Properties

Label 216.3.p.a.91.1
Level $216$
Weight $3$
Character 216.91
Analytic conductor $5.886$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(19,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.p (of order \(6\), 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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 91.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 216.91
Dual form 216.3.p.a.19.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +(-10.8485 - 18.7901i) q^{11} +(-8.00000 - 13.8564i) q^{16} -28.3939 q^{17} +2.30306 q^{19} +(-21.6969 + 37.5802i) q^{22} +(-12.5000 - 21.6506i) q^{25} +(-16.0000 + 27.7128i) q^{32} +(28.3939 + 49.1796i) q^{34} +(-2.30306 - 3.98902i) q^{38} +(-17.8939 + 30.9931i) q^{41} +(-33.2423 - 57.5774i) q^{43} +86.7878 q^{44} +(-24.5000 + 42.4352i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(57.2423 - 99.1467i) q^{59} +64.0000 q^{64} +(66.9393 - 115.942i) q^{67} +(56.7878 - 98.3593i) q^{68} +100.394 q^{73} +(-4.60612 + 7.97804i) q^{76} +71.5755 q^{82} +(79.0000 + 136.832i) q^{83} +(-66.4847 + 115.155i) q^{86} +(-86.7878 - 150.321i) q^{88} -146.000 q^{89} +(49.9847 + 86.5760i) q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 14 q^{11} - 32 q^{16} + 4 q^{17} + 68 q^{19} - 28 q^{22} - 50 q^{25} - 64 q^{32} - 4 q^{34} - 68 q^{38} + 46 q^{41} + 14 q^{43} + 112 q^{44} - 98 q^{49} - 100 q^{50} + 82 q^{59} + 256 q^{64} + 62 q^{67} - 8 q^{68} + 284 q^{73} - 136 q^{76} - 184 q^{82} + 316 q^{83} + 28 q^{86} - 112 q^{88} - 584 q^{89} - 94 q^{97} + 392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 1.73205i −0.500000 0.866025i
\(3\) 0 0
\(4\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) −10.8485 18.7901i −0.986224 1.70819i −0.636364 0.771389i \(-0.719562\pi\)
−0.349861 0.936802i \(-0.613771\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 13.8564i −0.500000 0.866025i
\(17\) −28.3939 −1.67023 −0.835114 0.550077i \(-0.814598\pi\)
−0.835114 + 0.550077i \(0.814598\pi\)
\(18\) 0 0
\(19\) 2.30306 0.121214 0.0606069 0.998162i \(-0.480696\pi\)
0.0606069 + 0.998162i \(0.480696\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −21.6969 + 37.5802i −0.986224 + 1.70819i
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −12.5000 21.6506i −0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −16.0000 + 27.7128i −0.500000 + 0.866025i
\(33\) 0 0
\(34\) 28.3939 + 49.1796i 0.835114 + 1.44646i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −2.30306 3.98902i −0.0606069 0.104974i
\(39\) 0 0
\(40\) 0 0
\(41\) −17.8939 + 30.9931i −0.436436 + 0.755929i −0.997412 0.0719030i \(-0.977093\pi\)
0.560976 + 0.827832i \(0.310426\pi\)
\(42\) 0 0
\(43\) −33.2423 57.5774i −0.773078 1.33901i −0.935869 0.352349i \(-0.885383\pi\)
0.162791 0.986661i \(-0.447950\pi\)
\(44\) 86.7878 1.97245
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) −24.5000 + 42.4352i −0.500000 + 0.866025i
\(50\) −25.0000 + 43.3013i −0.500000 + 0.866025i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 57.2423 99.1467i 0.970209 1.68045i 0.275294 0.961360i \(-0.411225\pi\)
0.694915 0.719092i \(-0.255442\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 66.9393 115.942i 0.999094 1.73048i 0.462687 0.886522i \(-0.346886\pi\)
0.536407 0.843959i \(-0.319781\pi\)
\(68\) 56.7878 98.3593i 0.835114 1.44646i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 100.394 1.37526 0.687629 0.726062i \(-0.258651\pi\)
0.687629 + 0.726062i \(0.258651\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.60612 + 7.97804i −0.0606069 + 0.104974i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 71.5755 0.872872
\(83\) 79.0000 + 136.832i 0.951807 + 1.64858i 0.741511 + 0.670941i \(0.234110\pi\)
0.210296 + 0.977638i \(0.432557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −66.4847 + 115.155i −0.773078 + 1.33901i
\(87\) 0 0
\(88\) −86.7878 150.321i −0.986224 1.70819i
\(89\) −146.000 −1.64045 −0.820225 0.572041i \(-0.806152\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 49.9847 + 86.5760i 0.515306 + 0.892536i 0.999842 + 0.0177651i \(0.00565510\pi\)
−0.484536 + 0.874771i \(0.661012\pi\)
\(98\) 98.0000 1.00000
\(99\) 0 0
\(100\) 100.000 1.00000
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.8786 0.129706 0.0648531 0.997895i \(-0.479342\pi\)
0.0648531 + 0.997895i \(0.479342\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 49.0000 84.8705i 0.433628 0.751066i −0.563554 0.826079i \(-0.690566\pi\)
0.997183 + 0.0750128i \(0.0238998\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −228.969 −1.94042
\(119\) 0 0
\(120\) 0 0
\(121\) −174.879 + 302.899i −1.44528 + 2.50329i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −64.0000 110.851i −0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 31.0000 53.6936i 0.236641 0.409875i −0.723107 0.690736i \(-0.757287\pi\)
0.959748 + 0.280861i \(0.0906201\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −267.757 −1.99819
\(135\) 0 0
\(136\) −227.151 −1.67023
\(137\) 118.288 + 204.880i 0.863414 + 1.49548i 0.868613 + 0.495491i \(0.165012\pi\)
−0.00519888 + 0.999986i \(0.501655\pi\)
\(138\) 0 0
\(139\) −29.3332 + 50.8065i −0.211030 + 0.365515i −0.952037 0.305983i \(-0.901015\pi\)
0.741007 + 0.671497i \(0.234348\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −100.394 173.887i −0.687629 1.19101i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 18.4245 0.121214
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −322.000 −1.97546 −0.987730 0.156171i \(-0.950085\pi\)
−0.987730 + 0.156171i \(0.950085\pi\)
\(164\) −71.5755 123.972i −0.436436 0.755929i
\(165\) 0 0
\(166\) 158.000 273.664i 0.951807 1.64858i
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −84.5000 146.358i −0.500000 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 265.939 1.54616
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −173.576 + 300.642i −0.986224 + 1.70819i
\(177\) 0 0
\(178\) 146.000 + 252.879i 0.820225 + 1.42067i
\(179\) 34.0000 0.189944 0.0949721 0.995480i \(-0.469724\pi\)
0.0949721 + 0.995480i \(0.469724\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 308.030 + 533.524i 1.64722 + 2.85307i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 186.166 322.450i 0.964592 1.67072i 0.253886 0.967234i \(-0.418291\pi\)
0.710706 0.703489i \(-0.248375\pi\)
\(194\) 99.9694 173.152i 0.515306 0.892536i
\(195\) 0 0
\(196\) −98.0000 169.741i −0.500000 0.866025i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −100.000 173.205i −0.500000 0.866025i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.9847 43.2748i −0.119544 0.207056i
\(210\) 0 0
\(211\) 113.000 195.722i 0.535545 0.927591i −0.463592 0.886049i \(-0.653440\pi\)
0.999137 0.0415423i \(-0.0132271\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −13.8786 24.0384i −0.0648531 0.112329i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −196.000 −0.867257
\(227\) −74.7577 129.484i −0.329329 0.570414i 0.653050 0.757315i \(-0.273489\pi\)
−0.982379 + 0.186900i \(0.940156\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 363.969 1.56210 0.781050 0.624468i \(-0.214684\pi\)
0.781050 + 0.624468i \(0.214684\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 228.969 + 396.587i 0.970209 + 1.68045i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) −142.560 246.922i −0.591536 1.02457i −0.994026 0.109146i \(-0.965188\pi\)
0.402490 0.915425i \(-0.368145\pi\)
\(242\) 699.514 2.89055
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −394.666 −1.57238 −0.786188 0.617988i \(-0.787948\pi\)
−0.786188 + 0.617988i \(0.787948\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −128.000 + 221.703i −0.500000 + 0.866025i
\(257\) 50.4694 87.4155i 0.196379 0.340138i −0.750973 0.660333i \(-0.770415\pi\)
0.947352 + 0.320195i \(0.103748\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −124.000 −0.473282
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 267.757 + 463.769i 0.999094 + 1.73048i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 227.151 + 393.437i 0.835114 + 1.44646i
\(273\) 0 0
\(274\) 236.576 409.761i 0.863414 1.49548i
\(275\) −271.212 + 469.752i −0.986224 + 1.70819i
\(276\) 0 0
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 117.333 0.422060
\(279\) 0 0
\(280\) 0 0
\(281\) −119.000 206.114i −0.423488 0.733502i 0.572790 0.819702i \(-0.305861\pi\)
−0.996278 + 0.0862000i \(0.972528\pi\)
\(282\) 0 0
\(283\) 41.0000 71.0141i 0.144876 0.250933i −0.784450 0.620191i \(-0.787055\pi\)
0.929327 + 0.369258i \(0.120388\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 517.212 1.78966
\(290\) 0 0
\(291\) 0 0
\(292\) −200.788 + 347.775i −0.687629 + 1.19101i
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −18.4245 31.9122i −0.0606069 0.104974i
\(305\) 0 0
\(306\) 0 0
\(307\) −21.1520 −0.0688992 −0.0344496 0.999406i \(-0.510968\pi\)
−0.0344496 + 0.999406i \(0.510968\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) −278.469 482.323i −0.889679 1.54097i −0.840256 0.542191i \(-0.817595\pi\)
−0.0494230 0.998778i \(-0.515738\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −65.3928 −0.202455
\(324\) 0 0
\(325\) 0 0
\(326\) 322.000 + 557.720i 0.987730 + 1.71080i
\(327\) 0 0
\(328\) −143.151 + 247.945i −0.436436 + 0.755929i
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 12.1244i −0.0211480 0.0366295i 0.855258 0.518203i \(-0.173399\pi\)
−0.876406 + 0.481573i \(0.840065\pi\)
\(332\) −632.000 −1.90361
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 86.2571 149.402i 0.255956 0.443329i −0.709199 0.705009i \(-0.750943\pi\)
0.965155 + 0.261680i \(0.0842765\pi\)
\(338\) −169.000 + 292.717i −0.500000 + 0.866025i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −265.939 460.619i −0.773078 1.33901i
\(345\) 0 0
\(346\) 0 0
\(347\) 68.9699 119.459i 0.198761 0.344263i −0.749366 0.662156i \(-0.769642\pi\)
0.948127 + 0.317892i \(0.102975\pi\)
\(348\) 0 0
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 694.302 1.97245
\(353\) −342.439 593.121i −0.970081 1.68023i −0.695294 0.718725i \(-0.744726\pi\)
−0.274788 0.961505i \(-0.588608\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 292.000 505.759i 0.820225 1.42067i
\(357\) 0 0
\(358\) −34.0000 58.8897i −0.0949721 0.164496i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −355.696 −0.985307
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 616.060 1067.05i 1.64722 2.85307i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −622.242 −1.64180 −0.820900 0.571073i \(-0.806527\pi\)
−0.820900 + 0.571073i \(0.806527\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −744.665 −1.92918
\(387\) 0 0
\(388\) −399.878 −1.03061
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −196.000 + 339.482i −0.500000 + 0.866025i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −200.000 + 346.410i −0.500000 + 0.866025i
\(401\) 294.379 509.879i 0.734111 1.27152i −0.221001 0.975274i \(-0.570932\pi\)
0.955112 0.296244i \(-0.0957342\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −406.833 + 704.655i −0.994701 + 1.72287i −0.408313 + 0.912842i \(0.633883\pi\)
−0.586388 + 0.810030i \(0.699451\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −49.9694 + 86.5495i −0.119544 + 0.207056i
\(419\) −257.000 + 445.137i −0.613365 + 1.06238i 0.377304 + 0.926090i \(0.376851\pi\)
−0.990669 + 0.136290i \(0.956482\pi\)
\(420\) 0 0
\(421\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) −452.000 −1.07109
\(423\) 0 0
\(424\) 0 0
\(425\) 354.923 + 614.745i 0.835114 + 1.44646i
\(426\) 0 0
\(427\) 0 0
\(428\) −27.7571 + 48.0768i −0.0648531 + 0.112329i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 269.484 0.622364 0.311182 0.950350i \(-0.399275\pi\)
0.311182 + 0.950350i \(0.399275\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −270.939 469.281i −0.611601 1.05932i −0.990971 0.134079i \(-0.957192\pi\)
0.379370 0.925245i \(-0.376141\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 227.243 0.506109 0.253054 0.967452i \(-0.418565\pi\)
0.253054 + 0.967452i \(0.418565\pi\)
\(450\) 0 0
\(451\) 776.485 1.72170
\(452\) 196.000 + 339.482i 0.433628 + 0.751066i
\(453\) 0 0
\(454\) −149.515 + 258.968i −0.329329 + 0.570414i
\(455\) 0 0
\(456\) 0 0
\(457\) 322.620 + 558.795i 0.705953 + 1.22275i 0.966347 + 0.257244i \(0.0828143\pi\)
−0.260394 + 0.965502i \(0.583852\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −363.969 630.413i −0.781050 1.35282i
\(467\) 791.332 1.69450 0.847250 0.531194i \(-0.178257\pi\)
0.847250 + 0.531194i \(0.178257\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 457.939 793.173i 0.970209 1.68045i
\(473\) −721.257 + 1249.25i −1.52486 + 2.64113i
\(474\) 0 0
\(475\) −28.7883 49.8627i −0.0606069 0.104974i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −285.120 + 493.843i −0.591536 + 1.02457i
\(483\) 0 0
\(484\) −699.514 1211.59i −1.44528 2.50329i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 61.6964 106.861i 0.125655 0.217640i −0.796334 0.604857i \(-0.793230\pi\)
0.921989 + 0.387217i \(0.126564\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −457.696 + 792.753i −0.917227 + 1.58868i −0.113620 + 0.993524i \(0.536245\pi\)
−0.803607 + 0.595160i \(0.797089\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 394.666 + 683.582i 0.786188 + 1.36172i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) −201.878 −0.392758
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −267.849 −0.514106 −0.257053 0.966397i \(-0.582751\pi\)
−0.257053 + 0.966397i \(0.582751\pi\)
\(522\) 0 0
\(523\) 398.000 0.760994 0.380497 0.924782i \(-0.375753\pi\)
0.380497 + 0.924782i \(0.375753\pi\)
\(524\) 124.000 + 214.774i 0.236641 + 0.409875i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −264.500 458.127i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 535.514 927.538i 0.999094 1.73048i
\(537\) 0 0
\(538\) 0 0
\(539\) 1063.15 1.97245
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 454.302 786.874i 0.835114 1.44646i
\(545\) 0 0
\(546\) 0 0
\(547\) 86.4852 + 149.797i 0.158108 + 0.273852i 0.934186 0.356785i \(-0.116127\pi\)
−0.776078 + 0.630637i \(0.782794\pi\)
\(548\) −946.302 −1.72683
\(549\) 0 0
\(550\) 1084.85 1.97245
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −117.333 203.226i −0.211030 0.365515i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −238.000 + 412.228i −0.423488 + 0.733502i
\(563\) 534.150 925.176i 0.948758 1.64330i 0.200710 0.979651i \(-0.435675\pi\)
0.748047 0.663646i \(-0.230992\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −164.000 −0.289753
\(567\) 0 0
\(568\) 0 0
\(569\) 255.014 + 441.698i 0.448180 + 0.776270i 0.998268 0.0588367i \(-0.0187391\pi\)
−0.550088 + 0.835107i \(0.685406\pi\)
\(570\) 0 0
\(571\) 239.666 415.113i 0.419730 0.726994i −0.576182 0.817321i \(-0.695458\pi\)
0.995912 + 0.0903277i \(0.0287914\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 998.392 1.73032 0.865158 0.501500i \(-0.167218\pi\)
0.865158 + 0.501500i \(0.167218\pi\)
\(578\) −517.212 895.838i −0.894831 1.54989i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 803.151 1.37526
\(585\) 0 0
\(586\) 0 0
\(587\) 409.424 + 709.143i 0.697485 + 1.20808i 0.969336 + 0.245741i \(0.0790312\pi\)
−0.271850 + 0.962340i \(0.587635\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 862.000 1.45363 0.726813 0.686836i \(-0.241001\pi\)
0.726813 + 0.686836i \(0.241001\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 566.530 + 981.258i 0.942645 + 1.63271i 0.760399 + 0.649456i \(0.225003\pi\)
0.182246 + 0.983253i \(0.441663\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) −36.8490 + 63.8243i −0.0606069 + 0.104974i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 21.1520 + 36.6364i 0.0344496 + 0.0596684i
\(615\) 0 0
\(616\) 0 0
\(617\) −430.893 + 746.328i −0.698368 + 1.20961i 0.270665 + 0.962674i \(0.412757\pi\)
−0.969032 + 0.246935i \(0.920577\pi\)
\(618\) 0 0
\(619\) −618.150 1070.67i −0.998628 1.72967i −0.544670 0.838651i \(-0.683345\pi\)
−0.453958 0.891023i \(-0.649988\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 + 541.266i −0.500000 + 0.866025i
\(626\) −556.939 + 964.646i −0.889679 + 1.54097i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −634.893 1099.67i −0.990472 1.71555i −0.614497 0.788919i \(-0.710641\pi\)
−0.375975 0.926630i \(-0.622692\pi\)
\(642\) 0 0
\(643\) 487.212 843.875i 0.757717 1.31240i −0.186296 0.982494i \(-0.559648\pi\)
0.944012 0.329910i \(-0.107018\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 65.3928 + 113.264i 0.101227 + 0.175331i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −2483.97 −3.82738
\(650\) 0 0
\(651\) 0 0
\(652\) 644.000 1115.44i 0.987730 1.71080i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 572.604 0.872872
\(657\) 0 0
\(658\) 0 0
\(659\) −497.000 860.829i −0.754173 1.30627i −0.945784 0.324795i \(-0.894705\pi\)
0.191611 0.981471i \(-0.438629\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) −14.0000 + 24.2487i −0.0211480 + 0.0366295i
\(663\) 0 0
\(664\) 632.000 + 1094.66i 0.951807 + 1.64858i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 623.000 + 1079.07i 0.925706 + 1.60337i 0.790422 + 0.612563i \(0.209861\pi\)
0.135284 + 0.990807i \(0.456805\pi\)
\(674\) −345.029 −0.511912
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −932.664 −1.36554 −0.682770 0.730633i \(-0.739225\pi\)
−0.682770 + 0.730633i \(0.739225\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −531.878 + 921.239i −0.773078 + 1.33901i
\(689\) 0 0
\(690\) 0 0
\(691\) −367.000 635.663i −0.531114 0.919917i −0.999341 0.0363084i \(-0.988440\pi\)
0.468226 0.883609i \(-0.344893\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −275.880 −0.397521
\(695\) 0 0
\(696\) 0 0
\(697\) 508.077 880.014i 0.728948 1.26257i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −694.302 1202.57i −0.986224 1.70819i
\(705\) 0 0
\(706\) −684.878 + 1186.24i −0.970081 + 1.68023i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1168.00 −1.64045
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −68.0000 + 117.779i −0.0949721 + 0.164496i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 355.696 + 616.083i 0.492654 + 0.853301i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 943.879 + 1634.85i 1.29122 + 2.23645i
\(732\) 0 0
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2904.76 −3.94132
\(738\) 0 0
\(739\) −1088.24 −1.47258 −0.736292 0.676664i \(-0.763425\pi\)
−0.736292 + 0.676664i \(0.763425\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) −2464.24 −3.29444
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 622.242 + 1077.75i 0.820900 + 1.42184i
\(759\) 0 0
\(760\) 0 0
\(761\) 697.000 1207.24i 0.915900 1.58639i 0.110322 0.993896i \(-0.464812\pi\)
0.805578 0.592490i \(-0.201855\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 527.000 912.791i 0.685306 1.18698i −0.288035 0.957620i \(-0.593002\pi\)
0.973341 0.229364i \(-0.0736647\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 744.665 + 1289.80i 0.964592 + 1.67072i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 399.878 + 692.608i 0.515306 + 0.892536i
\(777\) 0 0
\(778\) 0 0
\(779\) −41.2107 + 71.3790i −0.0529021 + 0.0916290i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −463.000 + 801.940i −0.588310 + 1.01898i 0.406144 + 0.913809i \(0.366873\pi\)
−0.994454 + 0.105174i \(0.966460\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 800.000 1.00000
\(801\) 0 0
\(802\) −1177.51 −1.46822
\(803\) −1089.12 1886.41i −1.35631 2.34920i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1084.94 −1.34109 −0.670543 0.741871i \(-0.733939\pi\)
−0.670543 + 0.741871i \(0.733939\pi\)
\(810\) 0 0
\(811\) 1485.21 1.83133 0.915665 0.401942i \(-0.131665\pi\)
0.915665 + 0.401942i \(0.131665\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −76.5592 132.604i −0.0937077 0.162306i
\(818\) 1627.33 1.98940
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1262.00 −1.52600 −0.762999 0.646400i \(-0.776274\pi\)
−0.762999 + 0.646400i \(0.776274\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 695.650 1204.90i 0.835114 1.44646i
\(834\) 0 0
\(835\) 0 0
\(836\) 199.878 0.239088
\(837\) 0 0
\(838\) 1028.00 1.22673
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) −420.500 + 728.327i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 452.000 + 782.887i 0.535545 + 0.927591i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 709.847 1229.49i 0.835114 1.44646i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 111.029 0.129706
\(857\) 601.000 + 1040.96i 0.701284 + 1.21466i 0.968016 + 0.250888i \(0.0807226\pi\)
−0.266733 + 0.963771i \(0.585944\pi\)
\(858\) 0 0
\(859\) 198.394 343.629i 0.230960 0.400034i −0.727131 0.686499i \(-0.759147\pi\)
0.958091 + 0.286465i \(0.0924801\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −269.484 466.759i −0.311182 0.538983i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1438.00 1.63224 0.816118 0.577885i \(-0.196122\pi\)
0.816118 + 0.577885i \(0.196122\pi\)
\(882\) 0 0
\(883\) 778.121 0.881225 0.440612 0.897697i \(-0.354761\pi\)
0.440612 + 0.897697i \(0.354761\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −541.879 + 938.561i −0.611601 + 1.05932i
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −227.243 393.596i −0.253054 0.438303i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −776.485 1344.91i −0.860848 1.49103i
\(903\) 0 0
\(904\) 392.000 678.964i 0.433628 0.751066i
\(905\) 0 0
\(906\) 0 0
\(907\) −685.696 1187.66i −0.756005 1.30944i −0.944873 0.327436i \(-0.893815\pi\)
0.188868 0.982002i \(-0.439518\pi\)
\(908\) 598.061 0.658658
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(912\) 0 0
\(913\) 1714.06 2968.84i 1.87739 3.25174i
\(914\) 645.241 1117.59i 0.705953 1.22275i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 529.000 + 916.255i 0.569429 + 0.986281i 0.996622 + 0.0821203i \(0.0261692\pi\)
−0.427193 + 0.904160i \(0.640497\pi\)
\(930\) 0 0
\(931\) −56.4250 + 97.7310i −0.0606069 + 0.104974i
\(932\) −727.939 + 1260.83i −0.781050 + 1.35282i
\(933\) 0 0
\(934\) −791.332 1370.63i −0.847250 1.46748i
\(935\) 0 0
\(936\) 0 0
\(937\) −718.000 −0.766275 −0.383138 0.923691i \(-0.625157\pi\)
−0.383138 + 0.923691i \(0.625157\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1831.76 −1.94042
\(945\) 0 0
\(946\) 2885.03 3.04971
\(947\) 946.605 + 1639.57i 0.999582 + 1.73133i 0.524815 + 0.851216i \(0.324134\pi\)
0.474767 + 0.880111i \(0.342532\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −57.5765 + 99.7255i −0.0606069 + 0.104974i
\(951\) 0 0
\(952\) 0 0
\(953\) 1575.06 1.65274 0.826368 0.563131i \(-0.190403\pi\)
0.826368 + 0.563131i \(0.190403\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −480.500 832.250i −0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 1140.48 1.18307
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) −1399.03 + 2423.19i −1.44528 + 2.50329i
\(969\) 0 0
\(970\) 0 0
\(971\) −974.000 −1.00309 −0.501545 0.865132i \(-0.667235\pi\)
−0.501545 + 0.865132i \(0.667235\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 317.834 550.504i 0.325316 0.563464i −0.656260 0.754535i \(-0.727863\pi\)
0.981576 + 0.191071i \(0.0611960\pi\)
\(978\) 0 0
\(979\) 1583.88 + 2743.35i 1.61785 + 2.80220i
\(980\) 0 0
\(981\) 0 0
\(982\) −246.786 −0.251309
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 1830.79 1.83445
\(999\) 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 216.3.p.a.91.1 4
3.2 odd 2 72.3.p.a.67.1 yes 4
4.3 odd 2 864.3.t.a.847.2 4
8.3 odd 2 CM 216.3.p.a.91.1 4
8.5 even 2 864.3.t.a.847.2 4
9.2 odd 6 72.3.p.a.43.1 4
9.4 even 3 648.3.b.b.163.2 2
9.5 odd 6 648.3.b.a.163.1 2
9.7 even 3 inner 216.3.p.a.19.1 4
12.11 even 2 288.3.t.a.175.2 4
24.5 odd 2 288.3.t.a.175.2 4
24.11 even 2 72.3.p.a.67.1 yes 4
36.7 odd 6 864.3.t.a.559.2 4
36.11 even 6 288.3.t.a.79.2 4
36.23 even 6 2592.3.b.b.1135.2 2
36.31 odd 6 2592.3.b.a.1135.1 2
72.5 odd 6 2592.3.b.b.1135.2 2
72.11 even 6 72.3.p.a.43.1 4
72.13 even 6 2592.3.b.a.1135.1 2
72.29 odd 6 288.3.t.a.79.2 4
72.43 odd 6 inner 216.3.p.a.19.1 4
72.59 even 6 648.3.b.a.163.1 2
72.61 even 6 864.3.t.a.559.2 4
72.67 odd 6 648.3.b.b.163.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.p.a.43.1 4 9.2 odd 6
72.3.p.a.43.1 4 72.11 even 6
72.3.p.a.67.1 yes 4 3.2 odd 2
72.3.p.a.67.1 yes 4 24.11 even 2
216.3.p.a.19.1 4 9.7 even 3 inner
216.3.p.a.19.1 4 72.43 odd 6 inner
216.3.p.a.91.1 4 1.1 even 1 trivial
216.3.p.a.91.1 4 8.3 odd 2 CM
288.3.t.a.79.2 4 36.11 even 6
288.3.t.a.79.2 4 72.29 odd 6
288.3.t.a.175.2 4 12.11 even 2
288.3.t.a.175.2 4 24.5 odd 2
648.3.b.a.163.1 2 9.5 odd 6
648.3.b.a.163.1 2 72.59 even 6
648.3.b.b.163.2 2 9.4 even 3
648.3.b.b.163.2 2 72.67 odd 6
864.3.t.a.559.2 4 36.7 odd 6
864.3.t.a.559.2 4 72.61 even 6
864.3.t.a.847.2 4 4.3 odd 2
864.3.t.a.847.2 4 8.5 even 2
2592.3.b.a.1135.1 2 36.31 odd 6
2592.3.b.a.1135.1 2 72.13 even 6
2592.3.b.b.1135.2 2 36.23 even 6
2592.3.b.b.1135.2 2 72.5 odd 6