Properties

Label 2592.2.i.o.1729.1
Level $2592$
Weight $2$
Character 2592.1729
Analytic conductor $20.697$
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] = [2592,2,Mod(865,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (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: \(20.6972242039\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1729.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2592.1729
Dual form 2592.2.i.o.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{13} -3.00000 q^{17} -2.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(0.500000 + 0.866025i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{35} +1.00000 q^{37} +(-1.00000 + 1.73205i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +10.0000 q^{53} -2.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(0.500000 + 0.866025i) q^{65} +(-7.00000 + 12.1244i) q^{67} +10.0000 q^{71} -9.00000 q^{73} +(2.00000 - 3.46410i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} -11.0000 q^{89} -2.00000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + 2 q^{7} - 2 q^{11} - q^{13} - 6 q^{17} - 4 q^{19} - 6 q^{23} + 4 q^{25} + q^{29} - 8 q^{31} + 4 q^{35} + 2 q^{37} - 2 q^{41} + 10 q^{43} - 4 q^{47} + 3 q^{49} + 20 q^{53} - 4 q^{55} - 4 q^{59} - 9 q^{61} + q^{65} - 14 q^{67} + 20 q^{71} - 18 q^{73} + 4 q^{77} + 10 q^{79} - 12 q^{83} - 3 q^{85} - 22 q^{89} - 4 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i 0.908708 0.417432i \(-0.137070\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 + 1.73205i −0.156174 + 0.270501i −0.933486 0.358614i \(-0.883249\pi\)
0.777312 + 0.629115i \(0.216583\pi\)
\(42\) 0 0
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 3.46410i −0.291730 0.505291i 0.682489 0.730896i \(-0.260898\pi\)
−0.974219 + 0.225605i \(0.927564\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) −4.50000 7.79423i −0.576166 0.997949i −0.995914 0.0903080i \(-0.971215\pi\)
0.419748 0.907641i \(-0.362118\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.500000 + 0.866025i 0.0620174 + 0.107417i
\(66\) 0 0
\(67\) −7.00000 + 12.1244i −0.855186 + 1.48123i 0.0212861 + 0.999773i \(0.493224\pi\)
−0.876472 + 0.481452i \(0.840109\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 3.46410i 0.227921 0.394771i
\(78\) 0 0
\(79\) 5.00000 + 8.66025i 0.562544 + 0.974355i 0.997274 + 0.0737937i \(0.0235106\pi\)
−0.434730 + 0.900561i \(0.643156\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 10.3923i −0.658586 1.14070i −0.980982 0.194099i \(-0.937822\pi\)
0.322396 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) −1.50000 + 2.59808i −0.162698 + 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 + 1.73205i −0.102598 + 0.177705i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 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
\(103\) −6.00000 + 10.3923i −0.591198 + 1.02398i 0.402874 + 0.915255i \(0.368011\pi\)
−0.994071 + 0.108729i \(0.965322\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.50000 + 14.7224i −0.799613 + 1.38497i 0.120256 + 0.992743i \(0.461629\pi\)
−0.919868 + 0.392227i \(0.871705\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 5.19615i −0.275010 0.476331i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.00000 + 12.1244i −0.611593 + 1.05931i 0.379379 + 0.925241i \(0.376138\pi\)
−0.990972 + 0.134069i \(0.957196\pi\)
\(132\) 0 0
\(133\) −2.00000 3.46410i −0.173422 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.50000 7.79423i −0.384461 0.665906i 0.607233 0.794524i \(-0.292279\pi\)
−0.991694 + 0.128618i \(0.958946\pi\)
\(138\) 0 0
\(139\) −10.0000 + 17.3205i −0.848189 + 1.46911i 0.0346338 + 0.999400i \(0.488974\pi\)
−0.882823 + 0.469706i \(0.844360\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.5000 18.1865i 0.860194 1.48990i −0.0115483 0.999933i \(-0.503676\pi\)
0.871742 0.489966i \(-0.162991\pi\)
\(150\) 0 0
\(151\) 6.00000 + 10.3923i 0.488273 + 0.845714i 0.999909 0.0134886i \(-0.00429367\pi\)
−0.511636 + 0.859202i \(0.670960\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 + 6.92820i 0.321288 + 0.556487i
\(156\) 0 0
\(157\) −8.50000 + 14.7224i −0.678374 + 1.17498i 0.297097 + 0.954847i \(0.403982\pi\)
−0.975470 + 0.220131i \(0.929352\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.73205i 0.0773823 0.134030i −0.824737 0.565516i \(-0.808677\pi\)
0.902120 + 0.431486i \(0.142010\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.5000 + 18.1865i 0.798300 + 1.38270i 0.920722 + 0.390218i \(0.127601\pi\)
−0.122422 + 0.992478i \(0.539066\pi\)
\(174\) 0 0
\(175\) −4.00000 + 6.92820i −0.302372 + 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.500000 0.866025i 0.0367607 0.0636715i
\(186\) 0 0
\(187\) 3.00000 + 5.19615i 0.219382 + 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 15.5885i −0.651217 1.12794i −0.982828 0.184525i \(-0.940925\pi\)
0.331611 0.943416i \(-0.392408\pi\)
\(192\) 0 0
\(193\) 4.50000 7.79423i 0.323917 0.561041i −0.657376 0.753563i \(-0.728333\pi\)
0.981293 + 0.192522i \(0.0616668\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0000 0.783718 0.391859 0.920025i \(-0.371832\pi\)
0.391859 + 0.920025i \(0.371832\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.00000 + 1.73205i −0.0701862 + 0.121566i
\(204\) 0 0
\(205\) 1.00000 + 1.73205i 0.0698430 + 0.120972i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.00000 + 3.46410i 0.138343 + 0.239617i
\(210\) 0 0
\(211\) 1.00000 1.73205i 0.0688428 0.119239i −0.829549 0.558433i \(-0.811403\pi\)
0.898392 + 0.439194i \(0.144736\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.50000 2.59808i 0.100901 0.174766i
\(222\) 0 0
\(223\) 1.00000 + 1.73205i 0.0669650 + 0.115987i 0.897564 0.440884i \(-0.145335\pi\)
−0.830599 + 0.556871i \(0.812002\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.00000 5.19615i −0.199117 0.344881i 0.749125 0.662428i \(-0.230474\pi\)
−0.948242 + 0.317547i \(0.897141\pi\)
\(228\) 0 0
\(229\) 3.50000 6.06218i 0.231287 0.400600i −0.726900 0.686743i \(-0.759040\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.0000 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 + 10.3923i −0.388108 + 0.672222i −0.992195 0.124696i \(-0.960204\pi\)
0.604087 + 0.796918i \(0.293538\pi\)
\(240\) 0 0
\(241\) 2.50000 + 4.33013i 0.161039 + 0.278928i 0.935242 0.354010i \(-0.115182\pi\)
−0.774202 + 0.632938i \(0.781849\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.50000 2.59808i −0.0958315 0.165985i
\(246\) 0 0
\(247\) 1.00000 1.73205i 0.0636285 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.5000 + 25.1147i −0.904485 + 1.56661i −0.0828783 + 0.996560i \(0.526411\pi\)
−0.821607 + 0.570055i \(0.806922\pi\)
\(258\) 0 0
\(259\) 1.00000 + 1.73205i 0.0621370 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 + 10.3923i 0.369976 + 0.640817i 0.989561 0.144112i \(-0.0460326\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(264\) 0 0
\(265\) 5.00000 8.66025i 0.307148 0.531995i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 6.92820i 0.241209 0.417786i
\(276\) 0 0
\(277\) 1.00000 + 1.73205i 0.0600842 + 0.104069i 0.894503 0.447062i \(-0.147530\pi\)
−0.834419 + 0.551131i \(0.814196\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.50000 7.79423i −0.268447 0.464965i 0.700014 0.714130i \(-0.253177\pi\)
−0.968461 + 0.249165i \(0.919844\pi\)
\(282\) 0 0
\(283\) 8.00000 13.8564i 0.475551 0.823678i −0.524057 0.851683i \(-0.675582\pi\)
0.999608 + 0.0280052i \(0.00891551\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.5000 18.1865i 0.613417 1.06247i −0.377244 0.926114i \(-0.623128\pi\)
0.990660 0.136355i \(-0.0435386\pi\)
\(294\) 0 0
\(295\) 2.00000 + 3.46410i 0.116445 + 0.201688i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 5.19615i −0.173494 0.300501i
\(300\) 0 0
\(301\) −10.0000 + 17.3205i −0.576390 + 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.00000 −0.515339
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −3.50000 6.06218i −0.197832 0.342655i 0.749993 0.661445i \(-0.230057\pi\)
−0.947825 + 0.318791i \(0.896723\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.50000 2.59808i −0.0842484 0.145922i 0.820822 0.571184i \(-0.193516\pi\)
−0.905071 + 0.425261i \(0.860182\pi\)
\(318\) 0 0
\(319\) 1.00000 1.73205i 0.0559893 0.0969762i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 6.92820i 0.220527 0.381964i
\(330\) 0 0
\(331\) −7.00000 12.1244i −0.384755 0.666415i 0.606980 0.794717i \(-0.292381\pi\)
−0.991735 + 0.128302i \(0.959047\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.00000 + 12.1244i 0.382451 + 0.662424i
\(336\) 0 0
\(337\) −7.00000 + 12.1244i −0.381314 + 0.660456i −0.991250 0.131995i \(-0.957862\pi\)
0.609936 + 0.792451i \(0.291195\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.00000 + 8.66025i −0.268414 + 0.464907i −0.968452 0.249198i \(-0.919833\pi\)
0.700038 + 0.714105i \(0.253166\pi\)
\(348\) 0 0
\(349\) −17.0000 29.4449i −0.909989 1.57615i −0.814076 0.580758i \(-0.802756\pi\)
−0.0959126 0.995390i \(-0.530577\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.00000 + 1.73205i 0.0532246 + 0.0921878i 0.891410 0.453197i \(-0.149717\pi\)
−0.838186 + 0.545385i \(0.816383\pi\)
\(354\) 0 0
\(355\) 5.00000 8.66025i 0.265372 0.459639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.50000 + 7.79423i −0.235541 + 0.407969i
\(366\) 0 0
\(367\) −12.0000 20.7846i −0.626395 1.08495i −0.988269 0.152721i \(-0.951196\pi\)
0.361874 0.932227i \(-0.382137\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.0000 + 17.3205i 0.519174 + 0.899236i
\(372\) 0 0
\(373\) 5.00000 8.66025i 0.258890 0.448411i −0.707055 0.707159i \(-0.749977\pi\)
0.965945 + 0.258748i \(0.0833099\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 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
\(383\) 7.00000 12.1244i 0.357683 0.619526i −0.629890 0.776684i \(-0.716900\pi\)
0.987573 + 0.157159i \(0.0502334\pi\)
\(384\) 0 0
\(385\) −2.00000 3.46410i −0.101929 0.176547i
\(386\) 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
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.5000 + 21.6506i −0.624220 + 1.08118i 0.364471 + 0.931215i \(0.381250\pi\)
−0.988691 + 0.149966i \(0.952083\pi\)
\(402\) 0 0
\(403\) −4.00000 6.92820i −0.199254 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 1.73205i −0.0495682 0.0858546i
\(408\) 0 0
\(409\) 14.5000 25.1147i 0.716979 1.24184i −0.245212 0.969469i \(-0.578858\pi\)
0.962191 0.272374i \(-0.0878089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0000 34.6410i 0.977064 1.69232i 0.304115 0.952635i \(-0.401639\pi\)
0.672949 0.739689i \(-0.265027\pi\)
\(420\) 0 0
\(421\) 3.50000 + 6.06218i 0.170580 + 0.295452i 0.938623 0.344946i \(-0.112103\pi\)
−0.768043 + 0.640398i \(0.778769\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 10.3923i −0.291043 0.504101i
\(426\) 0 0
\(427\) 9.00000 15.5885i 0.435541 0.754378i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.00000 10.3923i 0.287019 0.497131i
\(438\) 0 0
\(439\) −14.0000 24.2487i −0.668184 1.15733i −0.978412 0.206666i \(-0.933739\pi\)
0.310228 0.950662i \(-0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 10.3923i −0.285069 0.493753i 0.687557 0.726130i \(-0.258683\pi\)
−0.972626 + 0.232377i \(0.925350\pi\)
\(444\) 0 0
\(445\) −5.50000 + 9.52628i −0.260725 + 0.451589i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.00000 + 1.73205i −0.0468807 + 0.0811998i
\(456\) 0 0
\(457\) 18.5000 + 32.0429i 0.865393 + 1.49891i 0.866656 + 0.498906i \(0.166265\pi\)
−0.00126243 + 0.999999i \(0.500402\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.0000 + 25.9808i 0.698620 + 1.21004i 0.968945 + 0.247276i \(0.0795353\pi\)
−0.270326 + 0.962769i \(0.587131\pi\)
\(462\) 0 0
\(463\) 14.0000 24.2487i 0.650635 1.12693i −0.332334 0.943162i \(-0.607836\pi\)
0.982969 0.183771i \(-0.0588306\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000 17.3205i 0.459800 0.796398i
\(474\) 0 0
\(475\) −4.00000 6.92820i −0.183533 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.0000 + 36.3731i 0.959514 + 1.66193i 0.723681 + 0.690134i \(0.242449\pi\)
0.235833 + 0.971794i \(0.424218\pi\)
\(480\) 0 0
\(481\) −0.500000 + 0.866025i −0.0227980 + 0.0394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.0000 29.4449i 0.767199 1.32883i −0.171877 0.985118i \(-0.554983\pi\)
0.939076 0.343710i \(-0.111684\pi\)
\(492\) 0 0
\(493\) −1.50000 2.59808i −0.0675566 0.117011i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.0000 + 17.3205i 0.448561 + 0.776931i
\(498\) 0 0
\(499\) 3.00000 5.19615i 0.134298 0.232612i −0.791031 0.611776i \(-0.790455\pi\)
0.925329 + 0.379165i \(0.123789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.00000 + 15.5885i −0.398918 + 0.690946i −0.993593 0.113020i \(-0.963948\pi\)
0.594675 + 0.803966i \(0.297281\pi\)
\(510\) 0 0
\(511\) −9.00000 15.5885i −0.398137 0.689593i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.00000 + 10.3923i 0.264392 + 0.457940i
\(516\) 0 0
\(517\) −4.00000 + 6.92820i −0.175920 + 0.304702i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 30.0000 1.31181 0.655904 0.754844i \(-0.272288\pi\)
0.655904 + 0.754844i \(0.272288\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 20.7846i 0.522728 0.905392i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.00000 1.73205i −0.0433148 0.0750234i
\(534\) 0 0
\(535\) 8.00000 13.8564i 0.345870 0.599065i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 13.0000 0.558914 0.279457 0.960158i \(-0.409846\pi\)
0.279457 + 0.960158i \(0.409846\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.50000 4.33013i 0.107088 0.185482i
\(546\) 0 0
\(547\) −6.00000 10.3923i −0.256541 0.444343i 0.708772 0.705438i \(-0.249250\pi\)
−0.965313 + 0.261095i \(0.915916\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.00000 1.73205i −0.0426014 0.0737878i
\(552\) 0 0
\(553\) −10.0000 + 17.3205i −0.425243 + 0.736543i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.0000 0.466085 0.233042 0.972467i \(-0.425132\pi\)
0.233042 + 0.972467i \(0.425132\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 + 31.1769i −0.758610 + 1.31395i 0.184950 + 0.982748i \(0.440788\pi\)
−0.943560 + 0.331202i \(0.892546\pi\)
\(564\) 0 0
\(565\) 8.50000 + 14.7224i 0.357598 + 0.619377i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.50000 4.33013i −0.104805 0.181528i 0.808853 0.588011i \(-0.200089\pi\)
−0.913659 + 0.406482i \(0.866755\pi\)
\(570\) 0 0
\(571\) 10.0000 17.3205i 0.418487 0.724841i −0.577301 0.816532i \(-0.695894\pi\)
0.995788 + 0.0916910i \(0.0292272\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 20.7846i 0.497844 0.862291i
\(582\) 0 0
\(583\) −10.0000 17.3205i −0.414158 0.717342i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000 + 25.9808i 0.619116 + 1.07234i 0.989647 + 0.143521i \(0.0458424\pi\)
−0.370531 + 0.928820i \(0.620824\pi\)
\(588\) 0 0
\(589\) 8.00000 13.8564i 0.329634 0.570943i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 27.7128i 0.653742 1.13231i −0.328465 0.944516i \(-0.606531\pi\)
0.982208 0.187799i \(-0.0601353\pi\)
\(600\) 0 0
\(601\) 8.50000 + 14.7224i 0.346722 + 0.600541i 0.985665 0.168714i \(-0.0539613\pi\)
−0.638943 + 0.769254i \(0.720628\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) 11.0000 19.0526i 0.446476 0.773320i −0.551678 0.834058i \(-0.686012\pi\)
0.998154 + 0.0607380i \(0.0193454\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.50000 12.9904i 0.301939 0.522973i −0.674636 0.738150i \(-0.735700\pi\)
0.976575 + 0.215177i \(0.0690329\pi\)
\(618\) 0 0
\(619\) −24.0000 41.5692i −0.964641 1.67081i −0.710575 0.703621i \(-0.751565\pi\)
−0.254066 0.967187i \(-0.581768\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −11.0000 19.0526i −0.440706 0.763325i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00000 1.73205i 0.0396838 0.0687343i
\(636\) 0 0
\(637\) 1.50000 + 2.59808i 0.0594322 + 0.102940i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.5000 + 30.3109i 0.691208 + 1.19721i 0.971442 + 0.237276i \(0.0762547\pi\)
−0.280234 + 0.959932i \(0.590412\pi\)
\(642\) 0 0
\(643\) 18.0000 31.1769i 0.709851 1.22950i −0.255062 0.966925i \(-0.582096\pi\)
0.964912 0.262573i \(-0.0845709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17.0000 + 29.4449i −0.665261 + 1.15227i 0.313953 + 0.949439i \(0.398347\pi\)
−0.979214 + 0.202828i \(0.934987\pi\)
\(654\) 0 0
\(655\) 7.00000 + 12.1244i 0.273513 + 0.473738i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 15.0000 + 25.9808i 0.584317 + 1.01207i 0.994960 + 0.100271i \(0.0319709\pi\)
−0.410643 + 0.911796i \(0.634696\pi\)
\(660\) 0 0
\(661\) 15.5000 26.8468i 0.602880 1.04422i −0.389503 0.921025i \(-0.627353\pi\)
0.992383 0.123194i \(-0.0393136\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.00000 + 15.5885i −0.347441 + 0.601786i
\(672\) 0 0
\(673\) −11.5000 19.9186i −0.443292 0.767805i 0.554639 0.832091i \(-0.312856\pi\)
−0.997932 + 0.0642860i \(0.979523\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.0000 29.4449i −0.653363 1.13166i −0.982301 0.187307i \(-0.940024\pi\)
0.328938 0.944351i \(-0.393309\pi\)
\(678\) 0 0
\(679\) −2.00000 + 3.46410i −0.0767530 + 0.132940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.00000 + 8.66025i −0.190485 + 0.329929i
\(690\) 0 0
\(691\) −5.00000 8.66025i −0.190209 0.329452i 0.755110 0.655598i \(-0.227583\pi\)
−0.945319 + 0.326146i \(0.894250\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0000 + 17.3205i 0.379322 + 0.657004i
\(696\) 0 0
\(697\) 3.00000 5.19615i 0.113633 0.196818i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −45.0000 −1.69963 −0.849813 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 10.3923i 0.225653 0.390843i
\(708\) 0 0
\(709\) −0.500000 0.866025i −0.0187779 0.0325243i 0.856484 0.516174i \(-0.172644\pi\)
−0.875262 + 0.483650i \(0.839311\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.0000 41.5692i −0.898807 1.55678i
\(714\) 0 0
\(715\) 1.00000 1.73205i 0.0373979 0.0647750i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.0000 −0.522112 −0.261056 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.00000 + 3.46410i −0.0742781 + 0.128654i
\(726\) 0 0
\(727\) 3.00000 + 5.19615i 0.111264 + 0.192715i 0.916280 0.400538i \(-0.131177\pi\)
−0.805016 + 0.593253i \(0.797843\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.0000 25.9808i −0.554795 0.960933i
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.0000 1.03139
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 20.7846i 0.440237 0.762513i −0.557470 0.830197i \(-0.688228\pi\)
0.997707 + 0.0676840i \(0.0215610\pi\)
\(744\) 0 0
\(745\) −10.5000 18.1865i −0.384690 0.666303i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0000 + 27.7128i 0.584627 + 1.01260i
\(750\) 0 0
\(751\) 11.0000 19.0526i 0.401396 0.695238i −0.592499 0.805571i \(-0.701859\pi\)
0.993895 + 0.110333i \(0.0351919\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.5000 33.7750i 0.706874 1.22434i −0.259136 0.965841i \(-0.583438\pi\)
0.966011 0.258502i \(-0.0832288\pi\)
\(762\) 0 0
\(763\) 5.00000 + 8.66025i 0.181012 + 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.00000 3.46410i −0.0722158 0.125081i
\(768\) 0 0
\(769\) −15.5000 + 26.8468i −0.558944 + 0.968120i 0.438641 + 0.898663i \(0.355460\pi\)
−0.997585 + 0.0694574i \(0.977873\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.00000 3.46410i 0.0716574 0.124114i
\(780\) 0 0
\(781\) −10.0000 17.3205i −0.357828 0.619777i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.50000 + 14.7224i 0.303378 + 0.525466i
\(786\) 0 0
\(787\) 9.00000 15.5885i 0.320815 0.555668i −0.659841 0.751405i \(-0.729376\pi\)
0.980656 + 0.195737i \(0.0627098\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −34.0000 −1.20890
\(792\) 0 0
\(793\) 9.00000 0.319599
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.5000 + 33.7750i −0.690725 + 1.19637i 0.280875 + 0.959744i \(0.409375\pi\)
−0.971601 + 0.236627i \(0.923958\pi\)
\(798\) 0 0
\(799\) 6.00000 + 10.3923i 0.212265 + 0.367653i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.00000 + 15.5885i 0.317603 + 0.550105i
\(804\) 0 0
\(805\) −6.00000 + 10.3923i −0.211472 + 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 37.0000 1.30085 0.650425 0.759570i \(-0.274591\pi\)
0.650425 + 0.759570i \(0.274591\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 3.46410i 0.0700569 0.121342i
\(816\) 0 0
\(817\) −10.0000 17.3205i −0.349856 0.605968i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.5000 19.9186i −0.401353 0.695163i 0.592537 0.805543i \(-0.298127\pi\)
−0.993889 + 0.110380i \(0.964793\pi\)
\(822\) 0 0
\(823\) 4.00000 6.92820i 0.139431 0.241502i −0.787850 0.615867i \(-0.788806\pi\)
0.927281 + 0.374365i \(0.122139\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.50000 + 7.79423i −0.155916 + 0.270054i
\(834\) 0 0
\(835\) −1.00000 1.73205i −0.0346064 0.0599401i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.0000 19.0526i −0.379762 0.657767i 0.611265 0.791426i \(-0.290661\pi\)
−0.991027 + 0.133658i \(0.957328\pi\)
\(840\) 0 0
\(841\) 14.0000 24.2487i 0.482759 0.836162i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.00000 + 5.19615i −0.102839 + 0.178122i
\(852\) 0 0
\(853\) 21.0000 + 36.3731i 0.719026 + 1.24539i 0.961386 + 0.275204i \(0.0887453\pi\)
−0.242360 + 0.970186i \(0.577921\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.50000 11.2583i −0.222036 0.384577i 0.733390 0.679808i \(-0.237937\pi\)
−0.955426 + 0.295231i \(0.904604\pi\)
\(858\) 0 0
\(859\) −22.0000 + 38.1051i −0.750630 + 1.30013i 0.196887 + 0.980426i \(0.436917\pi\)
−0.947518 + 0.319704i \(0.896417\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) 0 0
\(865\) 21.0000 0.714021
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.0000 17.3205i 0.339227 0.587558i
\(870\) 0 0
\(871\) −7.00000 12.1244i −0.237186 0.410818i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.00000 + 15.5885i 0.304256 + 0.526986i
\(876\) 0 0
\(877\) 17.5000 30.3109i 0.590933 1.02353i −0.403174 0.915123i \(-0.632093\pi\)
0.994107 0.108403i \(-0.0345736\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 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
\(887\) −27.0000 + 46.7654i −0.906571 + 1.57023i −0.0877772 + 0.996140i \(0.527976\pi\)
−0.818794 + 0.574087i \(0.805357\pi\)
\(888\) 0 0
\(889\) 2.00000 + 3.46410i 0.0670778 + 0.116182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.00000 + 6.92820i 0.133855 + 0.231843i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −30.0000 −0.999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.00000 12.1244i 0.232688 0.403027i
\(906\) 0 0
\(907\) 4.00000 + 6.92820i 0.132818 + 0.230047i 0.924762 0.380547i \(-0.124264\pi\)
−0.791944 + 0.610594i \(0.790931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.0000 + 25.9808i 0.496972 + 0.860781i 0.999994 0.00349271i \(-0.00111177\pi\)
−0.503022 + 0.864274i \(0.667778\pi\)
\(912\) 0 0
\(913\) −12.0000 + 20.7846i −0.397142 + 0.687870i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28.0000 −0.924641
\(918\) 0 0
\(919\) −42.0000 −1.38545 −0.692726 0.721201i \(-0.743591\pi\)
−0.692726 + 0.721201i \(0.743591\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −5.00000 + 8.66025i −0.164577 + 0.285056i
\(924\) 0 0
\(925\) 2.00000 + 3.46410i 0.0657596 + 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.5000 + 23.3827i 0.442921 + 0.767161i 0.997905 0.0646999i \(-0.0206090\pi\)
−0.554984 + 0.831861i \(0.687276\pi\)
\(930\) 0 0
\(931\) −3.00000 + 5.19615i −0.0983210 + 0.170297i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21.5000 + 37.2391i −0.700880 + 1.21396i 0.267278 + 0.963619i \(0.413876\pi\)
−0.968158 + 0.250340i \(0.919458\pi\)
\(942\) 0 0
\(943\) −6.00000 10.3923i −0.195387 0.338420i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27.0000 46.7654i −0.877382 1.51967i −0.854203 0.519939i \(-0.825955\pi\)
−0.0231788 0.999731i \(-0.507379\pi\)
\(948\) 0 0
\(949\) 4.50000 7.79423i 0.146076 0.253011i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.0000 0.680257 0.340128 0.940379i \(-0.389529\pi\)
0.340128 + 0.940379i \(0.389529\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.00000 15.5885i 0.290625 0.503378i
\(960\) 0 0
\(961\) −16.5000 28.5788i −0.532258 0.921898i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.50000 7.79423i −0.144860 0.250905i
\(966\) 0 0
\(967\) −9.00000 + 15.5885i −0.289420 + 0.501291i −0.973672 0.227956i \(-0.926796\pi\)
0.684251 + 0.729247i \(0.260129\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 0 0
\(973\) −40.0000 −1.28234
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00000 + 1.73205i −0.0319928 + 0.0554132i −0.881579 0.472037i \(-0.843519\pi\)
0.849586 + 0.527451i \(0.176852\pi\)
\(978\) 0 0
\(979\) 11.0000 + 19.0526i 0.351562 + 0.608922i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.0000 20.7846i −0.382741 0.662926i 0.608712 0.793391i \(-0.291686\pi\)
−0.991453 + 0.130465i \(0.958353\pi\)
\(984\) 0 0
\(985\) 5.50000 9.52628i 0.175245 0.303533i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.00000 + 13.8564i −0.253617 + 0.439278i
\(996\) 0 0
\(997\) 7.50000 + 12.9904i 0.237527 + 0.411409i 0.960004 0.279986i \(-0.0903297\pi\)
−0.722477 + 0.691395i \(0.756996\pi\)
\(998\) 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 2592.2.i.o.1729.1 2
3.2 odd 2 2592.2.i.k.1729.1 2
4.3 odd 2 2592.2.i.n.1729.1 2
9.2 odd 6 2592.2.i.k.865.1 2
9.4 even 3 2592.2.a.c.1.1 1
9.5 odd 6 2592.2.a.e.1.1 yes 1
9.7 even 3 inner 2592.2.i.o.865.1 2
12.11 even 2 2592.2.i.j.1729.1 2
36.7 odd 6 2592.2.i.n.865.1 2
36.11 even 6 2592.2.i.j.865.1 2
36.23 even 6 2592.2.a.f.1.1 yes 1
36.31 odd 6 2592.2.a.d.1.1 yes 1
72.5 odd 6 5184.2.a.j.1.1 1
72.13 even 6 5184.2.a.t.1.1 1
72.59 even 6 5184.2.a.m.1.1 1
72.67 odd 6 5184.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.c.1.1 1 9.4 even 3
2592.2.a.d.1.1 yes 1 36.31 odd 6
2592.2.a.e.1.1 yes 1 9.5 odd 6
2592.2.a.f.1.1 yes 1 36.23 even 6
2592.2.i.j.865.1 2 36.11 even 6
2592.2.i.j.1729.1 2 12.11 even 2
2592.2.i.k.865.1 2 9.2 odd 6
2592.2.i.k.1729.1 2 3.2 odd 2
2592.2.i.n.865.1 2 36.7 odd 6
2592.2.i.n.1729.1 2 4.3 odd 2
2592.2.i.o.865.1 2 9.7 even 3 inner
2592.2.i.o.1729.1 2 1.1 even 1 trivial
5184.2.a.j.1.1 1 72.5 odd 6
5184.2.a.m.1.1 1 72.59 even 6
5184.2.a.t.1.1 1 72.13 even 6
5184.2.a.w.1.1 1 72.67 odd 6