Properties

Label 896.2.u.c.113.7
Level $896$
Weight $2$
Character 896.113
Analytic conductor $7.155$
Analytic rank $0$
Dimension $52$
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] = [896,2,Mod(113,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.u (of order \(8\), degree \(4\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 113.7
Character \(\chi\) \(=\) 896.113
Dual form 896.2.u.c.785.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0427122 - 0.103116i) q^{3} +(2.38883 + 0.989485i) q^{5} +(0.707107 + 0.707107i) q^{7} +(2.11251 - 2.11251i) q^{9} +O(q^{10})\) \(q+(-0.0427122 - 0.103116i) q^{3} +(2.38883 + 0.989485i) q^{5} +(0.707107 + 0.707107i) q^{7} +(2.11251 - 2.11251i) q^{9} +(-2.21011 + 5.33568i) q^{11} +(-0.786702 + 0.325863i) q^{13} -0.288590i q^{15} +4.93074i q^{17} +(4.25079 - 1.76074i) q^{19} +(0.0427122 - 0.103116i) q^{21} +(1.84321 - 1.84321i) q^{23} +(1.19188 + 1.19188i) q^{25} +(-0.617414 - 0.255741i) q^{27} +(1.76117 + 4.25184i) q^{29} -6.41557 q^{31} +0.644594 q^{33} +(0.989485 + 2.38883i) q^{35} +(-4.10729 - 1.70130i) q^{37} +(0.0672035 + 0.0672035i) q^{39} +(1.92263 - 1.92263i) q^{41} +(3.47055 - 8.37866i) q^{43} +(7.13672 - 2.95613i) q^{45} -2.58511i q^{47} +1.00000i q^{49} +(0.508440 - 0.210603i) q^{51} +(-2.99079 + 7.22041i) q^{53} +(-10.5591 + 10.5591i) q^{55} +(-0.363121 - 0.363121i) q^{57} +(7.67423 + 3.17877i) q^{59} +(5.77294 + 13.9371i) q^{61} +2.98754 q^{63} -2.20173 q^{65} +(3.75199 + 9.05810i) q^{67} +(-0.268792 - 0.111337i) q^{69} +(3.24880 + 3.24880i) q^{71} +(9.07125 - 9.07125i) q^{73} +(0.0719946 - 0.173810i) q^{75} +(-5.33568 + 2.21011i) q^{77} -8.45254i q^{79} -8.88804i q^{81} +(9.24766 - 3.83051i) q^{83} +(-4.87889 + 11.7787i) q^{85} +(0.363210 - 0.363210i) q^{87} +(-7.81906 - 7.81906i) q^{89} +(-0.786702 - 0.325863i) q^{91} +(0.274023 + 0.661550i) q^{93} +11.8966 q^{95} +8.58236 q^{97} +(6.60280 + 15.9406i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q+O(q^{10}) \) Copy content Toggle raw display \( 52 q + 20 q^{23} + 24 q^{27} - 48 q^{33} + 24 q^{39} + 44 q^{43} + 40 q^{45} - 16 q^{51} - 36 q^{53} - 32 q^{55} - 32 q^{61} - 68 q^{63} + 80 q^{65} - 28 q^{67} - 32 q^{69} - 32 q^{75} - 12 q^{77} + 64 q^{85} + 56 q^{87} + 64 q^{95} - 72 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{8}\right)\)

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
\(3\) −0.0427122 0.103116i −0.0246599 0.0595343i 0.911070 0.412252i \(-0.135258\pi\)
−0.935730 + 0.352718i \(0.885258\pi\)
\(4\) 0 0
\(5\) 2.38883 + 0.989485i 1.06832 + 0.442511i 0.846396 0.532554i \(-0.178768\pi\)
0.221920 + 0.975065i \(0.428768\pi\)
\(6\) 0 0
\(7\) 0.707107 + 0.707107i 0.267261 + 0.267261i
\(8\) 0 0
\(9\) 2.11251 2.11251i 0.704171 0.704171i
\(10\) 0 0
\(11\) −2.21011 + 5.33568i −0.666373 + 1.60877i 0.121258 + 0.992621i \(0.461307\pi\)
−0.787632 + 0.616146i \(0.788693\pi\)
\(12\) 0 0
\(13\) −0.786702 + 0.325863i −0.218192 + 0.0903780i −0.489102 0.872227i \(-0.662675\pi\)
0.270910 + 0.962605i \(0.412675\pi\)
\(14\) 0 0
\(15\) 0.288590i 0.0745137i
\(16\) 0 0
\(17\) 4.93074i 1.19588i 0.801541 + 0.597940i \(0.204014\pi\)
−0.801541 + 0.597940i \(0.795986\pi\)
\(18\) 0 0
\(19\) 4.25079 1.76074i 0.975198 0.403940i 0.162554 0.986700i \(-0.448027\pi\)
0.812645 + 0.582759i \(0.198027\pi\)
\(20\) 0 0
\(21\) 0.0427122 0.103116i 0.00932056 0.0225018i
\(22\) 0 0
\(23\) 1.84321 1.84321i 0.384336 0.384336i −0.488326 0.872661i \(-0.662392\pi\)
0.872661 + 0.488326i \(0.162392\pi\)
\(24\) 0 0
\(25\) 1.19188 + 1.19188i 0.238376 + 0.238376i
\(26\) 0 0
\(27\) −0.617414 0.255741i −0.118821 0.0492174i
\(28\) 0 0
\(29\) 1.76117 + 4.25184i 0.327041 + 0.789546i 0.998809 + 0.0487850i \(0.0155349\pi\)
−0.671769 + 0.740761i \(0.734465\pi\)
\(30\) 0 0
\(31\) −6.41557 −1.15227 −0.576135 0.817354i \(-0.695440\pi\)
−0.576135 + 0.817354i \(0.695440\pi\)
\(32\) 0 0
\(33\) 0.644594 0.112209
\(34\) 0 0
\(35\) 0.989485 + 2.38883i 0.167253 + 0.403785i
\(36\) 0 0
\(37\) −4.10729 1.70130i −0.675235 0.279691i 0.0185985 0.999827i \(-0.494080\pi\)
−0.693833 + 0.720136i \(0.744080\pi\)
\(38\) 0 0
\(39\) 0.0672035 + 0.0672035i 0.0107612 + 0.0107612i
\(40\) 0 0
\(41\) 1.92263 1.92263i 0.300264 0.300264i −0.540853 0.841117i \(-0.681899\pi\)
0.841117 + 0.540853i \(0.181899\pi\)
\(42\) 0 0
\(43\) 3.47055 8.37866i 0.529255 1.27773i −0.402758 0.915307i \(-0.631948\pi\)
0.932012 0.362427i \(-0.118052\pi\)
\(44\) 0 0
\(45\) 7.13672 2.95613i 1.06388 0.440673i
\(46\) 0 0
\(47\) 2.58511i 0.377076i −0.982066 0.188538i \(-0.939625\pi\)
0.982066 0.188538i \(-0.0603749\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 0.508440 0.210603i 0.0711958 0.0294903i
\(52\) 0 0
\(53\) −2.99079 + 7.22041i −0.410817 + 0.991799i 0.574102 + 0.818783i \(0.305351\pi\)
−0.984919 + 0.173016i \(0.944649\pi\)
\(54\) 0 0
\(55\) −10.5591 + 10.5591i −1.42379 + 1.42379i
\(56\) 0 0
\(57\) −0.363121 0.363121i −0.0480966 0.0480966i
\(58\) 0 0
\(59\) 7.67423 + 3.17877i 0.999100 + 0.413841i 0.821467 0.570256i \(-0.193156\pi\)
0.177633 + 0.984097i \(0.443156\pi\)
\(60\) 0 0
\(61\) 5.77294 + 13.9371i 0.739149 + 1.78446i 0.609314 + 0.792929i \(0.291445\pi\)
0.129836 + 0.991536i \(0.458555\pi\)
\(62\) 0 0
\(63\) 2.98754 0.376395
\(64\) 0 0
\(65\) −2.20173 −0.273091
\(66\) 0 0
\(67\) 3.75199 + 9.05810i 0.458378 + 1.10662i 0.969054 + 0.246850i \(0.0793954\pi\)
−0.510675 + 0.859774i \(0.670605\pi\)
\(68\) 0 0
\(69\) −0.268792 0.111337i −0.0323588 0.0134035i
\(70\) 0 0
\(71\) 3.24880 + 3.24880i 0.385562 + 0.385562i 0.873101 0.487539i \(-0.162105\pi\)
−0.487539 + 0.873101i \(0.662105\pi\)
\(72\) 0 0
\(73\) 9.07125 9.07125i 1.06171 1.06171i 0.0637427 0.997966i \(-0.479696\pi\)
0.997966 0.0637427i \(-0.0203037\pi\)
\(74\) 0 0
\(75\) 0.0719946 0.173810i 0.00831322 0.0200699i
\(76\) 0 0
\(77\) −5.33568 + 2.21011i −0.608057 + 0.251865i
\(78\) 0 0
\(79\) 8.45254i 0.950985i −0.879720 0.475492i \(-0.842270\pi\)
0.879720 0.475492i \(-0.157730\pi\)
\(80\) 0 0
\(81\) 8.88804i 0.987560i
\(82\) 0 0
\(83\) 9.24766 3.83051i 1.01506 0.420453i 0.187763 0.982214i \(-0.439876\pi\)
0.827299 + 0.561762i \(0.189876\pi\)
\(84\) 0 0
\(85\) −4.87889 + 11.7787i −0.529190 + 1.27758i
\(86\) 0 0
\(87\) 0.363210 0.363210i 0.0389402 0.0389402i
\(88\) 0 0
\(89\) −7.81906 7.81906i −0.828819 0.828819i 0.158534 0.987353i \(-0.449323\pi\)
−0.987353 + 0.158534i \(0.949323\pi\)
\(90\) 0 0
\(91\) −0.786702 0.325863i −0.0824688 0.0341597i
\(92\) 0 0
\(93\) 0.274023 + 0.661550i 0.0284149 + 0.0685996i
\(94\) 0 0
\(95\) 11.8966 1.22057
\(96\) 0 0
\(97\) 8.58236 0.871407 0.435704 0.900090i \(-0.356500\pi\)
0.435704 + 0.900090i \(0.356500\pi\)
\(98\) 0 0
\(99\) 6.60280 + 15.9406i 0.663606 + 1.60209i
\(100\) 0 0
\(101\) −16.7673 6.94523i −1.66841 0.691076i −0.669734 0.742601i \(-0.733592\pi\)
−0.998672 + 0.0515244i \(0.983592\pi\)
\(102\) 0 0
\(103\) −7.88000 7.88000i −0.776439 0.776439i 0.202784 0.979223i \(-0.435001\pi\)
−0.979223 + 0.202784i \(0.935001\pi\)
\(104\) 0 0
\(105\) 0.204064 0.204064i 0.0199146 0.0199146i
\(106\) 0 0
\(107\) −1.15732 + 2.79402i −0.111882 + 0.270108i −0.969896 0.243519i \(-0.921698\pi\)
0.858014 + 0.513627i \(0.171698\pi\)
\(108\) 0 0
\(109\) 9.12306 3.77890i 0.873831 0.361953i 0.0997298 0.995015i \(-0.468202\pi\)
0.774101 + 0.633062i \(0.218202\pi\)
\(110\) 0 0
\(111\) 0.496195i 0.0470968i
\(112\) 0 0
\(113\) 1.15922i 0.109050i −0.998512 0.0545252i \(-0.982635\pi\)
0.998512 0.0545252i \(-0.0173645\pi\)
\(114\) 0 0
\(115\) 6.22693 2.57928i 0.580664 0.240519i
\(116\) 0 0
\(117\) −0.973528 + 2.35031i −0.0900027 + 0.217286i
\(118\) 0 0
\(119\) −3.48656 + 3.48656i −0.319612 + 0.319612i
\(120\) 0 0
\(121\) −15.8067 15.8067i −1.43697 1.43697i
\(122\) 0 0
\(123\) −0.280374 0.116135i −0.0252805 0.0104715i
\(124\) 0 0
\(125\) −3.27957 7.91759i −0.293334 0.708171i
\(126\) 0 0
\(127\) −7.07162 −0.627505 −0.313752 0.949505i \(-0.601586\pi\)
−0.313752 + 0.949505i \(0.601586\pi\)
\(128\) 0 0
\(129\) −1.01221 −0.0891203
\(130\) 0 0
\(131\) −0.643982 1.55471i −0.0562650 0.135836i 0.893247 0.449566i \(-0.148421\pi\)
−0.949512 + 0.313730i \(0.898421\pi\)
\(132\) 0 0
\(133\) 4.25079 + 1.76074i 0.368590 + 0.152675i
\(134\) 0 0
\(135\) −1.22184 1.22184i −0.105159 0.105159i
\(136\) 0 0
\(137\) 1.58507 1.58507i 0.135422 0.135422i −0.636147 0.771568i \(-0.719473\pi\)
0.771568 + 0.636147i \(0.219473\pi\)
\(138\) 0 0
\(139\) 0.854676 2.06337i 0.0724927 0.175013i −0.883480 0.468469i \(-0.844806\pi\)
0.955972 + 0.293457i \(0.0948056\pi\)
\(140\) 0 0
\(141\) −0.266567 + 0.110416i −0.0224490 + 0.00929867i
\(142\) 0 0
\(143\) 4.91778i 0.411245i
\(144\) 0 0
\(145\) 11.8995i 0.988204i
\(146\) 0 0
\(147\) 0.103116 0.0427122i 0.00850489 0.00352284i
\(148\) 0 0
\(149\) 1.40745 3.39788i 0.115303 0.278366i −0.855684 0.517499i \(-0.826863\pi\)
0.970987 + 0.239133i \(0.0768632\pi\)
\(150\) 0 0
\(151\) −15.8351 + 15.8351i −1.28864 + 1.28864i −0.353026 + 0.935614i \(0.614847\pi\)
−0.935614 + 0.353026i \(0.885153\pi\)
\(152\) 0 0
\(153\) 10.4162 + 10.4162i 0.842103 + 0.842103i
\(154\) 0 0
\(155\) −15.3257 6.34811i −1.23099 0.509893i
\(156\) 0 0
\(157\) −7.08115 17.0954i −0.565137 1.36436i −0.905612 0.424108i \(-0.860588\pi\)
0.340475 0.940254i \(-0.389412\pi\)
\(158\) 0 0
\(159\) 0.872285 0.0691767
\(160\) 0 0
\(161\) 2.60669 0.205436
\(162\) 0 0
\(163\) 0.587312 + 1.41790i 0.0460018 + 0.111058i 0.945210 0.326462i \(-0.105857\pi\)
−0.899208 + 0.437521i \(0.855857\pi\)
\(164\) 0 0
\(165\) 1.53982 + 0.637816i 0.119875 + 0.0496539i
\(166\) 0 0
\(167\) −11.3464 11.3464i −0.878009 0.878009i 0.115320 0.993328i \(-0.463211\pi\)
−0.993328 + 0.115320i \(0.963211\pi\)
\(168\) 0 0
\(169\) −8.67967 + 8.67967i −0.667667 + 0.667667i
\(170\) 0 0
\(171\) 5.26027 12.6994i 0.402263 0.971149i
\(172\) 0 0
\(173\) −3.98159 + 1.64923i −0.302715 + 0.125388i −0.528870 0.848703i \(-0.677384\pi\)
0.226156 + 0.974091i \(0.427384\pi\)
\(174\) 0 0
\(175\) 1.68557i 0.127417i
\(176\) 0 0
\(177\) 0.927111i 0.0696860i
\(178\) 0 0
\(179\) −2.06700 + 0.856179i −0.154495 + 0.0639938i −0.458591 0.888648i \(-0.651646\pi\)
0.304096 + 0.952641i \(0.401646\pi\)
\(180\) 0 0
\(181\) −0.180356 + 0.435418i −0.0134058 + 0.0323643i −0.930441 0.366443i \(-0.880576\pi\)
0.917035 + 0.398807i \(0.130576\pi\)
\(182\) 0 0
\(183\) 1.19057 1.19057i 0.0880094 0.0880094i
\(184\) 0 0
\(185\) −8.12821 8.12821i −0.597598 0.597598i
\(186\) 0 0
\(187\) −26.3088 10.8975i −1.92389 0.796902i
\(188\) 0 0
\(189\) −0.255741 0.617414i −0.0186024 0.0449102i
\(190\) 0 0
\(191\) 11.8891 0.860267 0.430133 0.902765i \(-0.358466\pi\)
0.430133 + 0.902765i \(0.358466\pi\)
\(192\) 0 0
\(193\) −15.1747 −1.09230 −0.546148 0.837689i \(-0.683906\pi\)
−0.546148 + 0.837689i \(0.683906\pi\)
\(194\) 0 0
\(195\) 0.0940407 + 0.227034i 0.00673440 + 0.0162583i
\(196\) 0 0
\(197\) −1.00227 0.415154i −0.0714088 0.0295785i 0.346693 0.937979i \(-0.387305\pi\)
−0.418102 + 0.908400i \(0.637305\pi\)
\(198\) 0 0
\(199\) −0.912005 0.912005i −0.0646504 0.0646504i 0.674042 0.738693i \(-0.264557\pi\)
−0.738693 + 0.674042i \(0.764557\pi\)
\(200\) 0 0
\(201\) 0.773783 0.773783i 0.0545784 0.0545784i
\(202\) 0 0
\(203\) −1.76117 + 4.25184i −0.123610 + 0.298420i
\(204\) 0 0
\(205\) 6.49523 2.69041i 0.453647 0.187907i
\(206\) 0 0
\(207\) 7.78760i 0.541276i
\(208\) 0 0
\(209\) 26.5723i 1.83804i
\(210\) 0 0
\(211\) −12.6330 + 5.23275i −0.869689 + 0.360237i −0.772489 0.635028i \(-0.780989\pi\)
−0.0972000 + 0.995265i \(0.530989\pi\)
\(212\) 0 0
\(213\) 0.196241 0.473768i 0.0134462 0.0324621i
\(214\) 0 0
\(215\) 16.5811 16.5811i 1.13082 1.13082i
\(216\) 0 0
\(217\) −4.53650 4.53650i −0.307957 0.307957i
\(218\) 0 0
\(219\) −1.32285 0.547941i −0.0893897 0.0370264i
\(220\) 0 0
\(221\) −1.60674 3.87902i −0.108081 0.260931i
\(222\) 0 0
\(223\) 1.27134 0.0851354 0.0425677 0.999094i \(-0.486446\pi\)
0.0425677 + 0.999094i \(0.486446\pi\)
\(224\) 0 0
\(225\) 5.03573 0.335715
\(226\) 0 0
\(227\) −7.45381 17.9951i −0.494727 1.19438i −0.952289 0.305199i \(-0.901277\pi\)
0.457562 0.889178i \(-0.348723\pi\)
\(228\) 0 0
\(229\) −6.50872 2.69600i −0.430109 0.178157i 0.157117 0.987580i \(-0.449780\pi\)
−0.587226 + 0.809423i \(0.699780\pi\)
\(230\) 0 0
\(231\) 0.455797 + 0.455797i 0.0299892 + 0.0299892i
\(232\) 0 0
\(233\) 13.6132 13.6132i 0.891831 0.891831i −0.102864 0.994695i \(-0.532801\pi\)
0.994695 + 0.102864i \(0.0328008\pi\)
\(234\) 0 0
\(235\) 2.55792 6.17537i 0.166860 0.402837i
\(236\) 0 0
\(237\) −0.871595 + 0.361026i −0.0566162 + 0.0234512i
\(238\) 0 0
\(239\) 11.2693i 0.728950i −0.931213 0.364475i \(-0.881248\pi\)
0.931213 0.364475i \(-0.118752\pi\)
\(240\) 0 0
\(241\) 26.1689i 1.68568i −0.538161 0.842842i \(-0.680881\pi\)
0.538161 0.842842i \(-0.319119\pi\)
\(242\) 0 0
\(243\) −2.76874 + 1.14685i −0.177615 + 0.0735705i
\(244\) 0 0
\(245\) −0.989485 + 2.38883i −0.0632159 + 0.152617i
\(246\) 0 0
\(247\) −2.77035 + 2.77035i −0.176273 + 0.176273i
\(248\) 0 0
\(249\) −0.789976 0.789976i −0.0500627 0.0500627i
\(250\) 0 0
\(251\) 8.49612 + 3.51921i 0.536270 + 0.222130i 0.634347 0.773048i \(-0.281269\pi\)
−0.0980767 + 0.995179i \(0.531269\pi\)
\(252\) 0 0
\(253\) 5.76107 + 13.9085i 0.362196 + 0.874417i
\(254\) 0 0
\(255\) 1.42296 0.0891094
\(256\) 0 0
\(257\) 25.6035 1.59710 0.798552 0.601925i \(-0.205599\pi\)
0.798552 + 0.601925i \(0.205599\pi\)
\(258\) 0 0
\(259\) −1.70130 4.10729i −0.105713 0.255215i
\(260\) 0 0
\(261\) 12.7025 + 5.26156i 0.786267 + 0.325683i
\(262\) 0 0
\(263\) 7.13840 + 7.13840i 0.440173 + 0.440173i 0.892070 0.451897i \(-0.149253\pi\)
−0.451897 + 0.892070i \(0.649253\pi\)
\(264\) 0 0
\(265\) −14.2890 + 14.2890i −0.877764 + 0.877764i
\(266\) 0 0
\(267\) −0.472304 + 1.14024i −0.0289045 + 0.0697817i
\(268\) 0 0
\(269\) 15.1259 6.26534i 0.922241 0.382005i 0.129511 0.991578i \(-0.458659\pi\)
0.792730 + 0.609573i \(0.208659\pi\)
\(270\) 0 0
\(271\) 10.5616i 0.641573i −0.947151 0.320787i \(-0.896053\pi\)
0.947151 0.320787i \(-0.103947\pi\)
\(272\) 0 0
\(273\) 0.0950401i 0.00575209i
\(274\) 0 0
\(275\) −8.99368 + 3.72531i −0.542340 + 0.224644i
\(276\) 0 0
\(277\) −0.539748 + 1.30307i −0.0324303 + 0.0782937i −0.939265 0.343193i \(-0.888492\pi\)
0.906835 + 0.421487i \(0.138492\pi\)
\(278\) 0 0
\(279\) −13.5530 + 13.5530i −0.811395 + 0.811395i
\(280\) 0 0
\(281\) 17.4433 + 17.4433i 1.04058 + 1.04058i 0.999141 + 0.0414418i \(0.0131951\pi\)
0.0414418 + 0.999141i \(0.486805\pi\)
\(282\) 0 0
\(283\) −20.5289 8.50335i −1.22032 0.505472i −0.322806 0.946465i \(-0.604626\pi\)
−0.897510 + 0.440993i \(0.854626\pi\)
\(284\) 0 0
\(285\) −0.508131 1.22674i −0.0300991 0.0726656i
\(286\) 0 0
\(287\) 2.71900 0.160498
\(288\) 0 0
\(289\) −7.31217 −0.430127
\(290\) 0 0
\(291\) −0.366572 0.884982i −0.0214888 0.0518786i
\(292\) 0 0
\(293\) −25.7260 10.6561i −1.50293 0.622534i −0.528845 0.848719i \(-0.677375\pi\)
−0.974084 + 0.226185i \(0.927375\pi\)
\(294\) 0 0
\(295\) 15.1871 + 15.1871i 0.884226 + 0.884226i
\(296\) 0 0
\(297\) 2.72910 2.72910i 0.158359 0.158359i
\(298\) 0 0
\(299\) −0.849423 + 2.05069i −0.0491234 + 0.118594i
\(300\) 0 0
\(301\) 8.37866 3.47055i 0.482938 0.200039i
\(302\) 0 0
\(303\) 2.02563i 0.116369i
\(304\) 0 0
\(305\) 39.0056i 2.23345i
\(306\) 0 0
\(307\) 16.8674 6.98671i 0.962673 0.398752i 0.154693 0.987963i \(-0.450561\pi\)
0.807980 + 0.589210i \(0.200561\pi\)
\(308\) 0 0
\(309\) −0.475985 + 1.14913i −0.0270778 + 0.0653717i
\(310\) 0 0
\(311\) −2.94765 + 2.94765i −0.167146 + 0.167146i −0.785724 0.618578i \(-0.787709\pi\)
0.618578 + 0.785724i \(0.287709\pi\)
\(312\) 0 0
\(313\) −16.8460 16.8460i −0.952190 0.952190i 0.0467182 0.998908i \(-0.485124\pi\)
−0.998908 + 0.0467182i \(0.985124\pi\)
\(314\) 0 0
\(315\) 7.13672 + 2.95613i 0.402109 + 0.166559i
\(316\) 0 0
\(317\) 1.90426 + 4.59730i 0.106954 + 0.258210i 0.968291 0.249823i \(-0.0803726\pi\)
−0.861337 + 0.508034i \(0.830373\pi\)
\(318\) 0 0
\(319\) −26.5788 −1.48813
\(320\) 0 0
\(321\) 0.337541 0.0188397
\(322\) 0 0
\(323\) 8.68172 + 20.9595i 0.483064 + 1.16622i
\(324\) 0 0
\(325\) −1.32604 0.549266i −0.0735557 0.0304678i
\(326\) 0 0
\(327\) −0.779332 0.779332i −0.0430972 0.0430972i
\(328\) 0 0
\(329\) 1.82795 1.82795i 0.100778 0.100778i
\(330\) 0 0
\(331\) −4.83420 + 11.6708i −0.265711 + 0.641484i −0.999272 0.0381394i \(-0.987857\pi\)
0.733561 + 0.679624i \(0.237857\pi\)
\(332\) 0 0
\(333\) −12.2707 + 5.08270i −0.672431 + 0.278530i
\(334\) 0 0
\(335\) 25.3508i 1.38506i
\(336\) 0 0
\(337\) 7.15072i 0.389524i −0.980850 0.194762i \(-0.937606\pi\)
0.980850 0.194762i \(-0.0623935\pi\)
\(338\) 0 0
\(339\) −0.119535 + 0.0495129i −0.00649224 + 0.00268917i
\(340\) 0 0
\(341\) 14.1791 34.2314i 0.767843 1.85374i
\(342\) 0 0
\(343\) −0.707107 + 0.707107i −0.0381802 + 0.0381802i
\(344\) 0 0
\(345\) −0.531932 0.531932i −0.0286382 0.0286382i
\(346\) 0 0
\(347\) 26.2900 + 10.8897i 1.41132 + 0.584589i 0.952664 0.304024i \(-0.0983303\pi\)
0.458658 + 0.888613i \(0.348330\pi\)
\(348\) 0 0
\(349\) 0.908593 + 2.19354i 0.0486359 + 0.117417i 0.946330 0.323201i \(-0.104759\pi\)
−0.897694 + 0.440619i \(0.854759\pi\)
\(350\) 0 0
\(351\) 0.569057 0.0303740
\(352\) 0 0
\(353\) −21.1802 −1.12731 −0.563655 0.826010i \(-0.690605\pi\)
−0.563655 + 0.826010i \(0.690605\pi\)
\(354\) 0 0
\(355\) 4.54619 + 10.9755i 0.241287 + 0.582518i
\(356\) 0 0
\(357\) 0.508440 + 0.210603i 0.0269095 + 0.0111463i
\(358\) 0 0
\(359\) 14.7991 + 14.7991i 0.781068 + 0.781068i 0.980011 0.198943i \(-0.0637508\pi\)
−0.198943 + 0.980011i \(0.563751\pi\)
\(360\) 0 0
\(361\) 1.53400 1.53400i 0.0807371 0.0807371i
\(362\) 0 0
\(363\) −0.954790 + 2.30507i −0.0501135 + 0.120985i
\(364\) 0 0
\(365\) 30.6455 12.6938i 1.60406 0.664423i
\(366\) 0 0
\(367\) 30.1594i 1.57431i −0.616757 0.787153i \(-0.711554\pi\)
0.616757 0.787153i \(-0.288446\pi\)
\(368\) 0 0
\(369\) 8.12314i 0.422874i
\(370\) 0 0
\(371\) −7.22041 + 2.99079i −0.374865 + 0.155274i
\(372\) 0 0
\(373\) −6.61011 + 15.9582i −0.342259 + 0.826286i 0.655228 + 0.755431i \(0.272573\pi\)
−0.997487 + 0.0708544i \(0.977427\pi\)
\(374\) 0 0
\(375\) −0.676355 + 0.676355i −0.0349268 + 0.0349268i
\(376\) 0 0
\(377\) −2.77103 2.77103i −0.142715 0.142715i
\(378\) 0 0
\(379\) −11.5376 4.77902i −0.592646 0.245482i 0.0661427 0.997810i \(-0.478931\pi\)
−0.658788 + 0.752328i \(0.728931\pi\)
\(380\) 0 0
\(381\) 0.302044 + 0.729199i 0.0154742 + 0.0373580i
\(382\) 0 0
\(383\) 11.4990 0.587569 0.293785 0.955872i \(-0.405085\pi\)
0.293785 + 0.955872i \(0.405085\pi\)
\(384\) 0 0
\(385\) −14.9329 −0.761050
\(386\) 0 0
\(387\) −10.3684 25.0316i −0.527057 1.27243i
\(388\) 0 0
\(389\) −11.8548 4.91043i −0.601063 0.248968i 0.0613390 0.998117i \(-0.480463\pi\)
−0.662402 + 0.749149i \(0.730463\pi\)
\(390\) 0 0
\(391\) 9.08838 + 9.08838i 0.459619 + 0.459619i
\(392\) 0 0
\(393\) −0.132810 + 0.132810i −0.00669939 + 0.00669939i
\(394\) 0 0
\(395\) 8.36366 20.1917i 0.420821 1.01595i
\(396\) 0 0
\(397\) −3.31721 + 1.37403i −0.166486 + 0.0689607i −0.464370 0.885641i \(-0.653719\pi\)
0.297884 + 0.954602i \(0.403719\pi\)
\(398\) 0 0
\(399\) 0.513531i 0.0257087i
\(400\) 0 0
\(401\) 6.54378i 0.326781i −0.986561 0.163390i \(-0.947757\pi\)
0.986561 0.163390i \(-0.0522430\pi\)
\(402\) 0 0
\(403\) 5.04714 2.09060i 0.251416 0.104140i
\(404\) 0 0
\(405\) 8.79458 21.2320i 0.437006 1.05503i
\(406\) 0 0
\(407\) 18.1551 18.1551i 0.899917 0.899917i
\(408\) 0 0
\(409\) 20.4982 + 20.4982i 1.01357 + 1.01357i 0.999907 + 0.0136658i \(0.00435009\pi\)
0.0136658 + 0.999907i \(0.495650\pi\)
\(410\) 0 0
\(411\) −0.231149 0.0957449i −0.0114017 0.00472275i
\(412\) 0 0
\(413\) 3.17877 + 7.67423i 0.156417 + 0.377624i
\(414\) 0 0
\(415\) 25.8813 1.27046
\(416\) 0 0
\(417\) −0.249272 −0.0122069
\(418\) 0 0
\(419\) −1.78532 4.31013i −0.0872184 0.210564i 0.874252 0.485472i \(-0.161352\pi\)
−0.961470 + 0.274909i \(0.911352\pi\)
\(420\) 0 0
\(421\) −5.34745 2.21499i −0.260619 0.107952i 0.248548 0.968620i \(-0.420047\pi\)
−0.509167 + 0.860668i \(0.670047\pi\)
\(422\) 0 0
\(423\) −5.46107 5.46107i −0.265526 0.265526i
\(424\) 0 0
\(425\) −5.87685 + 5.87685i −0.285069 + 0.285069i
\(426\) 0 0
\(427\) −5.77294 + 13.9371i −0.279372 + 0.674464i
\(428\) 0 0
\(429\) −0.507104 + 0.210049i −0.0244832 + 0.0101413i
\(430\) 0 0
\(431\) 16.4021i 0.790063i 0.918668 + 0.395031i \(0.129266\pi\)
−0.918668 + 0.395031i \(0.870734\pi\)
\(432\) 0 0
\(433\) 35.7323i 1.71719i 0.512659 + 0.858593i \(0.328661\pi\)
−0.512659 + 0.858593i \(0.671339\pi\)
\(434\) 0 0
\(435\) 1.22704 0.508256i 0.0588320 0.0243690i
\(436\) 0 0
\(437\) 4.58969 11.0805i 0.219555 0.530052i
\(438\) 0 0
\(439\) 4.67257 4.67257i 0.223010 0.223010i −0.586755 0.809765i \(-0.699595\pi\)
0.809765 + 0.586755i \(0.199595\pi\)
\(440\) 0 0
\(441\) 2.11251 + 2.11251i 0.100596 + 0.100596i
\(442\) 0 0
\(443\) −12.6836 5.25371i −0.602615 0.249611i 0.0604524 0.998171i \(-0.480746\pi\)
−0.663067 + 0.748560i \(0.730746\pi\)
\(444\) 0 0
\(445\) −10.9415 26.4152i −0.518679 1.25220i
\(446\) 0 0
\(447\) −0.410493 −0.0194156
\(448\) 0 0
\(449\) −23.9736 −1.13139 −0.565693 0.824616i \(-0.691391\pi\)
−0.565693 + 0.824616i \(0.691391\pi\)
\(450\) 0 0
\(451\) 6.00930 + 14.5077i 0.282967 + 0.683143i
\(452\) 0 0
\(453\) 2.30920 + 0.956504i 0.108496 + 0.0449405i
\(454\) 0 0
\(455\) −1.55686 1.55686i −0.0729867 0.0729867i
\(456\) 0 0
\(457\) 0.390350 0.390350i 0.0182598 0.0182598i −0.697918 0.716178i \(-0.745890\pi\)
0.716178 + 0.697918i \(0.245890\pi\)
\(458\) 0 0
\(459\) 1.26099 3.04430i 0.0588581 0.142096i
\(460\) 0 0
\(461\) −9.71309 + 4.02330i −0.452384 + 0.187384i −0.597229 0.802071i \(-0.703732\pi\)
0.144845 + 0.989454i \(0.453732\pi\)
\(462\) 0 0
\(463\) 40.7995i 1.89612i 0.318099 + 0.948058i \(0.396956\pi\)
−0.318099 + 0.948058i \(0.603044\pi\)
\(464\) 0 0
\(465\) 1.85147i 0.0858599i
\(466\) 0 0
\(467\) 6.10562 2.52903i 0.282535 0.117030i −0.236916 0.971530i \(-0.576137\pi\)
0.519451 + 0.854500i \(0.326137\pi\)
\(468\) 0 0
\(469\) −3.75199 + 9.05810i −0.173251 + 0.418264i
\(470\) 0 0
\(471\) −1.46036 + 1.46036i −0.0672900 + 0.0672900i
\(472\) 0 0
\(473\) 37.0355 + 37.0355i 1.70290 + 1.70290i
\(474\) 0 0
\(475\) 7.16503 + 2.96785i 0.328754 + 0.136174i
\(476\) 0 0
\(477\) 8.93511 + 21.5713i 0.409111 + 0.987681i
\(478\) 0 0
\(479\) 30.0314 1.37217 0.686086 0.727521i \(-0.259327\pi\)
0.686086 + 0.727521i \(0.259327\pi\)
\(480\) 0 0
\(481\) 3.78560 0.172609
\(482\) 0 0
\(483\) −0.111337 0.268792i −0.00506603 0.0122305i
\(484\) 0 0
\(485\) 20.5018 + 8.49212i 0.930938 + 0.385607i
\(486\) 0 0
\(487\) 14.4987 + 14.4987i 0.657001 + 0.657001i 0.954669 0.297668i \(-0.0962090\pi\)
−0.297668 + 0.954669i \(0.596209\pi\)
\(488\) 0 0
\(489\) 0.121123 0.121123i 0.00547737 0.00547737i
\(490\) 0 0
\(491\) −15.6643 + 37.8170i −0.706921 + 1.70666i 0.000632562 1.00000i \(0.499799\pi\)
−0.707554 + 0.706659i \(0.750201\pi\)
\(492\) 0 0
\(493\) −20.9647 + 8.68385i −0.944202 + 0.391101i
\(494\) 0 0
\(495\) 44.6126i 2.00519i
\(496\) 0 0
\(497\) 4.59450i 0.206092i
\(498\) 0 0
\(499\) −20.1154 + 8.33206i −0.900488 + 0.372994i −0.784408 0.620246i \(-0.787033\pi\)
−0.116080 + 0.993240i \(0.537033\pi\)
\(500\) 0 0
\(501\) −0.685368 + 1.65463i −0.0306200 + 0.0739232i
\(502\) 0 0
\(503\) 29.0379 29.0379i 1.29474 1.29474i 0.362914 0.931823i \(-0.381782\pi\)
0.931823 0.362914i \(-0.118218\pi\)
\(504\) 0 0
\(505\) −33.1819 33.1819i −1.47658 1.47658i
\(506\) 0 0
\(507\) 1.26574 + 0.524288i 0.0562137 + 0.0232845i
\(508\) 0 0
\(509\) 12.2966 + 29.6866i 0.545036 + 1.31583i 0.921131 + 0.389253i \(0.127267\pi\)
−0.376095 + 0.926581i \(0.622733\pi\)
\(510\) 0 0
\(511\) 12.8287 0.567507
\(512\) 0 0
\(513\) −3.07479 −0.135755
\(514\) 0 0
\(515\) −11.0268 26.6211i −0.485900 1.17307i
\(516\) 0 0
\(517\) 13.7933 + 5.71337i 0.606628 + 0.251274i
\(518\) 0 0
\(519\) 0.340125 + 0.340125i 0.0149298 + 0.0149298i
\(520\) 0 0
\(521\) −17.0565 + 17.0565i −0.747257 + 0.747257i −0.973963 0.226706i \(-0.927204\pi\)
0.226706 + 0.973963i \(0.427204\pi\)
\(522\) 0 0
\(523\) −3.03979 + 7.33869i −0.132921 + 0.320899i −0.976301 0.216419i \(-0.930562\pi\)
0.843380 + 0.537318i \(0.180562\pi\)
\(524\) 0 0
\(525\) 0.173810 0.0719946i 0.00758570 0.00314210i
\(526\) 0 0
\(527\) 31.6335i 1.37798i
\(528\) 0 0
\(529\) 16.2052i 0.704572i
\(530\) 0 0
\(531\) 22.9271 9.49672i 0.994952 0.412122i
\(532\) 0 0
\(533\) −0.886022 + 2.13905i −0.0383779 + 0.0926524i
\(534\) 0 0
\(535\) −5.52928 + 5.52928i −0.239051 + 0.239051i
\(536\) 0 0
\(537\) 0.176572 + 0.176572i 0.00761965 + 0.00761965i
\(538\) 0 0
\(539\) −5.33568 2.21011i −0.229824 0.0951962i
\(540\) 0 0
\(541\) −16.3582 39.4922i −0.703293 1.69790i −0.716115 0.697983i \(-0.754081\pi\)
0.0128211 0.999918i \(-0.495919\pi\)
\(542\) 0 0
\(543\) 0.0526021 0.00225737
\(544\) 0 0
\(545\) 25.5326 1.09370
\(546\) 0 0
\(547\) −9.91004 23.9250i −0.423723 1.02296i −0.981240 0.192792i \(-0.938246\pi\)
0.557517 0.830166i \(-0.311754\pi\)
\(548\) 0 0
\(549\) 41.6377 + 17.2469i 1.77705 + 0.736080i
\(550\) 0 0
\(551\) 14.9727 + 14.9727i 0.637859 + 0.637859i
\(552\) 0 0
\(553\) 5.97685 5.97685i 0.254161 0.254161i
\(554\) 0 0
\(555\) −0.490978 + 1.18532i −0.0208408 + 0.0503142i
\(556\) 0 0
\(557\) 16.6979 6.91650i 0.707513 0.293061i 0.000237874 1.00000i \(-0.499924\pi\)
0.707275 + 0.706939i \(0.249924\pi\)
\(558\) 0 0
\(559\) 7.72243i 0.326624i
\(560\) 0 0
\(561\) 3.17832i 0.134189i
\(562\) 0 0
\(563\) −14.5187 + 6.01384i −0.611890 + 0.253453i −0.667037 0.745025i \(-0.732438\pi\)
0.0551461 + 0.998478i \(0.482438\pi\)
\(564\) 0 0
\(565\) 1.14703 2.76918i 0.0482560 0.116500i
\(566\) 0 0
\(567\) 6.28479 6.28479i 0.263936 0.263936i
\(568\) 0 0
\(569\) 29.2630 + 29.2630i 1.22677 + 1.22677i 0.965179 + 0.261589i \(0.0842464\pi\)
0.261589 + 0.965179i \(0.415754\pi\)
\(570\) 0 0
\(571\) 14.0802 + 5.83222i 0.589239 + 0.244071i 0.657323 0.753609i \(-0.271689\pi\)
−0.0680840 + 0.997680i \(0.521689\pi\)
\(572\) 0 0
\(573\) −0.507811 1.22596i −0.0212141 0.0512154i
\(574\) 0 0
\(575\) 4.39377 0.183233
\(576\) 0 0
\(577\) 33.0545 1.37608 0.688039 0.725674i \(-0.258472\pi\)
0.688039 + 0.725674i \(0.258472\pi\)
\(578\) 0 0
\(579\) 0.648143 + 1.56476i 0.0269359 + 0.0650290i
\(580\) 0 0
\(581\) 9.24766 + 3.83051i 0.383658 + 0.158916i
\(582\) 0 0
\(583\) −31.9158 31.9158i −1.32182 1.32182i
\(584\) 0 0
\(585\) −4.65118 + 4.65118i −0.192303 + 0.192303i
\(586\) 0 0
\(587\) 10.1001 24.3838i 0.416876 1.00643i −0.566371 0.824150i \(-0.691653\pi\)
0.983247 0.182277i \(-0.0583469\pi\)
\(588\) 0 0
\(589\) −27.2713 + 11.2961i −1.12369 + 0.465449i
\(590\) 0 0
\(591\) 0.121083i 0.00498067i
\(592\) 0 0
\(593\) 34.2521i 1.40656i −0.710911 0.703282i \(-0.751717\pi\)
0.710911 0.703282i \(-0.248283\pi\)
\(594\) 0 0
\(595\) −11.7787 + 4.87889i −0.482879 + 0.200015i
\(596\) 0 0
\(597\) −0.0550889 + 0.132996i −0.00225464 + 0.00544318i
\(598\) 0 0
\(599\) −6.01007 + 6.01007i −0.245565 + 0.245565i −0.819148 0.573583i \(-0.805553\pi\)
0.573583 + 0.819148i \(0.305553\pi\)
\(600\) 0 0
\(601\) 1.40420 + 1.40420i 0.0572784 + 0.0572784i 0.735166 0.677887i \(-0.237104\pi\)
−0.677887 + 0.735166i \(0.737104\pi\)
\(602\) 0 0
\(603\) 27.0615 + 11.2092i 1.10203 + 0.456475i
\(604\) 0 0
\(605\) −22.1190 53.4000i −0.899265 2.17102i
\(606\) 0 0
\(607\) −28.6512 −1.16292 −0.581458 0.813576i \(-0.697518\pi\)
−0.581458 + 0.813576i \(0.697518\pi\)
\(608\) 0 0
\(609\) 0.513657 0.0208144
\(610\) 0 0
\(611\) 0.842389 + 2.03371i 0.0340794 + 0.0822750i
\(612\) 0 0
\(613\) −0.128901 0.0533925i −0.00520625 0.00215650i 0.380079 0.924954i \(-0.375897\pi\)
−0.385285 + 0.922798i \(0.625897\pi\)
\(614\) 0 0
\(615\) −0.554851 0.554851i −0.0223738 0.0223738i
\(616\) 0 0
\(617\) −1.70245 + 1.70245i −0.0685380 + 0.0685380i −0.740545 0.672007i \(-0.765433\pi\)
0.672007 + 0.740545i \(0.265433\pi\)
\(618\) 0 0
\(619\) −5.85543 + 14.1363i −0.235349 + 0.568184i −0.996791 0.0800499i \(-0.974492\pi\)
0.761441 + 0.648234i \(0.224492\pi\)
\(620\) 0 0
\(621\) −1.60941 + 0.666638i −0.0645832 + 0.0267513i
\(622\) 0 0
\(623\) 11.0578i 0.443023i
\(624\) 0 0
\(625\) 30.5867i 1.22347i
\(626\) 0 0
\(627\) 2.74004 1.13496i 0.109426 0.0453259i
\(628\) 0 0
\(629\) 8.38865 20.2520i 0.334477 0.807499i
\(630\) 0 0
\(631\) 12.4333 12.4333i 0.494962 0.494962i −0.414903 0.909865i \(-0.636185\pi\)
0.909865 + 0.414903i \(0.136185\pi\)
\(632\) 0 0
\(633\) 1.07916 + 1.07916i 0.0428929 + 0.0428929i
\(634\) 0 0
\(635\) −16.8929 6.99726i −0.670373 0.277678i
\(636\) 0 0
\(637\) −0.325863 0.786702i −0.0129111 0.0311703i
\(638\) 0 0
\(639\) 13.7263 0.543003
\(640\) 0 0
\(641\) 21.9009 0.865033 0.432517 0.901626i \(-0.357626\pi\)
0.432517 + 0.901626i \(0.357626\pi\)
\(642\) 0 0
\(643\) 11.7001 + 28.2465i 0.461406 + 1.11393i 0.967820 + 0.251642i \(0.0809705\pi\)
−0.506415 + 0.862290i \(0.669030\pi\)
\(644\) 0 0
\(645\) −2.41800 1.00157i −0.0952086 0.0394367i
\(646\) 0 0
\(647\) −1.24919 1.24919i −0.0491107 0.0491107i 0.682125 0.731236i \(-0.261056\pi\)
−0.731236 + 0.682125i \(0.761056\pi\)
\(648\) 0 0
\(649\) −33.9218 + 33.9218i −1.33155 + 1.33155i
\(650\) 0 0
\(651\) −0.274023 + 0.661550i −0.0107398 + 0.0259282i
\(652\) 0 0
\(653\) −15.0420 + 6.23061i −0.588640 + 0.243823i −0.657065 0.753834i \(-0.728202\pi\)
0.0684256 + 0.997656i \(0.478202\pi\)
\(654\) 0 0
\(655\) 4.35115i 0.170013i
\(656\) 0 0
\(657\) 38.3262i 1.49525i
\(658\) 0 0
\(659\) −36.1878 + 14.9895i −1.40968 + 0.583908i −0.952245 0.305336i \(-0.901231\pi\)
−0.457433 + 0.889244i \(0.651231\pi\)
\(660\) 0 0
\(661\) −4.17095 + 10.0696i −0.162231 + 0.391660i −0.984002 0.178158i \(-0.942986\pi\)
0.821771 + 0.569818i \(0.192986\pi\)
\(662\) 0 0
\(663\) −0.331363 + 0.331363i −0.0128691 + 0.0128691i
\(664\) 0 0
\(665\) 8.41218 + 8.41218i 0.326210 + 0.326210i
\(666\) 0 0
\(667\) 11.0832 + 4.59082i 0.429144 + 0.177757i
\(668\) 0 0
\(669\) −0.0543018 0.131096i −0.00209943 0.00506847i
\(670\) 0 0
\(671\) −87.1228 −3.36334
\(672\) 0 0
\(673\) 25.4507 0.981052 0.490526 0.871426i \(-0.336805\pi\)
0.490526 + 0.871426i \(0.336805\pi\)
\(674\) 0 0
\(675\) −0.431071 1.04070i −0.0165919 0.0400564i
\(676\) 0 0
\(677\) −2.08597 0.864037i −0.0801703 0.0332076i 0.342238 0.939613i \(-0.388815\pi\)
−0.422408 + 0.906406i \(0.638815\pi\)
\(678\) 0 0
\(679\) 6.06865 + 6.06865i 0.232893 + 0.232893i
\(680\) 0 0
\(681\) −1.53722 + 1.53722i −0.0589064 + 0.0589064i
\(682\) 0 0
\(683\) 4.34610 10.4924i 0.166299 0.401481i −0.818658 0.574282i \(-0.805281\pi\)
0.984957 + 0.172800i \(0.0552815\pi\)
\(684\) 0 0
\(685\) 5.35486 2.21806i 0.204599 0.0847476i
\(686\) 0 0
\(687\) 0.786308i 0.0299995i
\(688\) 0 0
\(689\) 6.65489i 0.253531i
\(690\) 0 0
\(691\) 21.2685 8.80972i 0.809093 0.335137i 0.0605011 0.998168i \(-0.480730\pi\)
0.748592 + 0.663031i \(0.230730\pi\)
\(692\) 0 0
\(693\) −6.60280 + 15.9406i −0.250820 + 0.605532i
\(694\) 0 0
\(695\) 4.08335 4.08335i 0.154890 0.154890i
\(696\) 0 0
\(697\) 9.47997 + 9.47997i 0.359079 + 0.359079i
\(698\) 0 0
\(699\) −1.98519 0.822294i −0.0750869 0.0311020i
\(700\) 0 0
\(701\) 5.21862 + 12.5989i 0.197105 + 0.475853i 0.991270 0.131849i \(-0.0420915\pi\)
−0.794165 + 0.607702i \(0.792091\pi\)
\(702\) 0 0
\(703\) −20.4548 −0.771467
\(704\) 0 0
\(705\) −0.746036 −0.0280973
\(706\) 0 0
\(707\) −6.94523 16.7673i −0.261202 0.630598i
\(708\) 0 0
\(709\) 7.46044 + 3.09021i 0.280183 + 0.116055i 0.518350 0.855169i \(-0.326547\pi\)
−0.238167 + 0.971224i \(0.576547\pi\)
\(710\) 0 0
\(711\) −17.8561 17.8561i −0.669656 0.669656i
\(712\) 0 0
\(713\) −11.8252 + 11.8252i −0.442859 + 0.442859i
\(714\) 0 0
\(715\) 4.86607 11.7477i 0.181981 0.439340i
\(716\) 0 0
\(717\) −1.16205 + 0.481336i −0.0433975 + 0.0179758i
\(718\) 0 0
\(719\) 40.3483i 1.50474i 0.658742 + 0.752369i \(0.271089\pi\)
−0.658742 + 0.752369i \(0.728911\pi\)
\(720\) 0 0
\(721\) 11.1440i 0.415024i
\(722\) 0 0
\(723\) −2.69844 + 1.11773i −0.100356 + 0.0415688i
\(724\) 0 0
\(725\) −2.96858 + 7.16679i −0.110250 + 0.266168i
\(726\) 0 0
\(727\) −20.1134 + 20.1134i −0.745965 + 0.745965i −0.973719 0.227754i \(-0.926862\pi\)
0.227754 + 0.973719i \(0.426862\pi\)
\(728\) 0 0
\(729\) −18.6179 18.6179i −0.689550 0.689550i
\(730\) 0 0
\(731\) 41.3130 + 17.1124i 1.52802 + 0.632925i
\(732\) 0 0
\(733\) 1.98757 + 4.79842i 0.0734126 + 0.177234i 0.956326 0.292302i \(-0.0944213\pi\)
−0.882913 + 0.469536i \(0.844421\pi\)
\(734\) 0 0
\(735\) 0.288590 0.0106448
\(736\) 0 0
\(737\) −56.6234 −2.08575
\(738\) 0 0
\(739\) −16.2543 39.2413i −0.597923 1.44351i −0.875694 0.482867i \(-0.839596\pi\)
0.277771 0.960647i \(-0.410404\pi\)
\(740\) 0 0
\(741\) 0.403996 + 0.167340i 0.0148412 + 0.00614741i
\(742\) 0 0
\(743\) −19.2443 19.2443i −0.706005 0.706005i 0.259687 0.965693i \(-0.416380\pi\)
−0.965693 + 0.259687i \(0.916380\pi\)
\(744\) 0 0
\(745\) 6.72431 6.72431i 0.246360 0.246360i
\(746\) 0 0
\(747\) 11.4438 27.6278i 0.418707 1.01085i
\(748\) 0 0
\(749\) −2.79402 + 1.15732i −0.102091 + 0.0422876i
\(750\) 0 0
\(751\) 24.7356i 0.902617i −0.892368 0.451308i \(-0.850957\pi\)
0.892368 0.451308i \(-0.149043\pi\)
\(752\) 0 0
\(753\) 1.02640i 0.0374042i
\(754\) 0 0
\(755\) −53.4958 + 22.1587i −1.94691 + 0.806437i
\(756\) 0 0
\(757\) 0.956469 2.30912i 0.0347634 0.0839264i −0.905544 0.424252i \(-0.860537\pi\)
0.940308 + 0.340325i \(0.110537\pi\)
\(758\) 0 0
\(759\) 1.18812 1.18812i 0.0431261 0.0431261i
\(760\) 0 0
\(761\) −8.17156 8.17156i −0.296219 0.296219i 0.543312 0.839531i \(-0.317170\pi\)
−0.839531 + 0.543312i \(0.817170\pi\)
\(762\) 0 0
\(763\) 9.12306 + 3.77890i 0.330277 + 0.136805i
\(764\) 0 0
\(765\) 14.5759 + 35.1893i 0.526992 + 1.27227i
\(766\) 0 0
\(767\) −7.07318 −0.255398
\(768\) 0 0
\(769\) 6.39428 0.230584 0.115292 0.993332i \(-0.463220\pi\)
0.115292 + 0.993332i \(0.463220\pi\)
\(770\) 0 0
\(771\) −1.09358 2.64014i −0.0393844 0.0950824i
\(772\) 0 0
\(773\) 1.86511 + 0.772553i 0.0670833 + 0.0277868i 0.415973 0.909377i \(-0.363441\pi\)
−0.348890 + 0.937164i \(0.613441\pi\)
\(774\) 0 0
\(775\) −7.64660 7.64660i −0.274674 0.274674i
\(776\) 0 0
\(777\) −0.350863 + 0.350863i −0.0125871 + 0.0125871i
\(778\) 0 0
\(779\) 4.78745 11.5579i 0.171528 0.414105i
\(780\) 0 0
\(781\) −24.5148 + 10.1544i −0.877208 + 0.363351i
\(782\) 0 0
\(783\) 3.07554i 0.109911i
\(784\) 0 0
\(785\) 47.8446i 1.70765i
\(786\) 0 0
\(787\) 6.33063 2.62223i 0.225663 0.0934725i −0.266988 0.963700i \(-0.586028\pi\)
0.492650 + 0.870227i \(0.336028\pi\)
\(788\) 0 0
\(789\) 0.431189 1.04098i 0.0153507 0.0370600i
\(790\) 0 0
\(791\) 0.819694 0.819694i 0.0291450 0.0291450i
\(792\) 0 0
\(793\) −9.08317 9.08317i −0.322553 0.322553i
\(794\) 0 0
\(795\) 2.08374 + 0.863113i 0.0739026 + 0.0306115i
\(796\) 0 0
\(797\) −11.7831 28.4469i −0.417379 1.00764i −0.983104 0.183048i \(-0.941404\pi\)
0.565725 0.824594i \(-0.308596\pi\)
\(798\) 0 0
\(799\) 12.7465 0.450938
\(800\) 0 0
\(801\) −33.0357 −1.16726
\(802\) 0 0
\(803\) 28.3528 + 68.4497i 1.00055 + 2.41554i
\(804\) 0 0
\(805\) 6.22693 + 2.57928i 0.219471 + 0.0909077i
\(806\) 0 0
\(807\) −1.29212 1.29212i −0.0454847 0.0454847i
\(808\) 0 0
\(809\) 26.8872 26.8872i 0.945302 0.945302i −0.0532776 0.998580i \(-0.516967\pi\)
0.998580 + 0.0532776i \(0.0169668\pi\)
\(810\) 0 0
\(811\) 4.13213 9.97584i 0.145099 0.350299i −0.834576 0.550893i \(-0.814287\pi\)
0.979674 + 0.200594i \(0.0642872\pi\)
\(812\) 0 0
\(813\) −1.08908 + 0.451110i −0.0381956 + 0.0158211i
\(814\) 0 0
\(815\) 3.96824i 0.139002i
\(816\) 0 0
\(817\) 41.7267i 1.45983i
\(818\) 0 0
\(819\) −2.35031 + 0.973528i −0.0821263 + 0.0340178i
\(820\) 0 0
\(821\) −10.2842 + 24.8283i −0.358922 + 0.866514i 0.636530 + 0.771252i \(0.280369\pi\)
−0.995452 + 0.0952624i \(0.969631\pi\)
\(822\) 0 0
\(823\) −22.7571 + 22.7571i −0.793263 + 0.793263i −0.982023 0.188761i \(-0.939553\pi\)
0.188761 + 0.982023i \(0.439553\pi\)
\(824\) 0 0
\(825\) 0.768280 + 0.768280i 0.0267481 + 0.0267481i
\(826\) 0 0
\(827\) −18.0804 7.48915i −0.628718 0.260423i 0.0454904 0.998965i \(-0.485515\pi\)
−0.674208 + 0.738541i \(0.735515\pi\)
\(828\) 0 0
\(829\) 18.0301 + 43.5285i 0.626211 + 1.51181i 0.844296 + 0.535878i \(0.180019\pi\)
−0.218084 + 0.975930i \(0.569981\pi\)
\(830\) 0 0
\(831\) 0.157421 0.00546088
\(832\) 0 0
\(833\) −4.93074 −0.170840
\(834\) 0 0
\(835\) −15.8775 38.3316i −0.549462 1.32652i
\(836\) 0 0
\(837\) 3.96106 + 1.64073i 0.136914 + 0.0567118i
\(838\) 0 0
\(839\) 0.265001 + 0.265001i 0.00914885 + 0.00914885i 0.711666 0.702518i \(-0.247941\pi\)
−0.702518 + 0.711666i \(0.747941\pi\)
\(840\) 0 0
\(841\) 5.52971 5.52971i 0.190680 0.190680i
\(842\) 0 0
\(843\) 1.05365 2.54374i 0.0362897 0.0876110i
\(844\) 0 0
\(845\) −29.3226 + 12.1458i −1.00873 + 0.417830i
\(846\) 0 0
\(847\) 22.3540i 0.768094i
\(848\) 0 0
\(849\) 2.48006i 0.0851155i
\(850\) 0 0
\(851\) −10.7064 + 4.43475i −0.367012 + 0.152021i
\(852\) 0 0
\(853\) 9.71394 23.4515i 0.332599 0.802965i −0.665785 0.746143i \(-0.731903\pi\)
0.998384 0.0568218i \(-0.0180967\pi\)
\(854\) 0 0
\(855\) 25.1318 25.1318i 0.859488 0.859488i
\(856\) 0 0
\(857\) 1.89081 + 1.89081i 0.0645887 + 0.0645887i 0.738663 0.674075i \(-0.235457\pi\)
−0.674075 + 0.738663i \(0.735457\pi\)
\(858\) 0 0
\(859\) −11.9540 4.95152i −0.407866 0.168944i 0.169311 0.985563i \(-0.445846\pi\)
−0.577177 + 0.816619i \(0.695846\pi\)
\(860\) 0 0
\(861\) −0.116135 0.280374i −0.00395786 0.00955512i
\(862\) 0 0
\(863\) −4.09524 −0.139404 −0.0697018 0.997568i \(-0.522205\pi\)
−0.0697018 + 0.997568i \(0.522205\pi\)
\(864\) 0 0
\(865\) −11.1432 −0.378881
\(866\) 0 0
\(867\) 0.312319 + 0.754004i 0.0106069 + 0.0256073i
\(868\) 0 0
\(869\) 45.1000 + 18.6810i 1.52991 + 0.633711i
\(870\) 0 0
\(871\) −5.90339 5.90339i −0.200029 0.200029i
\(872\) 0 0
\(873\) 18.1303 18.1303i 0.613619 0.613619i
\(874\) 0 0
\(875\) 3.27957 7.91759i 0.110870 0.267663i
\(876\) 0 0
\(877\) 24.3419 10.0827i 0.821967 0.340470i 0.0682495 0.997668i \(-0.478259\pi\)
0.753718 + 0.657198i \(0.228259\pi\)
\(878\) 0 0
\(879\) 3.10792i 0.104827i
\(880\) 0 0
\(881\) 7.47198i 0.251737i −0.992047 0.125869i \(-0.959828\pi\)
0.992047 0.125869i \(-0.0401718\pi\)
\(882\) 0 0
\(883\) 50.0643 20.7373i 1.68480 0.697866i 0.685260 0.728298i \(-0.259688\pi\)
0.999537 + 0.0304324i \(0.00968844\pi\)
\(884\) 0 0
\(885\) 0.917362 2.21471i 0.0308368 0.0744466i
\(886\) 0 0
\(887\) 34.8539 34.8539i 1.17028 1.17028i 0.188137 0.982143i \(-0.439755\pi\)
0.982143 0.188137i \(-0.0602447\pi\)
\(888\) 0 0
\(889\) −5.00039 5.00039i −0.167708 0.167708i
\(890\) 0 0
\(891\) 47.4237 + 19.6435i 1.58875 + 0.658084i
\(892\) 0 0
\(893\) −4.55169 10.9887i −0.152316 0.367724i
\(894\) 0 0
\(895\) −5.78488 −0.193367
\(896\) 0 0
\(897\) 0.247740 0.00827180
\(898\) 0 0
\(899\) −11.2989 27.2780i −0.376839 0.909771i
\(900\) 0 0
\(901\) −35.6019 14.7468i −1.18607 0.491287i
\(902\) 0 0
\(903\) −0.715742 0.715742i −0.0238184 0.0238184i
\(904\) 0 0
\(905\) −0.861678 + 0.861678i −0.0286432 + 0.0286432i
\(906\) 0 0
\(907\) −7.83871 + 18.9243i −0.260280 + 0.628372i −0.998956 0.0456892i \(-0.985452\pi\)
0.738676 + 0.674061i \(0.235452\pi\)
\(908\) 0 0
\(909\) −50.0929 + 20.7492i −1.66148 + 0.688207i
\(910\) 0 0
\(911\) 6.83115i 0.226326i 0.993576 + 0.113163i \(0.0360982\pi\)
−0.993576 + 0.113163i \(0.963902\pi\)
\(912\) 0 0
\(913\) 57.8084i 1.91318i
\(914\) 0 0
\(915\) 4.02211 1.66601i 0.132967 0.0550767i
\(916\) 0 0
\(917\) 0.643982 1.55471i 0.0212662 0.0513411i
\(918\) 0 0
\(919\) 2.10525 2.10525i 0.0694458 0.0694458i −0.671531 0.740977i \(-0.734363\pi\)
0.740977 + 0.671531i \(0.234363\pi\)
\(920\) 0 0
\(921\) −1.44089 1.44089i −0.0474788 0.0474788i
\(922\) 0 0
\(923\) −3.61450 1.49718i −0.118973 0.0492802i
\(924\) 0 0
\(925\) −2.86766 6.92315i −0.0942882 0.227632i
\(926\) 0 0
\(927\) −33.2932 −1.09349
\(928\) 0 0
\(929\) 17.0256 0.558591 0.279295 0.960205i \(-0.409899\pi\)
0.279295 + 0.960205i \(0.409899\pi\)
\(930\) 0 0
\(931\) 1.76074 + 4.25079i 0.0577058 + 0.139314i
\(932\) 0 0
\(933\) 0.429852 + 0.178050i 0.0140727 + 0.00582911i
\(934\) 0 0
\(935\) −52.0644 52.0644i −1.70269 1.70269i
\(936\) 0 0
\(937\) 8.75833 8.75833i 0.286122 0.286122i −0.549423 0.835545i \(-0.685152\pi\)
0.835545 + 0.549423i \(0.185152\pi\)
\(938\) 0 0
\(939\) −1.01757 + 2.45662i −0.0332070 + 0.0801688i
\(940\) 0 0
\(941\) −37.1950 + 15.4067i −1.21252 + 0.502243i −0.895025 0.446016i \(-0.852842\pi\)
−0.317497 + 0.948259i \(0.602842\pi\)
\(942\) 0 0
\(943\) 7.08760i 0.230804i
\(944\) 0 0
\(945\) 1.72795i 0.0562101i
\(946\) 0 0
\(947\) 8.71240 3.60880i 0.283115 0.117270i −0.236607 0.971605i \(-0.576035\pi\)
0.519722 + 0.854335i \(0.326035\pi\)
\(948\) 0 0
\(949\) −4.18039 + 10.0923i −0.135701 + 0.327611i
\(950\) 0 0
\(951\) 0.392722 0.392722i 0.0127349 0.0127349i
\(952\) 0 0
\(953\) −5.07561 5.07561i −0.164415 0.164415i 0.620104 0.784519i \(-0.287090\pi\)
−0.784519 + 0.620104i \(0.787090\pi\)
\(954\) 0 0
\(955\) 28.4011 + 11.7641i 0.919037 + 0.380678i
\(956\) 0 0
\(957\) 1.13524 + 2.74071i 0.0366971 + 0.0885945i
\(958\) 0 0
\(959\) 2.24163 0.0723860
\(960\) 0 0
\(961\) 10.1596 0.327728
\(962\) 0 0
\(963\) 3.45754 + 8.34725i 0.111418 + 0.268986i
\(964\) 0 0
\(965\) −36.2496 15.0151i −1.16692 0.483353i
\(966\) 0 0
\(967\) −9.43070 9.43070i −0.303271 0.303271i 0.539021 0.842292i \(-0.318794\pi\)
−0.842292 + 0.539021i \(0.818794\pi\)
\(968\) 0 0
\(969\) 1.79045 1.79045i 0.0575177 0.0575177i
\(970\) 0 0
\(971\) −16.7488 + 40.4351i −0.537493 + 1.29762i 0.388974 + 0.921249i \(0.372829\pi\)
−0.926468 + 0.376375i \(0.877171\pi\)
\(972\) 0 0
\(973\) 2.06337 0.854676i 0.0661486 0.0273997i
\(974\) 0 0
\(975\) 0.160197i 0.00513042i
\(976\) 0 0
\(977\) 26.5281i 0.848708i −0.905496 0.424354i \(-0.860501\pi\)
0.905496 0.424354i \(-0.139499\pi\)
\(978\) 0 0
\(979\) 59.0010 24.4390i 1.88568 0.781074i
\(980\) 0 0
\(981\) 11.2896 27.2555i 0.360450 0.870202i
\(982\) 0 0
\(983\) −17.3494 + 17.3494i −0.553360 + 0.553360i −0.927409 0.374049i \(-0.877969\pi\)
0.374049 + 0.927409i \(0.377969\pi\)
\(984\) 0 0
\(985\) −1.98346 1.98346i −0.0631983 0.0631983i
\(986\) 0 0
\(987\) −0.266567 0.110416i −0.00848491 0.00351457i
\(988\) 0 0
\(989\) −9.04666 21.8406i −0.287667 0.694490i
\(990\) 0 0
\(991\) 28.9954 0.921069 0.460535 0.887642i \(-0.347658\pi\)
0.460535 + 0.887642i \(0.347658\pi\)
\(992\) 0 0
\(993\) 1.40993 0.0447427
\(994\) 0 0
\(995\) −1.27621 3.08104i −0.0404585 0.0976755i
\(996\) 0 0
\(997\) −6.34404 2.62779i −0.200918 0.0832228i 0.279956 0.960013i \(-0.409680\pi\)
−0.480873 + 0.876790i \(0.659680\pi\)
\(998\) 0 0
\(999\) 2.10081 + 2.10081i 0.0664666 + 0.0664666i
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 896.2.u.c.113.7 52
4.3 odd 2 224.2.u.c.197.8 yes 52
32.13 even 8 inner 896.2.u.c.785.7 52
32.19 odd 8 224.2.u.c.141.8 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.u.c.141.8 52 32.19 odd 8
224.2.u.c.197.8 yes 52 4.3 odd 2
896.2.u.c.113.7 52 1.1 even 1 trivial
896.2.u.c.785.7 52 32.13 even 8 inner