Properties

Label 2016.2.s.d.289.1
Level $2016$
Weight $2$
Character 2016.289
Analytic conductor $16.098$
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] = [2016,2,Mod(289,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, 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("2016.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.s (of order \(3\), 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: \(16.0978410475\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2016.289
Dual form 2016.2.s.d.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{5} +(0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{5} +(0.500000 - 2.59808i) q^{7} +(2.50000 + 4.33013i) q^{11} +2.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(3.00000 - 5.19615i) q^{19} +(1.00000 - 1.73205i) q^{23} +(-2.00000 - 3.46410i) q^{25} +1.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(6.00000 + 5.19615i) q^{35} +(-5.00000 + 8.66025i) q^{37} +4.00000 q^{41} +4.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{53} -15.0000 q^{55} +(6.50000 + 11.2583i) q^{59} +(4.00000 - 6.92820i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(7.00000 + 12.1244i) q^{67} +12.0000 q^{71} +(3.00000 + 5.19615i) q^{73} +(12.5000 - 4.33013i) q^{77} +(-5.50000 + 9.52628i) q^{79} +7.00000 q^{83} +6.00000 q^{85} +(3.00000 - 5.19615i) q^{89} +(1.00000 - 5.19615i) q^{91} +(9.00000 + 15.5885i) q^{95} +19.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + q^{7} + 5 q^{11} + 4 q^{13} - 2 q^{17} + 6 q^{19} + 2 q^{23} - 4 q^{25} + 2 q^{29} + q^{31} + 12 q^{35} - 10 q^{37} + 8 q^{41} + 8 q^{43} - 8 q^{47} - 13 q^{49} - 5 q^{53} - 30 q^{55} + 13 q^{59} + 8 q^{61} - 6 q^{65} + 14 q^{67} + 24 q^{71} + 6 q^{73} + 25 q^{77} - 11 q^{79} + 14 q^{83} + 12 q^{85} + 6 q^{89} + 2 q^{91} + 18 q^{95} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(5\) −1.50000 + 2.59808i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) 0.500000 2.59808i 0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.50000 + 4.33013i 0.753778 + 1.30558i 0.945979 + 0.324227i \(0.105104\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 3.00000 5.19615i 0.688247 1.19208i −0.284157 0.958778i \(-0.591714\pi\)
0.972404 0.233301i \(-0.0749529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 1.73205i 0.208514 0.361158i −0.742732 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.0898027 + 0.155543i 0.907428 0.420208i \(-0.138043\pi\)
−0.817625 + 0.575751i \(0.804710\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 + 5.19615i 1.01419 + 0.878310i
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 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
\(47\) −4.00000 + 6.92820i −0.583460 + 1.01058i 0.411606 + 0.911362i \(0.364968\pi\)
−0.995066 + 0.0992202i \(0.968365\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.50000 4.33013i −0.343401 0.594789i 0.641661 0.766989i \(-0.278246\pi\)
−0.985062 + 0.172200i \(0.944912\pi\)
\(54\) 0 0
\(55\) −15.0000 −2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.50000 + 11.2583i 0.846228 + 1.46571i 0.884551 + 0.466444i \(0.154465\pi\)
−0.0383226 + 0.999265i \(0.512201\pi\)
\(60\) 0 0
\(61\) 4.00000 6.92820i 0.512148 0.887066i −0.487753 0.872982i \(-0.662183\pi\)
0.999901 0.0140840i \(-0.00448323\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) 7.00000 + 12.1244i 0.855186 + 1.48123i 0.876472 + 0.481452i \(0.159891\pi\)
−0.0212861 + 0.999773i \(0.506776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 3.00000 + 5.19615i 0.351123 + 0.608164i 0.986447 0.164083i \(-0.0524664\pi\)
−0.635323 + 0.772246i \(0.719133\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.5000 4.33013i 1.42451 0.493464i
\(78\) 0 0
\(79\) −5.50000 + 9.52628i −0.618798 + 1.07179i 0.370907 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143120i \(0.954286\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.00000 0.768350 0.384175 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 5.19615i 0.317999 0.550791i −0.662071 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) 1.00000 5.19615i 0.104828 0.544705i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.00000 + 15.5885i 0.923381 + 1.59934i
\(96\) 0 0
\(97\) 19.0000 1.92916 0.964579 0.263795i \(-0.0849741\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −4.00000 + 6.92820i −0.394132 + 0.682656i −0.992990 0.118199i \(-0.962288\pi\)
0.598858 + 0.800855i \(0.295621\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.50000 + 11.2583i −0.628379 + 1.08838i 0.359498 + 0.933146i \(0.382948\pi\)
−0.987877 + 0.155238i \(0.950386\pi\)
\(108\) 0 0
\(109\) −5.00000 8.66025i −0.478913 0.829502i 0.520794 0.853682i \(-0.325636\pi\)
−0.999708 + 0.0241802i \(0.992302\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.00000 + 1.73205i −0.458349 + 0.158777i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.50000 7.79423i 0.393167 0.680985i −0.599699 0.800226i \(-0.704713\pi\)
0.992865 + 0.119241i \(0.0380462\pi\)
\(132\) 0 0
\(133\) −12.0000 10.3923i −1.04053 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0000 + 19.0526i 0.939793 + 1.62777i 0.765855 + 0.643013i \(0.222316\pi\)
0.173939 + 0.984757i \(0.444351\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.00000 + 8.66025i 0.418121 + 0.724207i
\(144\) 0 0
\(145\) −1.50000 + 2.59808i −0.124568 + 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) −5.50000 9.52628i −0.447584 0.775238i 0.550645 0.834740i \(-0.314382\pi\)
−0.998228 + 0.0595022i \(0.981049\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 6.00000 + 10.3923i 0.478852 + 0.829396i 0.999706 0.0242497i \(-0.00771967\pi\)
−0.520854 + 0.853646i \(0.674386\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 3.46410i −0.315244 0.273009i
\(162\) 0 0
\(163\) 7.00000 12.1244i 0.548282 0.949653i −0.450110 0.892973i \(-0.648615\pi\)
0.998392 0.0566798i \(-0.0180514\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 + 15.5885i −0.684257 + 1.18517i 0.289412 + 0.957205i \(0.406540\pi\)
−0.973670 + 0.227964i \(0.926793\pi\)
\(174\) 0 0
\(175\) −10.0000 + 3.46410i −0.755929 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 10.3923i −0.448461 0.776757i 0.549825 0.835280i \(-0.314694\pi\)
−0.998286 + 0.0585225i \(0.981361\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.0000 25.9808i −1.10282 1.91014i
\(186\) 0 0
\(187\) 5.00000 8.66025i 0.365636 0.633300i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 + 10.3923i −0.434145 + 0.751961i −0.997225 0.0744412i \(-0.976283\pi\)
0.563081 + 0.826402i \(0.309616\pi\)
\(192\) 0 0
\(193\) −8.50000 14.7224i −0.611843 1.05974i −0.990930 0.134382i \(-0.957095\pi\)
0.379086 0.925361i \(-0.376238\pi\)
\(194\) 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
\(199\) −8.00000 13.8564i −0.567105 0.982255i −0.996850 0.0793045i \(-0.974730\pi\)
0.429745 0.902950i \(-0.358603\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.500000 2.59808i 0.0350931 0.182349i
\(204\) 0 0
\(205\) −6.00000 + 10.3923i −0.419058 + 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 + 10.3923i −0.409197 + 0.708749i
\(216\) 0 0
\(217\) 2.50000 0.866025i 0.169711 0.0587896i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 3.46410i −0.134535 0.233021i
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.500000 0.866025i −0.0331862 0.0574801i 0.848955 0.528465i \(-0.177232\pi\)
−0.882141 + 0.470985i \(0.843899\pi\)
\(228\) 0 0
\(229\) 4.00000 6.92820i 0.264327 0.457829i −0.703060 0.711131i \(-0.748183\pi\)
0.967387 + 0.253302i \(0.0815167\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 + 5.19615i −0.196537 + 0.340411i −0.947403 0.320043i \(-0.896303\pi\)
0.750867 + 0.660454i \(0.229636\pi\)
\(234\) 0 0
\(235\) −12.0000 20.7846i −0.782794 1.35584i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.0322078 + 0.0557856i 0.881680 0.471848i \(-0.156413\pi\)
−0.849472 + 0.527633i \(0.823079\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 16.5000 12.9904i 1.05415 0.829925i
\(246\) 0 0
\(247\) 6.00000 10.3923i 0.381771 0.661247i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.0000 −1.07303 −0.536515 0.843891i \(-0.680260\pi\)
−0.536515 + 0.843891i \(0.680260\pi\)
\(252\) 0 0
\(253\) 10.0000 0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.00000 12.1244i 0.436648 0.756297i −0.560781 0.827964i \(-0.689499\pi\)
0.997429 + 0.0716680i \(0.0228322\pi\)
\(258\) 0 0
\(259\) 20.0000 + 17.3205i 1.24274 + 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.00000 13.8564i −0.493301 0.854423i 0.506669 0.862141i \(-0.330877\pi\)
−0.999970 + 0.00771799i \(0.997543\pi\)
\(264\) 0 0
\(265\) 15.0000 0.921443
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.5000 18.1865i −0.640196 1.10885i −0.985389 0.170321i \(-0.945520\pi\)
0.345192 0.938532i \(-0.387814\pi\)
\(270\) 0 0
\(271\) −5.50000 + 9.52628i −0.334101 + 0.578680i −0.983312 0.181928i \(-0.941766\pi\)
0.649211 + 0.760609i \(0.275099\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 17.3205i 0.603023 1.04447i
\(276\) 0 0
\(277\) 14.0000 + 24.2487i 0.841178 + 1.45696i 0.888899 + 0.458103i \(0.151471\pi\)
−0.0477206 + 0.998861i \(0.515196\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −2.00000 3.46410i −0.118888 0.205919i 0.800439 0.599414i \(-0.204600\pi\)
−0.919327 + 0.393494i \(0.871266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 10.3923i 0.118056 0.613438i
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.0000 1.46052 0.730258 0.683172i \(-0.239400\pi\)
0.730258 + 0.683172i \(0.239400\pi\)
\(294\) 0 0
\(295\) −39.0000 −2.27067
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 3.46410i 0.115663 0.200334i
\(300\) 0 0
\(301\) 2.00000 10.3923i 0.115278 0.599002i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 + 20.7846i 0.687118 + 1.19012i
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.0282617 + 0.0489506i −0.879810 0.475325i \(-0.842331\pi\)
0.851549 + 0.524276i \(0.175664\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.50000 + 11.2583i −0.365076 + 0.632331i −0.988788 0.149323i \(-0.952290\pi\)
0.623712 + 0.781654i \(0.285624\pi\)
\(318\) 0 0
\(319\) 2.50000 + 4.33013i 0.139973 + 0.242441i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −4.00000 6.92820i −0.221880 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000 + 13.8564i 0.882109 + 0.763928i
\(330\) 0 0
\(331\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −42.0000 −2.29471
\(336\) 0 0
\(337\) −3.00000 −0.163420 −0.0817102 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.50000 + 4.33013i −0.135383 + 0.234490i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.0000 20.7846i −0.638696 1.10625i −0.985719 0.168397i \(-0.946141\pi\)
0.347024 0.937856i \(-0.387192\pi\)
\(354\) 0 0
\(355\) −18.0000 + 31.1769i −0.955341 + 1.65470i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.0000 22.5167i 0.686114 1.18838i −0.286972 0.957939i \(-0.592649\pi\)
0.973085 0.230445i \(-0.0740181\pi\)
\(360\) 0 0
\(361\) −8.50000 14.7224i −0.447368 0.774865i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −18.0000 −0.942163
\(366\) 0 0
\(367\) 9.50000 + 16.4545i 0.495896 + 0.858917i 0.999989 0.00473247i \(-0.00150640\pi\)
−0.504093 + 0.863649i \(0.668173\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.5000 + 4.33013i −0.648968 + 0.224809i
\(372\) 0 0
\(373\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.0000 32.9090i 0.970855 1.68157i 0.277868 0.960619i \(-0.410372\pi\)
0.692987 0.720950i \(-0.256294\pi\)
\(384\) 0 0
\(385\) −7.50000 + 38.9711i −0.382235 + 1.98615i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.00000 15.5885i −0.456318 0.790366i 0.542445 0.840091i \(-0.317499\pi\)
−0.998763 + 0.0497253i \(0.984165\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.5000 28.5788i −0.830205 1.43796i
\(396\) 0 0
\(397\) −4.00000 + 6.92820i −0.200754 + 0.347717i −0.948772 0.315963i \(-0.897673\pi\)
0.748017 + 0.663679i \(0.231006\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.0000 + 27.7128i −0.799002 + 1.38391i 0.121265 + 0.992620i \(0.461305\pi\)
−0.920267 + 0.391292i \(0.872028\pi\)
\(402\) 0 0
\(403\) 1.00000 + 1.73205i 0.0498135 + 0.0862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −50.0000 −2.47841
\(408\) 0 0
\(409\) −9.50000 16.4545i −0.469745 0.813622i 0.529657 0.848212i \(-0.322321\pi\)
−0.999402 + 0.0345902i \(0.988987\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 32.5000 11.2583i 1.59922 0.553986i
\(414\) 0 0
\(415\) −10.5000 + 18.1865i −0.515425 + 0.892742i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 + 6.92820i −0.194029 + 0.336067i
\(426\) 0 0
\(427\) −16.0000 13.8564i −0.774294 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.0000 24.2487i −0.674356 1.16802i −0.976657 0.214807i \(-0.931088\pi\)
0.302300 0.953213i \(-0.402245\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 10.3923i −0.287019 0.497131i
\(438\) 0 0
\(439\) 10.5000 18.1865i 0.501138 0.867996i −0.498861 0.866682i \(-0.666248\pi\)
0.999999 0.00131415i \(-0.000418308\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.50000 7.79423i 0.213801 0.370315i −0.739100 0.673596i \(-0.764749\pi\)
0.952901 + 0.303281i \(0.0980821\pi\)
\(444\) 0 0
\(445\) 9.00000 + 15.5885i 0.426641 + 0.738964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 10.0000 + 17.3205i 0.470882 + 0.815591i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.0000 + 10.3923i 0.562569 + 0.487199i
\(456\) 0 0
\(457\) 14.5000 25.1147i 0.678281 1.17482i −0.297217 0.954810i \(-0.596058\pi\)
0.975498 0.220008i \(-0.0706083\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.0000 + 31.1769i −0.832941 + 1.44270i 0.0627555 + 0.998029i \(0.480011\pi\)
−0.895696 + 0.444667i \(0.853322\pi\)
\(468\) 0 0
\(469\) 35.0000 12.1244i 1.61615 0.559851i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000 + 17.3205i 0.459800 + 0.796398i
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0000 20.7846i −0.548294 0.949673i −0.998392 0.0566937i \(-0.981944\pi\)
0.450098 0.892979i \(-0.351389\pi\)
\(480\) 0 0
\(481\) −10.0000 + 17.3205i −0.455961 + 0.789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −28.5000 + 49.3634i −1.29412 + 2.24148i
\(486\) 0 0
\(487\) 1.50000 + 2.59808i 0.0679715 + 0.117730i 0.898008 0.439979i \(-0.145014\pi\)
−0.830037 + 0.557709i \(0.811681\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.0000 1.39901 0.699505 0.714628i \(-0.253404\pi\)
0.699505 + 0.714628i \(0.253404\pi\)
\(492\) 0 0
\(493\) −1.00000 1.73205i −0.0450377 0.0780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 31.1769i 0.269137 1.39848i
\(498\) 0 0
\(499\) −5.00000 + 8.66025i −0.223831 + 0.387686i −0.955968 0.293471i \(-0.905190\pi\)
0.732137 + 0.681157i \(0.238523\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.5000 + 21.6506i −0.554053 + 0.959648i 0.443924 + 0.896065i \(0.353586\pi\)
−0.997977 + 0.0635830i \(0.979747\pi\)
\(510\) 0 0
\(511\) 15.0000 5.19615i 0.663561 0.229864i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.0000 20.7846i −0.528783 0.915879i
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.0000 34.6410i −0.876216 1.51765i −0.855462 0.517866i \(-0.826727\pi\)
−0.0207541 0.999785i \(-0.506607\pi\)
\(522\) 0 0
\(523\) 13.0000 22.5167i 0.568450 0.984585i −0.428269 0.903651i \(-0.640876\pi\)
0.996719 0.0809336i \(-0.0257902\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00000 1.73205i 0.0435607 0.0754493i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −19.5000 33.7750i −0.843059 1.46022i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.00000 34.6410i −0.215365 1.49209i
\(540\) 0 0
\(541\) 12.0000 20.7846i 0.515920 0.893600i −0.483909 0.875118i \(-0.660783\pi\)
0.999829 0.0184818i \(-0.00588327\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.00000 5.19615i 0.127804 0.221364i
\(552\) 0 0
\(553\) 22.0000 + 19.0526i 0.935535 + 0.810197i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.5000 + 28.5788i 0.699127 + 1.21092i 0.968769 + 0.247964i \(0.0797613\pi\)
−0.269642 + 0.962961i \(0.586905\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.50000 9.52628i −0.231797 0.401485i 0.726540 0.687124i \(-0.241127\pi\)
−0.958337 + 0.285640i \(0.907794\pi\)
\(564\) 0 0
\(565\) 18.0000 31.1769i 0.757266 1.31162i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) 2.00000 + 3.46410i 0.0836974 + 0.144968i 0.904835 0.425762i \(-0.139994\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −3.50000 6.06218i −0.145707 0.252372i 0.783930 0.620850i \(-0.213212\pi\)
−0.929636 + 0.368478i \(0.879879\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.50000 18.1865i 0.145204 0.754505i
\(582\) 0 0
\(583\) 12.5000 21.6506i 0.517697 0.896678i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.00000 + 5.19615i −0.123195 + 0.213380i −0.921026 0.389501i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(594\) 0 0
\(595\) 3.00000 15.5885i 0.122988 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.00000 15.5885i −0.367730 0.636927i 0.621480 0.783430i \(-0.286532\pi\)
−0.989210 + 0.146503i \(0.953198\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.0000 36.3731i −0.853771 1.47878i
\(606\) 0 0
\(607\) 1.50000 2.59808i 0.0608831 0.105453i −0.833977 0.551799i \(-0.813942\pi\)
0.894860 + 0.446346i \(0.147275\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 + 13.8564i −0.323645 + 0.560570i
\(612\) 0 0
\(613\) 11.0000 + 19.0526i 0.444286 + 0.769526i 0.998002 0.0631797i \(-0.0201241\pi\)
−0.553716 + 0.832705i \(0.686791\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) −16.0000 27.7128i −0.643094 1.11387i −0.984738 0.174042i \(-0.944317\pi\)
0.341644 0.939829i \(-0.389016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 10.3923i −0.480770 0.416359i
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.5000 18.1865i 0.416680 0.721711i
\(636\) 0 0
\(637\) −13.0000 5.19615i −0.515079 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.00000 13.8564i −0.315981 0.547295i 0.663665 0.748030i \(-0.269000\pi\)
−0.979646 + 0.200735i \(0.935667\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 + 41.5692i 0.943537 + 1.63425i 0.758654 + 0.651494i \(0.225858\pi\)
0.184884 + 0.982760i \(0.440809\pi\)
\(648\) 0 0
\(649\) −32.5000 + 56.2917i −1.27574 + 2.20964i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.5000 21.6506i 0.489163 0.847255i −0.510759 0.859724i \(-0.670636\pi\)
0.999922 + 0.0124688i \(0.00396906\pi\)
\(654\) 0 0
\(655\) 13.5000 + 23.3827i 0.527489 + 0.913637i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 45.0000 15.5885i 1.74503 0.604494i
\(666\) 0 0
\(667\) 1.00000 1.73205i 0.0387202 0.0670653i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) −39.0000 −1.50334 −0.751670 0.659540i \(-0.770751\pi\)
−0.751670 + 0.659540i \(0.770751\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.50000 7.79423i 0.172949 0.299557i −0.766501 0.642244i \(-0.778004\pi\)
0.939450 + 0.342687i \(0.111337\pi\)
\(678\) 0 0
\(679\) 9.50000 49.3634i 0.364577 1.89440i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.50000 + 2.59808i 0.0573959 + 0.0994126i 0.893296 0.449469i \(-0.148387\pi\)
−0.835900 + 0.548882i \(0.815054\pi\)
\(684\) 0 0
\(685\) −66.0000 −2.52173
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.00000 8.66025i −0.190485 0.329929i
\(690\) 0 0
\(691\) 14.0000 24.2487i 0.532585 0.922464i −0.466691 0.884420i \(-0.654554\pi\)
0.999276 0.0380440i \(-0.0121127\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.0000 + 36.3731i −0.796575 + 1.37971i
\(696\) 0 0
\(697\) −4.00000 6.92820i −0.151511 0.262424i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) 30.0000 + 51.9615i 1.13147 + 1.95977i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.0000 + 5.19615i −0.564133 + 0.195421i
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −30.0000 −1.12194
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.0000 25.9808i 0.559406 0.968919i −0.438141 0.898906i \(-0.644363\pi\)
0.997546 0.0700124i \(-0.0223039\pi\)
\(720\) 0 0
\(721\) 16.0000 + 13.8564i 0.595871 + 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.00000 3.46410i −0.0742781 0.128654i
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.00000 6.92820i −0.147945 0.256249i
\(732\) 0 0
\(733\) 2.00000 3.46410i 0.0738717 0.127950i −0.826723 0.562609i \(-0.809798\pi\)
0.900595 + 0.434659i \(0.143131\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −35.0000 + 60.6218i −1.28924 + 2.23303i
\(738\) 0 0
\(739\) 25.0000 + 43.3013i 0.919640 + 1.59286i 0.799962 + 0.600050i \(0.204853\pi\)
0.119677 + 0.992813i \(0.461814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 0 0
\(745\) −27.0000 46.7654i −0.989203 1.71335i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 26.0000 + 22.5167i 0.950019 + 0.822741i
\(750\) 0 0
\(751\) 7.50000 12.9904i 0.273679 0.474026i −0.696122 0.717923i \(-0.745093\pi\)
0.969801 + 0.243898i \(0.0784261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 33.0000 1.20099
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) −25.0000 + 8.66025i −0.905061 + 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.0000 + 22.5167i 0.469403 + 0.813029i
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.0000 25.9808i −0.539513 0.934463i −0.998930 0.0462427i \(-0.985275\pi\)
0.459418 0.888220i \(-0.348058\pi\)
\(774\) 0 0
\(775\) 2.00000 3.46410i 0.0718421 0.124434i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 20.7846i 0.429945 0.744686i
\(780\) 0 0
\(781\) 30.0000 + 51.9615i 1.07348 + 1.85933i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 2.00000 + 3.46410i 0.0712923 + 0.123482i 0.899468 0.436987i \(-0.143954\pi\)
−0.828176 + 0.560469i \(0.810621\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 + 31.1769i −0.213335 + 1.10852i
\(792\) 0 0
\(793\) 8.00000 13.8564i 0.284088 0.492055i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.0000 0.673015 0.336507 0.941681i \(-0.390754\pi\)
0.336507 + 0.941681i \(0.390754\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −15.0000 + 25.9808i −0.529339 + 0.916841i
\(804\) 0 0
\(805\) 15.0000 5.19615i 0.528681 0.183140i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.00000 + 3.46410i 0.0703163 + 0.121791i 0.899040 0.437867i \(-0.144266\pi\)
−0.828724 + 0.559658i \(0.810932\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.0000 + 36.3731i 0.735598 + 1.27409i
\(816\) 0 0
\(817\) 12.0000 20.7846i 0.419827 0.727161i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.5000 + 35.5070i −0.715455 + 1.23920i 0.247329 + 0.968932i \(0.420447\pi\)
−0.962784 + 0.270273i \(0.912886\pi\)
\(822\) 0 0
\(823\) −16.0000 27.7128i −0.557725 0.966008i −0.997686 0.0679910i \(-0.978341\pi\)
0.439961 0.898017i \(-0.354992\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) 21.0000 + 36.3731i 0.729360 + 1.26329i 0.957154 + 0.289579i \(0.0935154\pi\)
−0.227794 + 0.973709i \(0.573151\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.00000 + 13.8564i 0.0692959 + 0.480096i
\(834\) 0 0
\(835\) −33.0000 + 57.1577i −1.14201 + 1.97802i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 13.5000 23.3827i 0.464414 0.804389i
\(846\) 0 0
\(847\) 28.0000 + 24.2487i 0.962091 + 0.833196i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.0000 + 17.3205i 0.342796 + 0.593739i
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) −20.0000 + 34.6410i −0.682391 + 1.18194i 0.291858 + 0.956462i \(0.405727\pi\)
−0.974249 + 0.225475i \(0.927607\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.00000 + 5.19615i −0.102121 + 0.176879i −0.912558 0.408946i \(-0.865896\pi\)
0.810437 + 0.585826i \(0.199230\pi\)
\(864\) 0 0
\(865\) −27.0000 46.7654i −0.918028 1.59007i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −55.0000 −1.86575
\(870\) 0 0
\(871\) 14.0000 + 24.2487i 0.474372 + 0.821636i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.50000 + 7.79423i −0.0507093 + 0.263493i
\(876\) 0 0
\(877\) 4.00000 6.92820i 0.135070 0.233949i −0.790554 0.612392i \(-0.790207\pi\)
0.925624 + 0.378444i \(0.123541\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −19.0000 + 32.9090i −0.637958 + 1.10497i 0.347923 + 0.937523i \(0.386887\pi\)
−0.985880 + 0.167452i \(0.946446\pi\)
\(888\) 0 0
\(889\) −3.50000 + 18.1865i −0.117386 + 0.609957i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.0000 + 41.5692i 0.803129 + 1.39106i
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.500000 + 0.866025i 0.0166759 + 0.0288836i
\(900\) 0 0
\(901\) −5.00000 + 8.66025i −0.166574 + 0.288515i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19.0000 32.9090i −0.630885 1.09272i −0.987371 0.158424i \(-0.949359\pi\)
0.356487 0.934300i \(-0.383975\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 17.5000 + 30.3109i 0.579165 + 1.00314i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −18.0000 15.5885i −0.594412 0.514776i
\(918\) 0 0
\(919\) −22.0000 + 38.1051i −0.725713 + 1.25697i 0.232967 + 0.972485i \(0.425157\pi\)
−0.958680 + 0.284487i \(0.908177\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 40.0000 1.31519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) −33.0000 + 25.9808i −1.08153 + 0.851485i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 15.0000 + 25.9808i 0.490552 + 0.849662i
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15.5000 26.8468i −0.505286 0.875180i −0.999981 0.00611403i \(-0.998054\pi\)
0.494696 0.869066i \(-0.335280\pi\)
\(942\) 0 0
\(943\) 4.00000 6.92820i 0.130258 0.225613i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 + 24.2487i −0.454939 + 0.787977i −0.998685 0.0512727i \(-0.983672\pi\)
0.543746 + 0.839250i \(0.317006\pi\)
\(948\) 0 0
\(949\) 6.00000 + 10.3923i 0.194768 + 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.00000 0.129573 0.0647864 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) 0 0
\(955\) −18.0000 31.1769i −0.582466 1.00886i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 55.0000 19.0526i 1.77604 0.615239i
\(960\) 0 0
\(961\) 15.0000 25.9808i 0.483871 0.838089i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 51.0000 1.64175
\(966\) 0 0
\(967\) 19.0000 0.610999 0.305499 0.952192i \(-0.401177\pi\)
0.305499 + 0.952192i \(0.401177\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.50000 12.9904i 0.240686 0.416881i −0.720224 0.693742i \(-0.755961\pi\)
0.960910 + 0.276861i \(0.0892941\pi\)
\(972\) 0 0
\(973\) 7.00000 36.3731i 0.224410 1.16607i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26.0000 45.0333i −0.831814 1.44074i −0.896599 0.442844i \(-0.853969\pi\)
0.0647848 0.997899i \(-0.479364\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.00000 10.3923i −0.191370 0.331463i 0.754334 0.656490i \(-0.227960\pi\)
−0.945705 + 0.325027i \(0.894626\pi\)
\(984\) 0 0
\(985\) −9.00000 + 15.5885i −0.286764 + 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 6.92820i 0.127193 0.220304i
\(990\) 0 0
\(991\) 18.5000 + 32.0429i 0.587672 + 1.01788i 0.994537 + 0.104389i \(0.0332887\pi\)
−0.406865 + 0.913488i \(0.633378\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) 0 0
\(997\) −19.0000 32.9090i −0.601736 1.04224i −0.992558 0.121771i \(-0.961143\pi\)
0.390822 0.920466i \(-0.372191\pi\)
\(998\) 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 2016.2.s.d.289.1 yes 2
3.2 odd 2 2016.2.s.m.289.1 yes 2
4.3 odd 2 2016.2.s.c.289.1 2
7.4 even 3 inner 2016.2.s.d.865.1 yes 2
12.11 even 2 2016.2.s.l.289.1 yes 2
21.11 odd 6 2016.2.s.m.865.1 yes 2
28.11 odd 6 2016.2.s.c.865.1 yes 2
84.11 even 6 2016.2.s.l.865.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.s.c.289.1 2 4.3 odd 2
2016.2.s.c.865.1 yes 2 28.11 odd 6
2016.2.s.d.289.1 yes 2 1.1 even 1 trivial
2016.2.s.d.865.1 yes 2 7.4 even 3 inner
2016.2.s.l.289.1 yes 2 12.11 even 2
2016.2.s.l.865.1 yes 2 84.11 even 6
2016.2.s.m.289.1 yes 2 3.2 odd 2
2016.2.s.m.865.1 yes 2 21.11 odd 6