Properties

Label 2268.2.t.b.1781.6
Level $2268$
Weight $2$
Character 2268.1781
Analytic conductor $18.110$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(1781,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.t (of order \(6\), degree \(2\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1781.6
Root \(-1.61108 - 0.635951i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1781
Dual form 2268.2.t.b.2105.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.09150 + 1.89054i) q^{5} +(-1.38614 + 2.25358i) q^{7} +(1.26889 + 0.732592i) q^{11} +3.38041i q^{13} +(1.32136 - 2.28866i) q^{17} +(6.87816 - 3.97111i) q^{19} +(3.47245 - 2.00482i) q^{23} +(0.117249 - 0.203081i) q^{25} +7.75105i q^{29} +(-0.612252 - 0.353484i) q^{31} +(-5.77345 - 0.160752i) q^{35} +(1.41738 + 2.45498i) q^{37} -7.48345 q^{41} +2.54224 q^{43} +(6.27538 + 10.8693i) q^{47} +(-3.15726 - 6.24754i) q^{49} +(-2.41675 - 1.39531i) q^{53} +3.19850i q^{55} +(-6.71650 + 11.6333i) q^{59} +(-6.75061 + 3.89747i) q^{61} +(-6.39079 + 3.68972i) q^{65} +(-2.92029 + 5.05809i) q^{67} +11.6854i q^{71} +(-3.95924 - 2.28587i) q^{73} +(-3.40980 + 1.84407i) q^{77} +(-4.69189 - 8.12659i) q^{79} -3.41695 q^{83} +5.76905 q^{85} +(4.61937 + 8.00099i) q^{89} +(-7.61803 - 4.68571i) q^{91} +(15.0150 + 8.66894i) q^{95} -7.37154i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{7} + 6 q^{11} + 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{31} - 15 q^{35} + q^{37} + 12 q^{41} + 4 q^{43} + 18 q^{47} - 8 q^{49} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - 48 q^{77} - q^{79}+ \cdots + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.09150 + 1.89054i 0.488134 + 0.845473i 0.999907 0.0136476i \(-0.00434429\pi\)
−0.511773 + 0.859121i \(0.671011\pi\)
\(6\) 0 0
\(7\) −1.38614 + 2.25358i −0.523910 + 0.851774i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.26889 + 0.732592i 0.382584 + 0.220885i 0.678942 0.734192i \(-0.262439\pi\)
−0.296358 + 0.955077i \(0.595772\pi\)
\(12\) 0 0
\(13\) 3.38041i 0.937557i 0.883316 + 0.468779i \(0.155306\pi\)
−0.883316 + 0.468779i \(0.844694\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.32136 2.28866i 0.320476 0.555081i −0.660110 0.751169i \(-0.729491\pi\)
0.980586 + 0.196088i \(0.0628238\pi\)
\(18\) 0 0
\(19\) 6.87816 3.97111i 1.57796 0.911034i 0.582813 0.812606i \(-0.301952\pi\)
0.995144 0.0984279i \(-0.0313814\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.47245 2.00482i 0.724056 0.418034i −0.0921879 0.995742i \(-0.529386\pi\)
0.816244 + 0.577708i \(0.196053\pi\)
\(24\) 0 0
\(25\) 0.117249 0.203081i 0.0234498 0.0406163i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.75105i 1.43933i 0.694319 + 0.719667i \(0.255706\pi\)
−0.694319 + 0.719667i \(0.744294\pi\)
\(30\) 0 0
\(31\) −0.612252 0.353484i −0.109964 0.0634876i 0.444009 0.896022i \(-0.353556\pi\)
−0.553973 + 0.832534i \(0.686889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.77345 0.160752i −0.975890 0.0271721i
\(36\) 0 0
\(37\) 1.41738 + 2.45498i 0.233016 + 0.403596i 0.958694 0.284438i \(-0.0918071\pi\)
−0.725678 + 0.688034i \(0.758474\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.48345 −1.16872 −0.584360 0.811495i \(-0.698654\pi\)
−0.584360 + 0.811495i \(0.698654\pi\)
\(42\) 0 0
\(43\) 2.54224 0.387688 0.193844 0.981032i \(-0.437904\pi\)
0.193844 + 0.981032i \(0.437904\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.27538 + 10.8693i 0.915358 + 1.58545i 0.806376 + 0.591403i \(0.201426\pi\)
0.108983 + 0.994044i \(0.465241\pi\)
\(48\) 0 0
\(49\) −3.15726 6.24754i −0.451036 0.892506i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.41675 1.39531i −0.331966 0.191661i 0.324748 0.945801i \(-0.394721\pi\)
−0.656714 + 0.754140i \(0.728054\pi\)
\(54\) 0 0
\(55\) 3.19850i 0.431286i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.71650 + 11.6333i −0.874414 + 1.51453i −0.0170287 + 0.999855i \(0.505421\pi\)
−0.857385 + 0.514675i \(0.827913\pi\)
\(60\) 0 0
\(61\) −6.75061 + 3.89747i −0.864327 + 0.499020i −0.865459 0.500980i \(-0.832973\pi\)
0.00113176 + 0.999999i \(0.499640\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.39079 + 3.68972i −0.792680 + 0.457654i
\(66\) 0 0
\(67\) −2.92029 + 5.05809i −0.356770 + 0.617945i −0.987419 0.158124i \(-0.949455\pi\)
0.630649 + 0.776068i \(0.282789\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.6854i 1.38680i 0.720554 + 0.693398i \(0.243887\pi\)
−0.720554 + 0.693398i \(0.756113\pi\)
\(72\) 0 0
\(73\) −3.95924 2.28587i −0.463394 0.267541i 0.250076 0.968226i \(-0.419544\pi\)
−0.713470 + 0.700685i \(0.752878\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.40980 + 1.84407i −0.388583 + 0.210151i
\(78\) 0 0
\(79\) −4.69189 8.12659i −0.527879 0.914312i −0.999472 0.0324963i \(-0.989654\pi\)
0.471593 0.881816i \(-0.343679\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.41695 −0.375059 −0.187529 0.982259i \(-0.560048\pi\)
−0.187529 + 0.982259i \(0.560048\pi\)
\(84\) 0 0
\(85\) 5.76905 0.625741
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.61937 + 8.00099i 0.489653 + 0.848103i 0.999929 0.0119070i \(-0.00379021\pi\)
−0.510276 + 0.860010i \(0.670457\pi\)
\(90\) 0 0
\(91\) −7.61803 4.68571i −0.798586 0.491196i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.0150 + 8.66894i 1.54051 + 0.889414i
\(96\) 0 0
\(97\) 7.37154i 0.748467i −0.927335 0.374233i \(-0.877906\pi\)
0.927335 0.374233i \(-0.122094\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.96357 6.86510i 0.394390 0.683103i −0.598633 0.801023i \(-0.704289\pi\)
0.993023 + 0.117920i \(0.0376226\pi\)
\(102\) 0 0
\(103\) 3.26825 1.88693i 0.322031 0.185924i −0.330267 0.943888i \(-0.607139\pi\)
0.652297 + 0.757963i \(0.273805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.88241 + 3.97356i −0.665347 + 0.384138i −0.794311 0.607511i \(-0.792168\pi\)
0.128964 + 0.991649i \(0.458835\pi\)
\(108\) 0 0
\(109\) 0.505142 0.874932i 0.0483838 0.0838033i −0.840819 0.541316i \(-0.817926\pi\)
0.889203 + 0.457513i \(0.151260\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.1886i 1.14661i −0.819342 0.573304i \(-0.805661\pi\)
0.819342 0.573304i \(-0.194339\pi\)
\(114\) 0 0
\(115\) 7.58037 + 4.37653i 0.706873 + 0.408113i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.32609 + 6.15017i 0.304902 + 0.563785i
\(120\) 0 0
\(121\) −4.42662 7.66713i −0.402420 0.697012i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.4269 1.02206
\(126\) 0 0
\(127\) 6.79350 0.602826 0.301413 0.953494i \(-0.402542\pi\)
0.301413 + 0.953494i \(0.402542\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.86790 + 11.8956i 0.600051 + 1.03932i 0.992813 + 0.119679i \(0.0381865\pi\)
−0.392761 + 0.919640i \(0.628480\pi\)
\(132\) 0 0
\(133\) −0.584850 + 21.0050i −0.0507130 + 1.82136i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.4028 10.0475i −1.48682 0.858416i −0.486933 0.873439i \(-0.661884\pi\)
−0.999887 + 0.0150235i \(0.995218\pi\)
\(138\) 0 0
\(139\) 9.83116i 0.833868i 0.908937 + 0.416934i \(0.136895\pi\)
−0.908937 + 0.416934i \(0.863105\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.47646 + 4.28936i −0.207092 + 0.358694i
\(144\) 0 0
\(145\) −14.6536 + 8.46029i −1.21692 + 0.702589i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.3512 + 10.0177i −1.42146 + 0.820682i −0.996424 0.0844939i \(-0.973073\pi\)
−0.425038 + 0.905175i \(0.639739\pi\)
\(150\) 0 0
\(151\) 11.1168 19.2549i 0.904675 1.56694i 0.0833218 0.996523i \(-0.473447\pi\)
0.821353 0.570420i \(-0.193220\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.54331i 0.123962i
\(156\) 0 0
\(157\) −6.95305 4.01435i −0.554914 0.320380i 0.196188 0.980566i \(-0.437144\pi\)
−0.751102 + 0.660187i \(0.770477\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.295263 + 10.6044i −0.0232700 + 0.835744i
\(162\) 0 0
\(163\) −6.22604 10.7838i −0.487661 0.844654i 0.512238 0.858844i \(-0.328817\pi\)
−0.999899 + 0.0141893i \(0.995483\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.7197 1.52596 0.762978 0.646425i \(-0.223737\pi\)
0.762978 + 0.646425i \(0.223737\pi\)
\(168\) 0 0
\(169\) 1.57282 0.120987
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.913733 1.58263i −0.0694699 0.120325i 0.829198 0.558955i \(-0.188797\pi\)
−0.898668 + 0.438629i \(0.855464\pi\)
\(174\) 0 0
\(175\) 0.295137 + 0.545728i 0.0223103 + 0.0412532i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.1182 + 6.99645i 0.905757 + 0.522939i 0.879064 0.476705i \(-0.158169\pi\)
0.0266934 + 0.999644i \(0.491502\pi\)
\(180\) 0 0
\(181\) 16.3594i 1.21599i 0.793942 + 0.607994i \(0.208025\pi\)
−0.793942 + 0.607994i \(0.791975\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.09415 + 5.35923i −0.227486 + 0.394018i
\(186\) 0 0
\(187\) 3.35330 1.93603i 0.245218 0.141577i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.8326 + 6.83153i −0.856173 + 0.494312i −0.862729 0.505667i \(-0.831247\pi\)
0.00655557 + 0.999979i \(0.497913\pi\)
\(192\) 0 0
\(193\) 2.18885 3.79119i 0.157557 0.272896i −0.776430 0.630203i \(-0.782972\pi\)
0.933987 + 0.357307i \(0.116305\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00603i 0.0716767i −0.999358 0.0358384i \(-0.988590\pi\)
0.999358 0.0358384i \(-0.0114101\pi\)
\(198\) 0 0
\(199\) −5.67639 3.27726i −0.402388 0.232319i 0.285126 0.958490i \(-0.407965\pi\)
−0.687514 + 0.726171i \(0.741298\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −17.4676 10.7440i −1.22599 0.754082i
\(204\) 0 0
\(205\) −8.16820 14.1477i −0.570492 0.988121i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.6368 0.804934
\(210\) 0 0
\(211\) 18.2240 1.25459 0.627297 0.778780i \(-0.284161\pi\)
0.627297 + 0.778780i \(0.284161\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.77486 + 4.80620i 0.189244 + 0.327780i
\(216\) 0 0
\(217\) 1.64527 0.889784i 0.111688 0.0604024i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.73660 + 4.46673i 0.520420 + 0.300464i
\(222\) 0 0
\(223\) 10.0656i 0.674041i 0.941497 + 0.337021i \(0.109419\pi\)
−0.941497 + 0.337021i \(0.890581\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.94372 17.2230i 0.659988 1.14313i −0.320630 0.947204i \(-0.603895\pi\)
0.980618 0.195928i \(-0.0627720\pi\)
\(228\) 0 0
\(229\) 15.3854 8.88275i 1.01669 0.586988i 0.103549 0.994624i \(-0.466980\pi\)
0.913145 + 0.407636i \(0.133647\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.9077 + 8.02962i −0.911124 + 0.526038i −0.880793 0.473502i \(-0.842990\pi\)
−0.0303317 + 0.999540i \(0.509656\pi\)
\(234\) 0 0
\(235\) −13.6992 + 23.7277i −0.893636 + 1.54782i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.21127i 0.531143i −0.964091 0.265572i \(-0.914439\pi\)
0.964091 0.265572i \(-0.0855607\pi\)
\(240\) 0 0
\(241\) 24.6614 + 14.2382i 1.58858 + 0.917166i 0.993542 + 0.113468i \(0.0361959\pi\)
0.595037 + 0.803698i \(0.297137\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.36505 12.7881i 0.534423 0.817002i
\(246\) 0 0
\(247\) 13.4240 + 23.2510i 0.854147 + 1.47943i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.656343 0.0414280 0.0207140 0.999785i \(-0.493406\pi\)
0.0207140 + 0.999785i \(0.493406\pi\)
\(252\) 0 0
\(253\) 5.87486 0.369349
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.82042 6.61716i −0.238311 0.412767i 0.721918 0.691978i \(-0.243261\pi\)
−0.960230 + 0.279211i \(0.909927\pi\)
\(258\) 0 0
\(259\) −7.49718 0.208747i −0.465852 0.0129709i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.73888 + 3.31334i 0.353874 + 0.204310i 0.666390 0.745603i \(-0.267838\pi\)
−0.312516 + 0.949913i \(0.601172\pi\)
\(264\) 0 0
\(265\) 6.09194i 0.374225i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.38347 7.59239i 0.267265 0.462916i −0.700890 0.713270i \(-0.747214\pi\)
0.968154 + 0.250354i \(0.0805469\pi\)
\(270\) 0 0
\(271\) 14.2608 8.23346i 0.866280 0.500147i 0.000169619 1.00000i \(-0.499946\pi\)
0.866110 + 0.499853i \(0.166613\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.297551 0.171791i 0.0179430 0.0103594i
\(276\) 0 0
\(277\) 8.88732 15.3933i 0.533987 0.924893i −0.465225 0.885193i \(-0.654026\pi\)
0.999212 0.0397001i \(-0.0126402\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.1950i 0.966111i 0.875590 + 0.483055i \(0.160473\pi\)
−0.875590 + 0.483055i \(0.839527\pi\)
\(282\) 0 0
\(283\) −24.5717 14.1865i −1.46063 0.843298i −0.461594 0.887091i \(-0.652722\pi\)
−0.999041 + 0.0437937i \(0.986056\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.3731 16.8646i 0.612304 0.995484i
\(288\) 0 0
\(289\) 5.00804 + 8.67417i 0.294590 + 0.510246i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.76520 0.512068 0.256034 0.966668i \(-0.417584\pi\)
0.256034 + 0.966668i \(0.417584\pi\)
\(294\) 0 0
\(295\) −29.3243 −1.70733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.77711 + 11.7383i 0.391931 + 0.678844i
\(300\) 0 0
\(301\) −3.52389 + 5.72914i −0.203114 + 0.330222i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.7366 8.50818i −0.843816 0.487177i
\(306\) 0 0
\(307\) 12.8497i 0.733372i −0.930345 0.366686i \(-0.880492\pi\)
0.930345 0.366686i \(-0.119508\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.29671 5.71007i 0.186939 0.323789i −0.757289 0.653080i \(-0.773477\pi\)
0.944228 + 0.329291i \(0.106810\pi\)
\(312\) 0 0
\(313\) 2.95711 1.70729i 0.167146 0.0965018i −0.414093 0.910234i \(-0.635901\pi\)
0.581239 + 0.813733i \(0.302568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.8003 + 16.0505i −1.56142 + 0.901485i −0.564304 + 0.825567i \(0.690855\pi\)
−0.997114 + 0.0759182i \(0.975811\pi\)
\(318\) 0 0
\(319\) −5.67836 + 9.83521i −0.317927 + 0.550666i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.9890i 1.16786i
\(324\) 0 0
\(325\) 0.686498 + 0.396350i 0.0380801 + 0.0219855i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −33.1933 0.924216i −1.83001 0.0509537i
\(330\) 0 0
\(331\) 14.4416 + 25.0137i 0.793784 + 1.37487i 0.923608 + 0.383338i \(0.125225\pi\)
−0.129824 + 0.991537i \(0.541441\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.7500 −0.696608
\(336\) 0 0
\(337\) −9.65137 −0.525743 −0.262872 0.964831i \(-0.584670\pi\)
−0.262872 + 0.964831i \(0.584670\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.517919 0.897063i −0.0280469 0.0485787i
\(342\) 0 0
\(343\) 18.4557 + 1.54481i 0.996515 + 0.0834117i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.6758 + 6.16367i 0.573106 + 0.330883i 0.758389 0.651802i \(-0.225987\pi\)
−0.185283 + 0.982685i \(0.559320\pi\)
\(348\) 0 0
\(349\) 11.8023i 0.631763i 0.948799 + 0.315881i \(0.102300\pi\)
−0.948799 + 0.315881i \(0.897700\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.59855 11.4290i 0.351205 0.608305i −0.635256 0.772302i \(-0.719105\pi\)
0.986461 + 0.163997i \(0.0524386\pi\)
\(354\) 0 0
\(355\) −22.0916 + 12.7546i −1.17250 + 0.676943i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.22483 + 3.01656i −0.275756 + 0.159208i −0.631501 0.775375i \(-0.717561\pi\)
0.355745 + 0.934583i \(0.384227\pi\)
\(360\) 0 0
\(361\) 22.0394 38.1733i 1.15997 2.00912i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.98012i 0.522384i
\(366\) 0 0
\(367\) −14.8755 8.58836i −0.776494 0.448309i 0.0586924 0.998276i \(-0.481307\pi\)
−0.835186 + 0.549967i \(0.814640\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.49439 3.51225i 0.337172 0.182347i
\(372\) 0 0
\(373\) −2.35902 4.08595i −0.122146 0.211562i 0.798468 0.602037i \(-0.205644\pi\)
−0.920614 + 0.390475i \(0.872311\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −26.2017 −1.34946
\(378\) 0 0
\(379\) 9.34015 0.479771 0.239886 0.970801i \(-0.422890\pi\)
0.239886 + 0.970801i \(0.422890\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.85036 + 4.93696i 0.145646 + 0.252267i 0.929614 0.368535i \(-0.120140\pi\)
−0.783968 + 0.620802i \(0.786807\pi\)
\(384\) 0 0
\(385\) −7.20808 4.43356i −0.367358 0.225955i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.63671 3.83171i −0.336495 0.194275i 0.322226 0.946663i \(-0.395569\pi\)
−0.658721 + 0.752387i \(0.728902\pi\)
\(390\) 0 0
\(391\) 10.5963i 0.535879i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.2424 17.7404i 0.515351 0.892615i
\(396\) 0 0
\(397\) −1.12810 + 0.651310i −0.0566178 + 0.0326883i −0.528042 0.849218i \(-0.677074\pi\)
0.471424 + 0.881907i \(0.343740\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.18778 4.72722i 0.408878 0.236066i −0.281429 0.959582i \(-0.590809\pi\)
0.690308 + 0.723516i \(0.257475\pi\)
\(402\) 0 0
\(403\) 1.19492 2.06966i 0.0595233 0.103097i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.15345i 0.205879i
\(408\) 0 0
\(409\) 16.5182 + 9.53678i 0.816771 + 0.471563i 0.849302 0.527908i \(-0.177023\pi\)
−0.0325304 + 0.999471i \(0.510357\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.9067 31.2615i −0.831922 1.53828i
\(414\) 0 0
\(415\) −3.72961 6.45987i −0.183079 0.317102i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.40005 0.410369 0.205185 0.978723i \(-0.434220\pi\)
0.205185 + 0.978723i \(0.434220\pi\)
\(420\) 0 0
\(421\) −39.4355 −1.92197 −0.960985 0.276599i \(-0.910792\pi\)
−0.960985 + 0.276599i \(0.910792\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.309855 0.536685i −0.0150302 0.0260331i
\(426\) 0 0
\(427\) 0.574005 20.6155i 0.0277781 0.997652i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3340 + 5.96634i 0.497772 + 0.287389i 0.727793 0.685797i \(-0.240546\pi\)
−0.230021 + 0.973186i \(0.573880\pi\)
\(432\) 0 0
\(433\) 12.2121i 0.586875i −0.955978 0.293437i \(-0.905201\pi\)
0.955978 0.293437i \(-0.0947992\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.9227 27.5789i 0.761686 1.31928i
\(438\) 0 0
\(439\) −14.4639 + 8.35076i −0.690326 + 0.398560i −0.803734 0.594989i \(-0.797157\pi\)
0.113408 + 0.993548i \(0.463823\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.2403 15.1499i 1.24672 0.719791i 0.276262 0.961082i \(-0.410904\pi\)
0.970453 + 0.241291i \(0.0775708\pi\)
\(444\) 0 0
\(445\) −10.0841 + 17.4662i −0.478033 + 0.827977i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 30.1253i 1.42170i −0.703343 0.710851i \(-0.748310\pi\)
0.703343 0.710851i \(-0.251690\pi\)
\(450\) 0 0
\(451\) −9.49566 5.48232i −0.447133 0.258152i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.543409 19.5166i 0.0254754 0.914953i
\(456\) 0 0
\(457\) −12.6159 21.8513i −0.590146 1.02216i −0.994212 0.107433i \(-0.965737\pi\)
0.404067 0.914730i \(-0.367596\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.6349 1.14736 0.573680 0.819079i \(-0.305515\pi\)
0.573680 + 0.819079i \(0.305515\pi\)
\(462\) 0 0
\(463\) 12.6643 0.588560 0.294280 0.955719i \(-0.404920\pi\)
0.294280 + 0.955719i \(0.404920\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.4723 + 18.1385i 0.484599 + 0.839350i 0.999843 0.0176932i \(-0.00563223\pi\)
−0.515245 + 0.857043i \(0.672299\pi\)
\(468\) 0 0
\(469\) −7.35090 13.5923i −0.339433 0.627635i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.22581 + 1.86242i 0.148323 + 0.0856344i
\(474\) 0 0
\(475\) 1.86243i 0.0854543i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.8852 27.5141i 0.725816 1.25715i −0.232822 0.972519i \(-0.574796\pi\)
0.958637 0.284630i \(-0.0918707\pi\)
\(480\) 0 0
\(481\) −8.29884 + 4.79134i −0.378394 + 0.218466i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.9362 8.04605i 0.632809 0.365352i
\(486\) 0 0
\(487\) −17.7821 + 30.7995i −0.805784 + 1.39566i 0.109977 + 0.993934i \(0.464922\pi\)
−0.915761 + 0.401724i \(0.868411\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.18390i 0.143687i 0.997416 + 0.0718437i \(0.0228883\pi\)
−0.997416 + 0.0718437i \(0.977112\pi\)
\(492\) 0 0
\(493\) 17.7395 + 10.2419i 0.798947 + 0.461272i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −26.3339 16.1975i −1.18124 0.726557i
\(498\) 0 0
\(499\) −16.0214 27.7498i −0.717215 1.24225i −0.962099 0.272700i \(-0.912083\pi\)
0.244884 0.969552i \(-0.421250\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.6608 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(504\) 0 0
\(505\) 17.3050 0.770061
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −13.4427 23.2834i −0.595836 1.03202i −0.993428 0.114457i \(-0.963487\pi\)
0.397592 0.917562i \(-0.369846\pi\)
\(510\) 0 0
\(511\) 10.6394 5.75395i 0.470661 0.254540i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.13461 + 4.11917i 0.314388 + 0.181512i
\(516\) 0 0
\(517\) 18.3892i 0.808755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.0385 29.5116i 0.746471 1.29293i −0.203033 0.979172i \(-0.565080\pi\)
0.949504 0.313754i \(-0.101587\pi\)
\(522\) 0 0
\(523\) 4.71003 2.71933i 0.205955 0.118908i −0.393475 0.919335i \(-0.628727\pi\)
0.599430 + 0.800427i \(0.295394\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.61801 + 0.934157i −0.0704815 + 0.0406925i
\(528\) 0 0
\(529\) −3.46140 + 5.99532i −0.150496 + 0.260666i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 25.2971i 1.09574i
\(534\) 0 0
\(535\) −15.0243 8.67429i −0.649558 0.375022i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.570698 10.2404i 0.0245817 0.441085i
\(540\) 0 0
\(541\) 11.8329 + 20.4952i 0.508737 + 0.881158i 0.999949 + 0.0101183i \(0.00322080\pi\)
−0.491212 + 0.871040i \(0.663446\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.20545 0.0944713
\(546\) 0 0
\(547\) 24.1648 1.03321 0.516606 0.856223i \(-0.327195\pi\)
0.516606 + 0.856223i \(0.327195\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 30.7803 + 53.3130i 1.31128 + 2.27121i
\(552\) 0 0
\(553\) 24.8175 + 0.691004i 1.05535 + 0.0293845i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.36315 4.25111i −0.311987 0.180126i 0.335829 0.941923i \(-0.390984\pi\)
−0.647815 + 0.761798i \(0.724317\pi\)
\(558\) 0 0
\(559\) 8.59381i 0.363480i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.473776 + 0.820605i −0.0199673 + 0.0345844i −0.875836 0.482608i \(-0.839690\pi\)
0.855869 + 0.517192i \(0.173023\pi\)
\(564\) 0 0
\(565\) 23.0430 13.3039i 0.969427 0.559699i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.7859 + 9.11401i −0.661781 + 0.382079i −0.792955 0.609280i \(-0.791458\pi\)
0.131175 + 0.991359i \(0.458125\pi\)
\(570\) 0 0
\(571\) 6.12121 10.6023i 0.256165 0.443691i −0.709046 0.705162i \(-0.750874\pi\)
0.965211 + 0.261471i \(0.0842077\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.940253i 0.0392112i
\(576\) 0 0
\(577\) −10.2500 5.91784i −0.426713 0.246363i 0.271232 0.962514i \(-0.412569\pi\)
−0.697945 + 0.716151i \(0.745902\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.73636 7.70037i 0.196497 0.319465i
\(582\) 0 0
\(583\) −2.04439 3.54098i −0.0846699 0.146653i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.15361 −0.295261 −0.147631 0.989043i \(-0.547165\pi\)
−0.147631 + 0.989043i \(0.547165\pi\)
\(588\) 0 0
\(589\) −5.61489 −0.231358
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.4811 + 23.3500i 0.553603 + 0.958869i 0.998011 + 0.0630442i \(0.0200809\pi\)
−0.444408 + 0.895825i \(0.646586\pi\)
\(594\) 0 0
\(595\) −7.99668 + 13.0010i −0.327832 + 0.532990i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −30.5223 17.6221i −1.24711 0.720018i −0.276576 0.960992i \(-0.589200\pi\)
−0.970532 + 0.240974i \(0.922533\pi\)
\(600\) 0 0
\(601\) 3.91750i 0.159798i −0.996803 0.0798991i \(-0.974540\pi\)
0.996803 0.0798991i \(-0.0254598\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.66332 16.7374i 0.392870 0.680470i
\(606\) 0 0
\(607\) 12.5377 7.23862i 0.508888 0.293807i −0.223488 0.974707i \(-0.571744\pi\)
0.732376 + 0.680900i \(0.238411\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −36.7426 + 21.2134i −1.48645 + 0.858201i
\(612\) 0 0
\(613\) −6.51761 + 11.2888i −0.263244 + 0.455952i −0.967102 0.254389i \(-0.918126\pi\)
0.703858 + 0.710341i \(0.251459\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.63143i 0.146196i −0.997325 0.0730979i \(-0.976711\pi\)
0.997325 0.0730979i \(-0.0232886\pi\)
\(618\) 0 0
\(619\) 14.2737 + 8.24091i 0.573708 + 0.331230i 0.758629 0.651523i \(-0.225870\pi\)
−0.184921 + 0.982753i \(0.559203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −24.4340 0.680325i −0.978926 0.0272567i
\(624\) 0 0
\(625\) 11.8863 + 20.5876i 0.475450 + 0.823504i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.49147 0.298704
\(630\) 0 0
\(631\) −34.8383 −1.38689 −0.693446 0.720508i \(-0.743909\pi\)
−0.693446 + 0.720508i \(0.743909\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.41512 + 12.8434i 0.294260 + 0.509673i
\(636\) 0 0
\(637\) 21.1192 10.6728i 0.836775 0.422873i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.25538 4.18889i −0.286570 0.165451i 0.349824 0.936815i \(-0.386241\pi\)
−0.636394 + 0.771364i \(0.719575\pi\)
\(642\) 0 0
\(643\) 20.7870i 0.819761i −0.912139 0.409881i \(-0.865570\pi\)
0.912139 0.409881i \(-0.134430\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.74770 8.22325i 0.186651 0.323289i −0.757480 0.652858i \(-0.773570\pi\)
0.944132 + 0.329568i \(0.106903\pi\)
\(648\) 0 0
\(649\) −17.0450 + 9.84091i −0.669073 + 0.386290i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.64747 3.83792i 0.260136 0.150189i −0.364261 0.931297i \(-0.618678\pi\)
0.624396 + 0.781108i \(0.285345\pi\)
\(654\) 0 0
\(655\) −14.9927 + 25.9680i −0.585811 + 1.01465i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 43.9356i 1.71149i −0.517400 0.855743i \(-0.673100\pi\)
0.517400 0.855743i \(-0.326900\pi\)
\(660\) 0 0
\(661\) 22.1649 + 12.7969i 0.862115 + 0.497742i 0.864720 0.502254i \(-0.167496\pi\)
−0.00260513 + 0.999997i \(0.500829\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −40.3490 + 21.8213i −1.56467 + 0.846193i
\(666\) 0 0
\(667\) 15.5395 + 26.9151i 0.601691 + 1.04216i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.4210 −0.440903
\(672\) 0 0
\(673\) 15.2934 0.589518 0.294759 0.955572i \(-0.404761\pi\)
0.294759 + 0.955572i \(0.404761\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.6459 39.2238i −0.870352 1.50749i −0.861633 0.507532i \(-0.830558\pi\)
−0.00871898 0.999962i \(-0.502775\pi\)
\(678\) 0 0
\(679\) 16.6124 + 10.2180i 0.637524 + 0.392129i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0891 + 13.9079i 0.921744 + 0.532169i 0.884191 0.467126i \(-0.154710\pi\)
0.0375529 + 0.999295i \(0.488044\pi\)
\(684\) 0 0
\(685\) 43.8674i 1.67609i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.71672 8.16961i 0.179693 0.311237i
\(690\) 0 0
\(691\) −14.1115 + 8.14729i −0.536828 + 0.309938i −0.743792 0.668411i \(-0.766975\pi\)
0.206965 + 0.978348i \(0.433641\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.5862 + 10.7307i −0.705013 + 0.407040i
\(696\) 0 0
\(697\) −9.88831 + 17.1271i −0.374546 + 0.648733i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.393403i 0.0148586i 0.999972 + 0.00742932i \(0.00236485\pi\)
−0.999972 + 0.00742932i \(0.997635\pi\)
\(702\) 0 0
\(703\) 19.4980 + 11.2572i 0.735379 + 0.424572i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.97702 + 18.4482i 0.375225 + 0.693816i
\(708\) 0 0
\(709\) 16.3183 + 28.2641i 0.612846 + 1.06148i 0.990758 + 0.135639i \(0.0433087\pi\)
−0.377912 + 0.925841i \(0.623358\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.83469 −0.106160
\(714\) 0 0
\(715\) −10.8122 −0.404355
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.106604 + 0.184643i 0.00397565 + 0.00688602i 0.868006 0.496553i \(-0.165401\pi\)
−0.864031 + 0.503439i \(0.832068\pi\)
\(720\) 0 0
\(721\) −0.277900 + 9.98081i −0.0103495 + 0.371705i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.57409 + 0.908804i 0.0584604 + 0.0337521i
\(726\) 0 0
\(727\) 36.7435i 1.36274i −0.731939 0.681370i \(-0.761385\pi\)
0.731939 0.681370i \(-0.238615\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.35920 5.81831i 0.124245 0.215198i
\(732\) 0 0
\(733\) −9.41829 + 5.43765i −0.347873 + 0.200844i −0.663748 0.747956i \(-0.731035\pi\)
0.315875 + 0.948801i \(0.397702\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.41104 + 4.27877i −0.272989 + 0.157610i
\(738\) 0 0
\(739\) 6.91282 11.9734i 0.254292 0.440447i −0.710411 0.703787i \(-0.751491\pi\)
0.964703 + 0.263340i \(0.0848242\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.3310i 0.672501i −0.941773 0.336250i \(-0.890841\pi\)
0.941773 0.336250i \(-0.109159\pi\)
\(744\) 0 0
\(745\) −37.8776 21.8687i −1.38773 0.801206i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.585212 21.0180i 0.0213832 0.767979i
\(750\) 0 0
\(751\) 9.97084 + 17.2700i 0.363841 + 0.630191i 0.988589 0.150635i \(-0.0481318\pi\)
−0.624748 + 0.780826i \(0.714798\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 48.5362 1.76641
\(756\) 0 0
\(757\) −46.9292 −1.70567 −0.852836 0.522178i \(-0.825119\pi\)
−0.852836 + 0.522178i \(0.825119\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.7769 + 46.3789i 0.970661 + 1.68123i 0.693568 + 0.720391i \(0.256038\pi\)
0.277093 + 0.960843i \(0.410629\pi\)
\(762\) 0 0
\(763\) 1.27153 + 2.35115i 0.0460326 + 0.0851175i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −39.3254 22.7045i −1.41996 0.819813i
\(768\) 0 0
\(769\) 40.1035i 1.44617i −0.690760 0.723085i \(-0.742724\pi\)
0.690760 0.723085i \(-0.257276\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.82375 + 13.5511i −0.281401 + 0.487400i −0.971730 0.236095i \(-0.924132\pi\)
0.690329 + 0.723495i \(0.257466\pi\)
\(774\) 0 0
\(775\) −0.143572 + 0.0828913i −0.00515726 + 0.00297755i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −51.4724 + 29.7176i −1.84419 + 1.06474i
\(780\) 0 0
\(781\) −8.56060 + 14.8274i −0.306322 + 0.530566i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.5267i 0.625553i
\(786\) 0 0
\(787\) 39.9920 + 23.0894i 1.42556 + 0.823048i 0.996766 0.0803536i \(-0.0256050\pi\)
0.428795 + 0.903402i \(0.358938\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.4680 + 16.8951i 0.976651 + 0.600720i
\(792\) 0 0
\(793\) −13.1750 22.8198i −0.467859 0.810356i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.8775 1.20000 0.600002 0.799998i \(-0.295166\pi\)
0.600002 + 0.799998i \(0.295166\pi\)
\(798\) 0 0
\(799\) 33.1680 1.17340
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.34922 5.80102i −0.118191 0.204714i
\(804\) 0 0
\(805\) −20.3703 + 11.0165i −0.717958 + 0.388281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −33.7873 19.5071i −1.18790 0.685834i −0.230070 0.973174i \(-0.573896\pi\)
−0.957829 + 0.287340i \(0.907229\pi\)
\(810\) 0 0
\(811\) 7.73397i 0.271577i 0.990738 + 0.135788i \(0.0433567\pi\)
−0.990738 + 0.135788i \(0.956643\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 13.5915 23.5411i 0.476088 0.824609i
\(816\) 0 0
\(817\) 17.4859 10.0955i 0.611755 0.353197i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.443638 + 0.256134i −0.0154831 + 0.00893915i −0.507722 0.861521i \(-0.669512\pi\)
0.492238 + 0.870460i \(0.336179\pi\)
\(822\) 0 0
\(823\) 24.1753 41.8728i 0.842698 1.45960i −0.0449080 0.998991i \(-0.514299\pi\)
0.887606 0.460604i \(-0.152367\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.3086i 0.601879i 0.953643 + 0.300940i \(0.0973003\pi\)
−0.953643 + 0.300940i \(0.902700\pi\)
\(828\) 0 0
\(829\) 33.0205 + 19.0644i 1.14685 + 0.662134i 0.948118 0.317919i \(-0.102984\pi\)
0.198733 + 0.980054i \(0.436317\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.4703 1.02935i −0.639959 0.0356649i
\(834\) 0 0
\(835\) 21.5241 + 37.2808i 0.744871 + 1.29015i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.4052 1.04970 0.524852 0.851193i \(-0.324121\pi\)
0.524852 + 0.851193i \(0.324121\pi\)
\(840\) 0 0
\(841\) −31.0789 −1.07168
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.71674 + 2.97348i 0.0590577 + 0.102291i
\(846\) 0 0
\(847\) 23.4144 + 0.651937i 0.804528 + 0.0224008i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.84358 + 5.68319i 0.337434 + 0.194817i
\(852\) 0 0
\(853\) 32.0017i 1.09572i −0.836571 0.547858i \(-0.815443\pi\)
0.836571 0.547858i \(-0.184557\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5774 + 39.1053i −0.771230 + 1.33581i 0.165659 + 0.986183i \(0.447025\pi\)
−0.936889 + 0.349627i \(0.886308\pi\)
\(858\) 0 0
\(859\) 15.7911 9.11701i 0.538786 0.311068i −0.205801 0.978594i \(-0.565980\pi\)
0.744587 + 0.667526i \(0.232647\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.61966 + 3.82186i −0.225336 + 0.130098i −0.608419 0.793616i \(-0.708196\pi\)
0.383083 + 0.923714i \(0.374862\pi\)
\(864\) 0 0
\(865\) 1.99468 3.45489i 0.0678213 0.117470i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.7490i 0.466401i
\(870\) 0 0
\(871\) −17.0984 9.87179i −0.579358 0.334493i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −15.8393 + 25.7515i −0.535465 + 0.870560i
\(876\) 0 0
\(877\) −8.01581 13.8838i −0.270675 0.468822i 0.698360 0.715747i \(-0.253913\pi\)
−0.969035 + 0.246924i \(0.920580\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.7532 0.833958 0.416979 0.908916i \(-0.363089\pi\)
0.416979 + 0.908916i \(0.363089\pi\)
\(882\) 0 0
\(883\) −11.6958 −0.393595 −0.196798 0.980444i \(-0.563054\pi\)
−0.196798 + 0.980444i \(0.563054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.5429 + 47.7058i 0.924801 + 1.60180i 0.791880 + 0.610676i \(0.209102\pi\)
0.132921 + 0.991127i \(0.457564\pi\)
\(888\) 0 0
\(889\) −9.41671 + 15.3097i −0.315826 + 0.513471i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 86.3261 + 49.8404i 2.88879 + 1.66785i
\(894\) 0 0
\(895\) 30.5465i 1.02106i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.73987 4.74560i 0.0913799 0.158275i
\(900\) 0 0
\(901\) −6.38677 + 3.68741i −0.212774 + 0.122845i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.9281 + 17.8563i −1.02808 + 0.593565i
\(906\) 0 0
\(907\) 12.9383 22.4098i 0.429610 0.744107i −0.567228 0.823560i \(-0.691984\pi\)
0.996839 + 0.0794540i \(0.0253177\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.46068i 0.147789i 0.997266 + 0.0738944i \(0.0235428\pi\)
−0.997266 + 0.0738944i \(0.976457\pi\)
\(912\) 0 0
\(913\) −4.33572 2.50323i −0.143491 0.0828448i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −36.3274 1.01148i −1.19964 0.0334020i
\(918\) 0 0
\(919\) −21.2352 36.7805i −0.700485 1.21328i −0.968296 0.249804i \(-0.919634\pi\)
0.267811 0.963471i \(-0.413700\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −39.5013 −1.30020
\(924\) 0 0
\(925\) 0.664747 0.0218567
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.72508 + 13.3802i 0.253452 + 0.438991i 0.964474 0.264178i \(-0.0851008\pi\)
−0.711022 + 0.703170i \(0.751767\pi\)
\(930\) 0 0
\(931\) −46.5257 30.4338i −1.52482 0.997426i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.32027 + 4.22636i 0.239398 + 0.138217i
\(936\) 0 0
\(937\) 46.1410i 1.50736i 0.657241 + 0.753680i \(0.271723\pi\)
−0.657241 + 0.753680i \(0.728277\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.5052 35.5161i 0.668451 1.15779i −0.309886 0.950774i \(-0.600291\pi\)
0.978337 0.207017i \(-0.0663756\pi\)
\(942\) 0 0
\(943\) −25.9859 + 15.0030i −0.846218 + 0.488564i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.50657 + 4.91127i −0.276426 + 0.159595i −0.631804 0.775128i \(-0.717685\pi\)
0.355378 + 0.934723i \(0.384352\pi\)
\(948\) 0 0
\(949\) 7.72718 13.3839i 0.250835 0.434459i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 17.0826i 0.553359i 0.960962 + 0.276679i \(0.0892340\pi\)
−0.960962 + 0.276679i \(0.910766\pi\)
\(954\) 0 0
\(955\) −25.8305 14.9132i −0.835855 0.482581i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 46.7655 25.2914i 1.51014 0.816701i
\(960\) 0 0
\(961\) −15.2501 26.4139i −0.491939 0.852063i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.55652 0.307635
\(966\) 0 0
\(967\) −42.6481 −1.37147 −0.685735 0.727852i \(-0.740519\pi\)
−0.685735 + 0.727852i \(0.740519\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 11.7562 + 20.3623i 0.377275 + 0.653459i 0.990665 0.136321i \(-0.0435280\pi\)
−0.613390 + 0.789780i \(0.710195\pi\)
\(972\) 0 0
\(973\) −22.1553 13.6273i −0.710267 0.436872i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.1944 17.4327i −0.966003 0.557722i −0.0679878 0.997686i \(-0.521658\pi\)
−0.898015 + 0.439964i \(0.854991\pi\)
\(978\) 0 0
\(979\) 13.5365i 0.432627i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.1804 + 22.8292i −0.420390 + 0.728137i −0.995978 0.0896033i \(-0.971440\pi\)
0.575587 + 0.817740i \(0.304773\pi\)
\(984\) 0 0
\(985\) 1.90194 1.09808i 0.0606008 0.0349879i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.82780 5.09673i 0.280708 0.162067i
\(990\) 0 0
\(991\) 0.0805213 0.139467i 0.00255784 0.00443031i −0.864744 0.502214i \(-0.832519\pi\)
0.867301 + 0.497783i \(0.165852\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.3085i 0.453611i
\(996\) 0 0
\(997\) −14.5820 8.41890i −0.461816 0.266629i 0.250992 0.967989i \(-0.419243\pi\)
−0.712807 + 0.701360i \(0.752577\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2268.2.t.b.1781.6 16
3.2 odd 2 2268.2.t.a.1781.3 16
7.5 odd 6 2268.2.t.a.2105.3 16
9.2 odd 6 252.2.bm.a.185.4 yes 16
9.4 even 3 252.2.w.a.101.2 yes 16
9.5 odd 6 756.2.w.a.521.3 16
9.7 even 3 756.2.bm.a.17.3 16
21.5 even 6 inner 2268.2.t.b.2105.6 16
36.7 odd 6 3024.2.df.d.17.3 16
36.11 even 6 1008.2.df.d.689.5 16
36.23 even 6 3024.2.ca.d.2033.3 16
36.31 odd 6 1008.2.ca.d.353.7 16
63.2 odd 6 1764.2.w.b.509.7 16
63.4 even 3 1764.2.x.a.1469.7 16
63.5 even 6 756.2.bm.a.89.3 16
63.11 odd 6 1764.2.x.b.293.2 16
63.13 odd 6 1764.2.w.b.1109.7 16
63.16 even 3 5292.2.w.b.1097.6 16
63.20 even 6 1764.2.bm.a.1697.5 16
63.23 odd 6 5292.2.bm.a.4625.6 16
63.25 even 3 5292.2.x.b.881.6 16
63.31 odd 6 1764.2.x.b.1469.2 16
63.32 odd 6 5292.2.x.a.4409.3 16
63.34 odd 6 5292.2.bm.a.2285.6 16
63.38 even 6 1764.2.x.a.293.7 16
63.40 odd 6 252.2.bm.a.173.4 yes 16
63.41 even 6 5292.2.w.b.521.6 16
63.47 even 6 252.2.w.a.5.2 16
63.52 odd 6 5292.2.x.a.881.3 16
63.58 even 3 1764.2.bm.a.1685.5 16
63.59 even 6 5292.2.x.b.4409.6 16
63.61 odd 6 756.2.w.a.341.3 16
252.47 odd 6 1008.2.ca.d.257.7 16
252.103 even 6 1008.2.df.d.929.5 16
252.131 odd 6 3024.2.df.d.1601.3 16
252.187 even 6 3024.2.ca.d.2609.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.2 16 63.47 even 6
252.2.w.a.101.2 yes 16 9.4 even 3
252.2.bm.a.173.4 yes 16 63.40 odd 6
252.2.bm.a.185.4 yes 16 9.2 odd 6
756.2.w.a.341.3 16 63.61 odd 6
756.2.w.a.521.3 16 9.5 odd 6
756.2.bm.a.17.3 16 9.7 even 3
756.2.bm.a.89.3 16 63.5 even 6
1008.2.ca.d.257.7 16 252.47 odd 6
1008.2.ca.d.353.7 16 36.31 odd 6
1008.2.df.d.689.5 16 36.11 even 6
1008.2.df.d.929.5 16 252.103 even 6
1764.2.w.b.509.7 16 63.2 odd 6
1764.2.w.b.1109.7 16 63.13 odd 6
1764.2.x.a.293.7 16 63.38 even 6
1764.2.x.a.1469.7 16 63.4 even 3
1764.2.x.b.293.2 16 63.11 odd 6
1764.2.x.b.1469.2 16 63.31 odd 6
1764.2.bm.a.1685.5 16 63.58 even 3
1764.2.bm.a.1697.5 16 63.20 even 6
2268.2.t.a.1781.3 16 3.2 odd 2
2268.2.t.a.2105.3 16 7.5 odd 6
2268.2.t.b.1781.6 16 1.1 even 1 trivial
2268.2.t.b.2105.6 16 21.5 even 6 inner
3024.2.ca.d.2033.3 16 36.23 even 6
3024.2.ca.d.2609.3 16 252.187 even 6
3024.2.df.d.17.3 16 36.7 odd 6
3024.2.df.d.1601.3 16 252.131 odd 6
5292.2.w.b.521.6 16 63.41 even 6
5292.2.w.b.1097.6 16 63.16 even 3
5292.2.x.a.881.3 16 63.52 odd 6
5292.2.x.a.4409.3 16 63.32 odd 6
5292.2.x.b.881.6 16 63.25 even 3
5292.2.x.b.4409.6 16 63.59 even 6
5292.2.bm.a.2285.6 16 63.34 odd 6
5292.2.bm.a.4625.6 16 63.23 odd 6