Properties

Label 864.2.i.b.577.1
Level $864$
Weight $2$
Character 864.577
Analytic conductor $6.899$
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] = [864,2,Mod(289,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, 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("864.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 864.577
Dual form 864.2.i.b.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(2.00000 - 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-2.50000 - 4.33013i) q^{11} +(1.00000 - 1.73205i) q^{13} +3.00000 q^{17} +1.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-1.00000 - 1.73205i) q^{29} +(2.00000 - 3.46410i) q^{31} +8.00000 q^{35} -8.00000 q^{37} +(0.500000 - 0.866025i) q^{41} +(3.50000 + 6.06218i) q^{43} +(1.00000 + 1.73205i) q^{47} +(1.50000 - 2.59808i) q^{49} +4.00000 q^{53} -20.0000 q^{55} +(2.50000 - 4.33013i) q^{59} +(-4.00000 - 6.92820i) q^{65} +(6.50000 - 11.2583i) q^{67} -8.00000 q^{71} +3.00000 q^{73} +(5.00000 - 8.66025i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(6.00000 + 10.3923i) q^{83} +(6.00000 - 10.3923i) q^{85} +10.0000 q^{89} +4.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(5.50000 + 9.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 2 q^{7} - 5 q^{11} + 2 q^{13} + 6 q^{17} + 2 q^{19} - 6 q^{23} - 11 q^{25} - 2 q^{29} + 4 q^{31} + 16 q^{35} - 16 q^{37} + q^{41} + 7 q^{43} + 2 q^{47} + 3 q^{49} + 8 q^{53} - 40 q^{55} + 5 q^{59} - 8 q^{65} + 13 q^{67} - 16 q^{71} + 6 q^{73} + 10 q^{77} - 8 q^{79} + 12 q^{83} + 12 q^{85} + 20 q^{89} + 8 q^{91} + 4 q^{95} + 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) 2.00000 3.46410i 0.894427 1.54919i 0.0599153 0.998203i \(-0.480917\pi\)
0.834512 0.550990i \(-0.185750\pi\)
\(6\) 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
\(11\) −2.50000 4.33013i −0.753778 1.30558i −0.945979 0.324227i \(-0.894896\pi\)
0.192201 0.981356i \(-0.438437\pi\)
\(12\) 0 0
\(13\) 1.00000 1.73205i 0.277350 0.480384i −0.693375 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 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
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 1.73205i −0.185695 0.321634i 0.758115 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) 2.00000 3.46410i 0.359211 0.622171i −0.628619 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 0.866025i 0.0780869 0.135250i −0.824338 0.566099i \(-0.808452\pi\)
0.902424 + 0.430848i \(0.141786\pi\)
\(42\) 0 0
\(43\) 3.50000 + 6.06218i 0.533745 + 0.924473i 0.999223 + 0.0394140i \(0.0125491\pi\)
−0.465478 + 0.885059i \(0.654118\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000 + 1.73205i 0.145865 + 0.252646i 0.929695 0.368329i \(-0.120070\pi\)
−0.783830 + 0.620975i \(0.786737\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 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
\(55\) −20.0000 −2.69680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.50000 4.33013i 0.325472 0.563735i −0.656136 0.754643i \(-0.727810\pi\)
0.981608 + 0.190909i \(0.0611434\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 6.92820i −0.496139 0.859338i
\(66\) 0 0
\(67\) 6.50000 11.2583i 0.794101 1.37542i −0.129307 0.991605i \(-0.541275\pi\)
0.923408 0.383819i \(-0.125391\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000 8.66025i 0.569803 0.986928i
\(78\) 0 0
\(79\) −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 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
\(85\) 6.00000 10.3923i 0.650791 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 3.46410i 0.205196 0.355409i
\(96\) 0 0
\(97\) 5.50000 + 9.52628i 0.558440 + 0.967247i 0.997627 + 0.0688512i \(0.0219334\pi\)
−0.439187 + 0.898396i \(0.644733\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 + 10.3923i 0.597022 + 1.03407i 0.993258 + 0.115924i \(0.0369830\pi\)
−0.396236 + 0.918149i \(0.629684\pi\)
\(102\) 0 0
\(103\) −3.00000 + 5.19615i −0.295599 + 0.511992i −0.975124 0.221660i \(-0.928852\pi\)
0.679525 + 0.733652i \(0.262186\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.00000 −0.483368 −0.241684 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.00000 + 1.73205i −0.0940721 + 0.162938i −0.909221 0.416314i \(-0.863322\pi\)
0.815149 + 0.579252i \(0.196655\pi\)
\(114\) 0 0
\(115\) 12.0000 + 20.7846i 1.11901 + 1.93817i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 + 5.19615i 0.275010 + 0.476331i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 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
\(131\) 8.00000 13.8564i 0.698963 1.21064i −0.269863 0.962899i \(-0.586978\pi\)
0.968826 0.247741i \(-0.0796882\pi\)
\(132\) 0 0
\(133\) 1.00000 + 1.73205i 0.0867110 + 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.50000 2.59808i −0.128154 0.221969i 0.794808 0.606861i \(-0.207572\pi\)
−0.922961 + 0.384893i \(0.874238\pi\)
\(138\) 0 0
\(139\) 6.50000 11.2583i 0.551323 0.954919i −0.446857 0.894606i \(-0.647457\pi\)
0.998179 0.0603135i \(-0.0192101\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) 9.00000 + 15.5885i 0.732410 + 1.26857i 0.955851 + 0.293853i \(0.0949377\pi\)
−0.223441 + 0.974717i \(0.571729\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 13.8564i −0.642575 1.11297i
\(156\) 0 0
\(157\) −10.0000 + 17.3205i −0.798087 + 1.38233i 0.122774 + 0.992435i \(0.460821\pi\)
−0.920860 + 0.389892i \(0.872512\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 + 13.8564i −0.619059 + 1.07224i 0.370599 + 0.928793i \(0.379152\pi\)
−0.989658 + 0.143448i \(0.954181\pi\)
\(168\) 0 0
\(169\) 4.50000 + 7.79423i 0.346154 + 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000 + 15.5885i 0.684257 + 1.18517i 0.973670 + 0.227964i \(0.0732068\pi\)
−0.289412 + 0.957205i \(0.593460\pi\)
\(174\) 0 0
\(175\) 11.0000 19.0526i 0.831522 1.44024i
\(176\) 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
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −16.0000 + 27.7128i −1.17634 + 2.03749i
\(186\) 0 0
\(187\) −7.50000 12.9904i −0.548454 0.949951i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 5.19615i −0.217072 0.375980i 0.736839 0.676068i \(-0.236317\pi\)
−0.953912 + 0.300088i \(0.902984\pi\)
\(192\) 0 0
\(193\) 1.50000 2.59808i 0.107972 0.187014i −0.806976 0.590584i \(-0.798898\pi\)
0.914949 + 0.403570i \(0.132231\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.00000 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.00000 3.46410i 0.140372 0.243132i
\(204\) 0 0
\(205\) −2.00000 3.46410i −0.139686 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.50000 4.33013i −0.172929 0.299521i
\(210\) 0 0
\(211\) −14.0000 + 24.2487i −0.963800 + 1.66935i −0.250994 + 0.967989i \(0.580757\pi\)
−0.712806 + 0.701361i \(0.752576\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 28.0000 1.90958
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 5.19615i 0.201802 0.349531i
\(222\) 0 0
\(223\) 7.00000 + 12.1244i 0.468755 + 0.811907i 0.999362 0.0357107i \(-0.0113695\pi\)
−0.530607 + 0.847618i \(0.678036\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.50000 + 2.59808i 0.0995585 + 0.172440i 0.911502 0.411296i \(-0.134924\pi\)
−0.811943 + 0.583736i \(0.801590\pi\)
\(228\) 0 0
\(229\) −7.00000 + 12.1244i −0.462573 + 0.801200i −0.999088 0.0426906i \(-0.986407\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.00000 5.19615i 0.194054 0.336111i −0.752536 0.658551i \(-0.771170\pi\)
0.946590 + 0.322440i \(0.104503\pi\)
\(240\) 0 0
\(241\) 11.5000 + 19.9186i 0.740780 + 1.28307i 0.952141 + 0.305661i \(0.0988773\pi\)
−0.211360 + 0.977408i \(0.567789\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.00000 10.3923i −0.383326 0.663940i
\(246\) 0 0
\(247\) 1.00000 1.73205i 0.0636285 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 30.0000 1.88608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.50000 + 4.33013i −0.155946 + 0.270106i −0.933403 0.358830i \(-0.883176\pi\)
0.777457 + 0.628936i \(0.216509\pi\)
\(258\) 0 0
\(259\) −8.00000 13.8564i −0.497096 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.00000 + 5.19615i 0.184988 + 0.320408i 0.943572 0.331166i \(-0.107442\pi\)
−0.758585 + 0.651575i \(0.774109\pi\)
\(264\) 0 0
\(265\) 8.00000 13.8564i 0.491436 0.851192i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −27.5000 + 47.6314i −1.65831 + 2.87228i
\(276\) 0 0
\(277\) 7.00000 + 12.1244i 0.420589 + 0.728482i 0.995997 0.0893846i \(-0.0284900\pi\)
−0.575408 + 0.817867i \(0.695157\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.00000 15.5885i −0.536895 0.929929i −0.999069 0.0431402i \(-0.986264\pi\)
0.462174 0.886789i \(-0.347070\pi\)
\(282\) 0 0
\(283\) 14.0000 24.2487i 0.832214 1.44144i −0.0640654 0.997946i \(-0.520407\pi\)
0.896279 0.443491i \(-0.146260\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000 25.9808i 0.876309 1.51781i 0.0209480 0.999781i \(-0.493332\pi\)
0.855361 0.518032i \(-0.173335\pi\)
\(294\) 0 0
\(295\) −10.0000 17.3205i −0.582223 1.00844i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) −7.00000 + 12.1244i −0.403473 + 0.698836i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.00000 −0.513657 −0.256829 0.966457i \(-0.582678\pi\)
−0.256829 + 0.966457i \(0.582678\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.00000 15.5885i 0.510343 0.883940i −0.489585 0.871956i \(-0.662852\pi\)
0.999928 0.0119847i \(-0.00381495\pi\)
\(312\) 0 0
\(313\) −6.50000 11.2583i −0.367402 0.636358i 0.621757 0.783210i \(-0.286419\pi\)
−0.989158 + 0.146852i \(0.953086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 + 15.5885i 0.505490 + 0.875535i 0.999980 + 0.00635137i \(0.00202172\pi\)
−0.494489 + 0.869184i \(0.664645\pi\)
\(318\) 0 0
\(319\) −5.00000 + 8.66025i −0.279946 + 0.484881i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.00000 + 3.46410i −0.110264 + 0.190982i
\(330\) 0 0
\(331\) 2.00000 + 3.46410i 0.109930 + 0.190404i 0.915742 0.401768i \(-0.131604\pi\)
−0.805812 + 0.592172i \(0.798271\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −26.0000 45.0333i −1.42053 2.46043i
\(336\) 0 0
\(337\) 12.5000 21.6506i 0.680918 1.17939i −0.293783 0.955872i \(-0.594914\pi\)
0.974701 0.223513i \(-0.0717525\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.500000 + 0.866025i −0.0268414 + 0.0464907i −0.879134 0.476575i \(-0.841878\pi\)
0.852293 + 0.523065i \(0.175212\pi\)
\(348\) 0 0
\(349\) −8.00000 13.8564i −0.428230 0.741716i 0.568486 0.822693i \(-0.307529\pi\)
−0.996716 + 0.0809766i \(0.974196\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.5000 + 30.3109i 0.931431 + 1.61329i 0.780878 + 0.624684i \(0.214772\pi\)
0.150553 + 0.988602i \(0.451894\pi\)
\(354\) 0 0
\(355\) −16.0000 + 27.7128i −0.849192 + 1.47084i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 10.3923i 0.314054 0.543958i
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 + 6.92820i 0.207670 + 0.359694i
\(372\) 0 0
\(373\) 11.0000 19.0526i 0.569558 0.986504i −0.427051 0.904227i \(-0.640448\pi\)
0.996610 0.0822766i \(-0.0262191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.0000 17.3205i 0.510976 0.885037i −0.488943 0.872316i \(-0.662617\pi\)
0.999919 0.0127209i \(-0.00404928\pi\)
\(384\) 0 0
\(385\) −20.0000 34.6410i −1.01929 1.76547i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.00000 5.19615i −0.152106 0.263455i 0.779895 0.625910i \(-0.215272\pi\)
−0.932002 + 0.362454i \(0.881939\pi\)
\(390\) 0 0
\(391\) −9.00000 + 15.5885i −0.455150 + 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −32.0000 −1.61009
\(396\) 0 0
\(397\) −32.0000 −1.60603 −0.803017 0.595956i \(-0.796773\pi\)
−0.803017 + 0.595956i \(0.796773\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.50000 4.33013i 0.124844 0.216236i −0.796828 0.604206i \(-0.793490\pi\)
0.921672 + 0.387970i \(0.126824\pi\)
\(402\) 0 0
\(403\) −4.00000 6.92820i −0.199254 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 + 34.6410i 0.991363 + 1.71709i
\(408\) 0 0
\(409\) −12.5000 + 21.6506i −0.618085 + 1.07056i 0.371750 + 0.928333i \(0.378758\pi\)
−0.989835 + 0.142222i \(0.954575\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0000 0.492068
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 3.46410i 0.0977064 0.169232i −0.813029 0.582224i \(-0.802183\pi\)
0.910735 + 0.412991i \(0.135516\pi\)
\(420\) 0 0
\(421\) −4.00000 6.92820i −0.194948 0.337660i 0.751935 0.659237i \(-0.229121\pi\)
−0.946883 + 0.321577i \(0.895787\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −16.5000 28.5788i −0.800368 1.38628i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00000 + 5.19615i −0.143509 + 0.248566i
\(438\) 0 0
\(439\) 4.00000 + 6.92820i 0.190910 + 0.330665i 0.945552 0.325471i \(-0.105523\pi\)
−0.754642 + 0.656136i \(0.772190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13.5000 23.3827i −0.641404 1.11094i −0.985119 0.171871i \(-0.945019\pi\)
0.343715 0.939074i \(-0.388315\pi\)
\(444\) 0 0
\(445\) 20.0000 34.6410i 0.948091 1.64214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.0000 1.46298 0.731490 0.681852i \(-0.238825\pi\)
0.731490 + 0.681852i \(0.238825\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 13.8564i 0.375046 0.649598i
\(456\) 0 0
\(457\) 3.50000 + 6.06218i 0.163723 + 0.283577i 0.936201 0.351465i \(-0.114316\pi\)
−0.772478 + 0.635042i \(0.780983\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 15.5885i −0.419172 0.726027i 0.576685 0.816967i \(-0.304346\pi\)
−0.995856 + 0.0909401i \(0.971013\pi\)
\(462\) 0 0
\(463\) 8.00000 13.8564i 0.371792 0.643962i −0.618050 0.786139i \(-0.712077\pi\)
0.989841 + 0.142177i \(0.0454103\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.00000 −0.323921 −0.161961 0.986797i \(-0.551782\pi\)
−0.161961 + 0.986797i \(0.551782\pi\)
\(468\) 0 0
\(469\) 26.0000 1.20057
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.5000 30.3109i 0.804651 1.39370i
\(474\) 0 0
\(475\) −5.50000 9.52628i −0.252357 0.437096i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.00000 + 15.5885i 0.411220 + 0.712255i 0.995023 0.0996406i \(-0.0317693\pi\)
−0.583803 + 0.811895i \(0.698436\pi\)
\(480\) 0 0
\(481\) −8.00000 + 13.8564i −0.364769 + 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 44.0000 1.99794
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.50000 6.06218i 0.157953 0.273582i −0.776178 0.630514i \(-0.782844\pi\)
0.934130 + 0.356932i \(0.116177\pi\)
\(492\) 0 0
\(493\) −3.00000 5.19615i −0.135113 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 13.8564i −0.358849 0.621545i
\(498\) 0 0
\(499\) −10.5000 + 18.1865i −0.470045 + 0.814141i −0.999413 0.0342508i \(-0.989095\pi\)
0.529369 + 0.848392i \(0.322429\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 48.0000 2.13597
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.0000 20.7846i 0.531891 0.921262i −0.467416 0.884037i \(-0.654815\pi\)
0.999307 0.0372243i \(-0.0118516\pi\)
\(510\) 0 0
\(511\) 3.00000 + 5.19615i 0.132712 + 0.229864i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.0000 + 20.7846i 0.528783 + 0.915879i
\(516\) 0 0
\(517\) 5.00000 8.66025i 0.219900 0.380878i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 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
\(535\) −10.0000 + 17.3205i −0.432338 + 0.748831i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −15.0000 −0.646096
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16.0000 27.7128i 0.685365 1.18709i
\(546\) 0 0
\(547\) −16.5000 28.5788i −0.705489 1.22194i −0.966515 0.256611i \(-0.917394\pi\)
0.261026 0.965332i \(-0.415939\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.00000 1.73205i −0.0426014 0.0737878i
\(552\) 0 0
\(553\) 8.00000 13.8564i 0.340195 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) 14.0000 0.592137
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.50000 12.9904i 0.316087 0.547479i −0.663581 0.748105i \(-0.730964\pi\)
0.979668 + 0.200625i \(0.0642974\pi\)
\(564\) 0 0
\(565\) 4.00000 + 6.92820i 0.168281 + 0.291472i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.50000 9.52628i −0.230572 0.399362i 0.727405 0.686209i \(-0.240726\pi\)
−0.957977 + 0.286846i \(0.907393\pi\)
\(570\) 0 0
\(571\) −6.50000 + 11.2583i −0.272017 + 0.471146i −0.969378 0.245573i \(-0.921024\pi\)
0.697362 + 0.716720i \(0.254357\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 66.0000 2.75239
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 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
\(587\) 4.50000 + 7.79423i 0.185735 + 0.321702i 0.943824 0.330449i \(-0.107200\pi\)
−0.758089 + 0.652151i \(0.773867\pi\)
\(588\) 0 0
\(589\) 2.00000 3.46410i 0.0824086 0.142736i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.0000 + 34.6410i −0.817178 + 1.41539i 0.0905757 + 0.995890i \(0.471129\pi\)
−0.907754 + 0.419504i \(0.862204\pi\)
\(600\) 0 0
\(601\) −9.50000 16.4545i −0.387513 0.671192i 0.604601 0.796528i \(-0.293332\pi\)
−0.992114 + 0.125336i \(0.959999\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 28.0000 + 48.4974i 1.13836 + 1.97170i
\(606\) 0 0
\(607\) −10.0000 + 17.3205i −0.405887 + 0.703018i −0.994424 0.105453i \(-0.966371\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.5000 + 38.9711i −0.905816 + 1.56892i −0.0859976 + 0.996295i \(0.527408\pi\)
−0.819818 + 0.572624i \(0.805926\pi\)
\(618\) 0 0
\(619\) −10.5000 18.1865i −0.422031 0.730978i 0.574107 0.818780i \(-0.305349\pi\)
−0.996138 + 0.0878015i \(0.972016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000 + 17.3205i 0.400642 + 0.693932i
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.00000 6.92820i 0.158735 0.274937i
\(636\) 0 0
\(637\) −3.00000 5.19615i −0.118864 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.50000 + 4.33013i 0.0987441 + 0.171030i 0.911165 0.412042i \(-0.135184\pi\)
−0.812421 + 0.583071i \(0.801851\pi\)
\(642\) 0 0
\(643\) 7.50000 12.9904i 0.295771 0.512291i −0.679393 0.733775i \(-0.737757\pi\)
0.975164 + 0.221484i \(0.0710901\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −25.0000 −0.981336
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.0000 + 19.0526i −0.430463 + 0.745584i −0.996913 0.0785119i \(-0.974983\pi\)
0.566450 + 0.824096i \(0.308316\pi\)
\(654\) 0 0
\(655\) −32.0000 55.4256i −1.25034 2.16566i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.00000 10.3923i −0.233727 0.404827i 0.725175 0.688565i \(-0.241759\pi\)
−0.958902 + 0.283738i \(0.908425\pi\)
\(660\) 0 0
\(661\) −16.0000 + 27.7128i −0.622328 + 1.07790i 0.366723 + 0.930330i \(0.380480\pi\)
−0.989051 + 0.147573i \(0.952854\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 1.73205i −0.0385472 0.0667657i 0.846108 0.533011i \(-0.178940\pi\)
−0.884655 + 0.466246i \(0.845606\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14.0000 24.2487i −0.538064 0.931954i −0.999008 0.0445248i \(-0.985823\pi\)
0.460945 0.887429i \(-0.347511\pi\)
\(678\) 0 0
\(679\) −11.0000 + 19.0526i −0.422141 + 0.731170i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.00000 6.92820i 0.152388 0.263944i
\(690\) 0 0
\(691\) 4.00000 + 6.92820i 0.152167 + 0.263561i 0.932024 0.362397i \(-0.118041\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −26.0000 45.0333i −0.986236 1.70821i
\(696\) 0 0
\(697\) 1.50000 2.59808i 0.0568166 0.0984092i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 + 20.7846i −0.451306 + 0.781686i
\(708\) 0 0
\(709\) 10.0000 + 17.3205i 0.375558 + 0.650485i 0.990410 0.138157i \(-0.0441178\pi\)
−0.614852 + 0.788642i \(0.710784\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 + 20.7846i 0.449404 + 0.778390i
\(714\) 0 0
\(715\) −20.0000 + 34.6410i −0.747958 + 1.29550i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −11.0000 + 19.0526i −0.408530 + 0.707594i
\(726\) 0 0
\(727\) 15.0000 + 25.9808i 0.556319 + 0.963573i 0.997800 + 0.0663022i \(0.0211201\pi\)
−0.441480 + 0.897271i \(0.645547\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.5000 + 18.1865i 0.388357 + 0.672653i
\(732\) 0 0
\(733\) 17.0000 29.4449i 0.627909 1.08757i −0.360061 0.932929i \(-0.617244\pi\)
0.987971 0.154642i \(-0.0494225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −65.0000 −2.39431
\(738\) 0 0
\(739\) 49.0000 1.80249 0.901247 0.433306i \(-0.142653\pi\)
0.901247 + 0.433306i \(0.142653\pi\)
\(740\) 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
\(745\) 36.0000 + 62.3538i 1.31894 + 2.28447i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.00000 8.66025i −0.182696 0.316439i
\(750\) 0 0
\(751\) 26.0000 45.0333i 0.948753 1.64329i 0.200698 0.979653i \(-0.435679\pi\)
0.748056 0.663636i \(-0.230988\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 72.0000 2.62035
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) 8.00000 + 13.8564i 0.289619 + 0.501636i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.00000 8.66025i −0.180540 0.312704i
\(768\) 0 0
\(769\) −5.00000 + 8.66025i −0.180305 + 0.312297i −0.941984 0.335657i \(-0.891042\pi\)
0.761680 + 0.647954i \(0.224375\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) −44.0000 −1.58053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.500000 0.866025i 0.0179144 0.0310286i
\(780\) 0 0
\(781\) 20.0000 + 34.6410i 0.715656 + 1.23955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 40.0000 + 69.2820i 1.42766 + 2.47278i
\(786\) 0 0
\(787\) 18.0000 31.1769i 0.641631 1.11134i −0.343438 0.939175i \(-0.611592\pi\)
0.985069 0.172162i \(-0.0550751\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 20.7846i 0.425062 0.736229i −0.571364 0.820696i \(-0.693586\pi\)
0.996426 + 0.0844678i \(0.0269190\pi\)
\(798\) 0 0
\(799\) 3.00000 + 5.19615i 0.106132 + 0.183827i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.50000 12.9904i −0.264669 0.458421i
\(804\) 0 0
\(805\) −24.0000 + 41.5692i −0.845889 + 1.46512i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −17.0000 −0.597688 −0.298844 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(810\) 0 0
\(811\) 55.0000 1.93131 0.965656 0.259825i \(-0.0836650\pi\)
0.965656 + 0.259825i \(0.0836650\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −40.0000 + 69.2820i −1.40114 + 2.42684i
\(816\) 0 0
\(817\) 3.50000 + 6.06218i 0.122449 + 0.212089i
\(818\) 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
\(823\) −14.0000 + 24.2487i −0.488009 + 0.845257i −0.999905 0.0137907i \(-0.995610\pi\)
0.511896 + 0.859048i \(0.328943\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.50000 7.79423i 0.155916 0.270054i
\(834\) 0 0
\(835\) 32.0000 + 55.4256i 1.10741 + 1.91808i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.00000 3.46410i −0.0690477 0.119594i 0.829435 0.558604i \(-0.188663\pi\)
−0.898482 + 0.439010i \(0.855329\pi\)
\(840\) 0 0
\(841\) 12.5000 21.6506i 0.431034 0.746574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000 41.5692i 0.822709 1.42497i
\(852\) 0 0
\(853\) −15.0000 25.9808i −0.513590 0.889564i −0.999876 0.0157644i \(-0.994982\pi\)
0.486286 0.873800i \(-0.338351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.00000 8.66025i −0.170797 0.295829i 0.767902 0.640567i \(-0.221301\pi\)
−0.938699 + 0.344739i \(0.887967\pi\)
\(858\) 0 0
\(859\) −2.50000 + 4.33013i −0.0852989 + 0.147742i −0.905519 0.424307i \(-0.860518\pi\)
0.820220 + 0.572049i \(0.193851\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 0 0
\(865\) 72.0000 2.44807
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −20.0000 + 34.6410i −0.678454 + 1.17512i
\(870\) 0 0
\(871\) −13.0000 22.5167i −0.440488 0.762948i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24.0000 41.5692i −0.811348 1.40530i
\(876\) 0 0
\(877\) −20.0000 + 34.6410i −0.675352 + 1.16974i 0.301014 + 0.953620i \(0.402675\pi\)
−0.976366 + 0.216124i \(0.930658\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −13.0000 −0.437485 −0.218742 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) 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
\(889\) 2.00000 + 3.46410i 0.0670778 + 0.116182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.00000 + 1.73205i 0.0334637 + 0.0579609i
\(894\) 0 0
\(895\) −24.0000 + 41.5692i −0.802232 + 1.38951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.00000 6.92820i 0.132964 0.230301i
\(906\) 0 0
\(907\) −3.50000 6.06218i −0.116216 0.201291i 0.802049 0.597258i \(-0.203743\pi\)
−0.918265 + 0.395966i \(0.870410\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 20.7846i −0.397578 0.688625i 0.595849 0.803097i \(-0.296816\pi\)
−0.993426 + 0.114472i \(0.963482\pi\)
\(912\) 0 0
\(913\) 30.0000 51.9615i 0.992855 1.71968i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) 54.0000 1.78130 0.890648 0.454694i \(-0.150251\pi\)
0.890648 + 0.454694i \(0.150251\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.00000 + 13.8564i −0.263323 + 0.456089i
\(924\) 0 0
\(925\) 44.0000 + 76.2102i 1.44671 + 2.50578i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.00000 5.19615i −0.0984268 0.170480i 0.812607 0.582812i \(-0.198048\pi\)
−0.911034 + 0.412332i \(0.864714\pi\)
\(930\) 0 0
\(931\) 1.50000 2.59808i 0.0491605 0.0851485i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −60.0000 −1.96221
\(936\) 0 0
\(937\) −58.0000 −1.89478 −0.947389 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.0000 + 24.2487i −0.456387 + 0.790485i −0.998767 0.0496480i \(-0.984190\pi\)
0.542380 + 0.840133i \(0.317523\pi\)
\(942\) 0 0
\(943\) 3.00000 + 5.19615i 0.0976934 + 0.169210i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.5000 23.3827i −0.438691 0.759835i 0.558898 0.829237i \(-0.311224\pi\)
−0.997589 + 0.0694014i \(0.977891\pi\)
\(948\) 0 0
\(949\) 3.00000 5.19615i 0.0973841 0.168674i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.00000 5.19615i 0.0968751 0.167793i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.00000 10.3923i −0.193147 0.334540i
\(966\) 0 0
\(967\) 21.0000 36.3731i 0.675314 1.16968i −0.301062 0.953604i \(-0.597341\pi\)
0.976377 0.216075i \(-0.0693254\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 26.0000 0.833522
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.50000 + 9.52628i −0.175961 + 0.304773i −0.940493 0.339812i \(-0.889636\pi\)
0.764533 + 0.644585i \(0.222970\pi\)
\(978\) 0 0
\(979\) −25.0000 43.3013i −0.799003 1.38391i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.0000 + 31.1769i 0.574111 + 0.994389i 0.996138 + 0.0878058i \(0.0279855\pi\)
−0.422027 + 0.906583i \(0.638681\pi\)
\(984\) 0 0
\(985\) −8.00000 + 13.8564i −0.254901 + 0.441502i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −42.0000 −1.33552
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.00000 6.92820i 0.126809 0.219639i
\(996\) 0 0
\(997\) −30.0000 51.9615i −0.950110 1.64564i −0.745182 0.666861i \(-0.767638\pi\)
−0.204927 0.978777i \(-0.565696\pi\)
\(998\) 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 864.2.i.b.577.1 2
3.2 odd 2 288.2.i.b.193.1 yes 2
4.3 odd 2 864.2.i.a.577.1 2
8.3 odd 2 1728.2.i.a.577.1 2
8.5 even 2 1728.2.i.b.577.1 2
9.2 odd 6 288.2.i.b.97.1 yes 2
9.4 even 3 2592.2.a.a.1.1 1
9.5 odd 6 2592.2.a.g.1.1 1
9.7 even 3 inner 864.2.i.b.289.1 2
12.11 even 2 288.2.i.a.193.1 yes 2
24.5 odd 2 576.2.i.b.193.1 2
24.11 even 2 576.2.i.h.193.1 2
36.7 odd 6 864.2.i.a.289.1 2
36.11 even 6 288.2.i.a.97.1 2
36.23 even 6 2592.2.a.h.1.1 1
36.31 odd 6 2592.2.a.b.1.1 1
72.5 odd 6 5184.2.a.a.1.1 1
72.11 even 6 576.2.i.h.385.1 2
72.13 even 6 5184.2.a.be.1.1 1
72.29 odd 6 576.2.i.b.385.1 2
72.43 odd 6 1728.2.i.a.1153.1 2
72.59 even 6 5184.2.a.b.1.1 1
72.61 even 6 1728.2.i.b.1153.1 2
72.67 odd 6 5184.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.a.97.1 2 36.11 even 6
288.2.i.a.193.1 yes 2 12.11 even 2
288.2.i.b.97.1 yes 2 9.2 odd 6
288.2.i.b.193.1 yes 2 3.2 odd 2
576.2.i.b.193.1 2 24.5 odd 2
576.2.i.b.385.1 2 72.29 odd 6
576.2.i.h.193.1 2 24.11 even 2
576.2.i.h.385.1 2 72.11 even 6
864.2.i.a.289.1 2 36.7 odd 6
864.2.i.a.577.1 2 4.3 odd 2
864.2.i.b.289.1 2 9.7 even 3 inner
864.2.i.b.577.1 2 1.1 even 1 trivial
1728.2.i.a.577.1 2 8.3 odd 2
1728.2.i.a.1153.1 2 72.43 odd 6
1728.2.i.b.577.1 2 8.5 even 2
1728.2.i.b.1153.1 2 72.61 even 6
2592.2.a.a.1.1 1 9.4 even 3
2592.2.a.b.1.1 1 36.31 odd 6
2592.2.a.g.1.1 1 9.5 odd 6
2592.2.a.h.1.1 1 36.23 even 6
5184.2.a.a.1.1 1 72.5 odd 6
5184.2.a.b.1.1 1 72.59 even 6
5184.2.a.be.1.1 1 72.13 even 6
5184.2.a.bf.1.1 1 72.67 odd 6