Properties

Label 1728.2.s.f.575.4
Level $1728$
Weight $2$
Character 1728.575
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
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] = [1728,2,Mod(575,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1728.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 575.4
Root \(1.41203 - 0.0786378i\) of defining polynomial
Character \(\chi\) \(=\) 1728.575
Dual form 1728.2.s.f.1151.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.686141 - 0.396143i) q^{5} +(2.35143 + 1.35760i) q^{7} +O(q^{10})\) \(q+(0.686141 - 0.396143i) q^{5} +(2.35143 + 1.35760i) q^{7} +(-1.71352 + 2.96790i) q^{11} +(1.68614 + 2.92048i) q^{13} -2.52434i q^{17} +2.20979i q^{19} +(1.07561 + 1.86301i) q^{23} +(-2.18614 + 3.78651i) q^{25} +(-0.686141 - 0.396143i) q^{29} +(-1.47603 + 0.852189i) q^{31} +2.15121 q^{35} -4.74456 q^{37} +(-0.127719 + 0.0737384i) q^{41} +(6.01594 + 3.47331i) q^{43} +(-5.77846 + 10.0086i) q^{47} +(0.186141 + 0.322405i) q^{49} -8.51278i q^{53} +2.71519i q^{55} +(2.58891 + 4.48412i) q^{59} +(1.68614 - 2.92048i) q^{61} +(2.31386 + 1.33591i) q^{65} +(6.01594 - 3.47331i) q^{67} +1.75079 q^{71} -2.37228 q^{73} +(-8.05842 + 4.65253i) q^{77} +(8.80507 + 5.08361i) q^{79} +(-3.62725 + 6.28258i) q^{83} +(-1.00000 - 1.73205i) q^{85} -5.34363i q^{89} +9.15640i q^{91} +(0.875393 + 1.51622i) q^{95} +(6.24456 - 10.8159i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} + 2 q^{13} - 6 q^{25} + 6 q^{29} + 8 q^{37} - 24 q^{41} - 10 q^{49} + 2 q^{61} + 30 q^{65} + 4 q^{73} - 30 q^{77} - 8 q^{85} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.686141 0.396143i 0.306851 0.177161i −0.338665 0.940907i \(-0.609975\pi\)
0.645517 + 0.763746i \(0.276642\pi\)
\(6\) 0 0
\(7\) 2.35143 + 1.35760i 0.888756 + 0.513124i 0.873535 0.486761i \(-0.161822\pi\)
0.0152206 + 0.999884i \(0.495155\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.71352 + 2.96790i −0.516645 + 0.894855i 0.483168 + 0.875527i \(0.339486\pi\)
−0.999813 + 0.0193276i \(0.993847\pi\)
\(12\) 0 0
\(13\) 1.68614 + 2.92048i 0.467651 + 0.809996i 0.999317 0.0369586i \(-0.0117670\pi\)
−0.531666 + 0.846954i \(0.678434\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.52434i 0.612242i −0.951993 0.306121i \(-0.900969\pi\)
0.951993 0.306121i \(-0.0990312\pi\)
\(18\) 0 0
\(19\) 2.20979i 0.506960i 0.967341 + 0.253480i \(0.0815752\pi\)
−0.967341 + 0.253480i \(0.918425\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.07561 + 1.86301i 0.224279 + 0.388463i 0.956103 0.293031i \(-0.0946638\pi\)
−0.731824 + 0.681494i \(0.761330\pi\)
\(24\) 0 0
\(25\) −2.18614 + 3.78651i −0.437228 + 0.757301i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.686141 0.396143i −0.127413 0.0735620i 0.434939 0.900460i \(-0.356770\pi\)
−0.562352 + 0.826898i \(0.690103\pi\)
\(30\) 0 0
\(31\) −1.47603 + 0.852189i −0.265104 + 0.153058i −0.626660 0.779292i \(-0.715579\pi\)
0.361557 + 0.932350i \(0.382245\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.15121 0.363621
\(36\) 0 0
\(37\) −4.74456 −0.780001 −0.390001 0.920815i \(-0.627525\pi\)
−0.390001 + 0.920815i \(0.627525\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.127719 + 0.0737384i −0.0199463 + 0.0115160i −0.509940 0.860210i \(-0.670332\pi\)
0.489994 + 0.871726i \(0.336999\pi\)
\(42\) 0 0
\(43\) 6.01594 + 3.47331i 0.917423 + 0.529674i 0.882812 0.469727i \(-0.155647\pi\)
0.0346108 + 0.999401i \(0.488981\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.77846 + 10.0086i −0.842875 + 1.45990i 0.0445785 + 0.999006i \(0.485806\pi\)
−0.887454 + 0.460897i \(0.847528\pi\)
\(48\) 0 0
\(49\) 0.186141 + 0.322405i 0.0265915 + 0.0460579i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.51278i 1.16932i −0.811278 0.584660i \(-0.801228\pi\)
0.811278 0.584660i \(-0.198772\pi\)
\(54\) 0 0
\(55\) 2.71519i 0.366117i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.58891 + 4.48412i 0.337047 + 0.583783i 0.983876 0.178852i \(-0.0572384\pi\)
−0.646829 + 0.762635i \(0.723905\pi\)
\(60\) 0 0
\(61\) 1.68614 2.92048i 0.215888 0.373929i −0.737659 0.675174i \(-0.764069\pi\)
0.953547 + 0.301244i \(0.0974020\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.31386 + 1.33591i 0.286999 + 0.165699i
\(66\) 0 0
\(67\) 6.01594 3.47331i 0.734964 0.424332i −0.0852711 0.996358i \(-0.527176\pi\)
0.820236 + 0.572026i \(0.193842\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.75079 0.207780 0.103890 0.994589i \(-0.466871\pi\)
0.103890 + 0.994589i \(0.466871\pi\)
\(72\) 0 0
\(73\) −2.37228 −0.277655 −0.138827 0.990317i \(-0.544333\pi\)
−0.138827 + 0.990317i \(0.544333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.05842 + 4.65253i −0.918342 + 0.530205i
\(78\) 0 0
\(79\) 8.80507 + 5.08361i 0.990647 + 0.571951i 0.905468 0.424415i \(-0.139520\pi\)
0.0851797 + 0.996366i \(0.472854\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.62725 + 6.28258i −0.398142 + 0.689603i −0.993497 0.113861i \(-0.963678\pi\)
0.595355 + 0.803463i \(0.297012\pi\)
\(84\) 0 0
\(85\) −1.00000 1.73205i −0.108465 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.34363i 0.566424i −0.959057 0.283212i \(-0.908600\pi\)
0.959057 0.283212i \(-0.0913999\pi\)
\(90\) 0 0
\(91\) 9.15640i 0.959852i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.875393 + 1.51622i 0.0898134 + 0.155561i
\(96\) 0 0
\(97\) 6.24456 10.8159i 0.634039 1.09819i −0.352679 0.935745i \(-0.614729\pi\)
0.986718 0.162444i \(-0.0519376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.4307 + 8.90892i 1.53541 + 0.886471i 0.999099 + 0.0424521i \(0.0135170\pi\)
0.536314 + 0.844019i \(0.319816\pi\)
\(102\) 0 0
\(103\) 12.6325 7.29339i 1.24472 0.718640i 0.274669 0.961539i \(-0.411432\pi\)
0.970051 + 0.242899i \(0.0780985\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.2332 −1.27930 −0.639650 0.768667i \(-0.720920\pi\)
−0.639650 + 0.768667i \(0.720920\pi\)
\(108\) 0 0
\(109\) 9.48913 0.908893 0.454447 0.890774i \(-0.349837\pi\)
0.454447 + 0.890774i \(0.349837\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.8030 + 7.96916i −1.29848 + 0.749675i −0.980141 0.198303i \(-0.936457\pi\)
−0.318335 + 0.947978i \(0.603124\pi\)
\(114\) 0 0
\(115\) 1.47603 + 0.852189i 0.137641 + 0.0794670i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.42703 5.93580i 0.314156 0.544134i
\(120\) 0 0
\(121\) −0.372281 0.644810i −0.0338438 0.0586191i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.42554i 0.664160i
\(126\) 0 0
\(127\) 16.2912i 1.44561i 0.691053 + 0.722804i \(0.257147\pi\)
−0.691053 + 0.722804i \(0.742853\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.62725 + 6.28258i 0.316914 + 0.548911i 0.979843 0.199771i \(-0.0640200\pi\)
−0.662928 + 0.748683i \(0.730687\pi\)
\(132\) 0 0
\(133\) −3.00000 + 5.19615i −0.260133 + 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.2446 + 9.37880i 1.38787 + 0.801285i 0.993075 0.117485i \(-0.0374833\pi\)
0.394792 + 0.918771i \(0.370817\pi\)
\(138\) 0 0
\(139\) −8.09262 + 4.67228i −0.686407 + 0.396297i −0.802265 0.596968i \(-0.796372\pi\)
0.115858 + 0.993266i \(0.463038\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.5569 −0.966438
\(144\) 0 0
\(145\) −0.627719 −0.0521292
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.31386 + 4.80001i −0.681098 + 0.393232i −0.800269 0.599642i \(-0.795310\pi\)
0.119171 + 0.992874i \(0.461976\pi\)
\(150\) 0 0
\(151\) −7.92967 4.57820i −0.645308 0.372569i 0.141348 0.989960i \(-0.454856\pi\)
−0.786656 + 0.617391i \(0.788190\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.675178 + 1.16944i −0.0542316 + 0.0939319i
\(156\) 0 0
\(157\) −8.43070 14.6024i −0.672843 1.16540i −0.977094 0.212807i \(-0.931739\pi\)
0.304251 0.952592i \(-0.401594\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.84096i 0.460332i
\(162\) 0 0
\(163\) 11.8716i 0.929855i −0.885349 0.464928i \(-0.846080\pi\)
0.885349 0.464928i \(-0.153920\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.60592 + 16.6379i 0.743329 + 1.28748i 0.950972 + 0.309278i \(0.100087\pi\)
−0.207643 + 0.978205i \(0.566579\pi\)
\(168\) 0 0
\(169\) 0.813859 1.40965i 0.0626046 0.108434i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.80298 2.77300i −0.365164 0.210828i 0.306180 0.951974i \(-0.400949\pi\)
−0.671344 + 0.741146i \(0.734283\pi\)
\(174\) 0 0
\(175\) −10.2811 + 5.93580i −0.777178 + 0.448704i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.8114 1.40603 0.703016 0.711174i \(-0.251836\pi\)
0.703016 + 0.711174i \(0.251836\pi\)
\(180\) 0 0
\(181\) 4.00000 0.297318 0.148659 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.25544 + 1.87953i −0.239345 + 0.138186i
\(186\) 0 0
\(187\) 7.49198 + 4.32550i 0.547868 + 0.316312i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.95100 + 3.37923i −0.141169 + 0.244512i −0.927937 0.372736i \(-0.878420\pi\)
0.786768 + 0.617249i \(0.211753\pi\)
\(192\) 0 0
\(193\) 1.87228 + 3.24289i 0.134770 + 0.233428i 0.925509 0.378724i \(-0.123637\pi\)
−0.790740 + 0.612153i \(0.790304\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.6873i 0.761436i −0.924691 0.380718i \(-0.875677\pi\)
0.924691 0.380718i \(-0.124323\pi\)
\(198\) 0 0
\(199\) 19.3236i 1.36981i −0.728630 0.684907i \(-0.759843\pi\)
0.728630 0.684907i \(-0.240157\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.07561 1.86301i −0.0754928 0.130757i
\(204\) 0 0
\(205\) −0.0584220 + 0.101190i −0.00408037 + 0.00706741i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.55842 3.78651i −0.453656 0.261918i
\(210\) 0 0
\(211\) 5.30350 3.06198i 0.365108 0.210795i −0.306211 0.951964i \(-0.599061\pi\)
0.671319 + 0.741169i \(0.265728\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.50371 0.375350
\(216\) 0 0
\(217\) −4.62772 −0.314150
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.37228 4.25639i 0.495913 0.286316i
\(222\) 0 0
\(223\) −1.47603 0.852189i −0.0988426 0.0570668i 0.449764 0.893147i \(-0.351508\pi\)
−0.548606 + 0.836081i \(0.684841\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.44298 16.3557i 0.626752 1.08557i −0.361447 0.932393i \(-0.617717\pi\)
0.988199 0.153174i \(-0.0489496\pi\)
\(228\) 0 0
\(229\) 4.68614 + 8.11663i 0.309669 + 0.536362i 0.978290 0.207241i \(-0.0664485\pi\)
−0.668621 + 0.743603i \(0.733115\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 28.0627i 1.83845i −0.393737 0.919223i \(-0.628818\pi\)
0.393737 0.919223i \(-0.371182\pi\)
\(234\) 0 0
\(235\) 9.15640i 0.597298i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.33010 14.4282i −0.538830 0.933280i −0.998967 0.0454327i \(-0.985533\pi\)
0.460138 0.887847i \(-0.347800\pi\)
\(240\) 0 0
\(241\) −8.24456 + 14.2800i −0.531079 + 0.919856i 0.468263 + 0.883589i \(0.344880\pi\)
−0.999342 + 0.0362667i \(0.988453\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.255437 + 0.147477i 0.0163193 + 0.00942195i
\(246\) 0 0
\(247\) −6.45364 + 3.72601i −0.410635 + 0.237080i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.7347 −1.05629 −0.528144 0.849155i \(-0.677112\pi\)
−0.528144 + 0.849155i \(0.677112\pi\)
\(252\) 0 0
\(253\) −7.37228 −0.463491
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.98913 5.76722i 0.623105 0.359750i −0.154972 0.987919i \(-0.549529\pi\)
0.778077 + 0.628169i \(0.216195\pi\)
\(258\) 0 0
\(259\) −11.1565 6.44121i −0.693231 0.400237i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.25343 10.8313i 0.385603 0.667884i −0.606250 0.795274i \(-0.707327\pi\)
0.991853 + 0.127391i \(0.0406602\pi\)
\(264\) 0 0
\(265\) −3.37228 5.84096i −0.207158 0.358807i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.9484i 1.52113i −0.649260 0.760566i \(-0.724921\pi\)
0.649260 0.760566i \(-0.275079\pi\)
\(270\) 0 0
\(271\) 1.38712i 0.0842618i −0.999112 0.0421309i \(-0.986585\pi\)
0.999112 0.0421309i \(-0.0134147\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.49198 12.9765i −0.451783 0.782512i
\(276\) 0 0
\(277\) 0.313859 0.543620i 0.0188580 0.0326630i −0.856442 0.516243i \(-0.827330\pi\)
0.875300 + 0.483580i \(0.160664\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.56930 + 3.21543i 0.332236 + 0.191817i 0.656834 0.754036i \(-0.271895\pi\)
−0.324597 + 0.945852i \(0.605229\pi\)
\(282\) 0 0
\(283\) −8.80507 + 5.08361i −0.523407 + 0.302189i −0.738327 0.674442i \(-0.764384\pi\)
0.214921 + 0.976632i \(0.431051\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.400428 −0.0236365
\(288\) 0 0
\(289\) 10.6277 0.625160
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.3139 + 9.99616i −1.01149 + 0.583982i −0.911627 0.411020i \(-0.865173\pi\)
−0.0998599 + 0.995002i \(0.531839\pi\)
\(294\) 0 0
\(295\) 3.55271 + 2.05116i 0.206847 + 0.119423i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.62725 + 6.28258i −0.209769 + 0.363331i
\(300\) 0 0
\(301\) 9.43070 + 16.3345i 0.543577 + 0.941502i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.67181i 0.152988i
\(306\) 0 0
\(307\) 25.9530i 1.48121i −0.671938 0.740607i \(-0.734538\pi\)
0.671938 0.740607i \(-0.265462\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.6591 27.1224i −0.887948 1.53797i −0.842297 0.539014i \(-0.818797\pi\)
−0.0456512 0.998957i \(-0.514536\pi\)
\(312\) 0 0
\(313\) −2.24456 + 3.88770i −0.126870 + 0.219746i −0.922462 0.386087i \(-0.873826\pi\)
0.795592 + 0.605832i \(0.207160\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.6644 + 11.9306i 1.16063 + 0.670089i 0.951454 0.307790i \(-0.0995893\pi\)
0.209173 + 0.977879i \(0.432923\pi\)
\(318\) 0 0
\(319\) 2.35143 1.35760i 0.131655 0.0760109i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.57825 0.310382
\(324\) 0 0
\(325\) −14.7446 −0.817881
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −27.1753 + 15.6896i −1.49822 + 0.864998i
\(330\) 0 0
\(331\) −24.1149 13.9228i −1.32548 0.765264i −0.340879 0.940107i \(-0.610725\pi\)
−0.984596 + 0.174843i \(0.944058\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.75186 4.76635i 0.150350 0.260414i
\(336\) 0 0
\(337\) 2.12772 + 3.68532i 0.115904 + 0.200752i 0.918141 0.396254i \(-0.129690\pi\)
−0.802237 + 0.597006i \(0.796357\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.84096i 0.316306i
\(342\) 0 0
\(343\) 17.9955i 0.971668i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.86473 6.69391i −0.207470 0.359348i 0.743447 0.668795i \(-0.233189\pi\)
−0.950917 + 0.309447i \(0.899856\pi\)
\(348\) 0 0
\(349\) −5.68614 + 9.84868i −0.304372 + 0.527188i −0.977121 0.212683i \(-0.931780\pi\)
0.672749 + 0.739871i \(0.265113\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.24456 + 2.45060i 0.225915 + 0.130432i 0.608686 0.793411i \(-0.291697\pi\)
−0.382771 + 0.923843i \(0.625030\pi\)
\(354\) 0 0
\(355\) 1.20128 0.693562i 0.0637576 0.0368105i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.9679 1.58165 0.790823 0.612045i \(-0.209653\pi\)
0.790823 + 0.612045i \(0.209653\pi\)
\(360\) 0 0
\(361\) 14.1168 0.742992
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.62772 + 0.939764i −0.0851987 + 0.0491895i
\(366\) 0 0
\(367\) −11.7571 6.78799i −0.613718 0.354330i 0.160701 0.987003i \(-0.448624\pi\)
−0.774419 + 0.632673i \(0.781958\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.5569 20.0172i 0.600006 1.03924i
\(372\) 0 0
\(373\) 4.43070 + 7.67420i 0.229413 + 0.397355i 0.957634 0.287987i \(-0.0929860\pi\)
−0.728221 + 0.685342i \(0.759653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.67181i 0.137605i
\(378\) 0 0
\(379\) 9.66181i 0.496294i 0.968722 + 0.248147i \(0.0798216\pi\)
−0.968722 + 0.248147i \(0.920178\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.200214 + 0.346781i 0.0102305 + 0.0177197i 0.871095 0.491114i \(-0.163410\pi\)
−0.860865 + 0.508834i \(0.830077\pi\)
\(384\) 0 0
\(385\) −3.68614 + 6.38458i −0.187863 + 0.325388i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.19702 + 4.15520i 0.364903 + 0.210677i 0.671229 0.741250i \(-0.265767\pi\)
−0.306326 + 0.951927i \(0.599100\pi\)
\(390\) 0 0
\(391\) 4.70285 2.71519i 0.237834 0.137313i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.05535 0.405309
\(396\) 0 0
\(397\) 7.25544 0.364140 0.182070 0.983286i \(-0.441720\pi\)
0.182070 + 0.983286i \(0.441720\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.9891 10.9634i 0.948272 0.547485i 0.0557281 0.998446i \(-0.482252\pi\)
0.892544 + 0.450961i \(0.148919\pi\)
\(402\) 0 0
\(403\) −4.97760 2.87382i −0.247952 0.143155i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.12989 14.0814i 0.402984 0.697988i
\(408\) 0 0
\(409\) −1.12772 1.95327i −0.0557621 0.0965828i 0.836797 0.547513i \(-0.184426\pi\)
−0.892559 + 0.450931i \(0.851092\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.0588i 0.691788i
\(414\) 0 0
\(415\) 5.74764i 0.282141i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.82639 4.89545i −0.138078 0.239159i 0.788691 0.614790i \(-0.210759\pi\)
−0.926769 + 0.375631i \(0.877426\pi\)
\(420\) 0 0
\(421\) 14.8030 25.6395i 0.721453 1.24959i −0.238964 0.971028i \(-0.576808\pi\)
0.960417 0.278565i \(-0.0898589\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.55842 + 5.51856i 0.463652 + 0.267689i
\(426\) 0 0
\(427\) 7.92967 4.57820i 0.383744 0.221555i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.6581 −0.706054 −0.353027 0.935613i \(-0.614848\pi\)
−0.353027 + 0.935613i \(0.614848\pi\)
\(432\) 0 0
\(433\) −14.3723 −0.690688 −0.345344 0.938476i \(-0.612238\pi\)
−0.345344 + 0.938476i \(0.612238\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.11684 + 2.37686i −0.196935 + 0.113701i
\(438\) 0 0
\(439\) −28.4919 16.4498i −1.35984 0.785106i −0.370241 0.928936i \(-0.620725\pi\)
−0.989602 + 0.143830i \(0.954058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.58891 + 4.48412i −0.123003 + 0.213047i −0.920950 0.389680i \(-0.872586\pi\)
0.797948 + 0.602727i \(0.205919\pi\)
\(444\) 0 0
\(445\) −2.11684 3.66648i −0.100348 0.173808i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.81396i 0.179992i −0.995942 0.0899959i \(-0.971315\pi\)
0.995942 0.0899959i \(-0.0286854\pi\)
\(450\) 0 0
\(451\) 0.505408i 0.0237987i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.62725 + 6.28258i 0.170048 + 0.294532i
\(456\) 0 0
\(457\) 2.98913 5.17732i 0.139825 0.242185i −0.787605 0.616180i \(-0.788679\pi\)
0.927430 + 0.373996i \(0.122013\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19.8030 11.4333i −0.922317 0.532500i −0.0379435 0.999280i \(-0.512081\pi\)
−0.884373 + 0.466780i \(0.845414\pi\)
\(462\) 0 0
\(463\) −18.2108 + 10.5140i −0.846327 + 0.488627i −0.859410 0.511288i \(-0.829169\pi\)
0.0130831 + 0.999914i \(0.495835\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.3897 1.12862 0.564310 0.825563i \(-0.309142\pi\)
0.564310 + 0.825563i \(0.309142\pi\)
\(468\) 0 0
\(469\) 18.8614 0.870939
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −20.6168 + 11.9031i −0.947963 + 0.547307i
\(474\) 0 0
\(475\) −8.36737 4.83090i −0.383921 0.221657i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.90307 + 8.49236i −0.224027 + 0.388026i −0.956027 0.293278i \(-0.905254\pi\)
0.732000 + 0.681304i \(0.238587\pi\)
\(480\) 0 0
\(481\) −8.00000 13.8564i −0.364769 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.89497i 0.449308i
\(486\) 0 0
\(487\) 17.9365i 0.812780i −0.913700 0.406390i \(-0.866787\pi\)
0.913700 0.406390i \(-0.133213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.3950 21.4689i −0.559381 0.968876i −0.997548 0.0699824i \(-0.977706\pi\)
0.438168 0.898893i \(-0.355628\pi\)
\(492\) 0 0
\(493\) −1.00000 + 1.73205i −0.0450377 + 0.0780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.11684 + 2.37686i 0.184666 + 0.106617i
\(498\) 0 0
\(499\) −0.437696 + 0.252704i −0.0195940 + 0.0113126i −0.509765 0.860314i \(-0.670268\pi\)
0.490171 + 0.871626i \(0.336934\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.9073 −0.575507 −0.287754 0.957704i \(-0.592908\pi\)
−0.287754 + 0.957704i \(0.592908\pi\)
\(504\) 0 0
\(505\) 14.1168 0.628191
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.6861 9.05640i 0.695276 0.401418i −0.110310 0.993897i \(-0.535184\pi\)
0.805586 + 0.592480i \(0.201851\pi\)
\(510\) 0 0
\(511\) −5.57825 3.22060i −0.246767 0.142471i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.77846 10.0086i 0.254629 0.441031i
\(516\) 0 0
\(517\) −19.8030 34.2998i −0.870934 1.50850i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.4472i 0.457700i 0.973462 + 0.228850i \(0.0734966\pi\)
−0.973462 + 0.228850i \(0.926503\pi\)
\(522\) 0 0
\(523\) 4.41957i 0.193254i −0.995321 0.0966272i \(-0.969195\pi\)
0.995321 0.0966272i \(-0.0308055\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.15121 + 3.72601i 0.0937083 + 0.162308i
\(528\) 0 0
\(529\) 9.18614 15.9109i 0.399397 0.691777i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.430703 0.248667i −0.0186558 0.0107709i
\(534\) 0 0
\(535\) −9.07982 + 5.24224i −0.392555 + 0.226642i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.27582 −0.0549535
\(540\) 0 0
\(541\) −36.9783 −1.58982 −0.794910 0.606728i \(-0.792482\pi\)
−0.794910 + 0.606728i \(0.792482\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.51087 3.75906i 0.278895 0.161020i
\(546\) 0 0
\(547\) 7.21723 + 4.16687i 0.308586 + 0.178162i 0.646294 0.763089i \(-0.276318\pi\)
−0.337707 + 0.941251i \(0.609651\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.875393 1.51622i 0.0372930 0.0645933i
\(552\) 0 0
\(553\) 13.8030 + 23.9075i 0.586963 + 1.01665i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.4897i 0.868175i 0.900871 + 0.434087i \(0.142929\pi\)
−0.900871 + 0.434087i \(0.857071\pi\)
\(558\) 0 0
\(559\) 23.4259i 0.990811i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.54098 + 9.59726i 0.233524 + 0.404476i 0.958843 0.283937i \(-0.0916408\pi\)
−0.725318 + 0.688414i \(0.758307\pi\)
\(564\) 0 0
\(565\) −6.31386 + 10.9359i −0.265626 + 0.460078i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.989125 0.571072i −0.0414663 0.0239406i 0.479124 0.877747i \(-0.340955\pi\)
−0.520590 + 0.853807i \(0.674288\pi\)
\(570\) 0 0
\(571\) −14.5463 + 8.39829i −0.608742 + 0.351457i −0.772473 0.635048i \(-0.780980\pi\)
0.163731 + 0.986505i \(0.447647\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.40571 −0.392245
\(576\) 0 0
\(577\) −42.8397 −1.78344 −0.891719 0.452589i \(-0.850500\pi\)
−0.891719 + 0.452589i \(0.850500\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −17.0584 + 9.84868i −0.707703 + 0.408592i
\(582\) 0 0
\(583\) 25.2651 + 14.5868i 1.04637 + 0.604123i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.41637 + 11.1135i −0.264832 + 0.458702i −0.967520 0.252796i \(-0.918650\pi\)
0.702688 + 0.711499i \(0.251983\pi\)
\(588\) 0 0
\(589\) −1.88316 3.26172i −0.0775941 0.134397i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.4410i 0.634085i 0.948411 + 0.317043i \(0.102690\pi\)
−0.948411 + 0.317043i \(0.897310\pi\)
\(594\) 0 0
\(595\) 5.43039i 0.222624i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.20550 15.9444i −0.376126 0.651470i 0.614369 0.789019i \(-0.289411\pi\)
−0.990495 + 0.137549i \(0.956077\pi\)
\(600\) 0 0
\(601\) 14.9891 25.9619i 0.611419 1.05901i −0.379582 0.925158i \(-0.623932\pi\)
0.991001 0.133851i \(-0.0427344\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.510875 0.294954i −0.0207700 0.0119916i
\(606\) 0 0
\(607\) 33.1947 19.1650i 1.34733 0.777883i 0.359462 0.933160i \(-0.382960\pi\)
0.987871 + 0.155277i \(0.0496270\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −38.9732 −1.57669
\(612\) 0 0
\(613\) 30.2337 1.22113 0.610564 0.791967i \(-0.290943\pi\)
0.610564 + 0.791967i \(0.290943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.24456 + 4.18265i −0.291655 + 0.168387i −0.638688 0.769466i \(-0.720523\pi\)
0.347033 + 0.937853i \(0.387189\pi\)
\(618\) 0 0
\(619\) 23.9520 + 13.8287i 0.962711 + 0.555821i 0.897006 0.442018i \(-0.145737\pi\)
0.0657046 + 0.997839i \(0.479070\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.25450 12.5652i 0.290645 0.503412i
\(624\) 0 0
\(625\) −7.98913 13.8376i −0.319565 0.553503i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.9769i 0.477549i
\(630\) 0 0
\(631\) 17.9365i 0.714040i 0.934097 + 0.357020i \(0.116207\pi\)
−0.934097 + 0.357020i \(0.883793\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.45364 + 11.1780i 0.256105 + 0.443587i
\(636\) 0 0
\(637\) −0.627719 + 1.08724i −0.0248711 + 0.0430780i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −23.8723 13.7827i −0.942898 0.544383i −0.0520307 0.998645i \(-0.516569\pi\)
−0.890868 + 0.454263i \(0.849903\pi\)
\(642\) 0 0
\(643\) −5.46644 + 3.15605i −0.215575 + 0.124463i −0.603900 0.797060i \(-0.706387\pi\)
0.388324 + 0.921523i \(0.373054\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.9711 1.45348 0.726741 0.686912i \(-0.241034\pi\)
0.726741 + 0.686912i \(0.241034\pi\)
\(648\) 0 0
\(649\) −17.7446 −0.696535
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.68614 3.86025i 0.261649 0.151063i −0.363438 0.931619i \(-0.618397\pi\)
0.625087 + 0.780555i \(0.285064\pi\)
\(654\) 0 0
\(655\) 4.97760 + 2.87382i 0.194491 + 0.112290i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.33010 + 14.4282i −0.324495 + 0.562041i −0.981410 0.191923i \(-0.938528\pi\)
0.656915 + 0.753964i \(0.271861\pi\)
\(660\) 0 0
\(661\) 10.6861 + 18.5089i 0.415643 + 0.719914i 0.995496 0.0948069i \(-0.0302234\pi\)
−0.579853 + 0.814721i \(0.696890\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.75372i 0.184341i
\(666\) 0 0
\(667\) 1.70438i 0.0659938i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.77846 + 10.0086i 0.223075 + 0.386377i
\(672\) 0 0
\(673\) −0.569297 + 0.986051i −0.0219448 + 0.0380095i −0.876789 0.480875i \(-0.840319\pi\)
0.854844 + 0.518884i \(0.173652\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.5475 + 16.4819i 1.09717 + 0.633452i 0.935477 0.353389i \(-0.114971\pi\)
0.161695 + 0.986841i \(0.448304\pi\)
\(678\) 0 0
\(679\) 29.3673 16.9552i 1.12701 0.650681i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.07668 −0.0794618 −0.0397309 0.999210i \(-0.512650\pi\)
−0.0397309 + 0.999210i \(0.512650\pi\)
\(684\) 0 0
\(685\) 14.8614 0.567825
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 24.8614 14.3537i 0.947144 0.546834i
\(690\) 0 0
\(691\) −10.0064 5.77717i −0.380660 0.219774i 0.297446 0.954739i \(-0.403865\pi\)
−0.678105 + 0.734965i \(0.737199\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.70178 + 6.41168i −0.140417 + 0.243209i
\(696\) 0 0
\(697\) 0.186141 + 0.322405i 0.00705058 + 0.0122120i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.1282i 0.986850i 0.869788 + 0.493425i \(0.164255\pi\)
−0.869788 + 0.493425i \(0.835745\pi\)
\(702\) 0 0
\(703\) 10.4845i 0.395429i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.1895 + 41.8974i 0.909738 + 1.57571i
\(708\) 0 0
\(709\) −2.43070 + 4.21010i −0.0912870 + 0.158114i −0.908053 0.418855i \(-0.862431\pi\)
0.816766 + 0.576969i \(0.195765\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.17527 1.83324i −0.118915 0.0686554i
\(714\) 0 0
\(715\) −7.92967 + 4.57820i −0.296553 + 0.171215i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.65492 −0.285481 −0.142740 0.989760i \(-0.545591\pi\)
−0.142740 + 0.989760i \(0.545591\pi\)
\(720\) 0 0
\(721\) 39.6060 1.47500
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.00000 1.73205i 0.111417 0.0643268i
\(726\) 0 0
\(727\) −2.67732 1.54575i −0.0992963 0.0573287i 0.449530 0.893265i \(-0.351592\pi\)
−0.548826 + 0.835937i \(0.684925\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.76780 15.1863i 0.324289 0.561685i
\(732\) 0 0
\(733\) −2.68614 4.65253i −0.0992149 0.171845i 0.812145 0.583456i \(-0.198300\pi\)
−0.911360 + 0.411610i \(0.864966\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.8063i 0.876916i
\(738\) 0 0
\(739\) 9.66181i 0.355415i −0.984083 0.177708i \(-0.943132\pi\)
0.984083 0.177708i \(-0.0568681\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.5132 + 38.9940i 0.825929 + 1.43055i 0.901207 + 0.433389i \(0.142682\pi\)
−0.0752776 + 0.997163i \(0.523984\pi\)
\(744\) 0 0
\(745\) −3.80298 + 6.58696i −0.139331 + 0.241328i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −31.1168 17.9653i −1.13698 0.656438i
\(750\) 0 0
\(751\) −39.9743 + 23.0792i −1.45868 + 0.842170i −0.998947 0.0458859i \(-0.985389\pi\)
−0.459735 + 0.888056i \(0.652056\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.25450 −0.264018
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.5475 9.55373i 0.599848 0.346322i −0.169134 0.985593i \(-0.554097\pi\)
0.768982 + 0.639271i \(0.220764\pi\)
\(762\) 0 0
\(763\) 22.3130 + 12.8824i 0.807784 + 0.466375i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.73053 + 15.1217i −0.315241 + 0.546014i
\(768\) 0 0
\(769\) 27.5475 + 47.7138i 0.993390 + 1.72060i 0.596105 + 0.802907i \(0.296714\pi\)
0.397285 + 0.917695i \(0.369952\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 45.7330i 1.64490i −0.568835 0.822451i \(-0.692606\pi\)
0.568835 0.822451i \(-0.307394\pi\)
\(774\) 0 0
\(775\) 7.45202i 0.267685i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.162946 0.282231i −0.00583815 0.0101120i
\(780\) 0 0
\(781\) −3.00000 + 5.19615i −0.107348 + 0.185933i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.5693 6.67954i −0.412926 0.238403i
\(786\) 0 0
\(787\) 17.0095 9.82043i 0.606323 0.350061i −0.165202 0.986260i \(-0.552828\pi\)
0.771525 + 0.636199i \(0.219494\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −43.2756 −1.53870
\(792\) 0 0
\(793\) 11.3723 0.403842
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.8030 23.5576i 1.44532 0.834454i 0.447119 0.894475i \(-0.352450\pi\)
0.998197 + 0.0600211i \(0.0191168\pi\)
\(798\) 0 0
\(799\) 25.2651 + 14.5868i 0.893814 + 0.516043i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.06494 7.04069i 0.143449 0.248461i
\(804\) 0 0
\(805\) 2.31386 + 4.00772i 0.0815528 + 0.141254i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.1778i 0.955521i −0.878490 0.477760i \(-0.841449\pi\)
0.878490 0.477760i \(-0.158551\pi\)
\(810\) 0 0
\(811\) 25.9530i 0.911332i 0.890151 + 0.455666i \(0.150599\pi\)
−0.890151 + 0.455666i \(0.849401\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.70285 8.14558i −0.164734 0.285327i
\(816\) 0 0
\(817\) −7.67527 + 13.2940i −0.268524 + 0.465096i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.6861 10.7884i −0.652151 0.376519i 0.137129 0.990553i \(-0.456213\pi\)
−0.789280 + 0.614034i \(0.789546\pi\)
\(822\) 0 0
\(823\) −25.8657 + 14.9336i −0.901622 + 0.520552i −0.877726 0.479163i \(-0.840940\pi\)
−0.0238957 + 0.999714i \(0.507607\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.00314 0.243523 0.121762 0.992559i \(-0.461146\pi\)
0.121762 + 0.992559i \(0.461146\pi\)
\(828\) 0 0
\(829\) 13.7663 0.478124 0.239062 0.971004i \(-0.423160\pi\)
0.239062 + 0.971004i \(0.423160\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.813859 0.469882i 0.0281986 0.0162804i
\(834\) 0 0
\(835\) 13.1820 + 7.61065i 0.456183 + 0.263377i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22.8391 + 39.5585i −0.788493 + 1.36571i 0.138397 + 0.990377i \(0.455805\pi\)
−0.926890 + 0.375333i \(0.877528\pi\)
\(840\) 0 0
\(841\) −14.1861 24.5711i −0.489177 0.847280i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.28962i 0.0443643i
\(846\) 0 0
\(847\) 2.02163i 0.0694641i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.10328 8.83915i −0.174938 0.303002i
\(852\) 0 0
\(853\) 2.19702 3.80534i 0.0752244 0.130292i −0.825959 0.563730i \(-0.809366\pi\)
0.901184 + 0.433437i \(0.142699\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.0367 + 12.7229i 0.752758 + 0.434605i 0.826690 0.562658i \(-0.190221\pi\)
−0.0739313 + 0.997263i \(0.523555\pi\)
\(858\) 0 0
\(859\) 30.4056 17.5547i 1.03743 0.598958i 0.118323 0.992975i \(-0.462248\pi\)
0.919103 + 0.394017i \(0.128915\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.15335 −0.141382 −0.0706909 0.997498i \(-0.522520\pi\)
−0.0706909 + 0.997498i \(0.522520\pi\)
\(864\) 0 0
\(865\) −4.39403 −0.149402
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −30.1753 + 17.4217i −1.02363 + 0.590991i
\(870\) 0 0
\(871\) 20.2875 + 11.7130i 0.687414 + 0.396879i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.0809 + 17.4606i −0.340796 + 0.590276i
\(876\) 0 0
\(877\) 13.6861 + 23.7051i 0.462148 + 0.800464i 0.999068 0.0431693i \(-0.0137455\pi\)
−0.536920 + 0.843633i \(0.680412\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.7899i 0.666740i 0.942796 + 0.333370i \(0.108186\pi\)
−0.942796 + 0.333370i \(0.891814\pi\)
\(882\) 0 0
\(883\) 34.7921i 1.17085i 0.810727 + 0.585424i \(0.199072\pi\)
−0.810727 + 0.585424i \(0.800928\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.75186 4.76635i −0.0923983 0.160039i 0.816122 0.577880i \(-0.196120\pi\)
−0.908520 + 0.417842i \(0.862787\pi\)
\(888\) 0 0
\(889\) −22.1168 + 38.3075i −0.741775 + 1.28479i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −22.1168 12.7692i −0.740112 0.427304i
\(894\) 0 0
\(895\) 12.9073 7.45202i 0.431443 0.249094i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.35036 0.0450369
\(900\) 0 0
\(901\) −21.4891 −0.715907
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.74456 1.58457i 0.0912323 0.0526730i
\(906\) 0 0
\(907\) 21.3258 + 12.3125i 0.708111 + 0.408828i 0.810361 0.585930i \(-0.199271\pi\)
−0.102250 + 0.994759i \(0.532604\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.57932 11.3957i 0.217983 0.377557i −0.736209 0.676755i \(-0.763386\pi\)
0.954191 + 0.299198i \(0.0967191\pi\)
\(912\) 0 0
\(913\) −12.4307 21.5306i −0.411396 0.712559i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19.6974i 0.650464i
\(918\) 0 0
\(919\) 46.0993i 1.52067i 0.649529 + 0.760337i \(0.274966\pi\)
−0.649529 + 0.760337i \(0.725034\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.95207 + 5.11313i 0.0971686 + 0.168301i
\(924\) 0 0
\(925\) 10.3723 17.9653i 0.341039 0.590696i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −30.4307 17.5692i −0.998399 0.576426i −0.0906248 0.995885i \(-0.528886\pi\)
−0.907774 + 0.419459i \(0.862220\pi\)
\(930\) 0 0
\(931\) −0.712446 + 0.411331i −0.0233495 + 0.0134808i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.85407 0.224152
\(936\) 0 0
\(937\) −11.7228 −0.382968 −0.191484 0.981496i \(-0.561330\pi\)
−0.191484 + 0.981496i \(0.561330\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.8030 18.3615i 1.03675 0.598567i 0.117837 0.993033i \(-0.462404\pi\)
0.918910 + 0.394466i \(0.129071\pi\)
\(942\) 0 0
\(943\) −0.274750 0.158627i −0.00894709 0.00516561i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.2757 + 38.5827i −0.723864 + 1.25377i 0.235576 + 0.971856i \(0.424302\pi\)
−0.959440 + 0.281913i \(0.909031\pi\)
\(948\) 0 0
\(949\) −4.00000 6.92820i −0.129845 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 32.8164i 1.06303i −0.847050 0.531514i \(-0.821623\pi\)
0.847050 0.531514i \(-0.178377\pi\)
\(954\) 0 0
\(955\) 3.09150i 0.100039i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 25.4653 + 44.1071i 0.822317 + 1.42429i
\(960\) 0 0
\(961\) −14.0475 + 24.3311i −0.453147 + 0.784873i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.56930 + 1.48338i 0.0827086 + 0.0477518i
\(966\) 0 0
\(967\) 48.7282 28.1332i 1.56699 0.904704i 0.570476 0.821314i \(-0.306759\pi\)
0.996517 0.0833895i \(-0.0265746\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.7156 0.793160 0.396580 0.918000i \(-0.370197\pi\)
0.396580 + 0.918000i \(0.370197\pi\)
\(972\) 0 0
\(973\) −25.3723 −0.813398
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.6386 10.7610i 0.596301 0.344275i −0.171284 0.985222i \(-0.554791\pi\)
0.767585 + 0.640947i \(0.221458\pi\)
\(978\) 0 0
\(979\) 15.8593 + 9.15640i 0.506867 + 0.292640i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −8.40464 + 14.5573i −0.268066 + 0.464305i −0.968362 0.249548i \(-0.919718\pi\)
0.700296 + 0.713852i \(0.253051\pi\)
\(984\) 0 0
\(985\) −4.23369 7.33296i −0.134897 0.233648i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.9436i 0.475180i
\(990\) 0 0
\(991\) 16.2912i 0.517506i −0.965944 0.258753i \(-0.916688\pi\)
0.965944 0.258753i \(-0.0833116\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.65492 13.2587i −0.242677 0.420330i
\(996\) 0 0
\(997\) −26.4307 + 45.7793i −0.837069 + 1.44985i 0.0552661 + 0.998472i \(0.482399\pi\)
−0.892335 + 0.451374i \(0.850934\pi\)
\(998\) 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 1728.2.s.f.575.4 8
3.2 odd 2 576.2.s.f.191.3 8
4.3 odd 2 inner 1728.2.s.f.575.3 8
8.3 odd 2 108.2.h.a.35.4 8
8.5 even 2 108.2.h.a.35.3 8
9.2 odd 6 5184.2.c.j.5183.3 8
9.4 even 3 576.2.s.f.383.2 8
9.5 odd 6 inner 1728.2.s.f.1151.3 8
9.7 even 3 5184.2.c.j.5183.5 8
12.11 even 2 576.2.s.f.191.2 8
24.5 odd 2 36.2.h.a.11.2 yes 8
24.11 even 2 36.2.h.a.11.1 8
36.7 odd 6 5184.2.c.j.5183.6 8
36.11 even 6 5184.2.c.j.5183.4 8
36.23 even 6 inner 1728.2.s.f.1151.4 8
36.31 odd 6 576.2.s.f.383.3 8
72.5 odd 6 108.2.h.a.71.4 8
72.11 even 6 324.2.b.b.323.6 8
72.13 even 6 36.2.h.a.23.1 yes 8
72.29 odd 6 324.2.b.b.323.4 8
72.43 odd 6 324.2.b.b.323.3 8
72.59 even 6 108.2.h.a.71.3 8
72.61 even 6 324.2.b.b.323.5 8
72.67 odd 6 36.2.h.a.23.2 yes 8
120.29 odd 2 900.2.r.c.551.3 8
120.53 even 4 900.2.o.a.299.2 16
120.59 even 2 900.2.r.c.551.4 8
120.77 even 4 900.2.o.a.299.7 16
120.83 odd 4 900.2.o.a.299.4 16
120.107 odd 4 900.2.o.a.299.5 16
360.13 odd 12 900.2.o.a.599.5 16
360.67 even 12 900.2.o.a.599.2 16
360.139 odd 6 900.2.r.c.851.3 8
360.157 odd 12 900.2.o.a.599.4 16
360.229 even 6 900.2.r.c.851.4 8
360.283 even 12 900.2.o.a.599.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.h.a.11.1 8 24.11 even 2
36.2.h.a.11.2 yes 8 24.5 odd 2
36.2.h.a.23.1 yes 8 72.13 even 6
36.2.h.a.23.2 yes 8 72.67 odd 6
108.2.h.a.35.3 8 8.5 even 2
108.2.h.a.35.4 8 8.3 odd 2
108.2.h.a.71.3 8 72.59 even 6
108.2.h.a.71.4 8 72.5 odd 6
324.2.b.b.323.3 8 72.43 odd 6
324.2.b.b.323.4 8 72.29 odd 6
324.2.b.b.323.5 8 72.61 even 6
324.2.b.b.323.6 8 72.11 even 6
576.2.s.f.191.2 8 12.11 even 2
576.2.s.f.191.3 8 3.2 odd 2
576.2.s.f.383.2 8 9.4 even 3
576.2.s.f.383.3 8 36.31 odd 6
900.2.o.a.299.2 16 120.53 even 4
900.2.o.a.299.4 16 120.83 odd 4
900.2.o.a.299.5 16 120.107 odd 4
900.2.o.a.299.7 16 120.77 even 4
900.2.o.a.599.2 16 360.67 even 12
900.2.o.a.599.4 16 360.157 odd 12
900.2.o.a.599.5 16 360.13 odd 12
900.2.o.a.599.7 16 360.283 even 12
900.2.r.c.551.3 8 120.29 odd 2
900.2.r.c.551.4 8 120.59 even 2
900.2.r.c.851.3 8 360.139 odd 6
900.2.r.c.851.4 8 360.229 even 6
1728.2.s.f.575.3 8 4.3 odd 2 inner
1728.2.s.f.575.4 8 1.1 even 1 trivial
1728.2.s.f.1151.3 8 9.5 odd 6 inner
1728.2.s.f.1151.4 8 36.23 even 6 inner
5184.2.c.j.5183.3 8 9.2 odd 6
5184.2.c.j.5183.4 8 36.11 even 6
5184.2.c.j.5183.5 8 9.7 even 3
5184.2.c.j.5183.6 8 36.7 odd 6