Properties

Label 2592.2.i.c.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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 864)
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.c.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{5} +(-1.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.00000 + 1.73205i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(1.50000 - 2.59808i) q^{13} -2.00000 q^{17} -3.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(4.00000 + 6.92820i) q^{29} +6.00000 q^{35} +7.00000 q^{37} +(-4.00000 + 6.92820i) q^{41} +(6.00000 + 10.3923i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-1.00000 + 1.73205i) q^{49} +4.00000 q^{53} +12.0000 q^{55} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(3.00000 + 5.19615i) q^{65} +(1.50000 - 2.59808i) q^{67} +12.0000 q^{71} -15.0000 q^{73} +(-9.00000 + 15.5885i) q^{77} +(-4.50000 - 7.79423i) q^{79} +(6.00000 + 10.3923i) q^{83} +(2.00000 - 3.46410i) q^{85} -10.0000 q^{89} -9.00000 q^{91} +(3.00000 - 5.19615i) q^{95} +(-4.50000 - 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 6 q^{19} - 6 q^{23} + q^{25} + 8 q^{29} + 12 q^{35} + 14 q^{37} - 8 q^{41} + 12 q^{43} - 6 q^{47} - 2 q^{49} + 8 q^{53} + 24 q^{55} - 6 q^{59} + q^{61} + 6 q^{65} + 3 q^{67} + 24 q^{71} - 30 q^{73} - 18 q^{77} - 9 q^{79} + 12 q^{83} + 4 q^{85} - 20 q^{89} - 18 q^{91} + 6 q^{95} - 9 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) −1.50000 2.59808i −0.566947 0.981981i −0.996866 0.0791130i \(-0.974791\pi\)
0.429919 0.902867i \(-0.358542\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) 1.50000 2.59808i 0.416025 0.720577i −0.579510 0.814965i \(-0.696756\pi\)
0.995535 + 0.0943882i \(0.0300895\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 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
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 + 6.92820i 0.742781 + 1.28654i 0.951224 + 0.308500i \(0.0998271\pi\)
−0.208443 + 0.978035i \(0.566840\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 + 6.92820i −0.624695 + 1.08200i 0.363905 + 0.931436i \(0.381443\pi\)
−0.988600 + 0.150567i \(0.951890\pi\)
\(42\) 0 0
\(43\) 6.00000 + 10.3923i 0.914991 + 1.58481i 0.806914 + 0.590669i \(0.201136\pi\)
0.108078 + 0.994142i \(0.465531\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 + 5.19615i −0.390567 + 0.676481i −0.992524 0.122047i \(-0.961054\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 1.50000 2.59808i 0.183254 0.317406i −0.759733 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) 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
\(73\) −15.0000 −1.75562 −0.877809 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.00000 + 15.5885i −1.02565 + 1.77647i
\(78\) 0 0
\(79\) −4.50000 7.79423i −0.506290 0.876919i −0.999974 0.00727784i \(-0.997683\pi\)
0.493684 0.869641i \(-0.335650\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) 2.00000 3.46410i 0.216930 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) −4.50000 7.79423i −0.456906 0.791384i 0.541890 0.840450i \(-0.317709\pi\)
−0.998796 + 0.0490655i \(0.984376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.00000 + 13.8564i 0.796030 + 1.37876i 0.922183 + 0.386753i \(0.126403\pi\)
−0.126153 + 0.992011i \(0.540263\pi\)
\(102\) 0 0
\(103\) −7.50000 + 12.9904i −0.738997 + 1.27998i 0.213950 + 0.976845i \(0.431367\pi\)
−0.952947 + 0.303136i \(0.901966\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.00000 + 12.1244i −0.658505 + 1.14056i 0.322498 + 0.946570i \(0.395477\pi\)
−0.981003 + 0.193993i \(0.937856\pi\)
\(114\) 0 0
\(115\) −6.00000 10.3923i −0.559503 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 + 5.19615i 0.275010 + 0.476331i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 4.50000 + 7.79423i 0.390199 + 0.675845i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000 + 12.1244i 0.598050 + 1.03585i 0.993109 + 0.117198i \(0.0373911\pi\)
−0.395058 + 0.918656i \(0.629276\pi\)
\(138\) 0 0
\(139\) 10.5000 18.1865i 0.890598 1.54256i 0.0514389 0.998676i \(-0.483619\pi\)
0.839159 0.543885i \(-0.183047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −18.0000 −1.50524
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.00000 + 3.46410i −0.163846 + 0.283790i −0.936245 0.351348i \(-0.885723\pi\)
0.772399 + 0.635138i \(0.219057\pi\)
\(150\) 0 0
\(151\) 1.50000 + 2.59808i 0.122068 + 0.211428i 0.920583 0.390547i \(-0.127714\pi\)
−0.798515 + 0.601975i \(0.794381\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 12.1244i 0.558661 0.967629i −0.438948 0.898513i \(-0.644649\pi\)
0.997609 0.0691164i \(-0.0220180\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 0 0
\(163\) 3.00000 0.234978 0.117489 0.993074i \(-0.462515\pi\)
0.117489 + 0.993074i \(0.462515\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 15.5885i 0.696441 1.20627i −0.273252 0.961943i \(-0.588099\pi\)
0.969693 0.244328i \(-0.0785675\pi\)
\(168\) 0 0
\(169\) 2.00000 + 3.46410i 0.153846 + 0.266469i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.00000 6.92820i −0.304114 0.526742i 0.672949 0.739689i \(-0.265027\pi\)
−0.977064 + 0.212947i \(0.931694\pi\)
\(174\) 0 0
\(175\) 1.50000 2.59808i 0.113389 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 9.00000 0.668965 0.334482 0.942402i \(-0.391439\pi\)
0.334482 + 0.942402i \(0.391439\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.00000 + 12.1244i −0.514650 + 0.891400i
\(186\) 0 0
\(187\) 6.00000 + 10.3923i 0.438763 + 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) 2.50000 4.33013i 0.179954 0.311689i −0.761911 0.647682i \(-0.775738\pi\)
0.941865 + 0.335993i \(0.109072\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 15.0000 1.06332 0.531661 0.846957i \(-0.321568\pi\)
0.531661 + 0.846957i \(0.321568\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 20.7846i 0.842235 1.45879i
\(204\) 0 0
\(205\) −8.00000 13.8564i −0.558744 0.967773i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.00000 + 15.5885i 0.622543 + 1.07828i
\(210\) 0 0
\(211\) −7.50000 + 12.9904i −0.516321 + 0.894295i 0.483499 + 0.875345i \(0.339366\pi\)
−0.999820 + 0.0189499i \(0.993968\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00000 + 5.19615i −0.201802 + 0.349531i
\(222\) 0 0
\(223\) 12.0000 + 20.7846i 0.803579 + 1.39184i 0.917246 + 0.398321i \(0.130407\pi\)
−0.113666 + 0.993519i \(0.536260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 10.3923i −0.398234 0.689761i 0.595274 0.803523i \(-0.297043\pi\)
−0.993508 + 0.113761i \(0.963710\pi\)
\(228\) 0 0
\(229\) 3.00000 5.19615i 0.198246 0.343371i −0.749714 0.661762i \(-0.769809\pi\)
0.947960 + 0.318390i \(0.103142\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 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
\(241\) 1.50000 + 2.59808i 0.0966235 + 0.167357i 0.910285 0.413982i \(-0.135862\pi\)
−0.813662 + 0.581339i \(0.802529\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 3.46410i −0.127775 0.221313i
\(246\) 0 0
\(247\) −4.50000 + 7.79423i −0.286328 + 0.495935i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000 17.3205i 0.623783 1.08042i −0.364992 0.931011i \(-0.618928\pi\)
0.988775 0.149413i \(-0.0477384\pi\)
\(258\) 0 0
\(259\) −10.5000 18.1865i −0.652438 1.13006i
\(260\) 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
\(265\) −4.00000 + 6.92820i −0.245718 + 0.425596i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 5.19615i 0.180907 0.313340i
\(276\) 0 0
\(277\) 9.00000 + 15.5885i 0.540758 + 0.936620i 0.998861 + 0.0477206i \(0.0151957\pi\)
−0.458103 + 0.888899i \(0.651471\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 6.92820i −0.238620 0.413302i 0.721699 0.692207i \(-0.243362\pi\)
−0.960319 + 0.278906i \(0.910028\pi\)
\(282\) 0 0
\(283\) −6.00000 + 10.3923i −0.356663 + 0.617758i −0.987401 0.158237i \(-0.949419\pi\)
0.630738 + 0.775996i \(0.282752\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.00000 1.73205i 0.0584206 0.101187i −0.835336 0.549740i \(-0.814727\pi\)
0.893757 + 0.448552i \(0.148060\pi\)
\(294\) 0 0
\(295\) −6.00000 10.3923i −0.349334 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.00000 + 15.5885i 0.520483 + 0.901504i
\(300\) 0 0
\(301\) 18.0000 31.1769i 1.03750 1.79701i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 + 15.5885i −0.510343 + 0.883940i 0.489585 + 0.871956i \(0.337148\pi\)
−0.999928 + 0.0119847i \(0.996185\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 + 3.46410i 0.112331 + 0.194563i 0.916710 0.399554i \(-0.130835\pi\)
−0.804379 + 0.594117i \(0.797502\pi\)
\(318\) 0 0
\(319\) 24.0000 41.5692i 1.34374 2.32743i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 + 15.5885i −0.496186 + 0.859419i
\(330\) 0 0
\(331\) −1.50000 2.59808i −0.0824475 0.142803i 0.821853 0.569699i \(-0.192940\pi\)
−0.904301 + 0.426896i \(0.859607\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 + 5.19615i 0.163908 + 0.283896i
\(336\) 0 0
\(337\) −4.50000 + 7.79423i −0.245131 + 0.424579i −0.962168 0.272456i \(-0.912164\pi\)
0.717038 + 0.697034i \(0.245498\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) 0.500000 + 0.866025i 0.0267644 + 0.0463573i 0.879097 0.476642i \(-0.158146\pi\)
−0.852333 + 0.523000i \(0.824813\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 17.3205i −0.532246 0.921878i −0.999291 0.0376440i \(-0.988015\pi\)
0.467045 0.884234i \(-0.345319\pi\)
\(354\) 0 0
\(355\) −12.0000 + 20.7846i −0.636894 + 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.0000 25.9808i 0.785136 1.35990i
\(366\) 0 0
\(367\) −1.50000 2.59808i −0.0782994 0.135618i 0.824217 0.566274i \(-0.191616\pi\)
−0.902516 + 0.430656i \(0.858282\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 10.3923i −0.311504 0.539542i
\(372\) 0 0
\(373\) −11.5000 + 19.9186i −0.595447 + 1.03135i 0.398036 + 0.917370i \(0.369692\pi\)
−0.993484 + 0.113975i \(0.963641\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −27.0000 −1.38690 −0.693448 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) −18.0000 31.1769i −0.917365 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.0000 29.4449i −0.861934 1.49291i −0.870059 0.492947i \(-0.835920\pi\)
0.00812520 0.999967i \(-0.497414\pi\)
\(390\) 0 0
\(391\) 6.00000 10.3923i 0.303433 0.525561i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 18.0000 0.905678
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 17.3205i 0.499376 0.864945i −0.500624 0.865665i \(-0.666896\pi\)
1.00000 0.000720188i \(0.000229243\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.0000 36.3731i −1.04093 1.80295i
\(408\) 0 0
\(409\) −10.5000 + 18.1865i −0.519192 + 0.899266i 0.480560 + 0.876962i \(0.340434\pi\)
−0.999751 + 0.0223042i \(0.992900\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.0000 + 25.9808i −0.732798 + 1.26924i 0.222885 + 0.974845i \(0.428453\pi\)
−0.955683 + 0.294398i \(0.904881\pi\)
\(420\) 0 0
\(421\) 7.50000 + 12.9904i 0.365528 + 0.633112i 0.988861 0.148844i \(-0.0475552\pi\)
−0.623333 + 0.781956i \(0.714222\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 1.73205i −0.0485071 0.0840168i
\(426\) 0 0
\(427\) 1.50000 2.59808i 0.0725901 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.00000 15.5885i 0.430528 0.745697i
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 + 10.3923i 0.285069 + 0.493753i 0.972626 0.232377i \(-0.0746503\pi\)
−0.687557 + 0.726130i \(0.741317\pi\)
\(444\) 0 0
\(445\) 10.0000 17.3205i 0.474045 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 48.0000 2.26023
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.00000 15.5885i 0.421927 0.730798i
\(456\) 0 0
\(457\) 15.0000 + 25.9808i 0.701670 + 1.21533i 0.967880 + 0.251414i \(0.0808954\pi\)
−0.266209 + 0.963915i \(0.585771\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00000 + 1.73205i 0.0465746 + 0.0806696i 0.888373 0.459123i \(-0.151836\pi\)
−0.841798 + 0.539792i \(0.818503\pi\)
\(462\) 0 0
\(463\) −7.50000 + 12.9904i −0.348555 + 0.603714i −0.985993 0.166787i \(-0.946661\pi\)
0.637438 + 0.770501i \(0.279994\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.0000 62.3538i 1.65528 2.86703i
\(474\) 0 0
\(475\) −1.50000 2.59808i −0.0688247 0.119208i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.00000 10.3923i −0.274147 0.474837i 0.695773 0.718262i \(-0.255062\pi\)
−0.969920 + 0.243426i \(0.921729\pi\)
\(480\) 0 0
\(481\) 10.5000 18.1865i 0.478759 0.829235i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) 15.0000 0.679715 0.339857 0.940477i \(-0.389621\pi\)
0.339857 + 0.940477i \(0.389621\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −21.0000 + 36.3731i −0.947717 + 1.64149i −0.197499 + 0.980303i \(0.563282\pi\)
−0.750218 + 0.661190i \(0.770052\pi\)
\(492\) 0 0
\(493\) −8.00000 13.8564i −0.360302 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.0000 31.1769i −0.807410 1.39848i
\(498\) 0 0
\(499\) −12.0000 + 20.7846i −0.537194 + 0.930447i 0.461860 + 0.886953i \(0.347182\pi\)
−0.999054 + 0.0434940i \(0.986151\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.00000 + 8.66025i −0.221621 + 0.383859i −0.955300 0.295637i \(-0.904468\pi\)
0.733679 + 0.679496i \(0.237801\pi\)
\(510\) 0 0
\(511\) 22.5000 + 38.9711i 0.995341 + 1.72398i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −15.0000 25.9808i −0.660979 1.14485i
\(516\) 0 0
\(517\) −18.0000 + 31.1769i −0.791639 + 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 9.00000 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 + 20.7846i 0.519778 + 0.900281i
\(534\) 0 0
\(535\) 6.00000 10.3923i 0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −15.0000 −0.644900 −0.322450 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.00000 + 10.3923i −0.257012 + 0.445157i
\(546\) 0 0
\(547\) 4.50000 + 7.79423i 0.192406 + 0.333257i 0.946047 0.324029i \(-0.105038\pi\)
−0.753641 + 0.657286i \(0.771704\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 20.7846i −0.511217 0.885454i
\(552\) 0 0
\(553\) −13.5000 + 23.3827i −0.574078 + 0.994333i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 20.7846i 0.505740 0.875967i −0.494238 0.869326i \(-0.664553\pi\)
0.999978 0.00664037i \(-0.00211371\pi\)
\(564\) 0 0
\(565\) −14.0000 24.2487i −0.588984 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −23.0000 39.8372i −0.964210 1.67006i −0.711722 0.702461i \(-0.752085\pi\)
−0.252488 0.967600i \(-0.581249\pi\)
\(570\) 0 0
\(571\) 1.50000 2.59808i 0.0627730 0.108726i −0.832931 0.553377i \(-0.813339\pi\)
0.895704 + 0.444651i \(0.146672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 18.0000 31.1769i 0.746766 1.29344i
\(582\) 0 0
\(583\) −12.0000 20.7846i −0.496989 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.00000 15.5885i −0.371470 0.643404i 0.618322 0.785925i \(-0.287813\pi\)
−0.989792 + 0.142520i \(0.954479\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.0000 1.14982 0.574911 0.818216i \(-0.305037\pi\)
0.574911 + 0.818216i \(0.305037\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 + 20.7846i −0.490307 + 0.849236i −0.999938 0.0111569i \(-0.996449\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(600\) 0 0
\(601\) 13.0000 + 22.5167i 0.530281 + 0.918474i 0.999376 + 0.0353259i \(0.0112469\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.0000 43.3013i −1.01639 1.76045i
\(606\) 0 0
\(607\) −1.50000 + 2.59808i −0.0608831 + 0.105453i −0.894860 0.446346i \(-0.852725\pi\)
0.833977 + 0.551799i \(0.186058\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00000 + 1.73205i −0.0402585 + 0.0697297i −0.885453 0.464730i \(-0.846151\pi\)
0.845194 + 0.534460i \(0.179485\pi\)
\(618\) 0 0
\(619\) −4.50000 7.79423i −0.180870 0.313276i 0.761307 0.648392i \(-0.224558\pi\)
−0.942177 + 0.335115i \(0.891225\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.0000 + 25.9808i 0.600962 + 1.04090i
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.0000 −0.558217
\(630\) 0 0
\(631\) −27.0000 −1.07485 −0.537427 0.843311i \(-0.680603\pi\)
−0.537427 + 0.843311i \(0.680603\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 20.7846i 0.476205 0.824812i
\(636\) 0 0
\(637\) 3.00000 + 5.19615i 0.118864 + 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 3.46410i −0.0789953 0.136824i 0.823821 0.566849i \(-0.191838\pi\)
−0.902817 + 0.430026i \(0.858505\pi\)
\(642\) 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
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 24.2487i 0.547862 0.948925i −0.450558 0.892747i \(-0.648775\pi\)
0.998421 0.0561784i \(-0.0178916\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 + 10.3923i 0.233727 + 0.404827i 0.958902 0.283738i \(-0.0915745\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(660\) 0 0
\(661\) 2.50000 4.33013i 0.0972387 0.168422i −0.813302 0.581842i \(-0.802332\pi\)
0.910541 + 0.413419i \(0.135666\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −18.0000 −0.698010
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 5.19615i 0.115814 0.200595i
\(672\) 0 0
\(673\) 0.500000 + 0.866025i 0.0192736 + 0.0333828i 0.875501 0.483216i \(-0.160531\pi\)
−0.856228 + 0.516599i \(0.827198\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.00000 1.73205i −0.0384331 0.0665681i 0.846169 0.532915i \(-0.178903\pi\)
−0.884602 + 0.466347i \(0.845570\pi\)
\(678\) 0 0
\(679\) −13.5000 + 23.3827i −0.518082 + 0.897345i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 10.3923i 0.228582 0.395915i
\(690\) 0 0
\(691\) −6.00000 10.3923i −0.228251 0.395342i 0.729039 0.684472i \(-0.239967\pi\)
−0.957290 + 0.289130i \(0.906634\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.0000 + 36.3731i 0.796575 + 1.37971i
\(696\) 0 0
\(697\) 8.00000 13.8564i 0.303022 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 41.5692i 0.902613 1.56337i
\(708\) 0 0
\(709\) 25.5000 + 44.1673i 0.957673 + 1.65874i 0.728130 + 0.685439i \(0.240390\pi\)
0.229543 + 0.973299i \(0.426277\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 18.0000 31.1769i 0.673162 1.16595i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 45.0000 1.67589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 + 6.92820i −0.148556 + 0.257307i
\(726\) 0 0
\(727\) −12.0000 20.7846i −0.445055 0.770859i 0.553001 0.833181i \(-0.313483\pi\)
−0.998056 + 0.0623223i \(0.980149\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 0 0
\(733\) 9.00000 15.5885i 0.332423 0.575773i −0.650564 0.759452i \(-0.725467\pi\)
0.982986 + 0.183679i \(0.0588007\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.0000 −0.663039
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.00000 + 5.19615i −0.110059 + 0.190628i −0.915794 0.401648i \(-0.868437\pi\)
0.805735 + 0.592277i \(0.201771\pi\)
\(744\) 0 0
\(745\) −4.00000 6.92820i −0.146549 0.253830i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 + 15.5885i 0.328853 + 0.569590i
\(750\) 0 0
\(751\) −22.5000 + 38.9711i −0.821037 + 1.42208i 0.0838743 + 0.996476i \(0.473271\pi\)
−0.904911 + 0.425601i \(0.860063\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −27.0000 −0.981332 −0.490666 0.871348i \(-0.663246\pi\)
−0.490666 + 0.871348i \(0.663246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.0000 29.4449i 0.616250 1.06738i −0.373914 0.927463i \(-0.621985\pi\)
0.990164 0.139912i \(-0.0446820\pi\)
\(762\) 0 0
\(763\) −9.00000 15.5885i −0.325822 0.564340i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.00000 + 15.5885i 0.324971 + 0.562867i
\(768\) 0 0
\(769\) 24.5000 42.4352i 0.883493 1.53025i 0.0360609 0.999350i \(-0.488519\pi\)
0.847432 0.530904i \(-0.178148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 20.7846i 0.429945 0.744686i
\(780\) 0 0
\(781\) −36.0000 62.3538i −1.28818 2.23120i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 + 24.2487i 0.499681 + 0.865474i
\(786\) 0 0
\(787\) 13.5000 23.3827i 0.481223 0.833503i −0.518545 0.855050i \(-0.673526\pi\)
0.999768 + 0.0215477i \(0.00685939\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 3.00000 0.106533
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.0000 34.6410i 0.708436 1.22705i −0.257001 0.966411i \(-0.582734\pi\)
0.965437 0.260637i \(-0.0839324\pi\)
\(798\) 0 0
\(799\) 6.00000 + 10.3923i 0.212265 + 0.367653i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.0000 + 77.9423i 1.58802 + 2.75052i
\(804\) 0 0
\(805\) −18.0000 + 31.1769i −0.634417 + 1.09884i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.00000 + 5.19615i −0.105085 + 0.182013i
\(816\) 0 0
\(817\) −18.0000 31.1769i −0.629740 1.09074i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.0000 43.3013i −0.872506 1.51122i −0.859396 0.511311i \(-0.829160\pi\)
−0.0131101 0.999914i \(-0.504173\pi\)
\(822\) 0 0
\(823\) −4.50000 + 7.79423i −0.156860 + 0.271690i −0.933735 0.357966i \(-0.883471\pi\)
0.776875 + 0.629655i \(0.216804\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 0 0
\(829\) 45.0000 1.56291 0.781457 0.623959i \(-0.214477\pi\)
0.781457 + 0.623959i \(0.214477\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.00000 3.46410i 0.0692959 0.120024i
\(834\) 0 0
\(835\) 18.0000 + 31.1769i 0.622916 + 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.0000 + 31.1769i 0.621429 + 1.07635i 0.989220 + 0.146438i \(0.0467809\pi\)
−0.367791 + 0.929909i \(0.619886\pi\)
\(840\) 0 0
\(841\) −17.5000 + 30.3109i −0.603448 + 1.04520i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.00000 −0.275208
\(846\) 0 0
\(847\) 75.0000 2.57703
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.0000 + 36.3731i −0.719871 + 1.24685i
\(852\) 0 0
\(853\) 26.5000 + 45.8993i 0.907343 + 1.57156i 0.817741 + 0.575586i \(0.195226\pi\)
0.0896015 + 0.995978i \(0.471441\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.00000 + 3.46410i 0.0683187 + 0.118331i 0.898161 0.439666i \(-0.144903\pi\)
−0.829843 + 0.557998i \(0.811570\pi\)
\(858\) 0 0
\(859\) −13.5000 + 23.3827i −0.460614 + 0.797807i −0.998992 0.0448968i \(-0.985704\pi\)
0.538378 + 0.842704i \(0.319037\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.0000 −1.42970 −0.714848 0.699280i \(-0.753504\pi\)
−0.714848 + 0.699280i \(0.753504\pi\)
\(864\) 0 0
\(865\) 16.0000 0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27.0000 + 46.7654i −0.915912 + 1.58641i
\(870\) 0 0
\(871\) −4.50000 7.79423i −0.152477 0.264097i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.0000 + 31.1769i 0.608511 + 1.05397i
\(876\) 0 0
\(877\) 6.50000 11.2583i 0.219489 0.380167i −0.735163 0.677891i \(-0.762894\pi\)
0.954652 + 0.297724i \(0.0962275\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000 31.1769i 0.604381 1.04682i −0.387768 0.921757i \(-0.626754\pi\)
0.992149 0.125061i \(-0.0399128\pi\)
\(888\) 0 0
\(889\) 18.0000 + 31.1769i 0.603701 + 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.00000 + 15.5885i 0.301174 + 0.521648i
\(894\) 0 0
\(895\) 12.0000 20.7846i 0.401116 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.00000 + 15.5885i −0.299170 + 0.518178i
\(906\) 0 0
\(907\) 19.5000 + 33.7750i 0.647487 + 1.12148i 0.983721 + 0.179702i \(0.0575133\pi\)
−0.336234 + 0.941778i \(0.609153\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 41.5692i −0.795155 1.37725i −0.922740 0.385422i \(-0.874056\pi\)
0.127585 0.991828i \(-0.459277\pi\)
\(912\) 0 0
\(913\) 36.0000 62.3538i 1.19143 2.06361i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18.0000 31.1769i 0.592477 1.02620i
\(924\) 0 0
\(925\) 3.50000 + 6.06218i 0.115079 + 0.199323i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.0000 24.2487i −0.459325 0.795574i 0.539600 0.841921i \(-0.318575\pi\)
−0.998925 + 0.0463469i \(0.985242\pi\)
\(930\) 0 0
\(931\) 3.00000 5.19615i 0.0983210 0.170297i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.0000 + 19.0526i −0.358590 + 0.621096i −0.987725 0.156200i \(-0.950076\pi\)
0.629136 + 0.777295i \(0.283409\pi\)
\(942\) 0 0
\(943\) −24.0000 41.5692i −0.781548 1.35368i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.00000 15.5885i −0.292461 0.506557i 0.681930 0.731417i \(-0.261141\pi\)
−0.974391 + 0.224860i \(0.927807\pi\)
\(948\) 0 0
\(949\) −22.5000 + 38.9711i −0.730381 + 1.26506i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 46.0000 1.49009 0.745043 0.667016i \(-0.232429\pi\)
0.745043 + 0.667016i \(0.232429\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 21.0000 36.3731i 0.678125 1.17455i
\(960\) 0 0
\(961\) 15.5000 + 26.8468i 0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.00000 + 8.66025i 0.160956 + 0.278783i
\(966\) 0 0
\(967\) −16.5000 + 28.5788i −0.530604 + 0.919033i 0.468758 + 0.883327i \(0.344702\pi\)
−0.999362 + 0.0357069i \(0.988632\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) −63.0000 −2.01969
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.0000 + 38.1051i −0.703842 + 1.21909i 0.263265 + 0.964723i \(0.415201\pi\)
−0.967108 + 0.254367i \(0.918133\pi\)
\(978\) 0 0
\(979\) 30.0000 + 51.9615i 0.958804 + 1.66070i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.00000 + 5.19615i 0.0956851 + 0.165732i 0.909894 0.414840i \(-0.136162\pi\)
−0.814209 + 0.580572i \(0.802829\pi\)
\(984\) 0 0
\(985\) −2.00000 + 3.46410i −0.0637253 + 0.110375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −72.0000 −2.28947
\(990\) 0 0
\(991\) −21.0000 −0.667087 −0.333543 0.942735i \(-0.608244\pi\)
−0.333543 + 0.942735i \(0.608244\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.0000 + 25.9808i −0.475532 + 0.823646i
\(996\) 0 0
\(997\) −13.0000 22.5167i −0.411714 0.713110i 0.583363 0.812211i \(-0.301736\pi\)
−0.995077 + 0.0991016i \(0.968403\pi\)
\(998\) 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.c.1729.1 2
3.2 odd 2 2592.2.i.r.1729.1 2
4.3 odd 2 2592.2.i.g.1729.1 2
9.2 odd 6 2592.2.i.r.865.1 2
9.4 even 3 864.2.a.l.1.1 yes 1
9.5 odd 6 864.2.a.d.1.1 yes 1
9.7 even 3 inner 2592.2.i.c.865.1 2
12.11 even 2 2592.2.i.v.1729.1 2
36.7 odd 6 2592.2.i.g.865.1 2
36.11 even 6 2592.2.i.v.865.1 2
36.23 even 6 864.2.a.a.1.1 1
36.31 odd 6 864.2.a.i.1.1 yes 1
72.5 odd 6 1728.2.a.x.1.1 1
72.13 even 6 1728.2.a.h.1.1 1
72.59 even 6 1728.2.a.u.1.1 1
72.67 odd 6 1728.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.a.1.1 1 36.23 even 6
864.2.a.d.1.1 yes 1 9.5 odd 6
864.2.a.i.1.1 yes 1 36.31 odd 6
864.2.a.l.1.1 yes 1 9.4 even 3
1728.2.a.e.1.1 1 72.67 odd 6
1728.2.a.h.1.1 1 72.13 even 6
1728.2.a.u.1.1 1 72.59 even 6
1728.2.a.x.1.1 1 72.5 odd 6
2592.2.i.c.865.1 2 9.7 even 3 inner
2592.2.i.c.1729.1 2 1.1 even 1 trivial
2592.2.i.g.865.1 2 36.7 odd 6
2592.2.i.g.1729.1 2 4.3 odd 2
2592.2.i.r.865.1 2 9.2 odd 6
2592.2.i.r.1729.1 2 3.2 odd 2
2592.2.i.v.865.1 2 36.11 even 6
2592.2.i.v.1729.1 2 12.11 even 2