Properties

Label 1053.2.b.j.649.5
Level $1053$
Weight $2$
Character 1053.649
Analytic conductor $8.408$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1053,2,Mod(649,1053)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1053, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1053.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.b (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: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.5
Root \(-0.905597i\) of defining polynomial
Character \(\chi\) \(=\) 1053.649
Dual form 1053.2.b.j.649.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.905597i q^{2} +1.17989 q^{4} -2.24306i q^{5} -3.43611i q^{7} -2.87970i q^{8} -2.03130 q^{10} +3.69657i q^{11} +(2.93184 - 2.09865i) q^{13} -3.11173 q^{14} -0.248059 q^{16} -4.21120 q^{17} -4.25298i q^{19} -2.64657i q^{20} +3.34760 q^{22} -3.78325 q^{23} -0.0313048 q^{25} +(-1.90053 - 2.65506i) q^{26} -4.05425i q^{28} +2.37891 q^{29} +7.35733i q^{31} -5.53476i q^{32} +3.81365i q^{34} -7.70739 q^{35} +5.49928i q^{37} -3.85149 q^{38} -6.45934 q^{40} +7.92223i q^{41} +0.900531 q^{43} +4.36157i q^{44} +3.42609i q^{46} -5.54325i q^{47} -4.80686 q^{49} +0.0283495i q^{50} +(3.45926 - 2.47619i) q^{52} +7.59566 q^{53} +8.29163 q^{55} -9.89498 q^{56} -2.15433i q^{58} -5.13125i q^{59} -13.0181 q^{61} +6.66277 q^{62} -5.50838 q^{64} +(-4.70739 - 6.57628i) q^{65} -13.5102i q^{67} -4.96877 q^{68} +6.97979i q^{70} -2.65506i q^{71} +5.45741i q^{73} +4.98013 q^{74} -5.01807i q^{76} +12.7018 q^{77} +10.9377 q^{79} +0.556410i q^{80} +7.17434 q^{82} +0.537568i q^{83} +9.44596i q^{85} -0.815518i q^{86} +10.6450 q^{88} -5.75227i q^{89} +(-7.21120 - 10.0741i) q^{91} -4.46383 q^{92} -5.01995 q^{94} -9.53968 q^{95} -6.78485i q^{97} +4.35308i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.905597i 0.640353i −0.947358 0.320177i \(-0.896258\pi\)
0.947358 0.320177i \(-0.103742\pi\)
\(3\) 0 0
\(4\) 1.17989 0.589947
\(5\) 2.24306i 1.00313i −0.865121 0.501563i \(-0.832759\pi\)
0.865121 0.501563i \(-0.167241\pi\)
\(6\) 0 0
\(7\) 3.43611i 1.29873i −0.760478 0.649364i \(-0.775035\pi\)
0.760478 0.649364i \(-0.224965\pi\)
\(8\) 2.87970i 1.01813i
\(9\) 0 0
\(10\) −2.03130 −0.642355
\(11\) 3.69657i 1.11456i 0.830325 + 0.557279i \(0.188155\pi\)
−0.830325 + 0.557279i \(0.811845\pi\)
\(12\) 0 0
\(13\) 2.93184 2.09865i 0.813145 0.582061i
\(14\) −3.11173 −0.831645
\(15\) 0 0
\(16\) −0.248059 −0.0620147
\(17\) −4.21120 −1.02137 −0.510683 0.859769i \(-0.670607\pi\)
−0.510683 + 0.859769i \(0.670607\pi\)
\(18\) 0 0
\(19\) 4.25298i 0.975701i −0.872927 0.487851i \(-0.837781\pi\)
0.872927 0.487851i \(-0.162219\pi\)
\(20\) 2.64657i 0.591791i
\(21\) 0 0
\(22\) 3.34760 0.713712
\(23\) −3.78325 −0.788861 −0.394431 0.918926i \(-0.629058\pi\)
−0.394431 + 0.918926i \(0.629058\pi\)
\(24\) 0 0
\(25\) −0.0313048 −0.00626096
\(26\) −1.90053 2.65506i −0.372725 0.520700i
\(27\) 0 0
\(28\) 4.05425i 0.766181i
\(29\) 2.37891 0.441752 0.220876 0.975302i \(-0.429108\pi\)
0.220876 + 0.975302i \(0.429108\pi\)
\(30\) 0 0
\(31\) 7.35733i 1.32141i 0.750644 + 0.660707i \(0.229744\pi\)
−0.750644 + 0.660707i \(0.770256\pi\)
\(32\) 5.53476i 0.978417i
\(33\) 0 0
\(34\) 3.81365i 0.654035i
\(35\) −7.70739 −1.30279
\(36\) 0 0
\(37\) 5.49928i 0.904076i 0.891999 + 0.452038i \(0.149303\pi\)
−0.891999 + 0.452038i \(0.850697\pi\)
\(38\) −3.85149 −0.624794
\(39\) 0 0
\(40\) −6.45934 −1.02131
\(41\) 7.92223i 1.23724i 0.785689 + 0.618622i \(0.212309\pi\)
−0.785689 + 0.618622i \(0.787691\pi\)
\(42\) 0 0
\(43\) 0.900531 0.137330 0.0686649 0.997640i \(-0.478126\pi\)
0.0686649 + 0.997640i \(0.478126\pi\)
\(44\) 4.36157i 0.657531i
\(45\) 0 0
\(46\) 3.42609i 0.505150i
\(47\) 5.54325i 0.808566i −0.914634 0.404283i \(-0.867521\pi\)
0.914634 0.404283i \(-0.132479\pi\)
\(48\) 0 0
\(49\) −4.80686 −0.686695
\(50\) 0.0283495i 0.00400923i
\(51\) 0 0
\(52\) 3.45926 2.47619i 0.479713 0.343385i
\(53\) 7.59566 1.04334 0.521672 0.853146i \(-0.325308\pi\)
0.521672 + 0.853146i \(0.325308\pi\)
\(54\) 0 0
\(55\) 8.29163 1.11804
\(56\) −9.89498 −1.32227
\(57\) 0 0
\(58\) 2.15433i 0.282878i
\(59\) 5.13125i 0.668032i −0.942567 0.334016i \(-0.891596\pi\)
0.942567 0.334016i \(-0.108404\pi\)
\(60\) 0 0
\(61\) −13.0181 −1.66680 −0.833401 0.552669i \(-0.813609\pi\)
−0.833401 + 0.552669i \(0.813609\pi\)
\(62\) 6.66277 0.846173
\(63\) 0 0
\(64\) −5.50838 −0.688547
\(65\) −4.70739 6.57628i −0.583880 0.815687i
\(66\) 0 0
\(67\) 13.5102i 1.65054i −0.564741 0.825269i \(-0.691024\pi\)
0.564741 0.825269i \(-0.308976\pi\)
\(68\) −4.96877 −0.602552
\(69\) 0 0
\(70\) 6.97979i 0.834244i
\(71\) 2.65506i 0.315098i −0.987511 0.157549i \(-0.949641\pi\)
0.987511 0.157549i \(-0.0503592\pi\)
\(72\) 0 0
\(73\) 5.45741i 0.638741i 0.947630 + 0.319371i \(0.103472\pi\)
−0.947630 + 0.319371i \(0.896528\pi\)
\(74\) 4.98013 0.578928
\(75\) 0 0
\(76\) 5.01807i 0.575612i
\(77\) 12.7018 1.44751
\(78\) 0 0
\(79\) 10.9377 1.23059 0.615294 0.788297i \(-0.289037\pi\)
0.615294 + 0.788297i \(0.289037\pi\)
\(80\) 0.556410i 0.0622085i
\(81\) 0 0
\(82\) 7.17434 0.792273
\(83\) 0.537568i 0.0590057i 0.999565 + 0.0295029i \(0.00939241\pi\)
−0.999565 + 0.0295029i \(0.990608\pi\)
\(84\) 0 0
\(85\) 9.44596i 1.02456i
\(86\) 0.815518i 0.0879396i
\(87\) 0 0
\(88\) 10.6450 1.13476
\(89\) 5.75227i 0.609739i −0.952394 0.304870i \(-0.901387\pi\)
0.952394 0.304870i \(-0.0986129\pi\)
\(90\) 0 0
\(91\) −7.21120 10.0741i −0.755939 1.05605i
\(92\) −4.46383 −0.465387
\(93\) 0 0
\(94\) −5.01995 −0.517768
\(95\) −9.53968 −0.978751
\(96\) 0 0
\(97\) 6.78485i 0.688897i −0.938805 0.344449i \(-0.888066\pi\)
0.938805 0.344449i \(-0.111934\pi\)
\(98\) 4.35308i 0.439727i
\(99\) 0 0
\(100\) −0.0369364 −0.00369364
\(101\) −4.96231 −0.493768 −0.246884 0.969045i \(-0.579407\pi\)
−0.246884 + 0.969045i \(0.579407\pi\)
\(102\) 0 0
\(103\) 2.11173 0.208075 0.104038 0.994573i \(-0.466824\pi\)
0.104038 + 0.994573i \(0.466824\pi\)
\(104\) −6.04349 8.44281i −0.592613 0.827886i
\(105\) 0 0
\(106\) 6.87861i 0.668109i
\(107\) −8.07404 −0.780547 −0.390274 0.920699i \(-0.627620\pi\)
−0.390274 + 0.920699i \(0.627620\pi\)
\(108\) 0 0
\(109\) 10.0020i 0.958021i 0.877809 + 0.479011i \(0.159004\pi\)
−0.877809 + 0.479011i \(0.840996\pi\)
\(110\) 7.50887i 0.715942i
\(111\) 0 0
\(112\) 0.852357i 0.0805402i
\(113\) 19.4347 1.82826 0.914130 0.405422i \(-0.132875\pi\)
0.914130 + 0.405422i \(0.132875\pi\)
\(114\) 0 0
\(115\) 8.48604i 0.791327i
\(116\) 2.80686 0.260611
\(117\) 0 0
\(118\) −4.64684 −0.427776
\(119\) 14.4702i 1.32648i
\(120\) 0 0
\(121\) −2.66466 −0.242242
\(122\) 11.7892i 1.06734i
\(123\) 0 0
\(124\) 8.68087i 0.779565i
\(125\) 11.1451i 0.996845i
\(126\) 0 0
\(127\) −1.76413 −0.156541 −0.0782704 0.996932i \(-0.524940\pi\)
−0.0782704 + 0.996932i \(0.524940\pi\)
\(128\) 6.08116i 0.537503i
\(129\) 0 0
\(130\) −5.95545 + 4.26300i −0.522328 + 0.373890i
\(131\) −7.12979 −0.622933 −0.311466 0.950257i \(-0.600820\pi\)
−0.311466 + 0.950257i \(0.600820\pi\)
\(132\) 0 0
\(133\) −14.6137 −1.26717
\(134\) −12.2348 −1.05693
\(135\) 0 0
\(136\) 12.1270i 1.03988i
\(137\) 4.52058i 0.386219i 0.981177 + 0.193110i \(0.0618573\pi\)
−0.981177 + 0.193110i \(0.938143\pi\)
\(138\) 0 0
\(139\) 14.1099 1.19679 0.598394 0.801202i \(-0.295806\pi\)
0.598394 + 0.801202i \(0.295806\pi\)
\(140\) −9.09391 −0.768576
\(141\) 0 0
\(142\) −2.40441 −0.201774
\(143\) 7.75782 + 10.8377i 0.648741 + 0.906298i
\(144\) 0 0
\(145\) 5.33603i 0.443133i
\(146\) 4.94221 0.409020
\(147\) 0 0
\(148\) 6.48857i 0.533357i
\(149\) 5.80236i 0.475348i 0.971345 + 0.237674i \(0.0763850\pi\)
−0.971345 + 0.237674i \(0.923615\pi\)
\(150\) 0 0
\(151\) 9.74588i 0.793108i 0.918011 + 0.396554i \(0.129794\pi\)
−0.918011 + 0.396554i \(0.870206\pi\)
\(152\) −12.2473 −0.993389
\(153\) 0 0
\(154\) 11.5027i 0.926917i
\(155\) 16.5029 1.32555
\(156\) 0 0
\(157\) −1.07960 −0.0861611 −0.0430806 0.999072i \(-0.513717\pi\)
−0.0430806 + 0.999072i \(0.513717\pi\)
\(158\) 9.90516i 0.788012i
\(159\) 0 0
\(160\) −12.4148 −0.981475
\(161\) 12.9997i 1.02452i
\(162\) 0 0
\(163\) 5.54857i 0.434598i −0.976105 0.217299i \(-0.930275\pi\)
0.976105 0.217299i \(-0.0697247\pi\)
\(164\) 9.34739i 0.729909i
\(165\) 0 0
\(166\) 0.486819 0.0377845
\(167\) 13.3385i 1.03216i 0.856539 + 0.516082i \(0.172610\pi\)
−0.856539 + 0.516082i \(0.827390\pi\)
\(168\) 0 0
\(169\) 4.19133 12.3058i 0.322410 0.946600i
\(170\) 8.55423 0.656079
\(171\) 0 0
\(172\) 1.06253 0.0810173
\(173\) 7.44160 0.565774 0.282887 0.959153i \(-0.408708\pi\)
0.282887 + 0.959153i \(0.408708\pi\)
\(174\) 0 0
\(175\) 0.107567i 0.00813129i
\(176\) 0.916967i 0.0691190i
\(177\) 0 0
\(178\) −5.20923 −0.390449
\(179\) 23.2047 1.73440 0.867202 0.497957i \(-0.165916\pi\)
0.867202 + 0.497957i \(0.165916\pi\)
\(180\) 0 0
\(181\) 5.18439 0.385353 0.192676 0.981262i \(-0.438283\pi\)
0.192676 + 0.981262i \(0.438283\pi\)
\(182\) −9.12309 + 6.53044i −0.676248 + 0.484068i
\(183\) 0 0
\(184\) 10.8946i 0.803162i
\(185\) 12.3352 0.906901
\(186\) 0 0
\(187\) 15.5670i 1.13837i
\(188\) 6.54045i 0.477012i
\(189\) 0 0
\(190\) 8.63911i 0.626747i
\(191\) 16.1411 1.16793 0.583966 0.811778i \(-0.301500\pi\)
0.583966 + 0.811778i \(0.301500\pi\)
\(192\) 0 0
\(193\) 3.27925i 0.236045i 0.993011 + 0.118023i \(0.0376556\pi\)
−0.993011 + 0.118023i \(0.962344\pi\)
\(194\) −6.14434 −0.441138
\(195\) 0 0
\(196\) −5.67159 −0.405114
\(197\) 7.11249i 0.506744i 0.967369 + 0.253372i \(0.0815396\pi\)
−0.967369 + 0.253372i \(0.918460\pi\)
\(198\) 0 0
\(199\) −13.8449 −0.981437 −0.490719 0.871318i \(-0.663266\pi\)
−0.490719 + 0.871318i \(0.663266\pi\)
\(200\) 0.0901485i 0.00637446i
\(201\) 0 0
\(202\) 4.49385i 0.316186i
\(203\) 8.17420i 0.573716i
\(204\) 0 0
\(205\) 17.7700 1.24111
\(206\) 1.91238i 0.133242i
\(207\) 0 0
\(208\) −0.727267 + 0.520589i −0.0504269 + 0.0360963i
\(209\) 15.7215 1.08748
\(210\) 0 0
\(211\) 14.4339 0.993671 0.496836 0.867845i \(-0.334495\pi\)
0.496836 + 0.867845i \(0.334495\pi\)
\(212\) 8.96208 0.615518
\(213\) 0 0
\(214\) 7.31182i 0.499826i
\(215\) 2.01994i 0.137759i
\(216\) 0 0
\(217\) 25.2806 1.71616
\(218\) 9.05781 0.613472
\(219\) 0 0
\(220\) 9.78325 0.659586
\(221\) −12.3465 + 8.83784i −0.830519 + 0.594497i
\(222\) 0 0
\(223\) 11.0038i 0.736871i −0.929653 0.368435i \(-0.879894\pi\)
0.929653 0.368435i \(-0.120106\pi\)
\(224\) −19.0181 −1.27070
\(225\) 0 0
\(226\) 17.6000i 1.17073i
\(227\) 8.11818i 0.538823i 0.963025 + 0.269411i \(0.0868291\pi\)
−0.963025 + 0.269411i \(0.913171\pi\)
\(228\) 0 0
\(229\) 5.37790i 0.355382i −0.984086 0.177691i \(-0.943137\pi\)
0.984086 0.177691i \(-0.0568627\pi\)
\(230\) 7.68493 0.506729
\(231\) 0 0
\(232\) 6.85055i 0.449761i
\(233\) 20.0992 1.31675 0.658373 0.752692i \(-0.271245\pi\)
0.658373 + 0.752692i \(0.271245\pi\)
\(234\) 0 0
\(235\) −12.4338 −0.811094
\(236\) 6.05433i 0.394104i
\(237\) 0 0
\(238\) 13.1041 0.849414
\(239\) 14.1607i 0.915979i −0.888958 0.457989i \(-0.848570\pi\)
0.888958 0.457989i \(-0.151430\pi\)
\(240\) 0 0
\(241\) 30.2578i 1.94908i 0.224217 + 0.974539i \(0.428018\pi\)
−0.224217 + 0.974539i \(0.571982\pi\)
\(242\) 2.41310i 0.155120i
\(243\) 0 0
\(244\) −15.3600 −0.983326
\(245\) 10.7821i 0.688841i
\(246\) 0 0
\(247\) −8.92553 12.4691i −0.567918 0.793387i
\(248\) 21.1869 1.34537
\(249\) 0 0
\(250\) −10.0929 −0.638333
\(251\) −17.5085 −1.10512 −0.552562 0.833472i \(-0.686350\pi\)
−0.552562 + 0.833472i \(0.686350\pi\)
\(252\) 0 0
\(253\) 13.9850i 0.879232i
\(254\) 1.59759i 0.100242i
\(255\) 0 0
\(256\) −16.5238 −1.03274
\(257\) 18.3042 1.14179 0.570893 0.821025i \(-0.306597\pi\)
0.570893 + 0.821025i \(0.306597\pi\)
\(258\) 0 0
\(259\) 18.8961 1.17415
\(260\) −5.55423 7.75931i −0.344459 0.481212i
\(261\) 0 0
\(262\) 6.45672i 0.398897i
\(263\) −8.39323 −0.517549 −0.258774 0.965938i \(-0.583319\pi\)
−0.258774 + 0.965938i \(0.583319\pi\)
\(264\) 0 0
\(265\) 17.0375i 1.04661i
\(266\) 13.2341i 0.811437i
\(267\) 0 0
\(268\) 15.9406i 0.973730i
\(269\) 21.3238 1.30014 0.650069 0.759875i \(-0.274740\pi\)
0.650069 + 0.759875i \(0.274740\pi\)
\(270\) 0 0
\(271\) 18.4587i 1.12129i 0.828057 + 0.560643i \(0.189446\pi\)
−0.828057 + 0.560643i \(0.810554\pi\)
\(272\) 1.04462 0.0633397
\(273\) 0 0
\(274\) 4.09382 0.247317
\(275\) 0.115721i 0.00697821i
\(276\) 0 0
\(277\) −8.62780 −0.518394 −0.259197 0.965824i \(-0.583458\pi\)
−0.259197 + 0.965824i \(0.583458\pi\)
\(278\) 12.7779i 0.766367i
\(279\) 0 0
\(280\) 22.1950i 1.32640i
\(281\) 3.47481i 0.207290i −0.994614 0.103645i \(-0.966949\pi\)
0.994614 0.103645i \(-0.0330506\pi\)
\(282\) 0 0
\(283\) −13.7836 −0.819348 −0.409674 0.912232i \(-0.634358\pi\)
−0.409674 + 0.912232i \(0.634358\pi\)
\(284\) 3.13269i 0.185891i
\(285\) 0 0
\(286\) 9.81463 7.02545i 0.580351 0.415424i
\(287\) 27.2217 1.60684
\(288\) 0 0
\(289\) 0.734202 0.0431884
\(290\) −4.83229 −0.283762
\(291\) 0 0
\(292\) 6.43917i 0.376824i
\(293\) 28.5108i 1.66562i 0.553559 + 0.832810i \(0.313269\pi\)
−0.553559 + 0.832810i \(0.686731\pi\)
\(294\) 0 0
\(295\) −11.5097 −0.670120
\(296\) 15.8363 0.920465
\(297\) 0 0
\(298\) 5.25460 0.304391
\(299\) −11.0919 + 7.93971i −0.641459 + 0.459166i
\(300\) 0 0
\(301\) 3.09433i 0.178354i
\(302\) 8.82583 0.507870
\(303\) 0 0
\(304\) 1.05499i 0.0605078i
\(305\) 29.2004i 1.67201i
\(306\) 0 0
\(307\) 21.8137i 1.24497i −0.782631 0.622486i \(-0.786123\pi\)
0.782631 0.622486i \(-0.213877\pi\)
\(308\) 14.9868 0.853954
\(309\) 0 0
\(310\) 14.9450i 0.848817i
\(311\) −6.96322 −0.394848 −0.197424 0.980318i \(-0.563258\pi\)
−0.197424 + 0.980318i \(0.563258\pi\)
\(312\) 0 0
\(313\) 22.4506 1.26898 0.634491 0.772930i \(-0.281210\pi\)
0.634491 + 0.772930i \(0.281210\pi\)
\(314\) 0.977678i 0.0551736i
\(315\) 0 0
\(316\) 12.9054 0.725983
\(317\) 10.2161i 0.573792i 0.957962 + 0.286896i \(0.0926235\pi\)
−0.957962 + 0.286896i \(0.907377\pi\)
\(318\) 0 0
\(319\) 8.79381i 0.492359i
\(320\) 12.3556i 0.690700i
\(321\) 0 0
\(322\) 11.7724 0.656053
\(323\) 17.9102i 0.996548i
\(324\) 0 0
\(325\) −0.0917806 + 0.0656979i −0.00509107 + 0.00364426i
\(326\) −5.02477 −0.278296
\(327\) 0 0
\(328\) 22.8136 1.25967
\(329\) −19.0472 −1.05011
\(330\) 0 0
\(331\) 5.40162i 0.296900i 0.988920 + 0.148450i \(0.0474284\pi\)
−0.988920 + 0.148450i \(0.952572\pi\)
\(332\) 0.634273i 0.0348103i
\(333\) 0 0
\(334\) 12.0793 0.660949
\(335\) −30.3042 −1.65570
\(336\) 0 0
\(337\) −32.9242 −1.79350 −0.896749 0.442540i \(-0.854078\pi\)
−0.896749 + 0.442540i \(0.854078\pi\)
\(338\) −11.1441 3.79565i −0.606159 0.206456i
\(339\) 0 0
\(340\) 11.1452i 0.604436i
\(341\) −27.1969 −1.47279
\(342\) 0 0
\(343\) 7.53586i 0.406898i
\(344\) 2.59326i 0.139819i
\(345\) 0 0
\(346\) 6.73908i 0.362295i
\(347\) 13.6278 0.731579 0.365789 0.930698i \(-0.380799\pi\)
0.365789 + 0.930698i \(0.380799\pi\)
\(348\) 0 0
\(349\) 11.2746i 0.603518i 0.953384 + 0.301759i \(0.0975738\pi\)
−0.953384 + 0.301759i \(0.902426\pi\)
\(350\) 0.0974121 0.00520690
\(351\) 0 0
\(352\) 20.4597 1.09050
\(353\) 26.6308i 1.41741i −0.705504 0.708706i \(-0.749279\pi\)
0.705504 0.708706i \(-0.250721\pi\)
\(354\) 0 0
\(355\) −5.95545 −0.316083
\(356\) 6.78707i 0.359714i
\(357\) 0 0
\(358\) 21.0141i 1.11063i
\(359\) 34.9036i 1.84214i 0.389396 + 0.921071i \(0.372684\pi\)
−0.389396 + 0.921071i \(0.627316\pi\)
\(360\) 0 0
\(361\) 0.912132 0.0480069
\(362\) 4.69497i 0.246762i
\(363\) 0 0
\(364\) −8.50846 11.8864i −0.445964 0.623017i
\(365\) 12.2413 0.640738
\(366\) 0 0
\(367\) −2.71433 −0.141687 −0.0708434 0.997487i \(-0.522569\pi\)
−0.0708434 + 0.997487i \(0.522569\pi\)
\(368\) 0.938467 0.0489210
\(369\) 0 0
\(370\) 11.1707i 0.580737i
\(371\) 26.0995i 1.35502i
\(372\) 0 0
\(373\) −1.75096 −0.0906613 −0.0453307 0.998972i \(-0.514434\pi\)
−0.0453307 + 0.998972i \(0.514434\pi\)
\(374\) −14.0974 −0.728961
\(375\) 0 0
\(376\) −15.9629 −0.823224
\(377\) 6.97457 4.99250i 0.359209 0.257127i
\(378\) 0 0
\(379\) 2.90941i 0.149446i −0.997204 0.0747231i \(-0.976193\pi\)
0.997204 0.0747231i \(-0.0238073\pi\)
\(380\) −11.2558 −0.577412
\(381\) 0 0
\(382\) 14.6174i 0.747890i
\(383\) 20.4457i 1.04473i −0.852723 0.522363i \(-0.825050\pi\)
0.852723 0.522363i \(-0.174950\pi\)
\(384\) 0 0
\(385\) 28.4910i 1.45203i
\(386\) 2.96968 0.151152
\(387\) 0 0
\(388\) 8.00541i 0.406413i
\(389\) 12.9632 0.657261 0.328631 0.944459i \(-0.393413\pi\)
0.328631 + 0.944459i \(0.393413\pi\)
\(390\) 0 0
\(391\) 15.9320 0.805716
\(392\) 13.8423i 0.699143i
\(393\) 0 0
\(394\) 6.44105 0.324495
\(395\) 24.5339i 1.23444i
\(396\) 0 0
\(397\) 29.1809i 1.46455i 0.681010 + 0.732274i \(0.261541\pi\)
−0.681010 + 0.732274i \(0.738459\pi\)
\(398\) 12.5379i 0.628467i
\(399\) 0 0
\(400\) 0.00776543 0.000388271
\(401\) 22.1724i 1.10723i −0.832771 0.553617i \(-0.813247\pi\)
0.832771 0.553617i \(-0.186753\pi\)
\(402\) 0 0
\(403\) 15.4405 + 21.5705i 0.769144 + 1.07450i
\(404\) −5.85500 −0.291297
\(405\) 0 0
\(406\) −7.40253 −0.367381
\(407\) −20.3285 −1.00765
\(408\) 0 0
\(409\) 2.09865i 0.103772i 0.998653 + 0.0518858i \(0.0165232\pi\)
−0.998653 + 0.0518858i \(0.983477\pi\)
\(410\) 16.0925i 0.794750i
\(411\) 0 0
\(412\) 2.49162 0.122753
\(413\) −17.6315 −0.867591
\(414\) 0 0
\(415\) 1.20579 0.0591901
\(416\) −11.6155 16.2270i −0.569499 0.795595i
\(417\) 0 0
\(418\) 14.2373i 0.696369i
\(419\) 0.391399 0.0191211 0.00956053 0.999954i \(-0.496957\pi\)
0.00956053 + 0.999954i \(0.496957\pi\)
\(420\) 0 0
\(421\) 11.0850i 0.540249i 0.962825 + 0.270124i \(0.0870648\pi\)
−0.962825 + 0.270124i \(0.912935\pi\)
\(422\) 13.0713i 0.636301i
\(423\) 0 0
\(424\) 21.8732i 1.06226i
\(425\) 0.131831 0.00639473
\(426\) 0 0
\(427\) 44.7318i 2.16472i
\(428\) −9.52652 −0.460482
\(429\) 0 0
\(430\) −1.82925 −0.0882144
\(431\) 36.4573i 1.75609i −0.478580 0.878044i \(-0.658848\pi\)
0.478580 0.878044i \(-0.341152\pi\)
\(432\) 0 0
\(433\) 8.82757 0.424226 0.212113 0.977245i \(-0.431966\pi\)
0.212113 + 0.977245i \(0.431966\pi\)
\(434\) 22.8940i 1.09895i
\(435\) 0 0
\(436\) 11.8013i 0.565182i
\(437\) 16.0901i 0.769693i
\(438\) 0 0
\(439\) −35.1344 −1.67688 −0.838438 0.544997i \(-0.816531\pi\)
−0.838438 + 0.544997i \(0.816531\pi\)
\(440\) 23.8774i 1.13831i
\(441\) 0 0
\(442\) 8.00352 + 11.1810i 0.380688 + 0.531826i
\(443\) 23.9052 1.13577 0.567885 0.823108i \(-0.307762\pi\)
0.567885 + 0.823108i \(0.307762\pi\)
\(444\) 0 0
\(445\) −12.9027 −0.611645
\(446\) −9.96503 −0.471858
\(447\) 0 0
\(448\) 18.9274i 0.894236i
\(449\) 13.9683i 0.659206i −0.944120 0.329603i \(-0.893085\pi\)
0.944120 0.329603i \(-0.106915\pi\)
\(450\) 0 0
\(451\) −29.2851 −1.37898
\(452\) 22.9309 1.07858
\(453\) 0 0
\(454\) 7.35180 0.345037
\(455\) −22.5968 + 16.1751i −1.05936 + 0.758302i
\(456\) 0 0
\(457\) 17.4212i 0.814927i 0.913221 + 0.407464i \(0.133587\pi\)
−0.913221 + 0.407464i \(0.866413\pi\)
\(458\) −4.87021 −0.227570
\(459\) 0 0
\(460\) 10.0126i 0.466841i
\(461\) 23.8667i 1.11158i 0.831322 + 0.555791i \(0.187584\pi\)
−0.831322 + 0.555791i \(0.812416\pi\)
\(462\) 0 0
\(463\) 10.5373i 0.489709i 0.969560 + 0.244854i \(0.0787402\pi\)
−0.969560 + 0.244854i \(0.921260\pi\)
\(464\) −0.590109 −0.0273951
\(465\) 0 0
\(466\) 18.2018i 0.843182i
\(467\) −8.40923 −0.389133 −0.194566 0.980889i \(-0.562330\pi\)
−0.194566 + 0.980889i \(0.562330\pi\)
\(468\) 0 0
\(469\) −46.4227 −2.14360
\(470\) 11.2600i 0.519387i
\(471\) 0 0
\(472\) −14.7765 −0.680142
\(473\) 3.32888i 0.153062i
\(474\) 0 0
\(475\) 0.133139i 0.00610883i
\(476\) 17.0733i 0.782551i
\(477\) 0 0
\(478\) −12.8239 −0.586550
\(479\) 15.7068i 0.717660i −0.933403 0.358830i \(-0.883176\pi\)
0.933403 0.358830i \(-0.116824\pi\)
\(480\) 0 0
\(481\) 11.5411 + 16.1230i 0.526227 + 0.735145i
\(482\) 27.4014 1.24810
\(483\) 0 0
\(484\) −3.14402 −0.142910
\(485\) −15.2188 −0.691051
\(486\) 0 0
\(487\) 37.8948i 1.71718i 0.512666 + 0.858588i \(0.328658\pi\)
−0.512666 + 0.858588i \(0.671342\pi\)
\(488\) 37.4884i 1.69702i
\(489\) 0 0
\(490\) 9.76420 0.441102
\(491\) −3.76054 −0.169711 −0.0848553 0.996393i \(-0.527043\pi\)
−0.0848553 + 0.996393i \(0.527043\pi\)
\(492\) 0 0
\(493\) −10.0181 −0.451191
\(494\) −11.2919 + 8.08293i −0.508048 + 0.363668i
\(495\) 0 0
\(496\) 1.82505i 0.0819471i
\(497\) −9.12309 −0.409226
\(498\) 0 0
\(499\) 8.26147i 0.369834i 0.982754 + 0.184917i \(0.0592016\pi\)
−0.982754 + 0.184917i \(0.940798\pi\)
\(500\) 13.1500i 0.588086i
\(501\) 0 0
\(502\) 15.8556i 0.707670i
\(503\) −12.9954 −0.579434 −0.289717 0.957112i \(-0.593561\pi\)
−0.289717 + 0.957112i \(0.593561\pi\)
\(504\) 0 0
\(505\) 11.1307i 0.495312i
\(506\) −12.6648 −0.563020
\(507\) 0 0
\(508\) −2.08148 −0.0923509
\(509\) 26.8713i 1.19105i −0.803338 0.595524i \(-0.796945\pi\)
0.803338 0.595524i \(-0.203055\pi\)
\(510\) 0 0
\(511\) 18.7523 0.829551
\(512\) 2.80162i 0.123815i
\(513\) 0 0
\(514\) 16.5762i 0.731146i
\(515\) 4.73673i 0.208725i
\(516\) 0 0
\(517\) 20.4910 0.901195
\(518\) 17.1123i 0.751870i
\(519\) 0 0
\(520\) −18.9377 + 13.5559i −0.830474 + 0.594465i
\(521\) 14.2080 0.622464 0.311232 0.950334i \(-0.399258\pi\)
0.311232 + 0.950334i \(0.399258\pi\)
\(522\) 0 0
\(523\) 7.07846 0.309519 0.154760 0.987952i \(-0.450540\pi\)
0.154760 + 0.987952i \(0.450540\pi\)
\(524\) −8.41241 −0.367498
\(525\) 0 0
\(526\) 7.60088i 0.331414i
\(527\) 30.9832i 1.34965i
\(528\) 0 0
\(529\) −8.68705 −0.377698
\(530\) −15.4291 −0.670198
\(531\) 0 0
\(532\) −17.2427 −0.747564
\(533\) 16.6260 + 23.2267i 0.720151 + 1.00606i
\(534\) 0 0
\(535\) 18.1105i 0.782987i
\(536\) −38.9054 −1.68046
\(537\) 0 0
\(538\) 19.3108i 0.832548i
\(539\) 17.7689i 0.765362i
\(540\) 0 0
\(541\) 7.18897i 0.309078i −0.987987 0.154539i \(-0.950611\pi\)
0.987987 0.154539i \(-0.0493892\pi\)
\(542\) 16.7161 0.718020
\(543\) 0 0
\(544\) 23.3080i 0.999322i
\(545\) 22.4351 0.961016
\(546\) 0 0
\(547\) −31.7775 −1.35871 −0.679353 0.733811i \(-0.737740\pi\)
−0.679353 + 0.733811i \(0.737740\pi\)
\(548\) 5.33381i 0.227849i
\(549\) 0 0
\(550\) −0.104796 −0.00446852
\(551\) 10.1175i 0.431018i
\(552\) 0 0
\(553\) 37.5832i 1.59820i
\(554\) 7.81331i 0.331956i
\(555\) 0 0
\(556\) 16.6482 0.706042
\(557\) 32.2223i 1.36530i 0.730745 + 0.682650i \(0.239173\pi\)
−0.730745 + 0.682650i \(0.760827\pi\)
\(558\) 0 0
\(559\) 2.64021 1.88990i 0.111669 0.0799343i
\(560\) 1.91189 0.0807919
\(561\) 0 0
\(562\) −3.14678 −0.132739
\(563\) −33.6514 −1.41824 −0.709119 0.705089i \(-0.750907\pi\)
−0.709119 + 0.705089i \(0.750907\pi\)
\(564\) 0 0
\(565\) 43.5931i 1.83397i
\(566\) 12.4824i 0.524672i
\(567\) 0 0
\(568\) −7.64578 −0.320810
\(569\) −11.3359 −0.475224 −0.237612 0.971360i \(-0.576365\pi\)
−0.237612 + 0.971360i \(0.576365\pi\)
\(570\) 0 0
\(571\) −0.254521 −0.0106514 −0.00532568 0.999986i \(-0.501695\pi\)
−0.00532568 + 0.999986i \(0.501695\pi\)
\(572\) 9.15341 + 12.7874i 0.382723 + 0.534668i
\(573\) 0 0
\(574\) 24.6518i 1.02895i
\(575\) 0.118434 0.00493903
\(576\) 0 0
\(577\) 1.91400i 0.0796807i 0.999206 + 0.0398403i \(0.0126849\pi\)
−0.999206 + 0.0398403i \(0.987315\pi\)
\(578\) 0.664891i 0.0276558i
\(579\) 0 0
\(580\) 6.29595i 0.261425i
\(581\) 1.84714 0.0766324
\(582\) 0 0
\(583\) 28.0779i 1.16287i
\(584\) 15.7157 0.650321
\(585\) 0 0
\(586\) 25.8193 1.06659
\(587\) 8.88440i 0.366699i −0.983048 0.183349i \(-0.941306\pi\)
0.983048 0.183349i \(-0.0586939\pi\)
\(588\) 0 0
\(589\) 31.2906 1.28931
\(590\) 10.4231i 0.429113i
\(591\) 0 0
\(592\) 1.36414i 0.0560659i
\(593\) 43.1136i 1.77046i −0.465152 0.885231i \(-0.654000\pi\)
0.465152 0.885231i \(-0.346000\pi\)
\(594\) 0 0
\(595\) 32.4574 1.33062
\(596\) 6.84618i 0.280430i
\(597\) 0 0
\(598\) 7.19018 + 10.0447i 0.294028 + 0.410760i
\(599\) −11.1974 −0.457512 −0.228756 0.973484i \(-0.573466\pi\)
−0.228756 + 0.973484i \(0.573466\pi\)
\(600\) 0 0
\(601\) 5.92545 0.241704 0.120852 0.992671i \(-0.461437\pi\)
0.120852 + 0.992671i \(0.461437\pi\)
\(602\) −2.80221 −0.114210
\(603\) 0 0
\(604\) 11.4991i 0.467892i
\(605\) 5.97698i 0.242999i
\(606\) 0 0
\(607\) 5.06940 0.205760 0.102880 0.994694i \(-0.467194\pi\)
0.102880 + 0.994694i \(0.467194\pi\)
\(608\) −23.5393 −0.954643
\(609\) 0 0
\(610\) 26.4438 1.07068
\(611\) −11.6334 16.2519i −0.470635 0.657482i
\(612\) 0 0
\(613\) 24.6393i 0.995173i −0.867414 0.497587i \(-0.834220\pi\)
0.867414 0.497587i \(-0.165780\pi\)
\(614\) −19.7544 −0.797222
\(615\) 0 0
\(616\) 36.5775i 1.47375i
\(617\) 43.9864i 1.77083i 0.464806 + 0.885413i \(0.346124\pi\)
−0.464806 + 0.885413i \(0.653876\pi\)
\(618\) 0 0
\(619\) 20.5512i 0.826024i −0.910726 0.413012i \(-0.864477\pi\)
0.910726 0.413012i \(-0.135523\pi\)
\(620\) 19.4717 0.782002
\(621\) 0 0
\(622\) 6.30587i 0.252842i
\(623\) −19.7654 −0.791885
\(624\) 0 0
\(625\) −25.1555 −1.00622
\(626\) 20.3312i 0.812597i
\(627\) 0 0
\(628\) −1.27381 −0.0508305
\(629\) 23.1586i 0.923392i
\(630\) 0 0
\(631\) 25.1460i 1.00105i −0.865723 0.500523i \(-0.833141\pi\)
0.865723 0.500523i \(-0.166859\pi\)
\(632\) 31.4974i 1.25290i
\(633\) 0 0
\(634\) 9.25165 0.367430
\(635\) 3.95704i 0.157030i
\(636\) 0 0
\(637\) −14.0929 + 10.0879i −0.558382 + 0.399698i
\(638\) 7.96365 0.315284
\(639\) 0 0
\(640\) −13.6404 −0.539183
\(641\) −33.3421 −1.31693 −0.658467 0.752609i \(-0.728795\pi\)
−0.658467 + 0.752609i \(0.728795\pi\)
\(642\) 0 0
\(643\) 31.3801i 1.23751i −0.785585 0.618754i \(-0.787638\pi\)
0.785585 0.618754i \(-0.212362\pi\)
\(644\) 15.3382i 0.604411i
\(645\) 0 0
\(646\) 16.2194 0.638143
\(647\) −47.5495 −1.86936 −0.934681 0.355486i \(-0.884315\pi\)
−0.934681 + 0.355486i \(0.884315\pi\)
\(648\) 0 0
\(649\) 18.9680 0.744561
\(650\) 0.0594958 + 0.0831162i 0.00233362 + 0.00326008i
\(651\) 0 0
\(652\) 6.54673i 0.256390i
\(653\) −27.7456 −1.08577 −0.542884 0.839808i \(-0.682668\pi\)
−0.542884 + 0.839808i \(0.682668\pi\)
\(654\) 0 0
\(655\) 15.9925i 0.624880i
\(656\) 1.96518i 0.0767273i
\(657\) 0 0
\(658\) 17.2491i 0.672440i
\(659\) 15.6188 0.608422 0.304211 0.952605i \(-0.401607\pi\)
0.304211 + 0.952605i \(0.401607\pi\)
\(660\) 0 0
\(661\) 17.9164i 0.696869i 0.937333 + 0.348434i \(0.113287\pi\)
−0.937333 + 0.348434i \(0.886713\pi\)
\(662\) 4.89169 0.190121
\(663\) 0 0
\(664\) 1.54803 0.0600754
\(665\) 32.7794i 1.27113i
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 15.7380i 0.608922i
\(669\) 0 0
\(670\) 27.4434i 1.06023i
\(671\) 48.1225i 1.85775i
\(672\) 0 0
\(673\) −5.89919 −0.227397 −0.113699 0.993515i \(-0.536270\pi\)
−0.113699 + 0.993515i \(0.536270\pi\)
\(674\) 29.8161i 1.14847i
\(675\) 0 0
\(676\) 4.94532 14.5196i 0.190205 0.558444i
\(677\) 27.4090 1.05341 0.526707 0.850047i \(-0.323427\pi\)
0.526707 + 0.850047i \(0.323427\pi\)
\(678\) 0 0
\(679\) −23.3135 −0.894690
\(680\) 27.2016 1.04313
\(681\) 0 0
\(682\) 24.6294i 0.943109i
\(683\) 16.0989i 0.616006i −0.951385 0.308003i \(-0.900339\pi\)
0.951385 0.308003i \(-0.0996607\pi\)
\(684\) 0 0
\(685\) 10.1399 0.387427
\(686\) −6.82445 −0.260559
\(687\) 0 0
\(688\) −0.223385 −0.00851646
\(689\) 22.2692 15.9406i 0.848390 0.607290i
\(690\) 0 0
\(691\) 27.6824i 1.05309i −0.850148 0.526543i \(-0.823488\pi\)
0.850148 0.526543i \(-0.176512\pi\)
\(692\) 8.78030 0.333777
\(693\) 0 0
\(694\) 12.3413i 0.468469i
\(695\) 31.6494i 1.20053i
\(696\) 0 0
\(697\) 33.3621i 1.26368i
\(698\) 10.2103 0.386465
\(699\) 0 0
\(700\) 0.126918i 0.00479703i
\(701\) −10.0776 −0.380624 −0.190312 0.981724i \(-0.560950\pi\)
−0.190312 + 0.981724i \(0.560950\pi\)
\(702\) 0 0
\(703\) 23.3883 0.882108
\(704\) 20.3621i 0.767427i
\(705\) 0 0
\(706\) −24.1167 −0.907645
\(707\) 17.0511i 0.641271i
\(708\) 0 0
\(709\) 3.22720i 0.121200i −0.998162 0.0606000i \(-0.980699\pi\)
0.998162 0.0606000i \(-0.0193014\pi\)
\(710\) 5.39324i 0.202405i
\(711\) 0 0
\(712\) −16.5648 −0.620793
\(713\) 27.8346i 1.04241i
\(714\) 0 0
\(715\) 24.3097 17.4012i 0.909131 0.650769i
\(716\) 27.3792 1.02321
\(717\) 0 0
\(718\) 31.6086 1.17962
\(719\) −30.0712 −1.12147 −0.560733 0.827996i \(-0.689481\pi\)
−0.560733 + 0.827996i \(0.689481\pi\)
\(720\) 0 0
\(721\) 7.25614i 0.270233i
\(722\) 0.826023i 0.0307414i
\(723\) 0 0
\(724\) 6.11704 0.227338
\(725\) −0.0744713 −0.00276579
\(726\) 0 0
\(727\) −19.3772 −0.718660 −0.359330 0.933210i \(-0.616995\pi\)
−0.359330 + 0.933210i \(0.616995\pi\)
\(728\) −29.0105 + 20.7661i −1.07520 + 0.769643i
\(729\) 0 0
\(730\) 11.0857i 0.410299i
\(731\) −3.79232 −0.140264
\(732\) 0 0
\(733\) 24.4725i 0.903911i 0.892041 + 0.451955i \(0.149273\pi\)
−0.892041 + 0.451955i \(0.850727\pi\)
\(734\) 2.45809i 0.0907297i
\(735\) 0 0
\(736\) 20.9394i 0.771835i
\(737\) 49.9416 1.83962
\(738\) 0 0
\(739\) 3.30687i 0.121645i 0.998149 + 0.0608226i \(0.0193724\pi\)
−0.998149 + 0.0608226i \(0.980628\pi\)
\(740\) 14.5542 0.535024
\(741\) 0 0
\(742\) −23.6357 −0.867692
\(743\) 17.4815i 0.641335i 0.947192 + 0.320667i \(0.103907\pi\)
−0.947192 + 0.320667i \(0.896093\pi\)
\(744\) 0 0
\(745\) 13.0150 0.476834
\(746\) 1.58566i 0.0580553i
\(747\) 0 0
\(748\) 18.3674i 0.671580i
\(749\) 27.7433i 1.01372i
\(750\) 0 0
\(751\) −28.5843 −1.04306 −0.521528 0.853234i \(-0.674638\pi\)
−0.521528 + 0.853234i \(0.674638\pi\)
\(752\) 1.37505i 0.0501430i
\(753\) 0 0
\(754\) −4.52119 6.31615i −0.164652 0.230021i
\(755\) 21.8606 0.795587
\(756\) 0 0
\(757\) 11.9611 0.434733 0.217367 0.976090i \(-0.430253\pi\)
0.217367 + 0.976090i \(0.430253\pi\)
\(758\) −2.63475 −0.0956983
\(759\) 0 0
\(760\) 27.4714i 0.996494i
\(761\) 2.12842i 0.0771553i 0.999256 + 0.0385777i \(0.0122827\pi\)
−0.999256 + 0.0385777i \(0.987717\pi\)
\(762\) 0 0
\(763\) 34.3681 1.24421
\(764\) 19.0449 0.689019
\(765\) 0 0
\(766\) −18.5156 −0.668994
\(767\) −10.7687 15.0440i −0.388835 0.543207i
\(768\) 0 0
\(769\) 7.65086i 0.275897i −0.990439 0.137949i \(-0.955949\pi\)
0.990439 0.137949i \(-0.0440509\pi\)
\(770\) −25.8013 −0.929815
\(771\) 0 0
\(772\) 3.86917i 0.139254i
\(773\) 24.6440i 0.886383i 0.896427 + 0.443191i \(0.146154\pi\)
−0.896427 + 0.443191i \(0.853846\pi\)
\(774\) 0 0
\(775\) 0.230320i 0.00827333i
\(776\) −19.5384 −0.701386
\(777\) 0 0
\(778\) 11.7394i 0.420880i
\(779\) 33.6931 1.20718
\(780\) 0 0
\(781\) 9.81463 0.351195
\(782\) 14.4280i 0.515943i
\(783\) 0 0
\(784\) 1.19238 0.0425851
\(785\) 2.42159i 0.0864304i
\(786\) 0 0
\(787\) 41.8679i 1.49243i −0.665706 0.746214i \(-0.731869\pi\)
0.665706 0.746214i \(-0.268131\pi\)
\(788\) 8.39199i 0.298952i
\(789\) 0 0
\(790\) −22.2178 −0.790475
\(791\) 66.7797i 2.37441i
\(792\) 0 0
\(793\) −38.1671 + 27.3205i −1.35535 + 0.970181i
\(794\) 26.4261 0.937828
\(795\) 0 0
\(796\) −16.3355 −0.578996
\(797\) −41.7563 −1.47908 −0.739541 0.673111i \(-0.764957\pi\)
−0.739541 + 0.673111i \(0.764957\pi\)
\(798\) 0 0
\(799\) 23.3437i 0.825842i
\(800\) 0.173265i 0.00612583i
\(801\) 0 0
\(802\) −20.0792 −0.709022
\(803\) −20.1737 −0.711915
\(804\) 0 0
\(805\) 29.1590 1.02772
\(806\) 19.5341 13.9828i 0.688061 0.492524i
\(807\) 0 0
\(808\) 14.2900i 0.502720i
\(809\) 45.2476 1.59082 0.795410 0.606072i \(-0.207256\pi\)
0.795410 + 0.606072i \(0.207256\pi\)
\(810\) 0 0
\(811\) 52.8820i 1.85694i 0.371410 + 0.928469i \(0.378874\pi\)
−0.371410 + 0.928469i \(0.621126\pi\)
\(812\) 9.64469i 0.338462i
\(813\) 0 0
\(814\) 18.4094i 0.645249i
\(815\) −12.4458 −0.435956
\(816\) 0 0
\(817\) 3.82994i 0.133993i
\(818\) 1.90053 0.0664505
\(819\) 0 0
\(820\) 20.9667 0.732190
\(821\) 34.4222i 1.20134i 0.799496 + 0.600671i \(0.205100\pi\)
−0.799496 + 0.600671i \(0.794900\pi\)
\(822\) 0 0
\(823\) −11.9971 −0.418191 −0.209096 0.977895i \(-0.567052\pi\)
−0.209096 + 0.977895i \(0.567052\pi\)
\(824\) 6.08116i 0.211847i
\(825\) 0 0
\(826\) 15.9671i 0.555565i
\(827\) 4.35092i 0.151296i 0.997135 + 0.0756482i \(0.0241026\pi\)
−0.997135 + 0.0756482i \(0.975897\pi\)
\(828\) 0 0
\(829\) 45.6204 1.58446 0.792230 0.610222i \(-0.208920\pi\)
0.792230 + 0.610222i \(0.208920\pi\)
\(830\) 1.09196i 0.0379026i
\(831\) 0 0
\(832\) −16.1497 + 11.5602i −0.559889 + 0.400777i
\(833\) 20.2427 0.701367
\(834\) 0 0
\(835\) 29.9190 1.03539
\(836\) 18.5497 0.641554
\(837\) 0 0
\(838\) 0.354449i 0.0122442i
\(839\) 24.6675i 0.851615i −0.904814 0.425808i \(-0.859990\pi\)
0.904814 0.425808i \(-0.140010\pi\)
\(840\) 0 0
\(841\) −23.3408 −0.804855
\(842\) 10.0385 0.345950
\(843\) 0 0
\(844\) 17.0305 0.586214
\(845\) −27.6026 9.40138i −0.949559 0.323417i
\(846\) 0 0
\(847\) 9.15606i 0.314606i
\(848\) −1.88417 −0.0647027
\(849\) 0 0
\(850\) 0.119386i 0.00409489i
\(851\) 20.8051i 0.713190i
\(852\) 0 0
\(853\) 26.5198i 0.908021i −0.890996 0.454010i \(-0.849993\pi\)
0.890996 0.454010i \(-0.150007\pi\)
\(854\) 40.5089 1.38619
\(855\) 0 0
\(856\) 23.2508i 0.794697i
\(857\) −6.40948 −0.218944 −0.109472 0.993990i \(-0.534916\pi\)
−0.109472 + 0.993990i \(0.534916\pi\)
\(858\) 0 0
\(859\) 25.5996 0.873446 0.436723 0.899596i \(-0.356139\pi\)
0.436723 + 0.899596i \(0.356139\pi\)
\(860\) 2.38332i 0.0812706i
\(861\) 0 0
\(862\) −33.0156 −1.12452
\(863\) 48.4823i 1.65036i 0.564873 + 0.825178i \(0.308925\pi\)
−0.564873 + 0.825178i \(0.691075\pi\)
\(864\) 0 0
\(865\) 16.6919i 0.567542i
\(866\) 7.99421i 0.271654i
\(867\) 0 0
\(868\) 29.8284 1.01244
\(869\) 40.4321i 1.37156i
\(870\) 0 0
\(871\) −28.3533 39.6098i −0.960713 1.34213i
\(872\) 28.8029 0.975389
\(873\) 0 0
\(874\) 14.5711 0.492876
\(875\) −38.2957 −1.29463
\(876\) 0 0
\(877\) 31.1795i 1.05286i 0.850219 + 0.526429i \(0.176469\pi\)
−0.850219 + 0.526429i \(0.823531\pi\)
\(878\) 31.8176i 1.07379i
\(879\) 0 0
\(880\) −2.05681 −0.0693350
\(881\) −49.6614 −1.67314 −0.836568 0.547863i \(-0.815442\pi\)
−0.836568 + 0.547863i \(0.815442\pi\)
\(882\) 0 0
\(883\) 30.3523 1.02144 0.510718 0.859748i \(-0.329380\pi\)
0.510718 + 0.859748i \(0.329380\pi\)
\(884\) −14.5676 + 10.4277i −0.489962 + 0.350722i
\(885\) 0 0
\(886\) 21.6485i 0.727294i
\(887\) 12.8432 0.431233 0.215616 0.976478i \(-0.430824\pi\)
0.215616 + 0.976478i \(0.430824\pi\)
\(888\) 0 0
\(889\) 6.06173i 0.203304i
\(890\) 11.6846i 0.391669i
\(891\) 0 0
\(892\) 12.9834i 0.434715i
\(893\) −23.5754 −0.788919
\(894\) 0 0
\(895\) 52.0496i 1.73982i
\(896\) −20.8955 −0.698071
\(897\) 0 0
\(898\) −12.6497 −0.422125
\(899\) 17.5024i 0.583738i
\(900\) 0 0
\(901\) −31.9869 −1.06564
\(902\) 26.5205i 0.883035i
\(903\) 0 0
\(904\) 55.9660i 1.86140i
\(905\) 11.6289i 0.386557i
\(906\) 0 0
\(907\) −20.5986 −0.683965 −0.341983 0.939706i \(-0.611098\pi\)
−0.341983 + 0.939706i \(0.611098\pi\)
\(908\) 9.57860i 0.317877i
\(909\) 0 0
\(910\) 14.6481 + 20.4636i 0.485581 + 0.678362i
\(911\) 42.3899 1.40444 0.702219 0.711961i \(-0.252193\pi\)
0.702219 + 0.711961i \(0.252193\pi\)
\(912\) 0 0
\(913\) −1.98716 −0.0657653
\(914\) 15.7765 0.521842
\(915\) 0 0
\(916\) 6.34535i 0.209656i
\(917\) 24.4988i 0.809021i
\(918\) 0 0
\(919\) −42.7987 −1.41180 −0.705899 0.708313i \(-0.749457\pi\)
−0.705899 + 0.708313i \(0.749457\pi\)
\(920\) 24.4373 0.805672
\(921\) 0 0
\(922\) 21.6136 0.711805
\(923\) −5.57205 7.78420i −0.183406 0.256220i
\(924\) 0 0
\(925\) 0.172154i 0.00566038i
\(926\) 9.54253 0.313587
\(927\) 0 0
\(928\) 13.1667i 0.432218i
\(929\) 11.8059i 0.387340i 0.981067 + 0.193670i \(0.0620391\pi\)
−0.981067 + 0.193670i \(0.937961\pi\)
\(930\) 0 0
\(931\) 20.4435i 0.670009i
\(932\) 23.7150 0.776810
\(933\) 0 0
\(934\) 7.61537i 0.249183i
\(935\) −34.9177 −1.14193
\(936\) 0 0
\(937\) 24.7925 0.809935 0.404967 0.914331i \(-0.367283\pi\)
0.404967 + 0.914331i \(0.367283\pi\)
\(938\) 42.0402i 1.37266i
\(939\) 0 0
\(940\) −14.6706 −0.478503
\(941\) 23.8292i 0.776810i −0.921489 0.388405i \(-0.873026\pi\)
0.921489 0.388405i \(-0.126974\pi\)
\(942\) 0 0
\(943\) 29.9717i 0.976014i
\(944\) 1.27285i 0.0414278i
\(945\) 0 0
\(946\) 3.01462 0.0980138
\(947\) 49.0208i 1.59296i 0.604664 + 0.796480i \(0.293307\pi\)
−0.604664 + 0.796480i \(0.706693\pi\)
\(948\) 0 0
\(949\) 11.4532 + 16.0002i 0.371786 + 0.519389i
\(950\) 0.120570 0.00391181
\(951\) 0 0
\(952\) 41.6697 1.35052
\(953\) 7.78372 0.252140 0.126070 0.992021i \(-0.459764\pi\)
0.126070 + 0.992021i \(0.459764\pi\)
\(954\) 0 0
\(955\) 36.2055i 1.17158i
\(956\) 16.7081i 0.540379i
\(957\) 0 0
\(958\) −14.2240 −0.459556
\(959\) 15.5332 0.501594
\(960\) 0 0
\(961\) −23.1303 −0.746137
\(962\) 14.6009 10.4515i 0.470752 0.336971i
\(963\) 0 0
\(964\) 35.7011i 1.14985i
\(965\) 7.35554 0.236783
\(966\) 0 0
\(967\) 37.5960i 1.20900i 0.796603 + 0.604502i \(0.206628\pi\)
−0.796603 + 0.604502i \(0.793372\pi\)
\(968\) 7.67342i 0.246633i
\(969\) 0 0
\(970\) 13.7821i 0.442517i
\(971\) −9.16512 −0.294123 −0.147061 0.989127i \(-0.546981\pi\)
−0.147061 + 0.989127i \(0.546981\pi\)
\(972\) 0 0
\(973\) 48.4833i 1.55430i
\(974\) 34.3174 1.09960
\(975\) 0 0
\(976\) 3.22926 0.103366
\(977\) 47.3477i 1.51479i 0.652959 + 0.757394i \(0.273528\pi\)
−0.652959 + 0.757394i \(0.726472\pi\)
\(978\) 0 0
\(979\) 21.2637 0.679590
\(980\) 12.7217i 0.406380i
\(981\) 0 0
\(982\) 3.40553i 0.108675i
\(983\) 16.2390i 0.517943i 0.965885 + 0.258972i \(0.0833836\pi\)
−0.965885 + 0.258972i \(0.916616\pi\)
\(984\) 0 0
\(985\) 15.9537 0.508328
\(986\) 9.07232i 0.288922i
\(987\) 0 0
\(988\) −10.5312 14.7122i −0.335042 0.468056i
\(989\) −3.40693 −0.108334
\(990\) 0 0
\(991\) −37.7634 −1.19959 −0.599797 0.800152i \(-0.704752\pi\)
−0.599797 + 0.800152i \(0.704752\pi\)
\(992\) 40.7211 1.29289
\(993\) 0 0
\(994\) 8.26184i 0.262049i
\(995\) 31.0548i 0.984505i
\(996\) 0 0
\(997\) 47.6566 1.50930 0.754649 0.656128i \(-0.227807\pi\)
0.754649 + 0.656128i \(0.227807\pi\)
\(998\) 7.48156 0.236824
\(999\) 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 1053.2.b.j.649.5 10
3.2 odd 2 1053.2.b.i.649.6 10
9.2 odd 6 351.2.t.c.64.6 20
9.4 even 3 117.2.t.c.25.6 yes 20
9.5 odd 6 351.2.t.c.181.5 20
9.7 even 3 117.2.t.c.103.5 yes 20
13.12 even 2 inner 1053.2.b.j.649.6 10
39.38 odd 2 1053.2.b.i.649.5 10
117.25 even 6 117.2.t.c.103.6 yes 20
117.38 odd 6 351.2.t.c.64.5 20
117.77 odd 6 351.2.t.c.181.6 20
117.103 even 6 117.2.t.c.25.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.5 20 117.103 even 6
117.2.t.c.25.6 yes 20 9.4 even 3
117.2.t.c.103.5 yes 20 9.7 even 3
117.2.t.c.103.6 yes 20 117.25 even 6
351.2.t.c.64.5 20 117.38 odd 6
351.2.t.c.64.6 20 9.2 odd 6
351.2.t.c.181.5 20 9.5 odd 6
351.2.t.c.181.6 20 117.77 odd 6
1053.2.b.i.649.5 10 39.38 odd 2
1053.2.b.i.649.6 10 3.2 odd 2
1053.2.b.j.649.5 10 1.1 even 1 trivial
1053.2.b.j.649.6 10 13.12 even 2 inner