Properties

Label 1008.2.s.g.865.1
Level $1008$
Weight $2$
Character 1008.865
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(289,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.865
Dual form 1008.2.s.g.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-1.50000 + 2.59808i) q^{11} -6.00000 q^{13} +(-2.50000 + 4.33013i) q^{17} +(0.500000 + 0.866025i) q^{19} +(3.50000 + 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} -2.00000 q^{29} +(-2.50000 + 4.33013i) q^{31} +(0.500000 - 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} +2.00000 q^{41} +4.00000 q^{43} +(-2.50000 - 4.33013i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-0.500000 + 0.866025i) q^{53} +3.00000 q^{55} +(-7.50000 + 12.9904i) q^{59} +(2.50000 + 4.33013i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-4.50000 + 7.79423i) q^{67} +(-3.50000 + 6.06218i) q^{73} +(-7.50000 + 2.59808i) q^{77} +(0.500000 + 0.866025i) q^{79} +12.0000 q^{83} +5.00000 q^{85} +(3.50000 + 6.06218i) q^{89} +(-12.0000 - 10.3923i) q^{91} +(0.500000 - 0.866025i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 4 q^{7} - 3 q^{11} - 12 q^{13} - 5 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} - 4 q^{29} - 5 q^{31} + q^{35} - 3 q^{37} + 4 q^{41} + 8 q^{43} - 5 q^{47} + 2 q^{49} - q^{53} + 6 q^{55} - 15 q^{59} + 5 q^{61} + 6 q^{65} - 9 q^{67} - 7 q^{73} - 15 q^{77} + q^{79} + 24 q^{83} + 10 q^{85} + 7 q^{89} - 24 q^{91} + q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) −0.500000 0.866025i −0.223607 0.387298i 0.732294 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.50000 + 4.33013i −0.606339 + 1.05021i 0.385499 + 0.922708i \(0.374029\pi\)
−0.991838 + 0.127502i \(0.959304\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.50000 + 6.06218i 0.729800 + 1.26405i 0.956967 + 0.290196i \(0.0937204\pi\)
−0.227167 + 0.973856i \(0.572946\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.500000 2.59808i 0.0845154 0.439155i
\(36\) 0 0
\(37\) −1.50000 2.59808i −0.246598 0.427121i 0.715981 0.698119i \(-0.245980\pi\)
−0.962580 + 0.270998i \(0.912646\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.50000 4.33013i −0.364662 0.631614i 0.624059 0.781377i \(-0.285482\pi\)
−0.988722 + 0.149763i \(0.952149\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.50000 + 12.9904i −0.976417 + 1.69120i −0.301239 + 0.953549i \(0.597400\pi\)
−0.675178 + 0.737655i \(0.735933\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) −4.50000 + 7.79423i −0.549762 + 0.952217i 0.448528 + 0.893769i \(0.351948\pi\)
−0.998290 + 0.0584478i \(0.981385\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.50000 + 6.06218i −0.409644 + 0.709524i −0.994850 0.101361i \(-0.967680\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.50000 + 2.59808i −0.854704 + 0.296078i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.50000 + 6.06218i 0.370999 + 0.642590i 0.989720 0.143022i \(-0.0456819\pi\)
−0.618720 + 0.785611i \(0.712349\pi\)
\(90\) 0 0
\(91\) −12.0000 10.3923i −1.25794 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.500000 0.866025i 0.0512989 0.0888523i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.50000 2.59808i 0.149256 0.258518i −0.781697 0.623658i \(-0.785646\pi\)
0.930953 + 0.365140i \(0.118979\pi\)
\(102\) 0 0
\(103\) −7.50000 12.9904i −0.738997 1.27998i −0.952947 0.303136i \(-0.901966\pi\)
0.213950 0.976845i \(-0.431367\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.50000 7.79423i −0.435031 0.753497i 0.562267 0.826956i \(-0.309929\pi\)
−0.997298 + 0.0734594i \(0.976596\pi\)
\(108\) 0 0
\(109\) 2.50000 4.33013i 0.239457 0.414751i −0.721102 0.692829i \(-0.756364\pi\)
0.960558 + 0.278078i \(0.0896974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 3.50000 6.06218i 0.326377 0.565301i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.5000 + 4.33013i −1.14587 + 0.396942i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.50000 4.33013i −0.218426 0.378325i 0.735901 0.677089i \(-0.236759\pi\)
−0.954327 + 0.298764i \(0.903426\pi\)
\(132\) 0 0
\(133\) −0.500000 + 2.59808i −0.0433555 + 0.225282i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.50000 9.52628i 0.469897 0.813885i −0.529511 0.848303i \(-0.677624\pi\)
0.999408 + 0.0344182i \(0.0109578\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.00000 15.5885i 0.752618 1.30357i
\(144\) 0 0
\(145\) 1.00000 + 1.73205i 0.0830455 + 0.143839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.50000 14.7224i −0.696347 1.20611i −0.969724 0.244202i \(-0.921474\pi\)
0.273377 0.961907i \(-0.411859\pi\)
\(150\) 0 0
\(151\) −2.50000 + 4.33013i −0.203447 + 0.352381i −0.949637 0.313353i \(-0.898548\pi\)
0.746190 + 0.665733i \(0.231881\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) 4.50000 7.79423i 0.359139 0.622047i −0.628678 0.777666i \(-0.716404\pi\)
0.987817 + 0.155618i \(0.0497370\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.50000 + 18.1865i −0.275839 + 1.43330i
\(162\) 0 0
\(163\) 6.50000 + 11.2583i 0.509119 + 0.881820i 0.999944 + 0.0105623i \(0.00336213\pi\)
−0.490825 + 0.871258i \(0.663305\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.50000 11.2583i −0.494186 0.855955i 0.505792 0.862656i \(-0.331200\pi\)
−0.999978 + 0.00670064i \(0.997867\pi\)
\(174\) 0 0
\(175\) 10.0000 3.46410i 0.755929 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.50000 11.2583i 0.485833 0.841487i −0.514035 0.857769i \(-0.671850\pi\)
0.999867 + 0.0162823i \(0.00518305\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.50000 + 2.59808i −0.110282 + 0.191014i
\(186\) 0 0
\(187\) −7.50000 12.9904i −0.548454 0.949951i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.50000 + 9.52628i 0.397966 + 0.689297i 0.993475 0.114051i \(-0.0363829\pi\)
−0.595509 + 0.803349i \(0.703050\pi\)
\(192\) 0 0
\(193\) −1.50000 + 2.59808i −0.107972 + 0.187014i −0.914949 0.403570i \(-0.867769\pi\)
0.806976 + 0.590584i \(0.201102\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −6.50000 + 11.2583i −0.460773 + 0.798082i −0.999000 0.0447181i \(-0.985761\pi\)
0.538227 + 0.842800i \(0.319094\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00000 3.46410i −0.280745 0.243132i
\(204\) 0 0
\(205\) −1.00000 1.73205i −0.0698430 0.120972i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.00000 3.46410i −0.136399 0.236250i
\(216\) 0 0
\(217\) −12.5000 + 4.33013i −0.848555 + 0.293948i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0000 25.9808i 1.00901 1.74766i
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.50000 + 9.52628i −0.365048 + 0.632281i −0.988784 0.149354i \(-0.952281\pi\)
0.623736 + 0.781635i \(0.285614\pi\)
\(228\) 0 0
\(229\) −11.5000 19.9186i −0.759941 1.31626i −0.942880 0.333133i \(-0.891894\pi\)
0.182939 0.983124i \(-0.441439\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.50000 + 9.52628i 0.360317 + 0.624087i 0.988013 0.154371i \(-0.0493352\pi\)
−0.627696 + 0.778459i \(0.716002\pi\)
\(234\) 0 0
\(235\) −2.50000 + 4.33013i −0.163082 + 0.282466i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 8.50000 14.7224i 0.547533 0.948355i −0.450910 0.892570i \(-0.648900\pi\)
0.998443 0.0557856i \(-0.0177663\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.50000 4.33013i 0.351382 0.276642i
\(246\) 0 0
\(247\) −3.00000 5.19615i −0.190885 0.330623i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) −21.0000 −1.32026
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.5000 18.1865i −0.654972 1.13444i −0.981901 0.189396i \(-0.939347\pi\)
0.326929 0.945049i \(-0.393986\pi\)
\(258\) 0 0
\(259\) 1.50000 7.79423i 0.0932055 0.484310i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.50000 7.79423i 0.277482 0.480613i −0.693276 0.720672i \(-0.743833\pi\)
0.970758 + 0.240059i \(0.0771668\pi\)
\(264\) 0 0
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.50000 + 7.79423i −0.274370 + 0.475223i −0.969976 0.243201i \(-0.921803\pi\)
0.695606 + 0.718423i \(0.255136\pi\)
\(270\) 0 0
\(271\) −3.50000 6.06218i −0.212610 0.368251i 0.739921 0.672694i \(-0.234863\pi\)
−0.952531 + 0.304443i \(0.901530\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 + 10.3923i 0.361814 + 0.626680i
\(276\) 0 0
\(277\) 8.50000 14.7224i 0.510716 0.884585i −0.489207 0.872167i \(-0.662714\pi\)
0.999923 0.0124177i \(-0.00395278\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −6.50000 + 11.2583i −0.386385 + 0.669238i −0.991960 0.126550i \(-0.959610\pi\)
0.605575 + 0.795788i \(0.292943\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 + 3.46410i 0.236113 + 0.204479i
\(288\) 0 0
\(289\) −4.00000 6.92820i −0.235294 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 15.0000 0.873334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −21.0000 36.3731i −1.21446 2.10351i
\(300\) 0 0
\(301\) 8.00000 + 6.92820i 0.461112 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.50000 4.33013i 0.143150 0.247942i
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.50000 + 12.9904i −0.425286 + 0.736617i −0.996447 0.0842210i \(-0.973160\pi\)
0.571161 + 0.820838i \(0.306493\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.50000 + 2.59808i 0.0842484 + 0.145922i 0.905071 0.425261i \(-0.139818\pi\)
−0.820822 + 0.571184i \(0.806484\pi\)
\(318\) 0 0
\(319\) 3.00000 5.19615i 0.167968 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) −12.0000 + 20.7846i −0.665640 + 1.15292i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.50000 12.9904i 0.137829 0.716183i
\(330\) 0 0
\(331\) 14.5000 + 25.1147i 0.796992 + 1.38043i 0.921567 + 0.388221i \(0.126910\pi\)
−0.124574 + 0.992210i \(0.539757\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.00000 0.491723
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.50000 12.9904i −0.406148 0.703469i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.50000 + 12.9904i −0.402621 + 0.697360i −0.994041 0.109003i \(-0.965234\pi\)
0.591420 + 0.806363i \(0.298567\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.50000 2.59808i 0.0798369 0.138282i −0.823343 0.567545i \(-0.807893\pi\)
0.903179 + 0.429263i \(0.141227\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.5000 18.1865i −0.554169 0.959849i −0.997968 0.0637221i \(-0.979703\pi\)
0.443799 0.896126i \(-0.353630\pi\)
\(360\) 0 0
\(361\) 9.00000 15.5885i 0.473684 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) 11.5000 19.9186i 0.600295 1.03974i −0.392481 0.919760i \(-0.628383\pi\)
0.992776 0.119982i \(-0.0382835\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.50000 + 0.866025i −0.129794 + 0.0449618i
\(372\) 0 0
\(373\) −5.50000 9.52628i −0.284779 0.493252i 0.687776 0.725923i \(-0.258587\pi\)
−0.972556 + 0.232671i \(0.925254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.50000 + 2.59808i 0.0766464 + 0.132755i 0.901801 0.432151i \(-0.142245\pi\)
−0.825155 + 0.564907i \(0.808912\pi\)
\(384\) 0 0
\(385\) 6.00000 + 5.19615i 0.305788 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.5000 + 25.1147i −0.735179 + 1.27337i 0.219465 + 0.975620i \(0.429569\pi\)
−0.954645 + 0.297747i \(0.903765\pi\)
\(390\) 0 0
\(391\) −35.0000 −1.77003
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.500000 0.866025i 0.0251577 0.0435745i
\(396\) 0 0
\(397\) 8.50000 + 14.7224i 0.426603 + 0.738898i 0.996569 0.0827707i \(-0.0263769\pi\)
−0.569966 + 0.821668i \(0.693044\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.5000 + 23.3827i 0.674158 + 1.16768i 0.976714 + 0.214544i \(0.0688266\pi\)
−0.302556 + 0.953131i \(0.597840\pi\)
\(402\) 0 0
\(403\) 15.0000 25.9808i 0.747203 1.29419i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.00000 0.446113
\(408\) 0 0
\(409\) −13.5000 + 23.3827i −0.667532 + 1.15620i 0.311060 + 0.950390i \(0.399316\pi\)
−0.978592 + 0.205809i \(0.934017\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −37.5000 + 12.9904i −1.84525 + 0.639215i
\(414\) 0 0
\(415\) −6.00000 10.3923i −0.294528 0.510138i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.0000 + 17.3205i 0.485071 + 0.840168i
\(426\) 0 0
\(427\) −2.50000 + 12.9904i −0.120983 + 0.628649i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.50000 + 2.59808i −0.0722525 + 0.125145i −0.899888 0.436121i \(-0.856352\pi\)
0.827636 + 0.561266i \(0.189685\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.50000 + 6.06218i −0.167428 + 0.289993i
\(438\) 0 0
\(439\) 10.5000 + 18.1865i 0.501138 + 0.867996i 0.999999 + 0.00131415i \(0.000418308\pi\)
−0.498861 + 0.866682i \(0.666248\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.5000 + 26.8468i 0.736427 + 1.27553i 0.954094 + 0.299506i \(0.0968220\pi\)
−0.217667 + 0.976023i \(0.569845\pi\)
\(444\) 0 0
\(445\) 3.50000 6.06218i 0.165916 0.287375i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) −3.00000 + 5.19615i −0.141264 + 0.244677i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.00000 + 15.5885i −0.140642 + 0.730798i
\(456\) 0 0
\(457\) 8.50000 + 14.7224i 0.397613 + 0.688686i 0.993431 0.114433i \(-0.0365053\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.50000 + 2.59808i 0.0694117 + 0.120225i 0.898642 0.438682i \(-0.144554\pi\)
−0.829231 + 0.558906i \(0.811221\pi\)
\(468\) 0 0
\(469\) −22.5000 + 7.79423i −1.03895 + 0.359904i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 + 10.3923i −0.275880 + 0.477839i
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.5000 + 26.8468i −0.708213 + 1.22666i 0.257306 + 0.966330i \(0.417165\pi\)
−0.965519 + 0.260331i \(0.916168\pi\)
\(480\) 0 0
\(481\) 9.00000 + 15.5885i 0.410365 + 0.710772i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 + 1.73205i 0.0454077 + 0.0786484i
\(486\) 0 0
\(487\) 3.50000 6.06218i 0.158600 0.274703i −0.775764 0.631023i \(-0.782635\pi\)
0.934364 + 0.356320i \(0.115969\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 5.00000 8.66025i 0.225189 0.390038i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17.5000 30.3109i −0.783408 1.35690i −0.929946 0.367697i \(-0.880146\pi\)
0.146538 0.989205i \(-0.453187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.50000 7.79423i −0.199459 0.345473i 0.748894 0.662690i \(-0.230585\pi\)
−0.948353 + 0.317217i \(0.897252\pi\)
\(510\) 0 0
\(511\) −17.5000 + 6.06218i −0.774154 + 0.268175i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.50000 + 12.9904i −0.330489 + 0.572425i
\(516\) 0 0
\(517\) 15.0000 0.659699
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16.5000 + 28.5788i −0.722878 + 1.25206i 0.236963 + 0.971519i \(0.423848\pi\)
−0.959841 + 0.280543i \(0.909485\pi\)
\(522\) 0 0
\(523\) −3.50000 6.06218i −0.153044 0.265081i 0.779301 0.626650i \(-0.215574\pi\)
−0.932345 + 0.361569i \(0.882241\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.5000 21.6506i −0.544509 0.943116i
\(528\) 0 0
\(529\) −13.0000 + 22.5167i −0.565217 + 0.978985i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −4.50000 + 7.79423i −0.194552 + 0.336974i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −19.5000 7.79423i −0.839924 0.335721i
\(540\) 0 0
\(541\) 20.5000 + 35.5070i 0.881364 + 1.52657i 0.849825 + 0.527064i \(0.176707\pi\)
0.0315385 + 0.999503i \(0.489959\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.00000 1.73205i −0.0426014 0.0737878i
\(552\) 0 0
\(553\) −0.500000 + 2.59808i −0.0212622 + 0.110481i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.50000 + 4.33013i −0.105928 + 0.183473i −0.914117 0.405450i \(-0.867115\pi\)
0.808189 + 0.588924i \(0.200448\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.5000 35.5070i 0.863972 1.49644i −0.00409232 0.999992i \(-0.501303\pi\)
0.868064 0.496452i \(-0.165364\pi\)
\(564\) 0 0
\(565\) −9.00000 15.5885i −0.378633 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.5000 28.5788i −0.691716 1.19809i −0.971275 0.237959i \(-0.923522\pi\)
0.279559 0.960128i \(-0.409812\pi\)
\(570\) 0 0
\(571\) −6.50000 + 11.2583i −0.272017 + 0.471146i −0.969378 0.245573i \(-0.921024\pi\)
0.697362 + 0.716720i \(0.254357\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.0000 1.16768
\(576\) 0 0
\(577\) 16.5000 28.5788i 0.686904 1.18975i −0.285930 0.958250i \(-0.592303\pi\)
0.972834 0.231502i \(-0.0743641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 + 20.7846i 0.995688 + 0.862291i
\(582\) 0 0
\(583\) −1.50000 2.59808i −0.0621237 0.107601i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.5000 38.9711i −0.923964 1.60035i −0.793219 0.608937i \(-0.791596\pi\)
−0.130746 0.991416i \(-0.541737\pi\)
\(594\) 0 0
\(595\) 10.0000 + 8.66025i 0.409960 + 0.355036i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.50000 14.7224i 0.347301 0.601542i −0.638468 0.769648i \(-0.720432\pi\)
0.985769 + 0.168106i \(0.0537650\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 1.73205i 0.0406558 0.0704179i
\(606\) 0 0
\(607\) 6.50000 + 11.2583i 0.263827 + 0.456962i 0.967256 0.253804i \(-0.0816819\pi\)
−0.703429 + 0.710766i \(0.748349\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 + 25.9808i 0.606835 + 1.05107i
\(612\) 0 0
\(613\) −21.5000 + 37.2391i −0.868377 + 1.50407i −0.00472215 + 0.999989i \(0.501503\pi\)
−0.863655 + 0.504084i \(0.831830\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −4.50000 + 7.79423i −0.180870 + 0.313276i −0.942177 0.335115i \(-0.891225\pi\)
0.761307 + 0.648392i \(0.224558\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.50000 + 18.1865i −0.140225 + 0.728628i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.00000 + 6.92820i 0.158735 + 0.274937i
\(636\) 0 0
\(637\) −6.00000 41.5692i −0.237729 1.64703i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.50000 + 14.7224i −0.335730 + 0.581501i −0.983625 0.180229i \(-0.942316\pi\)
0.647895 + 0.761730i \(0.275650\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.50000 + 12.9904i −0.294855 + 0.510705i −0.974951 0.222419i \(-0.928605\pi\)
0.680096 + 0.733123i \(0.261938\pi\)
\(648\) 0 0
\(649\) −22.5000 38.9711i −0.883202 1.52975i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.5000 + 30.3109i 0.684828 + 1.18616i 0.973491 + 0.228726i \(0.0734560\pi\)
−0.288663 + 0.957431i \(0.593211\pi\)
\(654\) 0 0
\(655\) −2.50000 + 4.33013i −0.0976831 + 0.169192i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) −11.5000 + 19.9186i −0.447298 + 0.774743i −0.998209 0.0598209i \(-0.980947\pi\)
0.550911 + 0.834564i \(0.314280\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.50000 0.866025i 0.0969458 0.0335830i
\(666\) 0 0
\(667\) −7.00000 12.1244i −0.271041 0.469457i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.5000 + 33.7750i 0.749446 + 1.29808i 0.948089 + 0.318006i \(0.103013\pi\)
−0.198643 + 0.980072i \(0.563653\pi\)
\(678\) 0 0
\(679\) −4.00000 3.46410i −0.153506 0.132940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −13.5000 + 23.3827i −0.516563 + 0.894714i 0.483252 + 0.875481i \(0.339456\pi\)
−0.999815 + 0.0192323i \(0.993878\pi\)
\(684\) 0 0
\(685\) −11.0000 −0.420288
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.00000 5.19615i 0.114291 0.197958i
\(690\) 0 0
\(691\) 2.50000 + 4.33013i 0.0951045 + 0.164726i 0.909652 0.415371i \(-0.136348\pi\)
−0.814548 + 0.580097i \(0.803015\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 10.3923i −0.227593 0.394203i
\(696\) 0 0
\(697\) −5.00000 + 8.66025i −0.189389 + 0.328031i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 1.50000 2.59808i 0.0565736 0.0979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.50000 2.59808i 0.282067 0.0977107i
\(708\) 0 0
\(709\) 14.5000 + 25.1147i 0.544559 + 0.943204i 0.998635 + 0.0522406i \(0.0166363\pi\)
−0.454076 + 0.890963i \(0.650030\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −35.0000 −1.31076
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19.5000 + 33.7750i 0.727227 + 1.25959i 0.958051 + 0.286599i \(0.0925247\pi\)
−0.230823 + 0.972996i \(0.574142\pi\)
\(720\) 0 0
\(721\) 7.50000 38.9711i 0.279315 1.45136i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 + 6.92820i −0.148556 + 0.257307i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0000 + 17.3205i −0.369863 + 0.640622i
\(732\) 0 0
\(733\) −9.50000 16.4545i −0.350891 0.607760i 0.635515 0.772088i \(-0.280788\pi\)
−0.986406 + 0.164328i \(0.947454\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.5000 23.3827i −0.497279 0.861312i
\(738\) 0 0
\(739\) −2.50000 + 4.33013i −0.0919640 + 0.159286i −0.908337 0.418238i \(-0.862648\pi\)
0.816373 + 0.577524i \(0.195981\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −8.50000 + 14.7224i −0.311416 + 0.539388i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.50000 23.3827i 0.164426 0.854385i
\(750\) 0 0
\(751\) 26.5000 + 45.8993i 0.966999 + 1.67489i 0.704146 + 0.710055i \(0.251330\pi\)
0.262852 + 0.964836i \(0.415337\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.50000 + 2.59808i 0.0543750 + 0.0941802i 0.891932 0.452170i \(-0.149350\pi\)
−0.837557 + 0.546350i \(0.816017\pi\)
\(762\) 0 0
\(763\) 12.5000 4.33013i 0.452530 0.156761i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 45.0000 77.9423i 1.62486 2.81433i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.50000 16.4545i 0.341691 0.591827i −0.643056 0.765819i \(-0.722334\pi\)
0.984747 + 0.173993i \(0.0556670\pi\)
\(774\) 0 0
\(775\) 10.0000 + 17.3205i 0.359211 + 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.00000 + 1.73205i 0.0358287 + 0.0620572i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.00000 −0.321224
\(786\) 0 0
\(787\) −8.50000 + 14.7224i −0.302992 + 0.524798i −0.976812 0.214097i \(-0.931319\pi\)
0.673820 + 0.738896i \(0.264652\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 + 31.1769i 1.28001 + 1.10852i
\(792\) 0 0
\(793\) −15.0000 25.9808i −0.532666 0.922604i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 25.0000 0.884436
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.5000 18.1865i −0.370537 0.641789i
\(804\) 0 0
\(805\) 17.5000 6.06218i 0.616794 0.213664i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.5000 33.7750i 0.685583 1.18747i −0.287670 0.957730i \(-0.592880\pi\)
0.973253 0.229736i \(-0.0737862\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.50000 11.2583i 0.227685 0.394362i
\(816\) 0 0
\(817\) 2.00000 + 3.46410i 0.0699711 + 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.50000 7.79423i −0.157051 0.272020i 0.776753 0.629805i \(-0.216865\pi\)
−0.933804 + 0.357785i \(0.883532\pi\)
\(822\) 0 0
\(823\) 23.5000 40.7032i 0.819159 1.41882i −0.0871445 0.996196i \(-0.527774\pi\)
0.906303 0.422628i \(-0.138892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −5.50000 + 9.52628i −0.191023 + 0.330861i −0.945589 0.325362i \(-0.894514\pi\)
0.754567 + 0.656223i \(0.227847\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −32.5000 12.9904i −1.12606 0.450090i
\(834\) 0 0
\(835\) 6.00000 + 10.3923i 0.207639 + 0.359641i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −11.5000 19.9186i −0.395612 0.685220i
\(846\) 0 0
\(847\) −1.00000 + 5.19615i −0.0343604 + 0.178542i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.5000 18.1865i 0.359935 0.623426i
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.5000 33.7750i 0.666107 1.15373i −0.312877 0.949794i \(-0.601293\pi\)
0.978984 0.203938i \(-0.0653741\pi\)
\(858\) 0 0
\(859\) 0.500000 + 0.866025i 0.0170598 + 0.0295484i 0.874429 0.485153i \(-0.161236\pi\)
−0.857369 + 0.514701i \(0.827903\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.50000 14.7224i −0.289343 0.501157i 0.684310 0.729191i \(-0.260104\pi\)
−0.973653 + 0.228034i \(0.926770\pi\)
\(864\) 0 0
\(865\) −6.50000 + 11.2583i −0.221007 + 0.382795i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 27.0000 46.7654i 0.914860 1.58458i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −18.0000 15.5885i −0.608511 0.526986i
\(876\) 0 0
\(877\) −11.5000 19.9186i −0.388327 0.672603i 0.603897 0.797062i \(-0.293614\pi\)
−0.992225 + 0.124459i \(0.960280\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.50000 + 2.59808i 0.0503651 + 0.0872349i 0.890109 0.455748i \(-0.150628\pi\)
−0.839744 + 0.542983i \(0.817295\pi\)
\(888\) 0 0
\(889\) −16.0000 13.8564i −0.536623 0.464729i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.50000 4.33013i 0.0836593 0.144902i
\(894\) 0 0
\(895\) −13.0000 −0.434542
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.00000 8.66025i 0.166759 0.288836i
\(900\) 0 0
\(901\) −2.50000 4.33013i −0.0832871 0.144257i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 + 8.66025i 0.166206 + 0.287877i
\(906\) 0 0
\(907\) −0.500000 + 0.866025i −0.0166022 + 0.0287559i −0.874207 0.485553i \(-0.838618\pi\)
0.857605 + 0.514309i \(0.171952\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −18.0000 + 31.1769i −0.595713 + 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.50000 12.9904i 0.0825573 0.428980i
\(918\) 0 0
\(919\) −5.50000 9.52628i −0.181428 0.314243i 0.760939 0.648824i \(-0.224739\pi\)
−0.942367 + 0.334581i \(0.891405\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.50000 + 6.06218i 0.114831 + 0.198894i 0.917712 0.397246i \(-0.130034\pi\)
−0.802881 + 0.596139i \(0.796701\pi\)
\(930\) 0 0
\(931\) −5.50000 + 4.33013i −0.180255 + 0.141914i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.50000 + 12.9904i −0.245276 + 0.424831i
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −16.5000 + 28.5788i −0.537885 + 0.931644i 0.461133 + 0.887331i \(0.347443\pi\)
−0.999018 + 0.0443125i \(0.985890\pi\)
\(942\) 0 0
\(943\) 7.00000 + 12.1244i 0.227951 + 0.394823i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.5000 32.0429i −0.601169 1.04126i −0.992644 0.121067i \(-0.961368\pi\)
0.391475 0.920189i \(-0.371965\pi\)
\(948\) 0 0
\(949\) 21.0000 36.3731i 0.681689 1.18072i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 5.50000 9.52628i 0.177976 0.308263i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.5000 9.52628i 0.888021 0.307620i
\(960\) 0 0
\(961\) 3.00000 + 5.19615i 0.0967742 + 0.167618i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.5000 + 30.3109i 0.561602 + 0.972723i 0.997357 + 0.0726575i \(0.0231480\pi\)
−0.435755 + 0.900065i \(0.643519\pi\)
\(972\) 0 0
\(973\) 24.0000 + 20.7846i 0.769405 + 0.666324i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.50000 2.59808i 0.0479893 0.0831198i −0.841033 0.540984i \(-0.818052\pi\)
0.889022 + 0.457864i \(0.151385\pi\)
\(978\) 0 0
\(979\) −21.0000 −0.671163
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 10.5000 18.1865i 0.334898 0.580060i −0.648567 0.761157i \(-0.724631\pi\)
0.983465 + 0.181097i \(0.0579648\pi\)
\(984\) 0 0
\(985\) −3.00000 5.19615i −0.0955879 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.0000 + 24.2487i 0.445174 + 0.771064i
\(990\) 0 0
\(991\) 7.50000 12.9904i 0.238245 0.412653i −0.721966 0.691929i \(-0.756761\pi\)
0.960211 + 0.279276i \(0.0900944\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 13.0000 0.412128
\(996\) 0 0
\(997\) 20.5000 35.5070i 0.649242 1.12452i −0.334063 0.942551i \(-0.608420\pi\)
0.983304 0.181968i \(-0.0582469\pi\)
\(998\) 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 1008.2.s.g.865.1 2
3.2 odd 2 112.2.i.a.81.1 2
4.3 odd 2 504.2.s.c.361.1 2
7.2 even 3 inner 1008.2.s.g.289.1 2
7.3 odd 6 7056.2.a.u.1.1 1
7.4 even 3 7056.2.a.bj.1.1 1
12.11 even 2 56.2.i.b.25.1 yes 2
21.2 odd 6 112.2.i.a.65.1 2
21.5 even 6 784.2.i.h.177.1 2
21.11 odd 6 784.2.a.h.1.1 1
21.17 even 6 784.2.a.c.1.1 1
21.20 even 2 784.2.i.h.753.1 2
24.5 odd 2 448.2.i.d.193.1 2
24.11 even 2 448.2.i.b.193.1 2
28.3 even 6 3528.2.a.j.1.1 1
28.11 odd 6 3528.2.a.p.1.1 1
28.19 even 6 3528.2.s.q.3313.1 2
28.23 odd 6 504.2.s.c.289.1 2
28.27 even 2 3528.2.s.q.361.1 2
60.23 odd 4 1400.2.bh.a.249.2 4
60.47 odd 4 1400.2.bh.a.249.1 4
60.59 even 2 1400.2.q.d.1201.1 2
84.11 even 6 392.2.a.c.1.1 1
84.23 even 6 56.2.i.b.9.1 2
84.47 odd 6 392.2.i.b.177.1 2
84.59 odd 6 392.2.a.e.1.1 1
84.83 odd 2 392.2.i.b.361.1 2
168.11 even 6 3136.2.a.u.1.1 1
168.53 odd 6 3136.2.a.j.1.1 1
168.59 odd 6 3136.2.a.i.1.1 1
168.101 even 6 3136.2.a.t.1.1 1
168.107 even 6 448.2.i.b.65.1 2
168.149 odd 6 448.2.i.d.65.1 2
420.23 odd 12 1400.2.bh.a.849.1 4
420.59 odd 6 9800.2.a.s.1.1 1
420.107 odd 12 1400.2.bh.a.849.2 4
420.179 even 6 9800.2.a.be.1.1 1
420.359 even 6 1400.2.q.d.401.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.i.b.9.1 2 84.23 even 6
56.2.i.b.25.1 yes 2 12.11 even 2
112.2.i.a.65.1 2 21.2 odd 6
112.2.i.a.81.1 2 3.2 odd 2
392.2.a.c.1.1 1 84.11 even 6
392.2.a.e.1.1 1 84.59 odd 6
392.2.i.b.177.1 2 84.47 odd 6
392.2.i.b.361.1 2 84.83 odd 2
448.2.i.b.65.1 2 168.107 even 6
448.2.i.b.193.1 2 24.11 even 2
448.2.i.d.65.1 2 168.149 odd 6
448.2.i.d.193.1 2 24.5 odd 2
504.2.s.c.289.1 2 28.23 odd 6
504.2.s.c.361.1 2 4.3 odd 2
784.2.a.c.1.1 1 21.17 even 6
784.2.a.h.1.1 1 21.11 odd 6
784.2.i.h.177.1 2 21.5 even 6
784.2.i.h.753.1 2 21.20 even 2
1008.2.s.g.289.1 2 7.2 even 3 inner
1008.2.s.g.865.1 2 1.1 even 1 trivial
1400.2.q.d.401.1 2 420.359 even 6
1400.2.q.d.1201.1 2 60.59 even 2
1400.2.bh.a.249.1 4 60.47 odd 4
1400.2.bh.a.249.2 4 60.23 odd 4
1400.2.bh.a.849.1 4 420.23 odd 12
1400.2.bh.a.849.2 4 420.107 odd 12
3136.2.a.i.1.1 1 168.59 odd 6
3136.2.a.j.1.1 1 168.53 odd 6
3136.2.a.t.1.1 1 168.101 even 6
3136.2.a.u.1.1 1 168.11 even 6
3528.2.a.j.1.1 1 28.3 even 6
3528.2.a.p.1.1 1 28.11 odd 6
3528.2.s.q.361.1 2 28.27 even 2
3528.2.s.q.3313.1 2 28.19 even 6
7056.2.a.u.1.1 1 7.3 odd 6
7056.2.a.bj.1.1 1 7.4 even 3
9800.2.a.s.1.1 1 420.59 odd 6
9800.2.a.be.1.1 1 420.179 even 6