6.50390i − 0.563960i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.6295 −0.993576 −0.496788 0.867872i \(-0.665487\pi\)
−0.496788 + 0.867872i \(0.665487\pi\)
\(138\) 0 0
\(139\) 17.3060i 1.46787i 0.679217 + 0.733937i \(0.262319\pi\)
−0.679217 + 0.733937i \(0.737681\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.64206 0.388189
\(144\) 0 0
\(145\) 27.6607 2.29709
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 16.7970i − 1.37606i −0.725682 0.688031i \(-0.758475\pi\)
0.725682 0.688031i \(-0.241525\pi\)
\(150\) 0 0
\(151\) 1.20051 0.0976964 0.0488482 0.998806i \(-0.484445\pi\)
0.0488482 + 0.998806i \(0.484445\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 18.4630i − 1.48298i
\(156\) 0 0
\(157\) − 6.81445i − 0.543852i −0.962318 0.271926i \(-0.912339\pi\)
0.962318 0.271926i \(-0.0876606\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −21.4068 −1.68709
\(162\) 0 0
\(163\) − 2.51916i − 0.197316i −0.995121 0.0986580i \(-0.968545\pi\)
0.995121 0.0986580i \(-0.0314550\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.53908 0.428627 0.214313 0.976765i \(-0.431249\pi\)
0.214313 + 0.976765i \(0.431249\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 16.9366i − 1.28766i −0.765167 0.643832i \(-0.777343\pi\)
0.765167 0.643832i \(-0.222657\pi\)
\(174\) 0 0
\(175\) 17.4465 1.31883
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 9.81321i − 0.733474i −0.930325 0.366737i \(-0.880475\pi\)
0.930325 0.366737i \(-0.119525\pi\)
\(180\) 0 0
\(181\) − 3.95056i − 0.293643i −0.989163 0.146821i \(-0.953096\pi\)
0.989163 0.146821i \(-0.0469042\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −18.3118 −1.34631
\(186\) 0 0
\(187\) − 11.7170i − 0.856835i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.91757 0.572895 0.286448 0.958096i \(-0.407526\pi\)
0.286448 + 0.958096i \(0.407526\pi\)
\(192\) 0 0
\(193\) 17.9334 1.29087 0.645436 0.763814i \(-0.276676\pi\)
0.645436 + 0.763814i \(0.276676\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.95469i − 0.139266i −0.997573 0.0696329i \(-0.977817\pi\)
0.997573 0.0696329i \(-0.0221828\pi\)
\(198\) 0 0
\(199\) −1.75434 −0.124362 −0.0621811 0.998065i \(-0.519806\pi\)
−0.0621811 + 0.998065i \(0.519806\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 24.9981i − 1.75453i
\(204\) 0 0
\(205\) − 23.3927i − 1.63381i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.1381 −0.701267
\(210\) 0 0
\(211\) − 14.9125i − 1.02662i −0.858204 0.513309i \(-0.828419\pi\)
0.858204 0.513309i \(-0.171581\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −20.0388 −1.36664
\(216\) 0 0
\(217\) −16.6858 −1.13271
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.52410i 0.169790i
\(222\) 0 0
\(223\) 9.51121 0.636918 0.318459 0.947937i \(-0.396835\pi\)
0.318459 + 0.947937i \(0.396835\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.2139i 1.00978i 0.863183 + 0.504891i \(0.168467\pi\)
−0.863183 + 0.504891i \(0.831533\pi\)
\(228\) 0 0
\(229\) 28.1184i 1.85812i 0.369930 + 0.929059i \(0.379382\pi\)
−0.369930 + 0.929059i \(0.620618\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.4284 1.01075 0.505375 0.862900i \(-0.331354\pi\)
0.505375 + 0.862900i \(0.331354\pi\)
\(234\) 0 0
\(235\) 14.0812i 0.918556i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.14724 −0.268263 −0.134131 0.990964i \(-0.542824\pi\)
−0.134131 + 0.990964i \(0.542824\pi\)
\(240\) 0 0
\(241\) 20.3323 1.30972 0.654860 0.755750i \(-0.272727\pi\)
0.654860 + 0.755750i \(0.272727\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 6.15753i − 0.393390i
\(246\) 0 0
\(247\) 2.18396 0.138962
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.7482i 1.05713i 0.848891 + 0.528567i \(0.177271\pi\)
−0.848891 + 0.528567i \(0.822729\pi\)
\(252\) 0 0
\(253\) 33.3684i 2.09785i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.6709 1.72606 0.863032 0.505149i \(-0.168563\pi\)
0.863032 + 0.505149i \(0.168563\pi\)
\(258\) 0 0
\(259\) 16.5492i 1.02831i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −29.4465 −1.81575 −0.907875 0.419241i \(-0.862296\pi\)
−0.907875 + 0.419241i \(0.862296\pi\)
\(264\) 0 0
\(265\) 7.59080 0.466299
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 2.86814i − 0.174874i −0.996170 0.0874368i \(-0.972132\pi\)
0.996170 0.0874368i \(-0.0278676\pi\)
\(270\) 0 0
\(271\) 2.33228 0.141676 0.0708379 0.997488i \(-0.477433\pi\)
0.0708379 + 0.997488i \(0.477433\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 27.1951i − 1.63992i
\(276\) 0 0
\(277\) − 3.26234i − 0.196015i −0.995186 0.0980076i \(-0.968753\pi\)
0.995186 0.0980076i \(-0.0312469\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 27.6350 1.64856 0.824282 0.566179i \(-0.191579\pi\)
0.824282 + 0.566179i \(0.191579\pi\)
\(282\) 0 0
\(283\) − 19.3872i − 1.15245i −0.817292 0.576224i \(-0.804526\pi\)
0.817292 0.576224i \(-0.195474\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −21.1410 −1.24791
\(288\) 0 0
\(289\) −10.6289 −0.625229
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.4928i 0.846679i 0.905971 + 0.423339i \(0.139142\pi\)
−0.905971 + 0.423339i \(0.860858\pi\)
\(294\) 0 0
\(295\) 45.6631 2.65861
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 7.18826i − 0.415708i
\(300\) 0 0
\(301\) 18.1100i 1.04384i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −42.8909 −2.45593
\(306\) 0 0
\(307\) 10.8749i 0.620663i 0.950628 + 0.310331i \(0.100440\pi\)
−0.950628 + 0.310331i \(0.899560\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.5186 1.56044 0.780219 0.625506i \(-0.215107\pi\)
0.780219 + 0.625506i \(0.215107\pi\)
\(312\) 0 0
\(313\) −31.7653 −1.79548 −0.897741 0.440523i \(-0.854793\pi\)
−0.897741 + 0.440523i \(0.854793\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.7231i 0.714599i 0.933990 + 0.357299i \(0.116302\pi\)
−0.933990 + 0.357299i \(0.883698\pi\)
\(318\) 0 0
\(319\) −38.9664 −2.18170
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 5.51255i − 0.306727i
\(324\) 0 0
\(325\) 5.85840i 0.324966i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.7258 0.701596
\(330\) 0 0
\(331\) 13.8265i 0.759975i 0.924992 + 0.379988i \(0.124072\pi\)
−0.924992 + 0.379988i \(0.875928\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.89873 0.431554
\(336\) 0 0
\(337\) 27.3659 1.49072 0.745358 0.666665i \(-0.232279\pi\)
0.745358 + 0.666665i \(0.232279\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 26.0094i 1.40849i
\(342\) 0 0
\(343\) 15.2813 0.825115
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.9603i 1.12521i 0.826726 + 0.562604i \(0.190200\pi\)
−0.826726 + 0.562604i \(0.809800\pi\)
\(348\) 0 0
\(349\) 9.49826i 0.508430i 0.967148 + 0.254215i \(0.0818171\pi\)
−0.967148 + 0.254215i \(0.918183\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.8275 0.948862 0.474431 0.880293i \(-0.342654\pi\)
0.474431 + 0.880293i \(0.342654\pi\)
\(354\) 0 0
\(355\) − 33.0330i − 1.75321i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.14652 −0.218845 −0.109423 0.993995i \(-0.534900\pi\)
−0.109423 + 0.993995i \(0.534900\pi\)
\(360\) 0 0
\(361\) 14.2303 0.748963
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 23.8287i − 1.24725i
\(366\) 0 0
\(367\) 6.05231 0.315928 0.157964 0.987445i \(-0.449507\pi\)
0.157964 + 0.987445i \(0.449507\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 6.86014i − 0.356161i
\(372\) 0 0
\(373\) 13.6264i 0.705550i 0.935708 + 0.352775i \(0.114762\pi\)
−0.935708 + 0.352775i \(0.885238\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.39420 0.432323
\(378\) 0 0
\(379\) − 4.49791i − 0.231042i −0.993305 0.115521i \(-0.963146\pi\)
0.993305 0.115521i \(-0.0368538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.8991 −1.52777 −0.763887 0.645351i \(-0.776711\pi\)
−0.763887 + 0.645351i \(0.776711\pi\)
\(384\) 0 0
\(385\) −45.5535 −2.32162
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.4869i 0.988025i 0.869455 + 0.494012i \(0.164470\pi\)
−0.869455 + 0.494012i \(0.835530\pi\)
\(390\) 0 0
\(391\) −18.1439 −0.917578
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 33.3680i 1.67893i
\(396\) 0 0
\(397\) 27.1686i 1.36355i 0.731560 + 0.681777i \(0.238793\pi\)
−0.731560 + 0.681777i \(0.761207\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.2577 1.01162 0.505810 0.862645i \(-0.331194\pi\)
0.505810 + 0.862645i \(0.331194\pi\)
\(402\) 0 0
\(403\) − 5.60297i − 0.279104i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.7964 1.27868
\(408\) 0 0
\(409\) 18.4798 0.913769 0.456885 0.889526i \(-0.348965\pi\)
0.456885 + 0.889526i \(0.348965\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 41.2678i − 2.03065i
\(414\) 0 0
\(415\) −37.7680 −1.85396
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 0.857313i − 0.0418825i −0.999781 0.0209412i \(-0.993334\pi\)
0.999781 0.0209412i \(-0.00666629\pi\)
\(420\) 0 0
\(421\) − 9.04512i − 0.440832i −0.975406 0.220416i \(-0.929258\pi\)
0.975406 0.220416i \(-0.0707415\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.7872 0.717285
\(426\) 0 0
\(427\) 38.7624i 1.87584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.6418 1.33146 0.665730 0.746193i \(-0.268120\pi\)
0.665730 + 0.746193i \(0.268120\pi\)
\(432\) 0 0
\(433\) 1.05911 0.0508975 0.0254488 0.999676i \(-0.491899\pi\)
0.0254488 + 0.999676i \(0.491899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.6989i 0.750981i
\(438\) 0 0
\(439\) −1.16146 −0.0554334 −0.0277167 0.999616i \(-0.508824\pi\)
−0.0277167 + 0.999616i \(0.508824\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.38160i 0.398222i 0.979977 + 0.199111i \(0.0638054\pi\)
−0.979977 + 0.199111i \(0.936195\pi\)
\(444\) 0 0
\(445\) − 23.3126i − 1.10512i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.6747 −0.503769 −0.251884 0.967757i \(-0.581050\pi\)
−0.251884 + 0.967757i \(0.581050\pi\)
\(450\) 0 0
\(451\) 32.9540i 1.55174i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.81321 0.460051
\(456\) 0 0
\(457\) 1.80521 0.0844440 0.0422220 0.999108i \(-0.486556\pi\)
0.0422220 + 0.999108i \(0.486556\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.6739i 1.19575i 0.801589 + 0.597876i \(0.203988\pi\)
−0.801589 + 0.597876i \(0.796012\pi\)
\(462\) 0 0
\(463\) 7.38853 0.343374 0.171687 0.985152i \(-0.445078\pi\)
0.171687 + 0.985152i \(0.445078\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.35257i − 0.155139i −0.996987 0.0775693i \(-0.975284\pi\)
0.996987 0.0775693i \(-0.0247159\pi\)
\(468\) 0 0
\(469\) − 7.13843i − 0.329622i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.2293 1.29798
\(474\) 0 0
\(475\) − 12.7945i − 0.587054i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.5015 −0.616900 −0.308450 0.951241i \(-0.599810\pi\)
−0.308450 + 0.951241i \(0.599810\pi\)
\(480\) 0 0
\(481\) −5.55709 −0.253382
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 11.8662i − 0.538816i
\(486\) 0 0
\(487\) 33.7023 1.52720 0.763599 0.645691i \(-0.223430\pi\)
0.763599 + 0.645691i \(0.223430\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 0.546467i − 0.0246617i −0.999924 0.0123309i \(-0.996075\pi\)
0.999924 0.0123309i \(-0.00392513\pi\)
\(492\) 0 0
\(493\) − 21.1878i − 0.954252i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −29.8534 −1.33911
\(498\) 0 0
\(499\) 4.10579i 0.183800i 0.995768 + 0.0919002i \(0.0292941\pi\)
−0.995768 + 0.0919002i \(0.970706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.8449 −0.751079 −0.375539 0.926806i \(-0.622543\pi\)
−0.375539 + 0.926806i \(0.622543\pi\)
\(504\) 0 0
\(505\) 27.2448 1.21238
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.65092i 0.294797i 0.989077 + 0.147398i \(0.0470899\pi\)
−0.989077 + 0.147398i \(0.952910\pi\)
\(510\) 0 0
\(511\) −21.5350 −0.952654
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 32.2810i − 1.42247i
\(516\) 0 0
\(517\) − 19.8366i − 0.872413i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.3712 1.19916 0.599578 0.800316i \(-0.295335\pi\)
0.599578 + 0.800316i \(0.295335\pi\)
\(522\) 0 0
\(523\) 6.59126i 0.288216i 0.989562 + 0.144108i \(0.0460312\pi\)
−0.989562 + 0.144108i \(0.953969\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.1425 −0.616056
\(528\) 0 0
\(529\) 28.6711 1.24657
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 7.09899i − 0.307492i
\(534\) 0 0
\(535\) −26.3062 −1.13731
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.67430i 0.373629i
\(540\) 0 0
\(541\) − 32.3743i − 1.39188i −0.718099 0.695941i \(-0.754988\pi\)
0.718099 0.695941i \(-0.245012\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −19.7375 −0.845463
\(546\) 0 0
\(547\) 18.4957i 0.790817i 0.918505 + 0.395408i \(0.129397\pi\)
−0.918505 + 0.395408i \(0.870603\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −18.3326 −0.780997
\(552\) 0 0
\(553\) 30.1561 1.28237
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 23.4875i − 0.995198i −0.867407 0.497599i \(-0.834215\pi\)
0.867407 0.497599i \(-0.165785\pi\)
\(558\) 0 0
\(559\) −6.08120 −0.257207
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 16.3616i − 0.689560i −0.938683 0.344780i \(-0.887953\pi\)
0.938683 0.344780i \(-0.112047\pi\)
\(564\) 0 0
\(565\) − 53.4770i − 2.24979i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22.0818 −0.925717 −0.462859 0.886432i \(-0.653176\pi\)
−0.462859 + 0.886432i \(0.653176\pi\)
\(570\) 0 0
\(571\) − 36.3287i − 1.52031i −0.649743 0.760154i \(-0.725124\pi\)
0.649743 0.760154i \(-0.274876\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −42.1117 −1.75618
\(576\) 0 0
\(577\) 32.7126 1.36184 0.680921 0.732357i \(-0.261580\pi\)
0.680921 + 0.732357i \(0.261580\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 34.1326i 1.41606i
\(582\) 0 0
\(583\) −10.6934 −0.442875
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.3134i 0.879699i 0.898071 + 0.439850i \(0.144968\pi\)
−0.898071 + 0.439850i \(0.855032\pi\)
\(588\) 0 0
\(589\) 12.2367i 0.504204i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.67265 0.315078 0.157539 0.987513i \(-0.449644\pi\)
0.157539 + 0.987513i \(0.449644\pi\)
\(594\) 0 0
\(595\) − 24.7696i − 1.01545i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.1225 0.944761 0.472381 0.881395i \(-0.343395\pi\)
0.472381 + 0.881395i \(0.343395\pi\)
\(600\) 0 0
\(601\) −24.4418 −0.997001 −0.498501 0.866889i \(-0.666116\pi\)
−0.498501 + 0.866889i \(0.666116\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.7603i 1.41321i
\(606\) 0 0
\(607\) −29.5925 −1.20112 −0.600561 0.799579i \(-0.705056\pi\)
−0.600561 + 0.799579i \(0.705056\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.27323i 0.172877i
\(612\) 0 0
\(613\) 15.3430i 0.619696i 0.950786 + 0.309848i \(0.100278\pi\)
−0.950786 + 0.309848i \(0.899722\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.365995 0.0147344 0.00736720 0.999973i \(-0.497655\pi\)
0.00736720 + 0.999973i \(0.497655\pi\)
\(618\) 0 0
\(619\) − 11.5651i − 0.464839i −0.972616 0.232419i \(-0.925336\pi\)
0.972616 0.232419i \(-0.0746642\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −21.0686 −0.844096
\(624\) 0 0
\(625\) −19.9711 −0.798846
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.0267i 0.559280i
\(630\) 0 0
\(631\) −22.2627 −0.886264 −0.443132 0.896456i \(-0.646133\pi\)
−0.443132 + 0.896456i \(0.646133\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.0048i 0.833549i
\(636\) 0 0
\(637\) − 1.86863i − 0.0740379i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.0104 −1.58032 −0.790159 0.612902i \(-0.790002\pi\)
−0.790159 + 0.612902i \(0.790002\pi\)
\(642\) 0 0
\(643\) − 37.1465i − 1.46492i −0.680813 0.732458i \(-0.738373\pi\)
0.680813 0.732458i \(-0.261627\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.1525 −1.61787 −0.808936 0.587897i \(-0.799956\pi\)
−0.808936 + 0.587897i \(0.799956\pi\)
\(648\) 0 0
\(649\) −64.3270 −2.52506
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.05283i 0.197732i 0.995101 + 0.0988662i \(0.0315216\pi\)
−0.995101 + 0.0988662i \(0.968478\pi\)
\(654\) 0 0
\(655\) 13.1253 0.512849
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 31.3892i − 1.22275i −0.791341 0.611375i \(-0.790617\pi\)
0.791341 0.611375i \(-0.209383\pi\)
\(660\) 0 0
\(661\) − 25.0378i − 0.973859i −0.873441 0.486929i \(-0.838117\pi\)
0.873441 0.486929i \(-0.161883\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −21.4317 −0.831086
\(666\) 0 0
\(667\) 60.3397i 2.33636i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 60.4217 2.33256
\(672\) 0 0
\(673\) −28.8507 −1.11211 −0.556056 0.831145i \(-0.687686\pi\)
−0.556056 + 0.831145i \(0.687686\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 25.8847i − 0.994830i −0.867513 0.497415i \(-0.834283\pi\)
0.867513 0.497415i \(-0.165717\pi\)
\(678\) 0 0
\(679\) −10.7240 −0.411549
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.0440i 1.11134i 0.831404 + 0.555668i \(0.187537\pi\)
−0.831404 + 0.555668i \(0.812463\pi\)
\(684\) 0 0
\(685\) 38.3217i 1.46420i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.30359 0.0877597
\(690\) 0 0
\(691\) 31.6523i 1.20411i 0.798454 + 0.602056i \(0.205652\pi\)
−0.798454 + 0.602056i \(0.794348\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 57.0268 2.16315
\(696\) 0 0
\(697\) −17.9186 −0.678715
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.8106i 1.46586i 0.680306 + 0.732928i \(0.261847\pi\)
−0.680306 + 0.732928i \(0.738153\pi\)
\(702\) 0 0
\(703\) 12.1365 0.457736
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24.6223i − 0.926016i
\(708\) 0 0
\(709\) 1.18538i 0.0445178i 0.999752 + 0.0222589i \(0.00708581\pi\)
−0.999752 + 0.0222589i \(0.992914\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 40.2757 1.50834
\(714\) 0 0
\(715\) − 15.2966i − 0.572059i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.27616 −0.122180 −0.0610902 0.998132i \(-0.519458\pi\)
−0.0610902 + 0.998132i \(0.519458\pi\)
\(720\) 0 0
\(721\) −29.1738 −1.08649
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 49.1766i − 1.82637i
\(726\) 0 0
\(727\) −21.4797 −0.796636 −0.398318 0.917247i \(-0.630406\pi\)
−0.398318 + 0.917247i \(0.630406\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.3496i 0.567724i
\(732\) 0 0
\(733\) 4.61402i 0.170423i 0.996363 + 0.0852114i \(0.0271566\pi\)
−0.996363 + 0.0852114i \(0.972843\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.1272 −0.409875
\(738\) 0 0
\(739\) − 44.9923i − 1.65507i −0.561416 0.827534i \(-0.689743\pi\)
0.561416 0.827534i \(-0.310257\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.1359 0.628654 0.314327 0.949315i \(-0.398221\pi\)
0.314327 + 0.949315i \(0.398221\pi\)
\(744\) 0 0
\(745\) −55.3495 −2.02785
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 23.7740i 0.868684i
\(750\) 0 0
\(751\) 10.5517 0.385037 0.192518 0.981293i \(-0.438335\pi\)
0.192518 + 0.981293i \(0.438335\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 3.95594i − 0.143972i
\(756\) 0 0
\(757\) − 45.4487i − 1.65186i −0.563773 0.825930i \(-0.690651\pi\)
0.563773 0.825930i \(-0.309349\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −33.3202 −1.20786 −0.603928 0.797039i \(-0.706399\pi\)
−0.603928 + 0.797039i \(0.706399\pi\)
\(762\) 0 0
\(763\) 17.8377i 0.645767i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.8574 0.500363
\(768\) 0 0
\(769\) −14.6076 −0.526762 −0.263381 0.964692i \(-0.584838\pi\)
−0.263381 + 0.964692i \(0.584838\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.9736i 0.682433i 0.939985 + 0.341217i \(0.110839\pi\)
−0.939985 + 0.341217i \(0.889161\pi\)
\(774\) 0 0
\(775\) −32.8245 −1.17909
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 15.5039i 0.555487i
\(780\) 0 0
\(781\) 46.5347i 1.66514i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −22.4550 −0.801455
\(786\) 0 0
\(787\) 27.1995i 0.969558i 0.874637 + 0.484779i \(0.161100\pi\)
−0.874637 + 0.484779i \(0.838900\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.3295 −1.71840
\(792\) 0 0
\(793\) −13.0161 −0.462217
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.9799i 0.601458i 0.953710 + 0.300729i \(0.0972300\pi\)
−0.953710 + 0.300729i \(0.902770\pi\)
\(798\) 0 0
\(799\) 10.7861 0.381584
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.5682i 1.18460i
\(804\) 0 0
\(805\) 70.5400i 2.48621i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26.1302 −0.918688 −0.459344 0.888259i \(-0.651915\pi\)
−0.459344 + 0.888259i \(0.651915\pi\)
\(810\) 0 0
\(811\) − 30.5983i − 1.07445i −0.843438 0.537226i \(-0.819472\pi\)
0.843438 0.537226i \(-0.180528\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.30116 −0.290777
\(816\) 0 0
\(817\) 13.2811 0.464647
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 13.2629i − 0.462878i −0.972849 0.231439i \(-0.925657\pi\)
0.972849 0.231439i \(-0.0743434\pi\)
\(822\) 0 0
\(823\) −26.7159 −0.931257 −0.465629 0.884980i \(-0.654172\pi\)
−0.465629 + 0.884980i \(0.654172\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.39923i 0.292070i 0.989279 + 0.146035i \(0.0466512\pi\)
−0.989279 + 0.146035i \(0.953349\pi\)
\(828\) 0 0
\(829\) 20.1364i 0.699365i 0.936868 + 0.349683i \(0.113711\pi\)
−0.936868 + 0.349683i \(0.886289\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.71662 −0.163421
\(834\) 0 0
\(835\) − 18.2524i − 0.631651i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.2636 1.11386 0.556931 0.830559i \(-0.311979\pi\)
0.556931 + 0.830559i \(0.311979\pi\)
\(840\) 0 0
\(841\) −41.4626 −1.42975
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.29521i 0.113359i
\(846\) 0 0
\(847\) 31.4144 1.07941
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 39.9458i − 1.36933i
\(852\) 0 0
\(853\) 32.5703i 1.11519i 0.830114 + 0.557593i \(0.188275\pi\)
−0.830114 + 0.557593i \(0.811725\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.7173 1.11760 0.558801 0.829302i \(-0.311262\pi\)
0.558801 + 0.829302i \(0.311262\pi\)
\(858\) 0 0
\(859\) 12.3546i 0.421533i 0.977536 + 0.210767i \(0.0675960\pi\)
−0.977536 + 0.210767i \(0.932404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.3760 1.17017 0.585086 0.810971i \(-0.301061\pi\)
0.585086 + 0.810971i \(0.301061\pi\)
\(864\) 0 0
\(865\) −55.8096 −1.89758
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 47.0065i − 1.59459i
\(870\) 0 0
\(871\) 2.39703 0.0812204
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 8.42367i − 0.284772i
\(876\) 0 0
\(877\) − 30.2395i − 1.02112i −0.859843 0.510558i \(-0.829439\pi\)
0.859843 0.510558i \(-0.170561\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 32.4097 1.09191 0.545956 0.837814i \(-0.316167\pi\)
0.545956 + 0.837814i \(0.316167\pi\)
\(882\) 0 0
\(883\) − 42.6147i − 1.43410i −0.697023 0.717049i \(-0.745492\pi\)
0.697023 0.717049i \(-0.254508\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.9294 −1.13924 −0.569619 0.821909i \(-0.692909\pi\)
−0.569619 + 0.821909i \(0.692909\pi\)
\(888\) 0 0
\(889\) 18.9829 0.636667
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 9.33259i − 0.312303i
\(894\) 0 0
\(895\) −32.3366 −1.08089
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 47.0325i 1.56862i
\(900\) 0 0
\(901\) − 5.81450i − 0.193709i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.0179 −0.432730
\(906\) 0 0
\(907\) 52.6048i 1.74671i 0.487081 + 0.873357i \(0.338062\pi\)
−0.487081 + 0.873357i \(0.661938\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.6215 −0.683222 −0.341611 0.939841i \(-0.610972\pi\)
−0.341611 + 0.939841i \(0.610972\pi\)
\(912\) 0 0
\(913\) 53.2050 1.76083
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 11.8619i − 0.391715i
\(918\) 0 0
\(919\) 11.0660 0.365034 0.182517 0.983203i \(-0.441576\pi\)
0.182517 + 0.983203i \(0.441576\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 10.0246i − 0.329963i
\(924\) 0 0
\(925\) 32.5557i 1.07042i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.4090 −0.866450 −0.433225 0.901286i \(-0.642624\pi\)
−0.433225 + 0.901286i \(0.642624\pi\)
\(930\) 0 0
\(931\) 4.08102i 0.133750i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −38.6101 −1.26269
\(936\) 0 0
\(937\) 56.6713 1.85137 0.925685 0.378295i \(-0.123489\pi\)
0.925685 + 0.378295i \(0.123489\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.8715i 0.647793i 0.946092 + 0.323897i \(0.104993\pi\)
−0.946092 + 0.323897i \(0.895007\pi\)
\(942\) 0 0
\(943\) 51.0294 1.66175
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 30.3284i − 0.985540i −0.870160 0.492770i \(-0.835984\pi\)
0.870160 0.492770i \(-0.164016\pi\)
\(948\) 0 0
\(949\) − 7.23131i − 0.234738i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.0687 −1.42753 −0.713763 0.700387i \(-0.753011\pi\)
−0.713763 + 0.700387i \(0.753011\pi\)
\(954\) 0 0
\(955\) − 26.0900i − 0.844254i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 34.6330 1.11836
\(960\) 0 0
\(961\) 0.393317 0.0126877
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 59.0942i − 1.90231i
\(966\) 0 0
\(967\) 9.31361 0.299506 0.149753 0.988723i \(-0.452152\pi\)
0.149753 + 0.988723i \(0.452152\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38.4042i 1.23245i 0.787570 + 0.616225i \(0.211339\pi\)
−0.787570 + 0.616225i \(0.788661\pi\)
\(972\) 0 0
\(973\) − 51.5376i − 1.65222i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.84116 0.122890 0.0614448 0.998110i \(-0.480429\pi\)
0.0614448 + 0.998110i \(0.480429\pi\)
\(978\) 0 0
\(979\) 32.8412i 1.04961i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32.0851 1.02336 0.511678 0.859177i \(-0.329024\pi\)
0.511678 + 0.859177i \(0.329024\pi\)
\(984\) 0 0
\(985\) −6.44111 −0.205231
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 43.7132i − 1.39000i
\(990\) 0 0
\(991\) 13.5890 0.431669 0.215835 0.976430i \(-0.430753\pi\)
0.215835 + 0.976430i \(0.430753\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.78093i 0.183268i
\(996\) 0 0
\(997\) 20.7619i 0.657535i 0.944411 + 0.328767i \(0.106633\pi\)
−0.944411 + 0.328767i \(0.893367\pi\)
\(998\) 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.g.e.1873.3 16
3.2 odd 2 1248.2.g.b.625.7 16
4.3 odd 2 936.2.g.e.469.8 16
8.3 odd 2 936.2.g.e.469.7 16
8.5 even 2 inner 3744.2.g.e.1873.14 16
12.11 even 2 312.2.g.b.157.9 16
24.5 odd 2 1248.2.g.b.625.10 16
24.11 even 2 312.2.g.b.157.10 yes 16
48.5 odd 4 9984.2.a.bs.1.2 8
48.11 even 4 9984.2.a.bu.1.2 8
48.29 odd 4 9984.2.a.bv.1.7 8
48.35 even 4 9984.2.a.bt.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.g.b.157.9 16 12.11 even 2
312.2.g.b.157.10 yes 16 24.11 even 2
936.2.g.e.469.7 16 8.3 odd 2
936.2.g.e.469.8 16 4.3 odd 2
1248.2.g.b.625.7 16 3.2 odd 2
1248.2.g.b.625.10 16 24.5 odd 2
3744.2.g.e.1873.3 16 1.1 even 1 trivial
3744.2.g.e.1873.14 16 8.5 even 2 inner
9984.2.a.bs.1.2 8 48.5 odd 4
9984.2.a.bt.1.7 8 48.35 even 4
9984.2.a.bu.1.2 8 48.11 even 4
9984.2.a.bv.1.7 8 48.29 odd 4