Properties

Label 1620.2.e.a.971.38
Level $1620$
Weight $2$
Character 1620.971
Analytic conductor $12.936$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(971,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1620.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.e (of order \(2\), degree \(1\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 971.38
Character \(\chi\) \(=\) 1620.971
Dual form 1620.2.e.a.971.37

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.03403 + 0.964765i) q^{2} +(0.138457 + 1.99520i) q^{4} +1.00000i q^{5} +3.54880i q^{7} +(-1.78173 + 2.19669i) q^{8} +O(q^{10})\) \(q+(1.03403 + 0.964765i) q^{2} +(0.138457 + 1.99520i) q^{4} +1.00000i q^{5} +3.54880i q^{7} +(-1.78173 + 2.19669i) q^{8} +(-0.964765 + 1.03403i) q^{10} -4.36571 q^{11} +4.76047 q^{13} +(-3.42376 + 3.66958i) q^{14} +(-3.96166 + 0.552498i) q^{16} +3.02111i q^{17} +1.16321i q^{19} +(-1.99520 + 0.138457i) q^{20} +(-4.51430 - 4.21189i) q^{22} +3.11010 q^{23} -1.00000 q^{25} +(4.92249 + 4.59273i) q^{26} +(-7.08057 + 0.491355i) q^{28} -8.14117i q^{29} -4.79418i q^{31} +(-4.62953 - 3.25077i) q^{32} +(-2.91466 + 3.12393i) q^{34} -3.54880 q^{35} -7.83140 q^{37} +(-1.12222 + 1.20280i) q^{38} +(-2.19669 - 1.78173i) q^{40} -4.61999i q^{41} +8.70155i q^{43} +(-0.604462 - 8.71048i) q^{44} +(3.21595 + 3.00052i) q^{46} -8.13862 q^{47} -5.59396 q^{49} +(-1.03403 - 0.964765i) q^{50} +(0.659119 + 9.49810i) q^{52} +11.9836i q^{53} -4.36571i q^{55} +(-7.79559 - 6.32300i) q^{56} +(7.85431 - 8.41825i) q^{58} +0.684441 q^{59} +10.5876 q^{61} +(4.62526 - 4.95735i) q^{62} +(-1.65086 - 7.82781i) q^{64} +4.76047i q^{65} +7.95024i q^{67} +(-6.02772 + 0.418292i) q^{68} +(-3.66958 - 3.42376i) q^{70} -10.6669 q^{71} +9.41743 q^{73} +(-8.09794 - 7.55546i) q^{74} +(-2.32083 + 0.161054i) q^{76} -15.4930i q^{77} +8.75803i q^{79} +(-0.552498 - 3.96166i) q^{80} +(4.45720 - 4.77723i) q^{82} +13.3807 q^{83} -3.02111 q^{85} +(-8.39496 + 8.99771i) q^{86} +(7.77853 - 9.59010i) q^{88} -4.93294i q^{89} +16.8939i q^{91} +(0.430614 + 6.20528i) q^{92} +(-8.41561 - 7.85185i) q^{94} -1.16321 q^{95} -6.64201 q^{97} +(-5.78435 - 5.39686i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 24 q^{16} - 24 q^{22} - 48 q^{25} + 24 q^{28} - 24 q^{34} - 24 q^{40} + 48 q^{46} - 48 q^{49} + 24 q^{58} + 24 q^{64} + 24 q^{76} + 24 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.03403 + 0.964765i 0.731173 + 0.682192i
\(3\) 0 0
\(4\) 0.138457 + 1.99520i 0.0692283 + 0.997601i
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.54880i 1.34132i 0.741765 + 0.670660i \(0.233989\pi\)
−0.741765 + 0.670660i \(0.766011\pi\)
\(8\) −1.78173 + 2.19669i −0.629937 + 0.776646i
\(9\) 0 0
\(10\) −0.964765 + 1.03403i −0.305086 + 0.326991i
\(11\) −4.36571 −1.31631 −0.658156 0.752882i \(-0.728663\pi\)
−0.658156 + 0.752882i \(0.728663\pi\)
\(12\) 0 0
\(13\) 4.76047 1.32032 0.660158 0.751127i \(-0.270489\pi\)
0.660158 + 0.751127i \(0.270489\pi\)
\(14\) −3.42376 + 3.66958i −0.915037 + 0.980737i
\(15\) 0 0
\(16\) −3.96166 + 0.552498i −0.990415 + 0.138124i
\(17\) 3.02111i 0.732726i 0.930472 + 0.366363i \(0.119397\pi\)
−0.930472 + 0.366363i \(0.880603\pi\)
\(18\) 0 0
\(19\) 1.16321i 0.266858i 0.991058 + 0.133429i \(0.0425988\pi\)
−0.991058 + 0.133429i \(0.957401\pi\)
\(20\) −1.99520 + 0.138457i −0.446141 + 0.0309599i
\(21\) 0 0
\(22\) −4.51430 4.21189i −0.962452 0.897977i
\(23\) 3.11010 0.648501 0.324251 0.945971i \(-0.394888\pi\)
0.324251 + 0.945971i \(0.394888\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 4.92249 + 4.59273i 0.965380 + 0.900709i
\(27\) 0 0
\(28\) −7.08057 + 0.491355i −1.33810 + 0.0928573i
\(29\) 8.14117i 1.51178i −0.654701 0.755888i \(-0.727205\pi\)
0.654701 0.755888i \(-0.272795\pi\)
\(30\) 0 0
\(31\) 4.79418i 0.861061i −0.902576 0.430530i \(-0.858327\pi\)
0.902576 0.430530i \(-0.141673\pi\)
\(32\) −4.62953 3.25077i −0.818392 0.574660i
\(33\) 0 0
\(34\) −2.91466 + 3.12393i −0.499860 + 0.535750i
\(35\) −3.54880 −0.599856
\(36\) 0 0
\(37\) −7.83140 −1.28747 −0.643737 0.765247i \(-0.722617\pi\)
−0.643737 + 0.765247i \(0.722617\pi\)
\(38\) −1.12222 + 1.20280i −0.182048 + 0.195119i
\(39\) 0 0
\(40\) −2.19669 1.78173i −0.347327 0.281717i
\(41\) 4.61999i 0.721521i −0.932658 0.360761i \(-0.882517\pi\)
0.932658 0.360761i \(-0.117483\pi\)
\(42\) 0 0
\(43\) 8.70155i 1.32697i 0.748188 + 0.663487i \(0.230924\pi\)
−0.748188 + 0.663487i \(0.769076\pi\)
\(44\) −0.604462 8.71048i −0.0911261 1.31315i
\(45\) 0 0
\(46\) 3.21595 + 3.00052i 0.474167 + 0.442402i
\(47\) −8.13862 −1.18714 −0.593570 0.804783i \(-0.702282\pi\)
−0.593570 + 0.804783i \(0.702282\pi\)
\(48\) 0 0
\(49\) −5.59396 −0.799137
\(50\) −1.03403 0.964765i −0.146235 0.136438i
\(51\) 0 0
\(52\) 0.659119 + 9.49810i 0.0914033 + 1.31715i
\(53\) 11.9836i 1.64607i 0.567991 + 0.823035i \(0.307721\pi\)
−0.567991 + 0.823035i \(0.692279\pi\)
\(54\) 0 0
\(55\) 4.36571i 0.588672i
\(56\) −7.79559 6.32300i −1.04173 0.844947i
\(57\) 0 0
\(58\) 7.85431 8.41825i 1.03132 1.10537i
\(59\) 0.684441 0.0891066 0.0445533 0.999007i \(-0.485814\pi\)
0.0445533 + 0.999007i \(0.485814\pi\)
\(60\) 0 0
\(61\) 10.5876 1.35560 0.677801 0.735245i \(-0.262933\pi\)
0.677801 + 0.735245i \(0.262933\pi\)
\(62\) 4.62526 4.95735i 0.587409 0.629585i
\(63\) 0 0
\(64\) −1.65086 7.82781i −0.206358 0.978477i
\(65\) 4.76047i 0.590463i
\(66\) 0 0
\(67\) 7.95024i 0.971276i 0.874160 + 0.485638i \(0.161413\pi\)
−0.874160 + 0.485638i \(0.838587\pi\)
\(68\) −6.02772 + 0.418292i −0.730968 + 0.0507254i
\(69\) 0 0
\(70\) −3.66958 3.42376i −0.438599 0.409217i
\(71\) −10.6669 −1.26593 −0.632964 0.774181i \(-0.718162\pi\)
−0.632964 + 0.774181i \(0.718162\pi\)
\(72\) 0 0
\(73\) 9.41743 1.10223 0.551114 0.834430i \(-0.314203\pi\)
0.551114 + 0.834430i \(0.314203\pi\)
\(74\) −8.09794 7.55546i −0.941367 0.878305i
\(75\) 0 0
\(76\) −2.32083 + 0.161054i −0.266217 + 0.0184741i
\(77\) 15.4930i 1.76559i
\(78\) 0 0
\(79\) 8.75803i 0.985356i 0.870212 + 0.492678i \(0.163982\pi\)
−0.870212 + 0.492678i \(0.836018\pi\)
\(80\) −0.552498 3.96166i −0.0617712 0.442927i
\(81\) 0 0
\(82\) 4.45720 4.77723i 0.492216 0.527557i
\(83\) 13.3807 1.46872 0.734361 0.678759i \(-0.237482\pi\)
0.734361 + 0.678759i \(0.237482\pi\)
\(84\) 0 0
\(85\) −3.02111 −0.327685
\(86\) −8.39496 + 8.99771i −0.905251 + 0.970248i
\(87\) 0 0
\(88\) 7.77853 9.59010i 0.829194 1.02231i
\(89\) 4.93294i 0.522891i −0.965218 0.261445i \(-0.915801\pi\)
0.965218 0.261445i \(-0.0841991\pi\)
\(90\) 0 0
\(91\) 16.8939i 1.77097i
\(92\) 0.430614 + 6.20528i 0.0448947 + 0.646945i
\(93\) 0 0
\(94\) −8.41561 7.85185i −0.868004 0.809857i
\(95\) −1.16321 −0.119342
\(96\) 0 0
\(97\) −6.64201 −0.674394 −0.337197 0.941434i \(-0.609479\pi\)
−0.337197 + 0.941434i \(0.609479\pi\)
\(98\) −5.78435 5.39686i −0.584308 0.545165i
\(99\) 0 0
\(100\) −0.138457 1.99520i −0.0138457 0.199520i
\(101\) 6.85118i 0.681718i 0.940115 + 0.340859i \(0.110718\pi\)
−0.940115 + 0.340859i \(0.889282\pi\)
\(102\) 0 0
\(103\) 14.3805i 1.41695i −0.705735 0.708476i \(-0.749383\pi\)
0.705735 0.708476i \(-0.250617\pi\)
\(104\) −8.48188 + 10.4573i −0.831717 + 1.02542i
\(105\) 0 0
\(106\) −11.5613 + 12.3914i −1.12294 + 1.20356i
\(107\) −0.485829 −0.0469669 −0.0234835 0.999724i \(-0.507476\pi\)
−0.0234835 + 0.999724i \(0.507476\pi\)
\(108\) 0 0
\(109\) 15.6731 1.50121 0.750607 0.660749i \(-0.229761\pi\)
0.750607 + 0.660749i \(0.229761\pi\)
\(110\) 4.21189 4.51430i 0.401588 0.430422i
\(111\) 0 0
\(112\) −1.96070 14.0591i −0.185269 1.32846i
\(113\) 1.16042i 0.109163i 0.998509 + 0.0545817i \(0.0173825\pi\)
−0.998509 + 0.0545817i \(0.982617\pi\)
\(114\) 0 0
\(115\) 3.11010i 0.290019i
\(116\) 16.2433 1.12720i 1.50815 0.104658i
\(117\) 0 0
\(118\) 0.707736 + 0.660325i 0.0651523 + 0.0607878i
\(119\) −10.7213 −0.982820
\(120\) 0 0
\(121\) 8.05944 0.732676
\(122\) 10.9479 + 10.2145i 0.991180 + 0.924781i
\(123\) 0 0
\(124\) 9.56536 0.663787i 0.858995 0.0596098i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 10.0882i 0.895180i 0.894239 + 0.447590i \(0.147718\pi\)
−0.894239 + 0.447590i \(0.852282\pi\)
\(128\) 5.84495 9.68693i 0.516625 0.856211i
\(129\) 0 0
\(130\) −4.59273 + 4.92249i −0.402809 + 0.431731i
\(131\) −4.79010 −0.418513 −0.209256 0.977861i \(-0.567104\pi\)
−0.209256 + 0.977861i \(0.567104\pi\)
\(132\) 0 0
\(133\) −4.12798 −0.357941
\(134\) −7.67011 + 8.22083i −0.662597 + 0.710171i
\(135\) 0 0
\(136\) −6.63642 5.38280i −0.569069 0.461572i
\(137\) 2.91386i 0.248948i 0.992223 + 0.124474i \(0.0397243\pi\)
−0.992223 + 0.124474i \(0.960276\pi\)
\(138\) 0 0
\(139\) 13.7267i 1.16428i 0.813087 + 0.582142i \(0.197785\pi\)
−0.813087 + 0.582142i \(0.802215\pi\)
\(140\) −0.491355 7.08057i −0.0415270 0.598417i
\(141\) 0 0
\(142\) −11.0300 10.2911i −0.925613 0.863606i
\(143\) −20.7828 −1.73795
\(144\) 0 0
\(145\) 8.14117 0.676087
\(146\) 9.73795 + 9.08561i 0.805919 + 0.751930i
\(147\) 0 0
\(148\) −1.08431 15.6252i −0.0891297 1.28439i
\(149\) 10.4866i 0.859094i −0.903045 0.429547i \(-0.858673\pi\)
0.903045 0.429547i \(-0.141327\pi\)
\(150\) 0 0
\(151\) 23.0452i 1.87539i 0.347455 + 0.937697i \(0.387046\pi\)
−0.347455 + 0.937697i \(0.612954\pi\)
\(152\) −2.55520 2.07252i −0.207254 0.168104i
\(153\) 0 0
\(154\) 14.9471 16.0203i 1.20447 1.29096i
\(155\) 4.79418 0.385078
\(156\) 0 0
\(157\) 14.5160 1.15851 0.579253 0.815148i \(-0.303344\pi\)
0.579253 + 0.815148i \(0.303344\pi\)
\(158\) −8.44944 + 9.05611i −0.672202 + 0.720466i
\(159\) 0 0
\(160\) 3.25077 4.62953i 0.256996 0.365996i
\(161\) 11.0371i 0.869847i
\(162\) 0 0
\(163\) 9.73510i 0.762512i −0.924469 0.381256i \(-0.875492\pi\)
0.924469 0.381256i \(-0.124508\pi\)
\(164\) 9.21781 0.639668i 0.719790 0.0499497i
\(165\) 0 0
\(166\) 13.8361 + 12.9092i 1.07389 + 1.00195i
\(167\) 13.5625 1.04950 0.524751 0.851256i \(-0.324159\pi\)
0.524751 + 0.851256i \(0.324159\pi\)
\(168\) 0 0
\(169\) 9.66206 0.743236
\(170\) −3.12393 2.91466i −0.239595 0.223544i
\(171\) 0 0
\(172\) −17.3614 + 1.20479i −1.32379 + 0.0918642i
\(173\) 13.4117i 1.01967i 0.860271 + 0.509837i \(0.170294\pi\)
−0.860271 + 0.509837i \(0.829706\pi\)
\(174\) 0 0
\(175\) 3.54880i 0.268264i
\(176\) 17.2955 2.41205i 1.30369 0.181815i
\(177\) 0 0
\(178\) 4.75913 5.10083i 0.356712 0.382324i
\(179\) −19.9570 −1.49166 −0.745830 0.666137i \(-0.767947\pi\)
−0.745830 + 0.666137i \(0.767947\pi\)
\(180\) 0 0
\(181\) −17.6204 −1.30971 −0.654857 0.755753i \(-0.727271\pi\)
−0.654857 + 0.755753i \(0.727271\pi\)
\(182\) −16.2987 + 17.4689i −1.20814 + 1.29488i
\(183\) 0 0
\(184\) −5.54137 + 6.83192i −0.408515 + 0.503656i
\(185\) 7.83140i 0.575776i
\(186\) 0 0
\(187\) 13.1893i 0.964496i
\(188\) −1.12685 16.2382i −0.0821837 1.18429i
\(189\) 0 0
\(190\) −1.20280 1.12222i −0.0872600 0.0814144i
\(191\) −17.0170 −1.23131 −0.615654 0.788016i \(-0.711108\pi\)
−0.615654 + 0.788016i \(0.711108\pi\)
\(192\) 0 0
\(193\) 10.7707 0.775292 0.387646 0.921808i \(-0.373288\pi\)
0.387646 + 0.921808i \(0.373288\pi\)
\(194\) −6.86807 6.40798i −0.493099 0.460066i
\(195\) 0 0
\(196\) −0.774521 11.1611i −0.0553229 0.797220i
\(197\) 0.458004i 0.0326314i 0.999867 + 0.0163157i \(0.00519369\pi\)
−0.999867 + 0.0163157i \(0.994806\pi\)
\(198\) 0 0
\(199\) 2.14750i 0.152232i −0.997099 0.0761162i \(-0.975748\pi\)
0.997099 0.0761162i \(-0.0242520\pi\)
\(200\) 1.78173 2.19669i 0.125987 0.155329i
\(201\) 0 0
\(202\) −6.60978 + 7.08436i −0.465062 + 0.498454i
\(203\) 28.8913 2.02777
\(204\) 0 0
\(205\) 4.61999 0.322674
\(206\) 13.8738 14.8699i 0.966633 1.03604i
\(207\) 0 0
\(208\) −18.8594 + 2.63015i −1.30766 + 0.182368i
\(209\) 5.07822i 0.351268i
\(210\) 0 0
\(211\) 2.17455i 0.149702i 0.997195 + 0.0748510i \(0.0238481\pi\)
−0.997195 + 0.0748510i \(0.976152\pi\)
\(212\) −23.9096 + 1.65920i −1.64212 + 0.113955i
\(213\) 0 0
\(214\) −0.502365 0.468711i −0.0343409 0.0320404i
\(215\) −8.70155 −0.593441
\(216\) 0 0
\(217\) 17.0136 1.15496
\(218\) 16.2066 + 15.1209i 1.09765 + 1.02412i
\(219\) 0 0
\(220\) 8.71048 0.604462i 0.587260 0.0407528i
\(221\) 14.3819i 0.967430i
\(222\) 0 0
\(223\) 21.4295i 1.43503i −0.696545 0.717513i \(-0.745280\pi\)
0.696545 0.717513i \(-0.254720\pi\)
\(224\) 11.5363 16.4292i 0.770803 1.09773i
\(225\) 0 0
\(226\) −1.11954 + 1.19992i −0.0744704 + 0.0798174i
\(227\) 8.83418 0.586345 0.293172 0.956060i \(-0.405289\pi\)
0.293172 + 0.956060i \(0.405289\pi\)
\(228\) 0 0
\(229\) 20.0779 1.32678 0.663392 0.748272i \(-0.269116\pi\)
0.663392 + 0.748272i \(0.269116\pi\)
\(230\) −3.00052 + 3.21595i −0.197848 + 0.212054i
\(231\) 0 0
\(232\) 17.8836 + 14.5054i 1.17412 + 0.952324i
\(233\) 8.52014i 0.558173i −0.960266 0.279086i \(-0.909968\pi\)
0.960266 0.279086i \(-0.0900316\pi\)
\(234\) 0 0
\(235\) 8.13862i 0.530905i
\(236\) 0.0947654 + 1.36560i 0.00616870 + 0.0888928i
\(237\) 0 0
\(238\) −11.0862 10.3435i −0.718611 0.670472i
\(239\) 2.93361 0.189760 0.0948798 0.995489i \(-0.469753\pi\)
0.0948798 + 0.995489i \(0.469753\pi\)
\(240\) 0 0
\(241\) 25.8849 1.66739 0.833695 0.552225i \(-0.186221\pi\)
0.833695 + 0.552225i \(0.186221\pi\)
\(242\) 8.33374 + 7.77547i 0.535713 + 0.499826i
\(243\) 0 0
\(244\) 1.46592 + 21.1244i 0.0938461 + 1.35235i
\(245\) 5.59396i 0.357385i
\(246\) 0 0
\(247\) 5.53740i 0.352337i
\(248\) 10.5313 + 8.54195i 0.668739 + 0.542414i
\(249\) 0 0
\(250\) 0.964765 1.03403i 0.0610171 0.0653981i
\(251\) 26.6272 1.68070 0.840348 0.542047i \(-0.182350\pi\)
0.840348 + 0.542047i \(0.182350\pi\)
\(252\) 0 0
\(253\) −13.5778 −0.853630
\(254\) −9.73271 + 10.4315i −0.610685 + 0.654532i
\(255\) 0 0
\(256\) 15.3895 4.37762i 0.961843 0.273601i
\(257\) 1.52448i 0.0950946i 0.998869 + 0.0475473i \(0.0151405\pi\)
−0.998869 + 0.0475473i \(0.984860\pi\)
\(258\) 0 0
\(259\) 27.7921i 1.72691i
\(260\) −9.49810 + 0.659119i −0.589047 + 0.0408768i
\(261\) 0 0
\(262\) −4.95313 4.62132i −0.306005 0.285506i
\(263\) 25.9018 1.59718 0.798588 0.601878i \(-0.205581\pi\)
0.798588 + 0.601878i \(0.205581\pi\)
\(264\) 0 0
\(265\) −11.9836 −0.736145
\(266\) −4.26848 3.98253i −0.261717 0.244185i
\(267\) 0 0
\(268\) −15.8623 + 1.10076i −0.968946 + 0.0672398i
\(269\) 15.0345i 0.916667i 0.888780 + 0.458334i \(0.151554\pi\)
−0.888780 + 0.458334i \(0.848446\pi\)
\(270\) 0 0
\(271\) 2.18871i 0.132955i −0.997788 0.0664773i \(-0.978824\pi\)
0.997788 0.0664773i \(-0.0211760\pi\)
\(272\) −1.66916 11.9686i −0.101207 0.725703i
\(273\) 0 0
\(274\) −2.81119 + 3.01303i −0.169830 + 0.182024i
\(275\) 4.36571 0.263262
\(276\) 0 0
\(277\) 3.05505 0.183560 0.0917800 0.995779i \(-0.470744\pi\)
0.0917800 + 0.995779i \(0.470744\pi\)
\(278\) −13.2431 + 14.1939i −0.794266 + 0.851294i
\(279\) 0 0
\(280\) 6.32300 7.79559i 0.377872 0.465876i
\(281\) 6.76617i 0.403636i −0.979423 0.201818i \(-0.935315\pi\)
0.979423 0.201818i \(-0.0646849\pi\)
\(282\) 0 0
\(283\) 5.92694i 0.352320i 0.984362 + 0.176160i \(0.0563676\pi\)
−0.984362 + 0.176160i \(0.943632\pi\)
\(284\) −1.47690 21.2826i −0.0876381 1.26289i
\(285\) 0 0
\(286\) −21.4902 20.0506i −1.27074 1.18561i
\(287\) 16.3954 0.967790
\(288\) 0 0
\(289\) 7.87291 0.463113
\(290\) 8.41825 + 7.85431i 0.494337 + 0.461221i
\(291\) 0 0
\(292\) 1.30391 + 18.7897i 0.0763053 + 1.09958i
\(293\) 13.5069i 0.789082i 0.918878 + 0.394541i \(0.129096\pi\)
−0.918878 + 0.394541i \(0.870904\pi\)
\(294\) 0 0
\(295\) 0.684441i 0.0398497i
\(296\) 13.9535 17.2031i 0.811028 0.999912i
\(297\) 0 0
\(298\) 10.1171 10.8435i 0.586067 0.628146i
\(299\) 14.8055 0.856227
\(300\) 0 0
\(301\) −30.8801 −1.77990
\(302\) −22.2332 + 23.8296i −1.27938 + 1.37124i
\(303\) 0 0
\(304\) −0.642669 4.60823i −0.0368596 0.264300i
\(305\) 10.5876i 0.606244i
\(306\) 0 0
\(307\) 21.6606i 1.23623i 0.786086 + 0.618117i \(0.212104\pi\)
−0.786086 + 0.618117i \(0.787896\pi\)
\(308\) 30.9117 2.14511i 1.76136 0.122229i
\(309\) 0 0
\(310\) 4.95735 + 4.62526i 0.281559 + 0.262697i
\(311\) 6.40381 0.363127 0.181563 0.983379i \(-0.441884\pi\)
0.181563 + 0.983379i \(0.441884\pi\)
\(312\) 0 0
\(313\) −2.85521 −0.161386 −0.0806930 0.996739i \(-0.525713\pi\)
−0.0806930 + 0.996739i \(0.525713\pi\)
\(314\) 15.0101 + 14.0046i 0.847068 + 0.790323i
\(315\) 0 0
\(316\) −17.4740 + 1.21261i −0.982992 + 0.0682145i
\(317\) 13.6790i 0.768290i −0.923273 0.384145i \(-0.874496\pi\)
0.923273 0.384145i \(-0.125504\pi\)
\(318\) 0 0
\(319\) 35.5420i 1.98997i
\(320\) 7.82781 1.65086i 0.437588 0.0922861i
\(321\) 0 0
\(322\) −10.6482 + 11.4128i −0.593403 + 0.636009i
\(323\) −3.51417 −0.195534
\(324\) 0 0
\(325\) −4.76047 −0.264063
\(326\) 9.39209 10.0664i 0.520180 0.557528i
\(327\) 0 0
\(328\) 10.1487 + 8.23158i 0.560366 + 0.454513i
\(329\) 28.8823i 1.59233i
\(330\) 0 0
\(331\) 26.9698i 1.48239i 0.671289 + 0.741196i \(0.265741\pi\)
−0.671289 + 0.741196i \(0.734259\pi\)
\(332\) 1.85265 + 26.6972i 0.101677 + 1.46520i
\(333\) 0 0
\(334\) 14.0241 + 13.0847i 0.767367 + 0.715961i
\(335\) −7.95024 −0.434368
\(336\) 0 0
\(337\) 0.591145 0.0322017 0.0161009 0.999870i \(-0.494875\pi\)
0.0161009 + 0.999870i \(0.494875\pi\)
\(338\) 9.99091 + 9.32162i 0.543434 + 0.507029i
\(339\) 0 0
\(340\) −0.418292 6.02772i −0.0226851 0.326899i
\(341\) 20.9300i 1.13342i
\(342\) 0 0
\(343\) 4.98975i 0.269421i
\(344\) −19.1146 15.5038i −1.03059 0.835911i
\(345\) 0 0
\(346\) −12.9391 + 13.8682i −0.695613 + 0.745558i
\(347\) 9.47925 0.508873 0.254436 0.967090i \(-0.418110\pi\)
0.254436 + 0.967090i \(0.418110\pi\)
\(348\) 0 0
\(349\) −22.0924 −1.18258 −0.591289 0.806460i \(-0.701381\pi\)
−0.591289 + 0.806460i \(0.701381\pi\)
\(350\) 3.42376 3.66958i 0.183007 0.196147i
\(351\) 0 0
\(352\) 20.2112 + 14.1919i 1.07726 + 0.756432i
\(353\) 0.355858i 0.0189404i −0.999955 0.00947020i \(-0.996985\pi\)
0.999955 0.00947020i \(-0.00301450\pi\)
\(354\) 0 0
\(355\) 10.6669i 0.566141i
\(356\) 9.84221 0.682999i 0.521636 0.0361989i
\(357\) 0 0
\(358\) −20.6363 19.2539i −1.09066 1.01760i
\(359\) 26.7105 1.40973 0.704864 0.709342i \(-0.251008\pi\)
0.704864 + 0.709342i \(0.251008\pi\)
\(360\) 0 0
\(361\) 17.6470 0.928787
\(362\) −18.2201 16.9996i −0.957628 0.893477i
\(363\) 0 0
\(364\) −33.7068 + 2.33908i −1.76672 + 0.122601i
\(365\) 9.41743i 0.492931i
\(366\) 0 0
\(367\) 9.41093i 0.491247i −0.969365 0.245623i \(-0.921007\pi\)
0.969365 0.245623i \(-0.0789926\pi\)
\(368\) −12.3212 + 1.71833i −0.642285 + 0.0895739i
\(369\) 0 0
\(370\) 7.55546 8.09794i 0.392790 0.420992i
\(371\) −42.5273 −2.20790
\(372\) 0 0
\(373\) 12.8467 0.665179 0.332590 0.943072i \(-0.392078\pi\)
0.332590 + 0.943072i \(0.392078\pi\)
\(374\) 12.7246 13.6382i 0.657971 0.705213i
\(375\) 0 0
\(376\) 14.5008 17.8780i 0.747823 0.921987i
\(377\) 38.7558i 1.99602i
\(378\) 0 0
\(379\) 22.5515i 1.15839i −0.815188 0.579196i \(-0.803366\pi\)
0.815188 0.579196i \(-0.196634\pi\)
\(380\) −0.161054 2.32083i −0.00826188 0.119056i
\(381\) 0 0
\(382\) −17.5962 16.4174i −0.900300 0.839989i
\(383\) −25.8689 −1.32184 −0.660920 0.750456i \(-0.729834\pi\)
−0.660920 + 0.750456i \(0.729834\pi\)
\(384\) 0 0
\(385\) 15.4930 0.789598
\(386\) 11.1373 + 10.3912i 0.566873 + 0.528898i
\(387\) 0 0
\(388\) −0.919630 13.2521i −0.0466872 0.672776i
\(389\) 16.2154i 0.822155i 0.911600 + 0.411077i \(0.134847\pi\)
−0.911600 + 0.411077i \(0.865153\pi\)
\(390\) 0 0
\(391\) 9.39595i 0.475174i
\(392\) 9.96694 12.2882i 0.503406 0.620647i
\(393\) 0 0
\(394\) −0.441866 + 0.473592i −0.0222609 + 0.0238592i
\(395\) −8.75803 −0.440664
\(396\) 0 0
\(397\) 10.4591 0.524925 0.262463 0.964942i \(-0.415465\pi\)
0.262463 + 0.964942i \(0.415465\pi\)
\(398\) 2.07184 2.22059i 0.103852 0.111308i
\(399\) 0 0
\(400\) 3.96166 0.552498i 0.198083 0.0276249i
\(401\) 2.17980i 0.108854i −0.998518 0.0544269i \(-0.982667\pi\)
0.998518 0.0544269i \(-0.0173332\pi\)
\(402\) 0 0
\(403\) 22.8226i 1.13687i
\(404\) −13.6695 + 0.948591i −0.680082 + 0.0471942i
\(405\) 0 0
\(406\) 29.8747 + 27.8734i 1.48265 + 1.38333i
\(407\) 34.1896 1.69472
\(408\) 0 0
\(409\) −25.6245 −1.26705 −0.633525 0.773722i \(-0.718393\pi\)
−0.633525 + 0.773722i \(0.718393\pi\)
\(410\) 4.77723 + 4.45720i 0.235931 + 0.220126i
\(411\) 0 0
\(412\) 28.6920 1.99107i 1.41355 0.0980932i
\(413\) 2.42894i 0.119520i
\(414\) 0 0
\(415\) 13.3807i 0.656833i
\(416\) −22.0387 15.4752i −1.08054 0.758733i
\(417\) 0 0
\(418\) 4.89929 5.25106i 0.239632 0.256838i
\(419\) 8.97928 0.438666 0.219333 0.975650i \(-0.429612\pi\)
0.219333 + 0.975650i \(0.429612\pi\)
\(420\) 0 0
\(421\) −33.7085 −1.64285 −0.821426 0.570315i \(-0.806821\pi\)
−0.821426 + 0.570315i \(0.806821\pi\)
\(422\) −2.09793 + 2.24856i −0.102126 + 0.109458i
\(423\) 0 0
\(424\) −26.3241 21.3515i −1.27841 1.03692i
\(425\) 3.02111i 0.146545i
\(426\) 0 0
\(427\) 37.5732i 1.81830i
\(428\) −0.0672663 0.969328i −0.00325144 0.0468542i
\(429\) 0 0
\(430\) −8.99771 8.39496i −0.433908 0.404841i
\(431\) 31.0229 1.49432 0.747160 0.664645i \(-0.231417\pi\)
0.747160 + 0.664645i \(0.231417\pi\)
\(432\) 0 0
\(433\) −34.5070 −1.65830 −0.829150 0.559026i \(-0.811175\pi\)
−0.829150 + 0.559026i \(0.811175\pi\)
\(434\) 17.5926 + 16.4141i 0.844474 + 0.787903i
\(435\) 0 0
\(436\) 2.17005 + 31.2711i 0.103927 + 1.49761i
\(437\) 3.61769i 0.173058i
\(438\) 0 0
\(439\) 2.45314i 0.117082i 0.998285 + 0.0585411i \(0.0186449\pi\)
−0.998285 + 0.0585411i \(0.981355\pi\)
\(440\) 9.59010 + 7.77853i 0.457190 + 0.370827i
\(441\) 0 0
\(442\) −13.8751 + 14.8714i −0.659973 + 0.707359i
\(443\) 7.51153 0.356883 0.178442 0.983950i \(-0.442894\pi\)
0.178442 + 0.983950i \(0.442894\pi\)
\(444\) 0 0
\(445\) 4.93294 0.233844
\(446\) 20.6744 22.1589i 0.978963 1.04925i
\(447\) 0 0
\(448\) 27.7793 5.85858i 1.31245 0.276792i
\(449\) 29.2553i 1.38064i −0.723504 0.690321i \(-0.757469\pi\)
0.723504 0.690321i \(-0.242531\pi\)
\(450\) 0 0
\(451\) 20.1695i 0.949747i
\(452\) −2.31528 + 0.160668i −0.108902 + 0.00755720i
\(453\) 0 0
\(454\) 9.13485 + 8.52290i 0.428720 + 0.400000i
\(455\) −16.8939 −0.792000
\(456\) 0 0
\(457\) −30.9220 −1.44647 −0.723235 0.690602i \(-0.757346\pi\)
−0.723235 + 0.690602i \(0.757346\pi\)
\(458\) 20.7612 + 19.3704i 0.970109 + 0.905121i
\(459\) 0 0
\(460\) −6.20528 + 0.430614i −0.289323 + 0.0200775i
\(461\) 24.4816i 1.14022i 0.821567 + 0.570112i \(0.193100\pi\)
−0.821567 + 0.570112i \(0.806900\pi\)
\(462\) 0 0
\(463\) 19.0943i 0.887387i −0.896179 0.443694i \(-0.853668\pi\)
0.896179 0.443694i \(-0.146332\pi\)
\(464\) 4.49798 + 32.2525i 0.208813 + 1.49729i
\(465\) 0 0
\(466\) 8.21993 8.81012i 0.380781 0.408121i
\(467\) 33.8471 1.56626 0.783129 0.621859i \(-0.213622\pi\)
0.783129 + 0.621859i \(0.213622\pi\)
\(468\) 0 0
\(469\) −28.2138 −1.30279
\(470\) 7.85185 8.41561i 0.362179 0.388183i
\(471\) 0 0
\(472\) −1.21949 + 1.50350i −0.0561316 + 0.0692043i
\(473\) 37.9885i 1.74671i
\(474\) 0 0
\(475\) 1.16321i 0.0533715i
\(476\) −1.48443 21.3911i −0.0680390 0.980462i
\(477\) 0 0
\(478\) 3.03346 + 2.83025i 0.138747 + 0.129452i
\(479\) −3.07957 −0.140709 −0.0703547 0.997522i \(-0.522413\pi\)
−0.0703547 + 0.997522i \(0.522413\pi\)
\(480\) 0 0
\(481\) −37.2811 −1.69987
\(482\) 26.7659 + 24.9728i 1.21915 + 1.13748i
\(483\) 0 0
\(484\) 1.11588 + 16.0802i 0.0507220 + 0.730919i
\(485\) 6.64201i 0.301598i
\(486\) 0 0
\(487\) 32.0600i 1.45278i −0.687285 0.726388i \(-0.741198\pi\)
0.687285 0.726388i \(-0.258802\pi\)
\(488\) −18.8642 + 23.2576i −0.853944 + 1.05282i
\(489\) 0 0
\(490\) 5.39686 5.78435i 0.243805 0.261310i
\(491\) −20.8359 −0.940313 −0.470157 0.882583i \(-0.655803\pi\)
−0.470157 + 0.882583i \(0.655803\pi\)
\(492\) 0 0
\(493\) 24.5953 1.10772
\(494\) −5.34229 + 5.72587i −0.240361 + 0.257619i
\(495\) 0 0
\(496\) 2.64878 + 18.9929i 0.118934 + 0.852807i
\(497\) 37.8547i 1.69801i
\(498\) 0 0
\(499\) 21.0503i 0.942340i −0.882043 0.471170i \(-0.843832\pi\)
0.882043 0.471170i \(-0.156168\pi\)
\(500\) 1.99520 0.138457i 0.0892281 0.00619197i
\(501\) 0 0
\(502\) 27.5335 + 25.6890i 1.22888 + 1.14656i
\(503\) −42.5778 −1.89845 −0.949225 0.314597i \(-0.898131\pi\)
−0.949225 + 0.314597i \(0.898131\pi\)
\(504\) 0 0
\(505\) −6.85118 −0.304873
\(506\) −14.0399 13.0994i −0.624151 0.582339i
\(507\) 0 0
\(508\) −20.1279 + 1.39677i −0.893033 + 0.0619718i
\(509\) 19.2768i 0.854431i 0.904150 + 0.427216i \(0.140505\pi\)
−0.904150 + 0.427216i \(0.859495\pi\)
\(510\) 0 0
\(511\) 33.4206i 1.47844i
\(512\) 20.1366 + 10.3206i 0.889922 + 0.456112i
\(513\) 0 0
\(514\) −1.47077 + 1.57637i −0.0648728 + 0.0695306i
\(515\) 14.3805 0.633680
\(516\) 0 0
\(517\) 35.5309 1.56265
\(518\) 26.8128 28.7380i 1.17809 1.26267i
\(519\) 0 0
\(520\) −10.4573 8.48188i −0.458581 0.371955i
\(521\) 26.9346i 1.18002i 0.807394 + 0.590012i \(0.200877\pi\)
−0.807394 + 0.590012i \(0.799123\pi\)
\(522\) 0 0
\(523\) 6.75789i 0.295502i −0.989025 0.147751i \(-0.952797\pi\)
0.989025 0.147751i \(-0.0472034\pi\)
\(524\) −0.663221 9.55721i −0.0289729 0.417509i
\(525\) 0 0
\(526\) 26.7834 + 24.9892i 1.16781 + 1.08958i
\(527\) 14.4837 0.630922
\(528\) 0 0
\(529\) −13.3273 −0.579446
\(530\) −12.3914 11.5613i −0.538249 0.502192i
\(531\) 0 0
\(532\) −0.571547 8.23616i −0.0247797 0.357083i
\(533\) 21.9933i 0.952636i
\(534\) 0 0
\(535\) 0.485829i 0.0210042i
\(536\) −17.4642 14.1652i −0.754338 0.611843i
\(537\) 0 0
\(538\) −14.5047 + 15.5462i −0.625343 + 0.670243i
\(539\) 24.4216 1.05191
\(540\) 0 0
\(541\) −15.3714 −0.660869 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(542\) 2.11159 2.26320i 0.0907005 0.0972128i
\(543\) 0 0
\(544\) 9.82092 13.9863i 0.421068 0.599657i
\(545\) 15.6731i 0.671363i
\(546\) 0 0
\(547\) 18.7918i 0.803478i −0.915754 0.401739i \(-0.868406\pi\)
0.915754 0.401739i \(-0.131594\pi\)
\(548\) −5.81373 + 0.403443i −0.248350 + 0.0172342i
\(549\) 0 0
\(550\) 4.51430 + 4.21189i 0.192490 + 0.179595i
\(551\) 9.46985 0.403429
\(552\) 0 0
\(553\) −31.0805 −1.32168
\(554\) 3.15903 + 2.94740i 0.134214 + 0.125223i
\(555\) 0 0
\(556\) −27.3876 + 1.90056i −1.16149 + 0.0806015i
\(557\) 36.4336i 1.54374i 0.635779 + 0.771871i \(0.280679\pi\)
−0.635779 + 0.771871i \(0.719321\pi\)
\(558\) 0 0
\(559\) 41.4235i 1.75203i
\(560\) 14.0591 1.96070i 0.594106 0.0828548i
\(561\) 0 0
\(562\) 6.52776 6.99646i 0.275357 0.295128i
\(563\) −18.5365 −0.781219 −0.390610 0.920556i \(-0.627736\pi\)
−0.390610 + 0.920556i \(0.627736\pi\)
\(564\) 0 0
\(565\) −1.16042 −0.0488194
\(566\) −5.71811 + 6.12867i −0.240350 + 0.257607i
\(567\) 0 0
\(568\) 19.0056 23.4318i 0.797456 0.983178i
\(569\) 5.79139i 0.242788i −0.992604 0.121394i \(-0.961264\pi\)
0.992604 0.121394i \(-0.0387364\pi\)
\(570\) 0 0
\(571\) 40.0776i 1.67720i −0.544751 0.838598i \(-0.683376\pi\)
0.544751 0.838598i \(-0.316624\pi\)
\(572\) −2.87752 41.4659i −0.120315 1.73378i
\(573\) 0 0
\(574\) 16.9534 + 15.8177i 0.707622 + 0.660219i
\(575\) −3.11010 −0.129700
\(576\) 0 0
\(577\) −9.44022 −0.393001 −0.196501 0.980504i \(-0.562958\pi\)
−0.196501 + 0.980504i \(0.562958\pi\)
\(578\) 8.14087 + 7.59551i 0.338615 + 0.315932i
\(579\) 0 0
\(580\) 1.12720 + 16.2433i 0.0468044 + 0.674465i
\(581\) 47.4854i 1.97003i
\(582\) 0 0
\(583\) 52.3168i 2.16674i
\(584\) −16.7793 + 20.6871i −0.694334 + 0.856040i
\(585\) 0 0
\(586\) −13.0310 + 13.9666i −0.538306 + 0.576956i
\(587\) −40.0701 −1.65387 −0.826934 0.562299i \(-0.809917\pi\)
−0.826934 + 0.562299i \(0.809917\pi\)
\(588\) 0 0
\(589\) 5.57662 0.229781
\(590\) −0.660325 + 0.707736i −0.0271851 + 0.0291370i
\(591\) 0 0
\(592\) 31.0254 4.32683i 1.27513 0.177832i
\(593\) 21.8552i 0.897486i −0.893661 0.448743i \(-0.851872\pi\)
0.893661 0.448743i \(-0.148128\pi\)
\(594\) 0 0
\(595\) 10.7213i 0.439530i
\(596\) 20.9228 1.45194i 0.857033 0.0594736i
\(597\) 0 0
\(598\) 15.3094 + 14.2839i 0.626050 + 0.584111i
\(599\) −3.02327 −0.123528 −0.0617638 0.998091i \(-0.519673\pi\)
−0.0617638 + 0.998091i \(0.519673\pi\)
\(600\) 0 0
\(601\) 33.9064 1.38307 0.691536 0.722342i \(-0.256934\pi\)
0.691536 + 0.722342i \(0.256934\pi\)
\(602\) −31.9311 29.7920i −1.30141 1.21423i
\(603\) 0 0
\(604\) −45.9799 + 3.19076i −1.87089 + 0.129830i
\(605\) 8.05944i 0.327663i
\(606\) 0 0
\(607\) 30.4571i 1.23622i 0.786093 + 0.618108i \(0.212101\pi\)
−0.786093 + 0.618108i \(0.787899\pi\)
\(608\) 3.78131 5.38509i 0.153352 0.218394i
\(609\) 0 0
\(610\) −10.2145 + 10.9479i −0.413575 + 0.443269i
\(611\) −38.7436 −1.56740
\(612\) 0 0
\(613\) 26.4631 1.06883 0.534417 0.845221i \(-0.320531\pi\)
0.534417 + 0.845221i \(0.320531\pi\)
\(614\) −20.8973 + 22.3978i −0.843348 + 0.903900i
\(615\) 0 0
\(616\) 34.0333 + 27.6044i 1.37124 + 1.11221i
\(617\) 15.7692i 0.634843i 0.948285 + 0.317421i \(0.102817\pi\)
−0.948285 + 0.317421i \(0.897183\pi\)
\(618\) 0 0
\(619\) 2.68614i 0.107965i −0.998542 0.0539825i \(-0.982808\pi\)
0.998542 0.0539825i \(-0.0171915\pi\)
\(620\) 0.663787 + 9.56536i 0.0266583 + 0.384154i
\(621\) 0 0
\(622\) 6.62176 + 6.17817i 0.265508 + 0.247722i
\(623\) 17.5060 0.701363
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −2.95239 2.75461i −0.118001 0.110096i
\(627\) 0 0
\(628\) 2.00984 + 28.9624i 0.0802014 + 1.15573i
\(629\) 23.6595i 0.943366i
\(630\) 0 0
\(631\) 39.2086i 1.56087i −0.625238 0.780434i \(-0.714998\pi\)
0.625238 0.780434i \(-0.285002\pi\)
\(632\) −19.2387 15.6045i −0.765273 0.620712i
\(633\) 0 0
\(634\) 13.1970 14.1446i 0.524122 0.561753i
\(635\) −10.0882 −0.400337
\(636\) 0 0
\(637\) −26.6299 −1.05511
\(638\) −34.2897 + 36.7517i −1.35754 + 1.45501i
\(639\) 0 0
\(640\) 9.68693 + 5.84495i 0.382909 + 0.231042i
\(641\) 31.5443i 1.24592i 0.782252 + 0.622962i \(0.214071\pi\)
−0.782252 + 0.622962i \(0.785929\pi\)
\(642\) 0 0
\(643\) 9.18421i 0.362190i −0.983466 0.181095i \(-0.942036\pi\)
0.983466 0.181095i \(-0.0579641\pi\)
\(644\) −22.0213 + 1.52816i −0.867760 + 0.0602181i
\(645\) 0 0
\(646\) −3.63377 3.39035i −0.142969 0.133391i
\(647\) 25.5421 1.00416 0.502081 0.864821i \(-0.332568\pi\)
0.502081 + 0.864821i \(0.332568\pi\)
\(648\) 0 0
\(649\) −2.98807 −0.117292
\(650\) −4.92249 4.59273i −0.193076 0.180142i
\(651\) 0 0
\(652\) 19.4235 1.34789i 0.760683 0.0527874i
\(653\) 10.4334i 0.408289i 0.978941 + 0.204145i \(0.0654413\pi\)
−0.978941 + 0.204145i \(0.934559\pi\)
\(654\) 0 0
\(655\) 4.79010i 0.187165i
\(656\) 2.55253 + 18.3028i 0.0996597 + 0.714605i
\(657\) 0 0
\(658\) 27.8646 29.8653i 1.08628 1.16427i
\(659\) 8.12696 0.316581 0.158291 0.987393i \(-0.449402\pi\)
0.158291 + 0.987393i \(0.449402\pi\)
\(660\) 0 0
\(661\) 8.66356 0.336974 0.168487 0.985704i \(-0.446112\pi\)
0.168487 + 0.985704i \(0.446112\pi\)
\(662\) −26.0195 + 27.8877i −1.01128 + 1.08389i
\(663\) 0 0
\(664\) −23.8408 + 29.3932i −0.925203 + 1.14068i
\(665\) 4.12798i 0.160076i
\(666\) 0 0
\(667\) 25.3199i 0.980389i
\(668\) 1.87783 + 27.0600i 0.0726552 + 1.04698i
\(669\) 0 0
\(670\) −8.22083 7.67011i −0.317598 0.296322i
\(671\) −46.2224 −1.78440
\(672\) 0 0
\(673\) −15.7922 −0.608746 −0.304373 0.952553i \(-0.598447\pi\)
−0.304373 + 0.952553i \(0.598447\pi\)
\(674\) 0.611265 + 0.570316i 0.0235450 + 0.0219678i
\(675\) 0 0
\(676\) 1.33778 + 19.2778i 0.0514530 + 0.741452i
\(677\) 29.2086i 1.12258i 0.827620 + 0.561289i \(0.189695\pi\)
−0.827620 + 0.561289i \(0.810305\pi\)
\(678\) 0 0
\(679\) 23.5711i 0.904577i
\(680\) 5.38280 6.63642i 0.206421 0.254495i
\(681\) 0 0
\(682\) −20.1926 + 21.6424i −0.773213 + 0.828730i
\(683\) 29.1095 1.11385 0.556923 0.830564i \(-0.311982\pi\)
0.556923 + 0.830564i \(0.311982\pi\)
\(684\) 0 0
\(685\) −2.91386 −0.111333
\(686\) −4.81393 + 5.15957i −0.183797 + 0.196993i
\(687\) 0 0
\(688\) −4.80759 34.4726i −0.183288 1.31426i
\(689\) 57.0474i 2.17333i
\(690\) 0 0
\(691\) 5.36149i 0.203961i 0.994786 + 0.101980i \(0.0325179\pi\)
−0.994786 + 0.101980i \(0.967482\pi\)
\(692\) −26.7591 + 1.85694i −1.01723 + 0.0705903i
\(693\) 0 0
\(694\) 9.80188 + 9.14525i 0.372074 + 0.347149i
\(695\) −13.7267 −0.520684
\(696\) 0 0
\(697\) 13.9575 0.528677
\(698\) −22.8443 21.3140i −0.864669 0.806745i
\(699\) 0 0
\(700\) 7.08057 0.491355i 0.267620 0.0185715i
\(701\) 26.2574i 0.991728i −0.868400 0.495864i \(-0.834852\pi\)
0.868400 0.495864i \(-0.165148\pi\)
\(702\) 0 0
\(703\) 9.10953i 0.343573i
\(704\) 7.20719 + 34.1740i 0.271631 + 1.28798i
\(705\) 0 0
\(706\) 0.343319 0.367969i 0.0129210 0.0138487i
\(707\) −24.3134 −0.914401
\(708\) 0 0
\(709\) −22.6687 −0.851342 −0.425671 0.904878i \(-0.639962\pi\)
−0.425671 + 0.904878i \(0.639962\pi\)
\(710\) 10.2911 11.0300i 0.386217 0.413947i
\(711\) 0 0
\(712\) 10.8361 + 8.78918i 0.406101 + 0.329388i
\(713\) 14.9104i 0.558399i
\(714\) 0 0
\(715\) 20.7828i 0.777234i
\(716\) −2.76318 39.8183i −0.103265 1.48808i
\(717\) 0 0
\(718\) 27.6196 + 25.7694i 1.03076 + 0.961705i
\(719\) 29.1172 1.08589 0.542945 0.839768i \(-0.317309\pi\)
0.542945 + 0.839768i \(0.317309\pi\)
\(720\) 0 0
\(721\) 51.0334 1.90058
\(722\) 18.2476 + 17.0252i 0.679104 + 0.633611i
\(723\) 0 0
\(724\) −2.43966 35.1563i −0.0906694 1.30657i
\(725\) 8.14117i 0.302355i
\(726\) 0 0
\(727\) 22.7265i 0.842879i −0.906856 0.421440i \(-0.861525\pi\)
0.906856 0.421440i \(-0.138475\pi\)
\(728\) −37.1107 30.1005i −1.37541 1.11560i
\(729\) 0 0
\(730\) −9.08561 + 9.73795i −0.336274 + 0.360418i
\(731\) −26.2883 −0.972309
\(732\) 0 0
\(733\) 6.03204 0.222799 0.111399 0.993776i \(-0.464467\pi\)
0.111399 + 0.993776i \(0.464467\pi\)
\(734\) 9.07934 9.73123i 0.335124 0.359186i
\(735\) 0 0
\(736\) −14.3983 10.1102i −0.530728 0.372668i
\(737\) 34.7085i 1.27850i
\(738\) 0 0
\(739\) 11.1343i 0.409582i −0.978806 0.204791i \(-0.934349\pi\)
0.978806 0.204791i \(-0.0656514\pi\)
\(740\) 15.6252 1.08431i 0.574395 0.0398600i
\(741\) 0 0
\(742\) −43.9747 41.0288i −1.61436 1.50621i
\(743\) 17.4343 0.639601 0.319801 0.947485i \(-0.396384\pi\)
0.319801 + 0.947485i \(0.396384\pi\)
\(744\) 0 0
\(745\) 10.4866 0.384198
\(746\) 13.2840 + 12.3941i 0.486361 + 0.453780i
\(747\) 0 0
\(748\) 26.3153 1.82614i 0.962182 0.0667704i
\(749\) 1.72411i 0.0629976i
\(750\) 0 0
\(751\) 43.7231i 1.59548i −0.603002 0.797740i \(-0.706029\pi\)
0.603002 0.797740i \(-0.293971\pi\)
\(752\) 32.2424 4.49657i 1.17576 0.163973i
\(753\) 0 0
\(754\) 37.3902 40.0748i 1.36167 1.45944i
\(755\) −23.0452 −0.838701
\(756\) 0 0
\(757\) −28.7104 −1.04350 −0.521749 0.853099i \(-0.674720\pi\)
−0.521749 + 0.853099i \(0.674720\pi\)
\(758\) 21.7569 23.3190i 0.790246 0.846985i
\(759\) 0 0
\(760\) 2.07252 2.55520i 0.0751782 0.0926868i
\(761\) 14.3521i 0.520265i 0.965573 + 0.260132i \(0.0837662\pi\)
−0.965573 + 0.260132i \(0.916234\pi\)
\(762\) 0 0
\(763\) 55.6208i 2.01361i
\(764\) −2.35612 33.9524i −0.0852414 1.22835i
\(765\) 0 0
\(766\) −26.7494 24.9574i −0.966494 0.901748i
\(767\) 3.25826 0.117649
\(768\) 0 0
\(769\) 20.7832 0.749463 0.374732 0.927133i \(-0.377735\pi\)
0.374732 + 0.927133i \(0.377735\pi\)
\(770\) 16.0203 + 14.9471i 0.577333 + 0.538657i
\(771\) 0 0
\(772\) 1.49128 + 21.4897i 0.0536722 + 0.773432i
\(773\) 33.9470i 1.22099i −0.792020 0.610495i \(-0.790971\pi\)
0.792020 0.610495i \(-0.209029\pi\)
\(774\) 0 0
\(775\) 4.79418i 0.172212i
\(776\) 11.8343 14.5904i 0.424826 0.523765i
\(777\) 0 0
\(778\) −15.6441 + 16.7673i −0.560867 + 0.601138i
\(779\) 5.37400 0.192543
\(780\) 0 0
\(781\) 46.5686 1.66636
\(782\) −9.06489 + 9.71574i −0.324160 + 0.347434i
\(783\) 0 0
\(784\) 22.1614 3.09065i 0.791477 0.110380i
\(785\) 14.5160i 0.518099i
\(786\) 0 0
\(787\) 35.8401i 1.27756i −0.769388 0.638782i \(-0.779439\pi\)
0.769388 0.638782i \(-0.220561\pi\)
\(788\) −0.913811 + 0.0634137i −0.0325532 + 0.00225902i
\(789\) 0 0
\(790\) −9.05611 8.44944i −0.322202 0.300618i
\(791\) −4.11811 −0.146423
\(792\) 0 0
\(793\) 50.4019 1.78982
\(794\) 10.8150 + 10.0905i 0.383811 + 0.358100i
\(795\) 0 0
\(796\) 4.28470 0.297336i 0.151867 0.0105388i
\(797\) 8.52002i 0.301795i −0.988549 0.150897i \(-0.951784\pi\)
0.988549 0.150897i \(-0.0482163\pi\)
\(798\) 0 0
\(799\) 24.5876i 0.869848i
\(800\) 4.62953 + 3.25077i 0.163678 + 0.114932i
\(801\) 0 0
\(802\) 2.10299 2.25399i 0.0742592 0.0795910i
\(803\) −41.1138 −1.45087
\(804\) 0 0
\(805\) −11.0371 −0.389007
\(806\) 22.0184 23.5993i 0.775565 0.831251i
\(807\) 0 0
\(808\) −15.0499 12.2070i −0.529453 0.429439i
\(809\) 6.70486i 0.235730i 0.993030 + 0.117865i \(0.0376051\pi\)
−0.993030 + 0.117865i \(0.962395\pi\)
\(810\) 0 0
\(811\) 24.0578i 0.844784i −0.906413 0.422392i \(-0.861191\pi\)
0.906413 0.422392i \(-0.138809\pi\)
\(812\) 4.00020 + 57.6441i 0.140379 + 2.02291i
\(813\) 0 0
\(814\) 35.3533 + 32.9850i 1.23913 + 1.15612i
\(815\) 9.73510 0.341006
\(816\) 0 0
\(817\) −10.1217 −0.354113
\(818\) −26.4966 24.7216i −0.926433 0.864371i
\(819\) 0 0
\(820\) 0.639668 + 9.21781i 0.0223382 + 0.321900i
\(821\) 25.7881i 0.900013i 0.893025 + 0.450006i \(0.148578\pi\)
−0.893025 + 0.450006i \(0.851422\pi\)
\(822\) 0 0
\(823\) 11.6163i 0.404920i 0.979291 + 0.202460i \(0.0648936\pi\)
−0.979291 + 0.202460i \(0.935106\pi\)
\(824\) 31.5894 + 25.6222i 1.10047 + 0.892590i
\(825\) 0 0
\(826\) −2.34336 + 2.51161i −0.0815358 + 0.0873901i
\(827\) 2.17283 0.0755567 0.0377783 0.999286i \(-0.487972\pi\)
0.0377783 + 0.999286i \(0.487972\pi\)
\(828\) 0 0
\(829\) −37.9307 −1.31739 −0.658693 0.752412i \(-0.728891\pi\)
−0.658693 + 0.752412i \(0.728891\pi\)
\(830\) −12.9092 + 13.8361i −0.448086 + 0.480258i
\(831\) 0 0
\(832\) −7.85888 37.2641i −0.272458 1.29190i
\(833\) 16.9000i 0.585549i
\(834\) 0 0
\(835\) 13.5625i 0.469351i
\(836\) 10.1321 0.703114i 0.350425 0.0243177i
\(837\) 0 0
\(838\) 9.28489 + 8.66289i 0.320741 + 0.299255i
\(839\) 8.91430 0.307756 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(840\) 0 0
\(841\) −37.2786 −1.28547
\(842\) −34.8558 32.5208i −1.20121 1.12074i
\(843\) 0 0
\(844\) −4.33866 + 0.301081i −0.149343 + 0.0103636i
\(845\) 9.66206i 0.332385i
\(846\) 0 0
\(847\) 28.6013i 0.982753i
\(848\) −6.62090 47.4748i −0.227363 1.63029i
\(849\) 0 0
\(850\) 2.91466 3.12393i 0.0999720 0.107150i
\(851\) −24.3565 −0.834929
\(852\) 0 0
\(853\) 8.49722 0.290939 0.145470 0.989363i \(-0.453531\pi\)
0.145470 + 0.989363i \(0.453531\pi\)
\(854\) −36.2493 + 38.8520i −1.24043 + 1.32949i
\(855\) 0 0
\(856\) 0.865618 1.06722i 0.0295862 0.0364767i
\(857\) 40.8364i 1.39495i −0.716611 0.697473i \(-0.754308\pi\)
0.716611 0.697473i \(-0.245692\pi\)
\(858\) 0 0
\(859\) 44.5472i 1.51993i −0.649964 0.759965i \(-0.725216\pi\)
0.649964 0.759965i \(-0.274784\pi\)
\(860\) −1.20479 17.3614i −0.0410829 0.592017i
\(861\) 0 0
\(862\) 32.0787 + 29.9298i 1.09261 + 1.01941i
\(863\) −22.6193 −0.769971 −0.384985 0.922923i \(-0.625794\pi\)
−0.384985 + 0.922923i \(0.625794\pi\)
\(864\) 0 0
\(865\) −13.4117 −0.456012
\(866\) −35.6814 33.2911i −1.21250 1.13128i
\(867\) 0 0
\(868\) 2.35564 + 33.9455i 0.0799558 + 1.15219i
\(869\) 38.2350i 1.29704i
\(870\) 0 0
\(871\) 37.8469i 1.28239i
\(872\) −27.9253 + 34.4290i −0.945670 + 1.16591i
\(873\) 0 0
\(874\) −3.49022 + 3.74082i −0.118058 + 0.126535i
\(875\) 3.54880 0.119971
\(876\) 0 0
\(877\) 7.37149 0.248917 0.124459 0.992225i \(-0.460281\pi\)
0.124459 + 0.992225i \(0.460281\pi\)
\(878\) −2.36671 + 2.53664i −0.0798725 + 0.0856073i
\(879\) 0 0
\(880\) 2.41205 + 17.2955i 0.0813101 + 0.583030i
\(881\) 41.0770i 1.38392i −0.721937 0.691959i \(-0.756748\pi\)
0.721937 0.691959i \(-0.243252\pi\)
\(882\) 0 0
\(883\) 23.9030i 0.804401i −0.915552 0.402201i \(-0.868245\pi\)
0.915552 0.402201i \(-0.131755\pi\)
\(884\) −28.6948 + 1.99127i −0.965109 + 0.0669736i
\(885\) 0 0
\(886\) 7.76718 + 7.24686i 0.260944 + 0.243463i
\(887\) −37.8953 −1.27240 −0.636200 0.771524i \(-0.719495\pi\)
−0.636200 + 0.771524i \(0.719495\pi\)
\(888\) 0 0
\(889\) −35.8009 −1.20072
\(890\) 5.10083 + 4.75913i 0.170980 + 0.159526i
\(891\) 0 0
\(892\) 42.7562 2.96706i 1.43158 0.0993444i
\(893\) 9.46689i 0.316797i
\(894\) 0 0
\(895\) 19.9570i 0.667090i
\(896\) 34.3769 + 20.7425i 1.14845 + 0.692960i
\(897\) 0 0
\(898\) 28.2245 30.2510i 0.941862 1.00949i
\(899\) −39.0302 −1.30173
\(900\) 0 0
\(901\) −36.2036 −1.20612
\(902\) −19.4589 + 20.8560i −0.647910 + 0.694429i
\(903\) 0 0
\(904\) −2.54909 2.06756i −0.0847813 0.0687661i
\(905\) 17.6204i 0.585722i
\(906\) 0 0
\(907\) 22.5057i 0.747291i 0.927572 + 0.373646i \(0.121892\pi\)
−0.927572 + 0.373646i \(0.878108\pi\)
\(908\) 1.22315 + 17.6260i 0.0405917 + 0.584938i
\(909\) 0 0
\(910\) −17.4689 16.2987i −0.579089 0.540296i
\(911\) 7.91542 0.262250 0.131125 0.991366i \(-0.458141\pi\)
0.131125 + 0.991366i \(0.458141\pi\)
\(912\) 0 0
\(913\) −58.4163 −1.93330
\(914\) −31.9744 29.8325i −1.05762 0.986770i
\(915\) 0 0
\(916\) 2.77992 + 40.0594i 0.0918511 + 1.32360i
\(917\) 16.9991i 0.561359i
\(918\) 0 0
\(919\) 1.24778i 0.0411605i −0.999788 0.0205802i \(-0.993449\pi\)
0.999788 0.0205802i \(-0.00655136\pi\)
\(920\) −6.83192 5.54137i −0.225242 0.182693i
\(921\) 0 0
\(922\) −23.6190 + 25.3149i −0.777852 + 0.833701i
\(923\) −50.7795 −1.67143
\(924\) 0 0
\(925\) 7.83140 0.257495
\(926\) 18.4215 19.7442i 0.605368 0.648834i
\(927\) 0 0
\(928\) −26.4650 + 37.6897i −0.868758 + 1.23723i
\(929\) 36.0809i 1.18378i −0.806021 0.591888i \(-0.798383\pi\)
0.806021 0.591888i \(-0.201617\pi\)
\(930\) 0 0
\(931\) 6.50693i 0.213256i
\(932\) 16.9994 1.17967i 0.556834 0.0386414i
\(933\) 0 0
\(934\) 34.9991 + 32.6545i 1.14521 + 1.06849i
\(935\) 13.1893 0.431336
\(936\) 0 0
\(937\) 9.59638 0.313500 0.156750 0.987638i \(-0.449898\pi\)
0.156750 + 0.987638i \(0.449898\pi\)
\(938\) −29.1740 27.2197i −0.952566 0.888754i
\(939\) 0 0
\(940\) 16.2382 1.12685i 0.529631 0.0367537i
\(941\) 10.8397i 0.353365i −0.984268 0.176683i \(-0.943463\pi\)
0.984268 0.176683i \(-0.0565366\pi\)
\(942\) 0 0
\(943\) 14.3686i 0.467907i
\(944\) −2.71152 + 0.378152i −0.0882525 + 0.0123078i
\(945\) 0 0
\(946\) 36.6500 39.2814i 1.19159 1.27715i
\(947\) 35.1208 1.14127 0.570636 0.821203i \(-0.306697\pi\)
0.570636 + 0.821203i \(0.306697\pi\)
\(948\) 0 0
\(949\) 44.8314 1.45529
\(950\) 1.12222 1.20280i 0.0364096 0.0390238i
\(951\) 0 0
\(952\) 19.1025 23.5513i 0.619115 0.763303i
\(953\) 23.5204i 0.761900i −0.924596 0.380950i \(-0.875597\pi\)
0.924596 0.380950i \(-0.124403\pi\)
\(954\) 0 0
\(955\) 17.0170i 0.550658i
\(956\) 0.406178 + 5.85315i 0.0131367 + 0.189304i
\(957\) 0 0
\(958\) −3.18439 2.97107i −0.102883 0.0959908i
\(959\) −10.3407 −0.333918
\(960\) 0 0
\(961\) 8.01580 0.258574
\(962\) −38.5500 35.9675i −1.24290 1.15964i
\(963\) 0 0
\(964\) 3.58393 + 51.6455i 0.115431 + 1.66339i
\(965\) 10.7707i 0.346721i
\(966\) 0 0
\(967\) 20.2517i 0.651252i 0.945499 + 0.325626i \(0.105575\pi\)
−0.945499 + 0.325626i \(0.894425\pi\)
\(968\) −14.3598 + 17.7041i −0.461540 + 0.569030i
\(969\) 0 0
\(970\) 6.40798 6.86807i 0.205748 0.220520i
\(971\) −29.6220 −0.950616 −0.475308 0.879819i \(-0.657663\pi\)
−0.475308 + 0.879819i \(0.657663\pi\)
\(972\) 0 0
\(973\) −48.7133 −1.56168
\(974\) 30.9303 33.1511i 0.991072 1.06223i
\(975\) 0 0
\(976\) −41.9444 + 5.84962i −1.34261 + 0.187242i
\(977\) 35.3726i 1.13167i −0.824518 0.565835i \(-0.808554\pi\)
0.824518 0.565835i \(-0.191446\pi\)
\(978\) 0 0
\(979\) 21.5358i 0.688287i
\(980\) 11.1611 0.774521i 0.356528 0.0247412i
\(981\) 0 0
\(982\) −21.5451 20.1018i −0.687532 0.641474i
\(983\) 0.0488705 0.00155873 0.000779364 1.00000i \(-0.499752\pi\)
0.000779364 1.00000i \(0.499752\pi\)
\(984\) 0 0
\(985\) −0.458004 −0.0145932
\(986\) 25.4324 + 23.7287i 0.809934 + 0.755676i
\(987\) 0 0
\(988\) −11.0482 + 0.766691i −0.351491 + 0.0243917i
\(989\) 27.0627i 0.860545i
\(990\) 0 0
\(991\) 12.0395i 0.382446i −0.981547 0.191223i \(-0.938755\pi\)
0.981547 0.191223i \(-0.0612454\pi\)
\(992\) −15.5848 + 22.1948i −0.494817 + 0.704685i
\(993\) 0 0
\(994\) 36.5209 39.1431i 1.15837 1.24154i
\(995\) 2.14750 0.0680804
\(996\) 0 0
\(997\) −48.8679 −1.54766 −0.773831 0.633392i \(-0.781662\pi\)
−0.773831 + 0.633392i \(0.781662\pi\)
\(998\) 20.3086 21.7667i 0.642856 0.689013i
\(999\) 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 1620.2.e.a.971.38 yes 48
3.2 odd 2 inner 1620.2.e.a.971.11 48
4.3 odd 2 inner 1620.2.e.a.971.12 yes 48
12.11 even 2 inner 1620.2.e.a.971.37 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.e.a.971.11 48 3.2 odd 2 inner
1620.2.e.a.971.12 yes 48 4.3 odd 2 inner
1620.2.e.a.971.37 yes 48 12.11 even 2 inner
1620.2.e.a.971.38 yes 48 1.1 even 1 trivial