Properties

Label 2016.2.s.r.289.1
Level $2016$
Weight $2$
Character 2016.289
Analytic conductor $16.098$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +(-2.59808 - 4.50000i) q^{11} +(2.50000 + 4.33013i) q^{17} +(0.866025 - 1.50000i) q^{19} +(0.866025 - 1.50000i) q^{23} +(2.00000 + 3.46410i) q^{25} -8.00000 q^{29} +(-4.33013 - 7.50000i) q^{31} +(0.866025 + 2.50000i) q^{35} +(2.50000 - 4.33013i) q^{37} -4.00000 q^{41} -6.92820 q^{43} +(-4.33013 + 7.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-0.500000 - 0.866025i) q^{53} -5.19615 q^{55} +(-0.866025 - 1.50000i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(-6.06218 - 10.5000i) q^{67} -13.8564 q^{71} +(-7.50000 - 12.9904i) q^{73} +(13.5000 + 2.59808i) q^{77} +(-0.866025 + 1.50000i) q^{79} +6.92820 q^{83} +5.00000 q^{85} +(3.50000 - 6.06218i) q^{89} +(-0.866025 - 1.50000i) q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 10 q^{17} + 8 q^{25} - 32 q^{29} + 10 q^{37} - 16 q^{41} - 4 q^{49} - 2 q^{53} - 22 q^{61} - 30 q^{73} + 54 q^{77} + 20 q^{85} + 14 q^{89} + 48 q^{97}+O(q^{100}) \) 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
\(5\) 0.500000 0.866025i 0.223607 0.387298i −0.732294 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −1.73205 + 2.00000i −0.654654 + 0.755929i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.59808 4.50000i −0.783349 1.35680i −0.929980 0.367610i \(-0.880176\pi\)
0.146631 0.989191i \(-0.453157\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.50000 + 4.33013i 0.606339 + 1.05021i 0.991838 + 0.127502i \(0.0406959\pi\)
−0.385499 + 0.922708i \(0.625971\pi\)
\(18\) 0 0
\(19\) 0.866025 1.50000i 0.198680 0.344124i −0.749421 0.662094i \(-0.769668\pi\)
0.948101 + 0.317970i \(0.103001\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.866025 1.50000i 0.180579 0.312772i −0.761499 0.648166i \(-0.775536\pi\)
0.942078 + 0.335394i \(0.108870\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −4.33013 7.50000i −0.777714 1.34704i −0.933257 0.359211i \(-0.883046\pi\)
0.155543 0.987829i \(-0.450287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.866025 + 2.50000i 0.146385 + 0.422577i
\(36\) 0 0
\(37\) 2.50000 4.33013i 0.410997 0.711868i −0.584002 0.811752i \(-0.698514\pi\)
0.994999 + 0.0998840i \(0.0318472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.33013 + 7.50000i −0.631614 + 1.09399i 0.355608 + 0.934635i \(0.384274\pi\)
−0.987222 + 0.159352i \(0.949059\pi\)
\(48\) 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 0.829640 0.558298i \(-0.188546\pi\)
−0.898321 + 0.439340i \(0.855212\pi\)
\(54\) 0 0
\(55\) −5.19615 −0.700649
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.866025 1.50000i −0.112747 0.195283i 0.804130 0.594454i \(-0.202632\pi\)
−0.916877 + 0.399170i \(0.869298\pi\)
\(60\) 0 0
\(61\) −5.50000 + 9.52628i −0.704203 + 1.21972i 0.262776 + 0.964857i \(0.415362\pi\)
−0.966978 + 0.254858i \(0.917971\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.06218 10.5000i −0.740613 1.28278i −0.952217 0.305424i \(-0.901202\pi\)
0.211604 0.977356i \(-0.432131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) −7.50000 12.9904i −0.877809 1.52041i −0.853740 0.520699i \(-0.825671\pi\)
−0.0240681 0.999710i \(-0.507662\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.5000 + 2.59808i 1.53847 + 0.296078i
\(78\) 0 0
\(79\) −0.866025 + 1.50000i −0.0974355 + 0.168763i −0.910622 0.413239i \(-0.864397\pi\)
0.813187 + 0.582003i \(0.197731\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 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 −0.618720 0.785611i \(-0.712349\pi\)
0.989720 + 0.143022i \(0.0456819\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.866025 1.50000i −0.0888523 0.153897i
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.50000 9.52628i −0.547270 0.947900i −0.998460 0.0554722i \(-0.982334\pi\)
0.451190 0.892428i \(-0.351000\pi\)
\(102\) 0 0
\(103\) −7.79423 + 13.5000i −0.767988 + 1.33019i 0.170664 + 0.985329i \(0.445409\pi\)
−0.938652 + 0.344865i \(0.887925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.06218 10.5000i 0.586053 1.01507i −0.408690 0.912673i \(-0.634014\pi\)
0.994743 0.102400i \(-0.0326522\pi\)
\(108\) 0 0
\(109\) 1.50000 + 2.59808i 0.143674 + 0.248851i 0.928877 0.370387i \(-0.120775\pi\)
−0.785203 + 0.619238i \(0.787442\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −0.866025 1.50000i −0.0807573 0.139876i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.9904 2.50000i −1.19083 0.229175i
\(120\) 0 0
\(121\) −8.00000 + 13.8564i −0.727273 + 1.25967i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.33013 7.50000i 0.378325 0.655278i −0.612494 0.790475i \(-0.709834\pi\)
0.990819 + 0.135197i \(0.0431669\pi\)
\(132\) 0 0
\(133\) 1.50000 + 4.33013i 0.130066 + 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.50000 + 4.33013i 0.213589 + 0.369948i 0.952835 0.303488i \(-0.0981512\pi\)
−0.739246 + 0.673436i \(0.764818\pi\)
\(138\) 0 0
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 + 6.92820i −0.332182 + 0.575356i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.50000 + 6.06218i −0.286731 + 0.496633i −0.973028 0.230689i \(-0.925902\pi\)
0.686296 + 0.727322i \(0.259235\pi\)
\(150\) 0 0
\(151\) −4.33013 7.50000i −0.352381 0.610341i 0.634285 0.773099i \(-0.281294\pi\)
−0.986666 + 0.162758i \(0.947961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.66025 −0.695608
\(156\) 0 0
\(157\) −3.50000 6.06218i −0.279330 0.483814i 0.691888 0.722005i \(-0.256779\pi\)
−0.971219 + 0.238190i \(0.923446\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.50000 + 4.33013i 0.118217 + 0.341262i
\(162\) 0 0
\(163\) −4.33013 + 7.50000i −0.339162 + 0.587445i −0.984275 0.176641i \(-0.943477\pi\)
0.645114 + 0.764087i \(0.276810\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.50000 + 4.33013i −0.190071 + 0.329213i −0.945274 0.326278i \(-0.894205\pi\)
0.755202 + 0.655492i \(0.227539\pi\)
\(174\) 0 0
\(175\) −10.3923 2.00000i −0.785584 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.33013 + 7.50000i 0.323649 + 0.560576i 0.981238 0.192800i \(-0.0617570\pi\)
−0.657589 + 0.753377i \(0.728424\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.50000 4.33013i −0.183804 0.318357i
\(186\) 0 0
\(187\) 12.9904 22.5000i 0.949951 1.64536i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.2583 + 19.5000i −0.814624 + 1.41097i 0.0949733 + 0.995480i \(0.469723\pi\)
−0.909597 + 0.415491i \(0.863610\pi\)
\(192\) 0 0
\(193\) 5.50000 + 9.52628i 0.395899 + 0.685717i 0.993215 0.116289i \(-0.0370998\pi\)
−0.597317 + 0.802005i \(0.703766\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −4.33013 7.50000i −0.306955 0.531661i 0.670740 0.741693i \(-0.265977\pi\)
−0.977695 + 0.210032i \(0.932643\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.8564 16.0000i 0.972529 1.12298i
\(204\) 0 0
\(205\) −2.00000 + 3.46410i −0.139686 + 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) 20.7846 1.43087 0.715436 0.698679i \(-0.246228\pi\)
0.715436 + 0.698679i \(0.246228\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.46410 + 6.00000i −0.236250 + 0.409197i
\(216\) 0 0
\(217\) 22.5000 + 4.33013i 1.52740 + 0.293948i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −13.8564 −0.927894 −0.463947 0.885863i \(-0.653567\pi\)
−0.463947 + 0.885863i \(0.653567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.33013 + 7.50000i 0.287401 + 0.497792i 0.973189 0.230009i \(-0.0738756\pi\)
−0.685788 + 0.727801i \(0.740542\pi\)
\(228\) 0 0
\(229\) 7.50000 12.9904i 0.495614 0.858429i −0.504373 0.863486i \(-0.668276\pi\)
0.999987 + 0.00505719i \(0.00160976\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.50000 4.33013i 0.163780 0.283676i −0.772441 0.635086i \(-0.780964\pi\)
0.936222 + 0.351410i \(0.114298\pi\)
\(234\) 0 0
\(235\) 4.33013 + 7.50000i 0.282466 + 0.489246i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.3205 1.12037 0.560185 0.828367i \(-0.310730\pi\)
0.560185 + 0.828367i \(0.310730\pi\)
\(240\) 0 0
\(241\) −7.50000 12.9904i −0.483117 0.836784i 0.516695 0.856170i \(-0.327162\pi\)
−0.999812 + 0.0193858i \(0.993829\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.50000 2.59808i −0.415270 0.165985i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.50000 4.33013i 0.155946 0.270106i −0.777457 0.628936i \(-0.783491\pi\)
0.933403 + 0.358830i \(0.116824\pi\)
\(258\) 0 0
\(259\) 4.33013 + 12.5000i 0.269061 + 0.776712i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.866025 1.50000i −0.0534014 0.0924940i 0.838089 0.545534i \(-0.183673\pi\)
−0.891490 + 0.453040i \(0.850340\pi\)
\(264\) 0 0
\(265\) −1.00000 −0.0614295
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.50000 14.7224i −0.518254 0.897643i −0.999775 0.0212079i \(-0.993249\pi\)
0.481521 0.876435i \(-0.340085\pi\)
\(270\) 0 0
\(271\) 6.06218 10.5000i 0.368251 0.637830i −0.621041 0.783778i \(-0.713290\pi\)
0.989292 + 0.145948i \(0.0466233\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3923 18.0000i 0.626680 1.08544i
\(276\) 0 0
\(277\) −7.50000 12.9904i −0.450631 0.780516i 0.547794 0.836613i \(-0.315468\pi\)
−0.998425 + 0.0560969i \(0.982134\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) −4.33013 7.50000i −0.257399 0.445829i 0.708145 0.706067i \(-0.249532\pi\)
−0.965544 + 0.260238i \(0.916199\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.92820 8.00000i 0.408959 0.472225i
\(288\) 0 0
\(289\) −4.00000 + 6.92820i −0.235294 + 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −1.73205 −0.100844
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 13.8564i 0.691669 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.50000 + 9.52628i 0.314929 + 0.545473i
\(306\) 0 0
\(307\) 20.7846 1.18624 0.593120 0.805114i \(-0.297896\pi\)
0.593120 + 0.805114i \(0.297896\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.06218 + 10.5000i 0.343755 + 0.595400i 0.985127 0.171830i \(-0.0549678\pi\)
−0.641372 + 0.767230i \(0.721635\pi\)
\(312\) 0 0
\(313\) 12.5000 21.6506i 0.706542 1.22377i −0.259590 0.965719i \(-0.583588\pi\)
0.966132 0.258047i \(-0.0830791\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.50000 + 4.33013i −0.140414 + 0.243204i −0.927653 0.373444i \(-0.878177\pi\)
0.787239 + 0.616649i \(0.211510\pi\)
\(318\) 0 0
\(319\) 20.7846 + 36.0000i 1.16371 + 2.01561i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.66025 0.481869
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.50000 21.6506i −0.413488 1.19364i
\(330\) 0 0
\(331\) −4.33013 + 7.50000i −0.238005 + 0.412237i −0.960142 0.279513i \(-0.909827\pi\)
0.722137 + 0.691751i \(0.243160\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.1244 −0.662424
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −22.5000 + 38.9711i −1.21844 + 2.11041i
\(342\) 0 0
\(343\) 15.5885 + 10.0000i 0.841698 + 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.06218 + 10.5000i 0.325435 + 0.563670i 0.981600 0.190947i \(-0.0611560\pi\)
−0.656165 + 0.754617i \(0.727823\pi\)
\(348\) 0 0
\(349\) 32.0000 1.71292 0.856460 0.516213i \(-0.172659\pi\)
0.856460 + 0.516213i \(0.172659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.50000 9.52628i −0.292735 0.507033i 0.681720 0.731613i \(-0.261232\pi\)
−0.974456 + 0.224580i \(0.927899\pi\)
\(354\) 0 0
\(355\) −6.92820 + 12.0000i −0.367711 + 0.636894i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.33013 + 7.50000i −0.228535 + 0.395835i −0.957374 0.288850i \(-0.906727\pi\)
0.728839 + 0.684685i \(0.240060\pi\)
\(360\) 0 0
\(361\) 8.00000 + 13.8564i 0.421053 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) 0 0
\(367\) 14.7224 + 25.5000i 0.768505 + 1.33109i 0.938374 + 0.345623i \(0.112332\pi\)
−0.169869 + 0.985467i \(0.554334\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.59808 + 0.500000i 0.134885 + 0.0259587i
\(372\) 0 0
\(373\) −17.5000 + 30.3109i −0.906116 + 1.56944i −0.0867031 + 0.996234i \(0.527633\pi\)
−0.819413 + 0.573204i \(0.805700\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.3923 0.533817 0.266908 0.963722i \(-0.413998\pi\)
0.266908 + 0.963722i \(0.413998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.52628 16.5000i 0.486770 0.843111i −0.513114 0.858320i \(-0.671508\pi\)
0.999884 + 0.0152097i \(0.00484160\pi\)
\(384\) 0 0
\(385\) 9.00000 10.3923i 0.458682 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.50000 + 4.33013i 0.126755 + 0.219546i 0.922418 0.386194i \(-0.126210\pi\)
−0.795663 + 0.605740i \(0.792877\pi\)
\(390\) 0 0
\(391\) 8.66025 0.437968
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.866025 + 1.50000i 0.0435745 + 0.0754732i
\(396\) 0 0
\(397\) −8.50000 + 14.7224i −0.426603 + 0.738898i −0.996569 0.0827707i \(-0.973623\pi\)
0.569966 + 0.821668i \(0.306956\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.50000 + 16.4545i −0.474407 + 0.821698i −0.999571 0.0293039i \(-0.990671\pi\)
0.525163 + 0.851002i \(0.324004\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.9808 −1.28782
\(408\) 0 0
\(409\) 1.50000 + 2.59808i 0.0741702 + 0.128467i 0.900725 0.434389i \(-0.143036\pi\)
−0.826555 + 0.562856i \(0.809703\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.50000 + 0.866025i 0.221431 + 0.0426143i
\(414\) 0 0
\(415\) 3.46410 6.00000i 0.170046 0.294528i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 34.6410 1.69232 0.846162 0.532925i \(-0.178907\pi\)
0.846162 + 0.532925i \(0.178907\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −10.0000 + 17.3205i −0.485071 + 0.840168i
\(426\) 0 0
\(427\) −9.52628 27.5000i −0.461009 1.33082i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.2583 + 19.5000i 0.542295 + 0.939282i 0.998772 + 0.0495468i \(0.0157777\pi\)
−0.456477 + 0.889735i \(0.650889\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.50000 2.59808i −0.0717547 0.124283i
\(438\) 0 0
\(439\) 4.33013 7.50000i 0.206666 0.357955i −0.743996 0.668184i \(-0.767072\pi\)
0.950662 + 0.310228i \(0.100405\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.79423 + 13.5000i −0.370315 + 0.641404i −0.989614 0.143751i \(-0.954084\pi\)
0.619299 + 0.785155i \(0.287417\pi\)
\(444\) 0 0
\(445\) −3.50000 6.06218i −0.165916 0.287375i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.0000 −1.32140 −0.660701 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) 10.3923 + 18.0000i 0.489355 + 0.847587i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.50000 + 12.9904i −0.350835 + 0.607664i −0.986396 0.164386i \(-0.947436\pi\)
0.635561 + 0.772051i \(0.280769\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) 27.7128 1.28792 0.643962 0.765058i \(-0.277290\pi\)
0.643962 + 0.765058i \(0.277290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.59808 + 4.50000i −0.120225 + 0.208235i −0.919856 0.392256i \(-0.871695\pi\)
0.799632 + 0.600491i \(0.205028\pi\)
\(468\) 0 0
\(469\) 31.5000 + 6.06218i 1.45453 + 0.279925i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.0000 + 31.1769i 0.827641 + 1.43352i
\(474\) 0 0
\(475\) 6.92820 0.317888
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.9904 + 22.5000i 0.593546 + 1.02805i 0.993750 + 0.111625i \(0.0356057\pi\)
−0.400205 + 0.916426i \(0.631061\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 10.3923i 0.272446 0.471890i
\(486\) 0 0
\(487\) −12.9904 22.5000i −0.588650 1.01957i −0.994410 0.105592i \(-0.966326\pi\)
0.405759 0.913980i \(-0.367007\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.3205 0.781664 0.390832 0.920462i \(-0.372187\pi\)
0.390832 + 0.920462i \(0.372187\pi\)
\(492\) 0 0
\(493\) −20.0000 34.6410i −0.900755 1.56015i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 27.7128i 1.07655 1.24309i
\(498\) 0 0
\(499\) 9.52628 16.5000i 0.426455 0.738641i −0.570100 0.821575i \(-0.693096\pi\)
0.996555 + 0.0829337i \(0.0264290\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.7128 1.23565 0.617827 0.786314i \(-0.288013\pi\)
0.617827 + 0.786314i \(0.288013\pi\)
\(504\) 0 0
\(505\) −11.0000 −0.489494
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.50000 14.7224i 0.376756 0.652560i −0.613832 0.789436i \(-0.710373\pi\)
0.990588 + 0.136876i \(0.0437062\pi\)
\(510\) 0 0
\(511\) 38.9711 + 7.50000i 1.72398 + 0.331780i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.79423 + 13.5000i 0.343455 + 0.594881i
\(516\) 0 0
\(517\) 45.0000 1.97910
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.5000 21.6506i −0.547635 0.948532i −0.998436 0.0559071i \(-0.982195\pi\)
0.450801 0.892624i \(-0.351138\pi\)
\(522\) 0 0
\(523\) −12.9904 + 22.5000i −0.568030 + 0.983856i 0.428731 + 0.903432i \(0.358961\pi\)
−0.996761 + 0.0804241i \(0.974373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.6506 37.5000i 0.943116 1.63353i
\(528\) 0 0
\(529\) 10.0000 + 17.3205i 0.434783 + 0.753066i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.06218 10.5000i −0.262091 0.453955i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −28.5788 + 22.5000i −1.23098 + 0.969144i
\(540\) 0 0
\(541\) −4.50000 + 7.79423i −0.193470 + 0.335100i −0.946398 0.323003i \(-0.895308\pi\)
0.752928 + 0.658103i \(0.228641\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 3.46410 0.148114 0.0740571 0.997254i \(-0.476405\pi\)
0.0740571 + 0.997254i \(0.476405\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.92820 + 12.0000i −0.295151 + 0.511217i
\(552\) 0 0
\(553\) −1.50000 4.33013i −0.0637865 0.184136i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21.5000 37.2391i −0.910984 1.57787i −0.812677 0.582715i \(-0.801990\pi\)
−0.0983076 0.995156i \(-0.531343\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.79423 13.5000i −0.328488 0.568957i 0.653724 0.756733i \(-0.273206\pi\)
−0.982212 + 0.187776i \(0.939872\pi\)
\(564\) 0 0
\(565\) −2.00000 + 3.46410i −0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.5000 + 21.6506i −0.524027 + 0.907642i 0.475581 + 0.879672i \(0.342238\pi\)
−0.999609 + 0.0279702i \(0.991096\pi\)
\(570\) 0 0
\(571\) 2.59808 + 4.50000i 0.108726 + 0.188319i 0.915254 0.402876i \(-0.131990\pi\)
−0.806528 + 0.591195i \(0.798656\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.92820 0.288926
\(576\) 0 0
\(577\) −11.5000 19.9186i −0.478751 0.829222i 0.520952 0.853586i \(-0.325577\pi\)
−0.999703 + 0.0243645i \(0.992244\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 + 13.8564i −0.497844 + 0.574861i
\(582\) 0 0
\(583\) −2.59808 + 4.50000i −0.107601 + 0.186371i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −34.6410 −1.42979 −0.714894 0.699233i \(-0.753525\pi\)
−0.714894 + 0.699233i \(0.753525\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14.5000 25.1147i 0.595444 1.03134i −0.398040 0.917368i \(-0.630310\pi\)
0.993484 0.113971i \(-0.0363570\pi\)
\(594\) 0 0
\(595\) −8.66025 + 10.0000i −0.355036 + 0.409960i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −21.6506 37.5000i −0.884621 1.53221i −0.846147 0.532949i \(-0.821084\pi\)
−0.0384735 0.999260i \(-0.512250\pi\)
\(600\) 0 0
\(601\) −20.0000 −0.815817 −0.407909 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.00000 + 13.8564i 0.325246 + 0.563343i
\(606\) 0 0
\(607\) −2.59808 + 4.50000i −0.105453 + 0.182649i −0.913923 0.405887i \(-0.866962\pi\)
0.808470 + 0.588537i \(0.200296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.50000 4.33013i −0.100974 0.174892i 0.811112 0.584891i \(-0.198863\pi\)
−0.912086 + 0.409998i \(0.865529\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.0000 0.805170 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 0 0
\(619\) 7.79423 + 13.5000i 0.313276 + 0.542611i 0.979070 0.203526i \(-0.0652400\pi\)
−0.665793 + 0.746136i \(0.731907\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.06218 + 17.5000i 0.242876 + 0.701123i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.0000 0.996815
\(630\) 0 0
\(631\) −13.8564 −0.551615 −0.275807 0.961213i \(-0.588945\pi\)
−0.275807 + 0.961213i \(0.588945\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.500000 0.866025i −0.0197488 0.0342059i 0.855982 0.517005i \(-0.172953\pi\)
−0.875731 + 0.482800i \(0.839620\pi\)
\(642\) 0 0
\(643\) 34.6410 1.36611 0.683054 0.730368i \(-0.260651\pi\)
0.683054 + 0.730368i \(0.260651\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.7224 25.5000i −0.578799 1.00251i −0.995618 0.0935189i \(-0.970188\pi\)
0.416819 0.908990i \(-0.363145\pi\)
\(648\) 0 0
\(649\) −4.50000 + 7.79423i −0.176640 + 0.305950i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.50000 + 4.33013i −0.0978326 + 0.169451i −0.910787 0.412876i \(-0.864524\pi\)
0.812955 + 0.582327i \(0.197858\pi\)
\(654\) 0 0
\(655\) −4.33013 7.50000i −0.169192 0.293049i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.92820 −0.269884 −0.134942 0.990853i \(-0.543085\pi\)
−0.134942 + 0.990853i \(0.543085\pi\)
\(660\) 0 0
\(661\) −15.5000 26.8468i −0.602880 1.04422i −0.992383 0.123194i \(-0.960686\pi\)
0.389503 0.921025i \(-0.372647\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.50000 + 0.866025i 0.174503 + 0.0335830i
\(666\) 0 0
\(667\) −6.92820 + 12.0000i −0.268261 + 0.464642i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 57.1577 2.20655
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.5000 21.6506i 0.480414 0.832102i −0.519333 0.854572i \(-0.673820\pi\)
0.999748 + 0.0224702i \(0.00715308\pi\)
\(678\) 0 0
\(679\) −20.7846 + 24.0000i −0.797640 + 0.921035i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.4545 28.5000i −0.629613 1.09052i −0.987629 0.156807i \(-0.949880\pi\)
0.358016 0.933715i \(-0.383453\pi\)
\(684\) 0 0
\(685\) 5.00000 0.191040
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2.59808 4.50000i 0.0988355 0.171188i −0.812367 0.583146i \(-0.801822\pi\)
0.911203 + 0.411958i \(0.135155\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.46410 + 6.00000i −0.131401 + 0.227593i
\(696\) 0 0
\(697\) −10.0000 17.3205i −0.378777 0.656061i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) −4.33013 7.50000i −0.163314 0.282868i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.5788 + 5.50000i 1.07482 + 0.206849i
\(708\) 0 0
\(709\) −13.5000 + 23.3827i −0.507003 + 0.878155i 0.492964 + 0.870050i \(0.335913\pi\)
−0.999967 + 0.00810550i \(0.997420\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.0000 −0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.79423 13.5000i 0.290676 0.503465i −0.683294 0.730143i \(-0.739453\pi\)
0.973970 + 0.226678i \(0.0727866\pi\)
\(720\) 0 0
\(721\) −13.5000 38.9711i −0.502766 1.45136i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −16.0000 27.7128i −0.594225 1.02923i
\(726\) 0 0
\(727\) 13.8564 0.513906 0.256953 0.966424i \(-0.417281\pi\)
0.256953 + 0.966424i \(0.417281\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.3205 30.0000i −0.640622 1.10959i
\(732\) 0 0
\(733\) −25.5000 + 44.1673i −0.941864 + 1.63136i −0.179952 + 0.983675i \(0.557594\pi\)
−0.761912 + 0.647681i \(0.775739\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31.5000 + 54.5596i −1.16032 + 2.00973i
\(738\) 0 0
\(739\) 9.52628 + 16.5000i 0.350430 + 0.606962i 0.986325 0.164813i \(-0.0527022\pi\)
−0.635895 + 0.771776i \(0.719369\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.7128 1.01668 0.508342 0.861155i \(-0.330258\pi\)
0.508342 + 0.861155i \(0.330258\pi\)
\(744\) 0 0
\(745\) 3.50000 + 6.06218i 0.128230 + 0.222101i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.5000 + 30.3109i 0.383662 + 1.10754i
\(750\) 0 0
\(751\) 4.33013 7.50000i 0.158009 0.273679i −0.776142 0.630558i \(-0.782826\pi\)
0.934150 + 0.356879i \(0.116159\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.66025 −0.315179
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.50000 11.2583i 0.235625 0.408114i −0.723829 0.689979i \(-0.757620\pi\)
0.959454 + 0.281865i \(0.0909530\pi\)
\(762\) 0 0
\(763\) −7.79423 1.50000i −0.282170 0.0543036i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.50000 + 4.33013i 0.0899188 + 0.155744i 0.907477 0.420103i \(-0.138006\pi\)
−0.817558 + 0.575846i \(0.804673\pi\)
\(774\) 0 0
\(775\) 17.3205 30.0000i 0.622171 1.07763i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.46410 + 6.00000i −0.124114 + 0.214972i
\(780\) 0 0
\(781\) 36.0000 + 62.3538i 1.28818 + 2.23120i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) −19.9186 34.5000i −0.710021 1.22979i −0.964849 0.262806i \(-0.915352\pi\)
0.254828 0.966986i \(-0.417981\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.92820 8.00000i 0.246339 0.284447i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −40.0000 −1.41687 −0.708436 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 0 0
\(799\) −43.3013 −1.53189
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −38.9711 + 67.5000i −1.37526 + 2.38202i
\(804\) 0 0
\(805\) 4.50000 + 0.866025i 0.158604 + 0.0305234i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.50000 + 6.06218i 0.123053 + 0.213135i 0.920970 0.389633i \(-0.127398\pi\)
−0.797917 + 0.602767i \(0.794065\pi\)
\(810\) 0 0
\(811\) −34.6410 −1.21641 −0.608205 0.793780i \(-0.708110\pi\)
−0.608205 + 0.793780i \(0.708110\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.33013 + 7.50000i 0.151678 + 0.262714i
\(816\) 0 0
\(817\) −6.00000 + 10.3923i −0.209913 + 0.363581i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.5000 21.6506i 0.436253 0.755612i −0.561144 0.827718i \(-0.689639\pi\)
0.997397 + 0.0721058i \(0.0229719\pi\)
\(822\) 0 0
\(823\) 7.79423 + 13.5000i 0.271690 + 0.470580i 0.969295 0.245902i \(-0.0790842\pi\)
−0.697605 + 0.716483i \(0.745751\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.7846 −0.722752 −0.361376 0.932420i \(-0.617693\pi\)
−0.361376 + 0.932420i \(0.617693\pi\)
\(828\) 0 0
\(829\) −22.5000 38.9711i −0.781457 1.35352i −0.931093 0.364783i \(-0.881143\pi\)
0.149635 0.988741i \(-0.452190\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 27.5000 21.6506i 0.952819 0.750150i
\(834\) 0 0
\(835\) −1.73205 + 3.00000i −0.0599401 + 0.103819i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −41.5692 −1.43513 −0.717564 0.696492i \(-0.754743\pi\)
−0.717564 + 0.696492i \(0.754743\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.50000 + 11.2583i −0.223607 + 0.387298i
\(846\) 0 0
\(847\) −13.8564 40.0000i −0.476112 1.37442i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.33013 7.50000i −0.148435 0.257097i
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.50000 + 6.06218i 0.119558 + 0.207080i 0.919592 0.392874i \(-0.128519\pi\)
−0.800035 + 0.599954i \(0.795186\pi\)
\(858\) 0 0
\(859\) −12.9904 + 22.5000i −0.443226 + 0.767690i −0.997927 0.0643599i \(-0.979499\pi\)
0.554701 + 0.832050i \(0.312833\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.8468 + 46.5000i −0.913875 + 1.58288i −0.105336 + 0.994437i \(0.533592\pi\)
−0.808540 + 0.588442i \(0.799742\pi\)
\(864\) 0 0
\(865\) 2.50000 + 4.33013i 0.0850026 + 0.147229i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −15.5885 + 18.0000i −0.526986 + 0.608511i
\(876\) 0 0
\(877\) 3.50000 6.06218i 0.118187 0.204705i −0.800862 0.598848i \(-0.795625\pi\)
0.919049 + 0.394143i \(0.128959\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −34.6410 −1.16576 −0.582882 0.812557i \(-0.698075\pi\)
−0.582882 + 0.812557i \(0.698075\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.3827 40.5000i 0.785114 1.35986i −0.143817 0.989604i \(-0.545938\pi\)
0.928931 0.370253i \(-0.120729\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.50000 + 12.9904i 0.250978 + 0.434707i
\(894\) 0 0
\(895\) 8.66025 0.289480
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 34.6410 + 60.0000i 1.15534 + 2.00111i
\(900\) 0 0
\(901\) 2.50000 4.33013i 0.0832871 0.144257i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.00000 + 5.19615i −0.0997234 + 0.172726i
\(906\) 0 0
\(907\) −19.9186 34.5000i −0.661386 1.14555i −0.980252 0.197754i \(-0.936635\pi\)
0.318866 0.947800i \(-0.396698\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 13.8564 0.459083 0.229542 0.973299i \(-0.426277\pi\)
0.229542 + 0.973299i \(0.426277\pi\)
\(912\) 0 0
\(913\) −18.0000 31.1769i −0.595713 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.50000 + 21.6506i 0.247672 + 0.714967i
\(918\) 0 0
\(919\) 18.1865 31.5000i 0.599918 1.03909i −0.392914 0.919575i \(-0.628533\pi\)
0.992833 0.119514i \(-0.0381336\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23.5000 40.7032i 0.771010 1.33543i −0.166000 0.986126i \(-0.553085\pi\)
0.937010 0.349303i \(-0.113582\pi\)
\(930\) 0 0
\(931\) −11.2583 4.50000i −0.368977 0.147482i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.9904 22.5000i −0.424831 0.735829i
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.5000 21.6506i −0.407488 0.705791i 0.587119 0.809500i \(-0.300262\pi\)
−0.994608 + 0.103710i \(0.966929\pi\)
\(942\) 0 0
\(943\) −3.46410 + 6.00000i −0.112807 + 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.59808 + 4.50000i −0.0844261 + 0.146230i −0.905147 0.425100i \(-0.860239\pi\)
0.820720 + 0.571330i \(0.193572\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −20.0000 −0.647864 −0.323932 0.946080i \(-0.605005\pi\)
−0.323932 + 0.946080i \(0.605005\pi\)
\(954\) 0 0
\(955\) 11.2583 + 19.5000i 0.364311 + 0.631005i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.9904 2.50000i −0.419481 0.0807292i
\(960\) 0 0
\(961\) −22.0000 + 38.1051i −0.709677 + 1.22920i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) −13.8564 −0.445592 −0.222796 0.974865i \(-0.571518\pi\)
−0.222796 + 0.974865i \(0.571518\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.33013 7.50000i 0.138960 0.240686i −0.788143 0.615492i \(-0.788957\pi\)
0.927103 + 0.374806i \(0.122291\pi\)
\(972\) 0 0
\(973\) 12.0000 13.8564i 0.384702 0.444216i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.5000 + 32.0429i 0.591867 + 1.02514i 0.993981 + 0.109555i \(0.0349424\pi\)
−0.402113 + 0.915590i \(0.631724\pi\)
\(978\) 0 0
\(979\) −36.3731 −1.16249
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.4545 28.5000i −0.524816 0.909009i −0.999582 0.0288967i \(-0.990801\pi\)
0.474766 0.880112i \(-0.342533\pi\)
\(984\) 0 0
\(985\) 4.00000 6.92820i 0.127451 0.220751i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 + 10.3923i −0.190789 + 0.330456i
\(990\) 0 0
\(991\) −12.9904 22.5000i −0.412653 0.714736i 0.582526 0.812812i \(-0.302064\pi\)
−0.995179 + 0.0980761i \(0.968731\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.66025 −0.274549
\(996\) 0 0
\(997\) −11.5000 19.9186i −0.364209 0.630828i 0.624440 0.781073i \(-0.285327\pi\)
−0.988649 + 0.150245i \(0.951994\pi\)
\(998\) 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.r.289.1 4
3.2 odd 2 224.2.i.b.65.2 yes 4
4.3 odd 2 inner 2016.2.s.r.289.2 4
7.4 even 3 inner 2016.2.s.r.865.1 4
12.11 even 2 224.2.i.b.65.1 4
21.2 odd 6 1568.2.a.s.1.1 2
21.5 even 6 1568.2.a.n.1.2 2
21.11 odd 6 224.2.i.b.193.2 yes 4
21.17 even 6 1568.2.i.u.1537.1 4
21.20 even 2 1568.2.i.u.961.1 4
24.5 odd 2 448.2.i.i.65.1 4
24.11 even 2 448.2.i.i.65.2 4
28.11 odd 6 inner 2016.2.s.r.865.2 4
84.11 even 6 224.2.i.b.193.1 yes 4
84.23 even 6 1568.2.a.s.1.2 2
84.47 odd 6 1568.2.a.n.1.1 2
84.59 odd 6 1568.2.i.u.1537.2 4
84.83 odd 2 1568.2.i.u.961.2 4
168.5 even 6 3136.2.a.bu.1.1 2
168.11 even 6 448.2.i.i.193.2 4
168.53 odd 6 448.2.i.i.193.1 4
168.107 even 6 3136.2.a.bh.1.1 2
168.131 odd 6 3136.2.a.bu.1.2 2
168.149 odd 6 3136.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.i.b.65.1 4 12.11 even 2
224.2.i.b.65.2 yes 4 3.2 odd 2
224.2.i.b.193.1 yes 4 84.11 even 6
224.2.i.b.193.2 yes 4 21.11 odd 6
448.2.i.i.65.1 4 24.5 odd 2
448.2.i.i.65.2 4 24.11 even 2
448.2.i.i.193.1 4 168.53 odd 6
448.2.i.i.193.2 4 168.11 even 6
1568.2.a.n.1.1 2 84.47 odd 6
1568.2.a.n.1.2 2 21.5 even 6
1568.2.a.s.1.1 2 21.2 odd 6
1568.2.a.s.1.2 2 84.23 even 6
1568.2.i.u.961.1 4 21.20 even 2
1568.2.i.u.961.2 4 84.83 odd 2
1568.2.i.u.1537.1 4 21.17 even 6
1568.2.i.u.1537.2 4 84.59 odd 6
2016.2.s.r.289.1 4 1.1 even 1 trivial
2016.2.s.r.289.2 4 4.3 odd 2 inner
2016.2.s.r.865.1 4 7.4 even 3 inner
2016.2.s.r.865.2 4 28.11 odd 6 inner
3136.2.a.bh.1.1 2 168.107 even 6
3136.2.a.bh.1.2 2 168.149 odd 6
3136.2.a.bu.1.1 2 168.5 even 6
3136.2.a.bu.1.2 2 168.131 odd 6