Properties

Label 2646.2.d.a.2645.1
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.a.2645.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +1.00000i q^{8} +1.73205i q^{10} +1.00000 q^{16} +3.46410 q^{17} -6.92820i q^{19} +1.73205 q^{20} +6.00000i q^{23} -2.00000 q^{25} +9.00000i q^{29} +3.46410i q^{31} -1.00000i q^{32} -3.46410i q^{34} +4.00000 q^{37} -6.92820 q^{38} -1.73205i q^{40} -3.46410 q^{41} -8.00000 q^{43} +6.00000 q^{46} +3.46410 q^{47} +2.00000i q^{50} -3.00000i q^{53} +9.00000 q^{58} +12.1244 q^{59} -3.46410i q^{61} +3.46410 q^{62} -1.00000 q^{64} +14.0000 q^{67} -3.46410 q^{68} -6.00000i q^{71} -12.1244i q^{73} -4.00000i q^{74} +6.92820i q^{76} +11.0000 q^{79} -1.73205 q^{80} +3.46410i q^{82} +17.3205 q^{83} -6.00000 q^{85} +8.00000i q^{86} +10.3923 q^{89} -6.00000i q^{92} -3.46410i q^{94} +12.0000i q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{16} - 8q^{25} + 16q^{37} - 32q^{43} + 24q^{46} + 36q^{58} - 4q^{64} + 56q^{67} + 44q^{79} - 24q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.73205i 0.547723i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) − 6.92820i − 1.58944i −0.606977 0.794719i \(-0.707618\pi\)
0.606977 0.794719i \(-0.292382\pi\)
\(20\) 1.73205 0.387298
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.00000i 1.67126i 0.549294 + 0.835629i \(0.314897\pi\)
−0.549294 + 0.835629i \(0.685103\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 3.46410i − 0.594089i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −6.92820 −1.12390
\(39\) 0 0
\(40\) − 1.73205i − 0.273861i
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.00000i 0.282843i
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.00000i − 0.412082i −0.978543 0.206041i \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) 12.1244 1.57846 0.789228 0.614100i \(-0.210481\pi\)
0.789228 + 0.614100i \(0.210481\pi\)
\(60\) 0 0
\(61\) − 3.46410i − 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) 0 0
\(71\) − 6.00000i − 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) 0 0
\(73\) − 12.1244i − 1.41905i −0.704681 0.709524i \(-0.748910\pi\)
0.704681 0.709524i \(-0.251090\pi\)
\(74\) − 4.00000i − 0.464991i
\(75\) 0 0
\(76\) 6.92820i 0.794719i
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −1.73205 −0.193649
\(81\) 0 0
\(82\) 3.46410i 0.382546i
\(83\) 17.3205 1.90117 0.950586 0.310460i \(-0.100483\pi\)
0.950586 + 0.310460i \(0.100483\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 8.00000i 0.862662i
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) − 3.46410i − 0.357295i
\(95\) 12.0000i 1.23117i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −5.19615 −0.517036 −0.258518 0.966006i \(-0.583234\pi\)
−0.258518 + 0.966006i \(0.583234\pi\)
\(102\) 0 0
\(103\) − 10.3923i − 1.02398i −0.858990 0.511992i \(-0.828908\pi\)
0.858990 0.511992i \(-0.171092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) − 10.3923i − 0.969087i
\(116\) − 9.00000i − 0.835629i
\(117\) 0 0
\(118\) − 12.1244i − 1.11614i
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) −3.46410 −0.313625
\(123\) 0 0
\(124\) − 3.46410i − 0.311086i
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −10.3923 −0.907980 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 14.0000i − 1.20942i
\(135\) 0 0
\(136\) 3.46410i 0.297044i
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) − 10.3923i − 0.881464i −0.897639 0.440732i \(-0.854719\pi\)
0.897639 0.440732i \(-0.145281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) − 15.5885i − 1.29455i
\(146\) −12.1244 −1.00342
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) − 15.0000i − 1.22885i −0.788976 0.614424i \(-0.789388\pi\)
0.788976 0.614424i \(-0.210612\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.92820 0.561951
\(153\) 0 0
\(154\) 0 0
\(155\) − 6.00000i − 0.481932i
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) − 11.0000i − 0.875113i
\(159\) 0 0
\(160\) 1.73205i 0.136931i
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) − 17.3205i − 1.34433i
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 6.00000i 0.460179i
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 5.19615 0.395056 0.197528 0.980297i \(-0.436709\pi\)
0.197528 + 0.980297i \(0.436709\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) − 10.3923i − 0.778936i
\(179\) 15.0000i 1.12115i 0.828103 + 0.560576i \(0.189420\pi\)
−0.828103 + 0.560576i \(0.810580\pi\)
\(180\) 0 0
\(181\) 17.3205i 1.28742i 0.765268 + 0.643712i \(0.222606\pi\)
−0.765268 + 0.643712i \(0.777394\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 0 0
\(188\) −3.46410 −0.252646
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) 6.00000i 0.434145i 0.976156 + 0.217072i \(0.0696508\pi\)
−0.976156 + 0.217072i \(0.930349\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −6.92820 −0.497416
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) − 2.00000i − 0.141421i
\(201\) 0 0
\(202\) 5.19615i 0.365600i
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −10.3923 −0.724066
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 3.00000i 0.206041i
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) 13.8564 0.944999
\(216\) 0 0
\(217\) 0 0
\(218\) − 16.0000i − 1.08366i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 15.5885i − 1.04388i −0.852982 0.521940i \(-0.825208\pi\)
0.852982 0.521940i \(-0.174792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 25.9808 1.72440 0.862202 0.506565i \(-0.169085\pi\)
0.862202 + 0.506565i \(0.169085\pi\)
\(228\) 0 0
\(229\) 3.46410i 0.228914i 0.993428 + 0.114457i \(0.0365129\pi\)
−0.993428 + 0.114457i \(0.963487\pi\)
\(230\) −10.3923 −0.685248
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) − 30.0000i − 1.96537i −0.185296 0.982683i \(-0.559325\pi\)
0.185296 0.982683i \(-0.440675\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −12.1244 −0.789228
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000i 1.55243i 0.630468 + 0.776215i \(0.282863\pi\)
−0.630468 + 0.776215i \(0.717137\pi\)
\(240\) 0 0
\(241\) 1.73205i 0.111571i 0.998443 + 0.0557856i \(0.0177663\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 0 0
\(244\) 3.46410i 0.221766i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −3.46410 −0.219971
\(249\) 0 0
\(250\) − 12.1244i − 0.766812i
\(251\) 8.66025 0.546630 0.273315 0.961925i \(-0.411880\pi\)
0.273315 + 0.961925i \(0.411880\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 7.00000i 0.439219i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.7846 −1.29651 −0.648254 0.761424i \(-0.724501\pi\)
−0.648254 + 0.761424i \(0.724501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 10.3923i 0.642039i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) 1.73205 0.105605 0.0528025 0.998605i \(-0.483185\pi\)
0.0528025 + 0.998605i \(0.483185\pi\)
\(270\) 0 0
\(271\) − 12.1244i − 0.736502i −0.929726 0.368251i \(-0.879957\pi\)
0.929726 0.368251i \(-0.120043\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −10.3923 −0.623289
\(279\) 0 0
\(280\) 0 0
\(281\) − 18.0000i − 1.07379i −0.843649 0.536895i \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 0 0
\(283\) 6.92820i 0.411839i 0.978569 + 0.205919i \(0.0660185\pi\)
−0.978569 + 0.205919i \(0.933982\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −15.5885 −0.915386
\(291\) 0 0
\(292\) 12.1244i 0.709524i
\(293\) −29.4449 −1.72019 −0.860094 0.510136i \(-0.829595\pi\)
−0.860094 + 0.510136i \(0.829595\pi\)
\(294\) 0 0
\(295\) −21.0000 −1.22267
\(296\) 4.00000i 0.232495i
\(297\) 0 0
\(298\) −15.0000 −0.868927
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) − 6.92820i − 0.397360i
\(305\) 6.00000i 0.343559i
\(306\) 0 0
\(307\) − 6.92820i − 0.395413i −0.980261 0.197707i \(-0.936651\pi\)
0.980261 0.197707i \(-0.0633494\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.00000 −0.340777
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 0 0
\(313\) 22.5167i 1.27272i 0.771393 + 0.636358i \(0.219560\pi\)
−0.771393 + 0.636358i \(0.780440\pi\)
\(314\) 10.3923 0.586472
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.73205 0.0968246
\(321\) 0 0
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) − 2.00000i − 0.110770i
\(327\) 0 0
\(328\) − 3.46410i − 0.191273i
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −17.3205 −0.950586
\(333\) 0 0
\(334\) 17.3205i 0.947736i
\(335\) −24.2487 −1.32485
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) − 8.00000i − 0.431331i
\(345\) 0 0
\(346\) − 5.19615i − 0.279347i
\(347\) 33.0000i 1.77153i 0.464131 + 0.885766i \(0.346367\pi\)
−0.464131 + 0.885766i \(0.653633\pi\)
\(348\) 0 0
\(349\) − 3.46410i − 0.185429i −0.995693 0.0927146i \(-0.970446\pi\)
0.995693 0.0927146i \(-0.0295544\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.46410 0.184376 0.0921878 0.995742i \(-0.470614\pi\)
0.0921878 + 0.995742i \(0.470614\pi\)
\(354\) 0 0
\(355\) 10.3923i 0.551566i
\(356\) −10.3923 −0.550791
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) 24.0000i 1.26667i 0.773877 + 0.633336i \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 17.3205 0.910346
\(363\) 0 0
\(364\) 0 0
\(365\) 21.0000i 1.09919i
\(366\) 0 0
\(367\) − 8.66025i − 0.452062i −0.974120 0.226031i \(-0.927425\pi\)
0.974120 0.226031i \(-0.0725750\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 6.92820i 0.360180i
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.46410i 0.178647i
\(377\) 0 0
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) − 12.0000i − 0.615587i
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) 17.3205 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 10.0000i − 0.508987i
\(387\) 0 0
\(388\) 6.92820i 0.351726i
\(389\) − 21.0000i − 1.06474i −0.846511 0.532371i \(-0.821301\pi\)
0.846511 0.532371i \(-0.178699\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 0 0
\(393\) 0 0
\(394\) 3.00000 0.151138
\(395\) −19.0526 −0.958638
\(396\) 0 0
\(397\) 27.7128i 1.39087i 0.718591 + 0.695433i \(0.244787\pi\)
−0.718591 + 0.695433i \(0.755213\pi\)
\(398\) 8.66025 0.434099
\(399\) 0 0
\(400\) −2.00000 −0.100000
\(401\) 6.00000i 0.299626i 0.988714 + 0.149813i \(0.0478671\pi\)
−0.988714 + 0.149813i \(0.952133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 5.19615 0.258518
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 15.5885i − 0.770800i −0.922750 0.385400i \(-0.874064\pi\)
0.922750 0.385400i \(-0.125936\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) 0 0
\(412\) 10.3923i 0.511992i
\(413\) 0 0
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.46410 0.169232 0.0846162 0.996414i \(-0.473034\pi\)
0.0846162 + 0.996414i \(0.473034\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) −6.92820 −0.336067
\(426\) 0 0
\(427\) 0 0
\(428\) 3.00000i 0.145010i
\(429\) 0 0
\(430\) − 13.8564i − 0.668215i
\(431\) − 36.0000i − 1.73406i −0.498257 0.867029i \(-0.666026\pi\)
0.498257 0.867029i \(-0.333974\pi\)
\(432\) 0 0
\(433\) 1.73205i 0.0832370i 0.999134 + 0.0416185i \(0.0132514\pi\)
−0.999134 + 0.0416185i \(0.986749\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 41.5692 1.98853
\(438\) 0 0
\(439\) 31.1769i 1.48799i 0.668184 + 0.743996i \(0.267072\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −15.5885 −0.738135
\(447\) 0 0
\(448\) 0 0
\(449\) − 6.00000i − 0.283158i −0.989927 0.141579i \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) 0 0
\(454\) − 25.9808i − 1.21934i
\(455\) 0 0
\(456\) 0 0
\(457\) −41.0000 −1.91790 −0.958950 0.283577i \(-0.908479\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 3.46410 0.161867
\(459\) 0 0
\(460\) 10.3923i 0.484544i
\(461\) 22.5167 1.04871 0.524353 0.851501i \(-0.324307\pi\)
0.524353 + 0.851501i \(0.324307\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 9.00000i 0.417815i
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) 5.19615 0.240449 0.120225 0.992747i \(-0.461639\pi\)
0.120225 + 0.992747i \(0.461639\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 6.00000i 0.276759i
\(471\) 0 0
\(472\) 12.1244i 0.558069i
\(473\) 0 0
\(474\) 0 0
\(475\) 13.8564i 0.635776i
\(476\) 0 0
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.73205 0.0788928
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 12.0000i 0.544892i
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 3.46410 0.156813
\(489\) 0 0
\(490\) 0 0
\(491\) − 3.00000i − 0.135388i −0.997706 0.0676941i \(-0.978436\pi\)
0.997706 0.0676941i \(-0.0215642\pi\)
\(492\) 0 0
\(493\) 31.1769i 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 3.46410i 0.155543i
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) −12.1244 −0.542218
\(501\) 0 0
\(502\) − 8.66025i − 0.386526i
\(503\) 27.7128 1.23565 0.617827 0.786314i \(-0.288013\pi\)
0.617827 + 0.786314i \(0.288013\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) −34.6410 −1.53544 −0.767718 0.640788i \(-0.778608\pi\)
−0.767718 + 0.640788i \(0.778608\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 20.7846i 0.916770i
\(515\) 18.0000i 0.793175i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.46410 −0.151765 −0.0758825 0.997117i \(-0.524177\pi\)
−0.0758825 + 0.997117i \(0.524177\pi\)
\(522\) 0 0
\(523\) − 20.7846i − 0.908848i −0.890786 0.454424i \(-0.849845\pi\)
0.890786 0.454424i \(-0.150155\pi\)
\(524\) 10.3923 0.453990
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 5.19615 0.225706
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.19615i 0.224649i
\(536\) 14.0000i 0.604708i
\(537\) 0 0
\(538\) − 1.73205i − 0.0746740i
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −12.1244 −0.520786
\(543\) 0 0
\(544\) − 3.46410i − 0.148522i
\(545\) −27.7128 −1.18709
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 62.3538 2.65636
\(552\) 0 0
\(553\) 0 0
\(554\) − 8.00000i − 0.339887i
\(555\) 0 0
\(556\) 10.3923i 0.440732i
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −5.19615 −0.218992 −0.109496 0.993987i \(-0.534924\pi\)
−0.109496 + 0.993987i \(0.534924\pi\)
\(564\) 0 0
\(565\) − 31.1769i − 1.31162i
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) − 42.0000i − 1.76073i −0.474295 0.880366i \(-0.657297\pi\)
0.474295 0.880366i \(-0.342703\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 12.0000i − 0.500435i
\(576\) 0 0
\(577\) − 15.5885i − 0.648956i −0.945893 0.324478i \(-0.894811\pi\)
0.945893 0.324478i \(-0.105189\pi\)
\(578\) 5.00000i 0.207973i
\(579\) 0 0
\(580\) 15.5885i 0.647275i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 12.1244 0.501709
\(585\) 0 0
\(586\) 29.4449i 1.21636i
\(587\) 36.3731 1.50128 0.750639 0.660713i \(-0.229746\pi\)
0.750639 + 0.660713i \(0.229746\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 21.0000i 0.864556i
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) −24.2487 −0.995775 −0.497888 0.867242i \(-0.665891\pi\)
−0.497888 + 0.867242i \(0.665891\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.0000i 0.614424i
\(597\) 0 0
\(598\) 0 0
\(599\) − 24.0000i − 0.980613i −0.871550 0.490307i \(-0.836885\pi\)
0.871550 0.490307i \(-0.163115\pi\)
\(600\) 0 0
\(601\) − 1.73205i − 0.0706518i −0.999376 0.0353259i \(-0.988753\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −19.0526 −0.774597
\(606\) 0 0
\(607\) 32.9090i 1.33573i 0.744281 + 0.667867i \(0.232792\pi\)
−0.744281 + 0.667867i \(0.767208\pi\)
\(608\) −6.92820 −0.280976
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −6.92820 −0.279600
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 20.7846i 0.835404i 0.908584 + 0.417702i \(0.137164\pi\)
−0.908584 + 0.417702i \(0.862836\pi\)
\(620\) 6.00000i 0.240966i
\(621\) 0 0
\(622\) − 13.8564i − 0.555591i
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 22.5167 0.899947
\(627\) 0 0
\(628\) − 10.3923i − 0.414698i
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 11.0000i 0.437557i
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 12.1244 0.481140
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) − 1.73205i − 0.0684653i
\(641\) − 12.0000i − 0.473972i −0.971513 0.236986i \(-0.923841\pi\)
0.971513 0.236986i \(-0.0761595\pi\)
\(642\) 0 0
\(643\) 27.7128i 1.09289i 0.837496 + 0.546443i \(0.184019\pi\)
−0.837496 + 0.546443i \(0.815981\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 10.3923 0.408564 0.204282 0.978912i \(-0.434514\pi\)
0.204282 + 0.978912i \(0.434514\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 15.0000i 0.586995i 0.955960 + 0.293498i \(0.0948193\pi\)
−0.955960 + 0.293498i \(0.905181\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) −3.46410 −0.135250
\(657\) 0 0
\(658\) 0 0
\(659\) 27.0000i 1.05177i 0.850555 + 0.525885i \(0.176266\pi\)
−0.850555 + 0.525885i \(0.823734\pi\)
\(660\) 0 0
\(661\) − 13.8564i − 0.538952i −0.963007 0.269476i \(-0.913150\pi\)
0.963007 0.269476i \(-0.0868504\pi\)
\(662\) − 10.0000i − 0.388661i
\(663\) 0 0
\(664\) 17.3205i 0.672166i
\(665\) 0 0
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 17.3205 0.670151
\(669\) 0 0
\(670\) 24.2487i 0.936809i
\(671\) 0 0
\(672\) 0 0
\(673\) −5.00000 −0.192736 −0.0963679 0.995346i \(-0.530723\pi\)
−0.0963679 + 0.995346i \(0.530723\pi\)
\(674\) 23.0000i 0.885927i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −25.9808 −0.998522 −0.499261 0.866452i \(-0.666395\pi\)
−0.499261 + 0.866452i \(0.666395\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 6.00000i − 0.230089i
\(681\) 0 0
\(682\) 0 0
\(683\) 9.00000i 0.344375i 0.985064 + 0.172188i \(0.0550836\pi\)
−0.985064 + 0.172188i \(0.944916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 27.7128i 1.05425i 0.849789 + 0.527123i \(0.176729\pi\)
−0.849789 + 0.527123i \(0.823271\pi\)
\(692\) −5.19615 −0.197528
\(693\) 0 0
\(694\) 33.0000 1.25266
\(695\) 18.0000i 0.682779i
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −3.46410 −0.131118
\(699\) 0 0
\(700\) 0 0
\(701\) 27.0000i 1.01978i 0.860241 + 0.509888i \(0.170313\pi\)
−0.860241 + 0.509888i \(0.829687\pi\)
\(702\) 0 0
\(703\) − 27.7128i − 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) − 3.46410i − 0.130373i
\(707\) 0 0
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 10.3923 0.390016
\(711\) 0 0
\(712\) 10.3923i 0.389468i
\(713\) −20.7846 −0.778390
\(714\) 0 0
\(715\) 0 0
\(716\) − 15.0000i − 0.560576i
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 48.4974 1.80865 0.904324 0.426846i \(-0.140375\pi\)
0.904324 + 0.426846i \(0.140375\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 29.0000i 1.07927i
\(723\) 0 0
\(724\) − 17.3205i − 0.643712i
\(725\) − 18.0000i − 0.668503i
\(726\) 0 0
\(727\) 12.1244i 0.449667i 0.974397 + 0.224834i \(0.0721839\pi\)
−0.974397 + 0.224834i \(0.927816\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 21.0000 0.777245
\(731\) −27.7128 −1.02500
\(732\) 0 0
\(733\) 6.92820i 0.255899i 0.991781 + 0.127950i \(0.0408395\pi\)
−0.991781 + 0.127950i \(0.959160\pi\)
\(734\) −8.66025 −0.319656
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 6.92820 0.254686
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) 0 0
\(745\) 25.9808i 0.951861i
\(746\) − 14.0000i − 0.512576i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 3.46410 0.126323
\(753\) 0 0
\(754\) 0 0
\(755\) 13.8564 0.504286
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 14.0000i 0.508503i
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 6.00000i − 0.217072i
\(765\) 0 0
\(766\) − 17.3205i − 0.625815i
\(767\) 0 0
\(768\) 0 0
\(769\) − 48.4974i − 1.74886i −0.485150 0.874431i \(-0.661235\pi\)
0.485150 0.874431i \(-0.338765\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 13.8564 0.498380 0.249190 0.968455i \(-0.419836\pi\)
0.249190 + 0.968455i \(0.419836\pi\)
\(774\) 0 0
\(775\) − 6.92820i − 0.248868i
\(776\) 6.92820 0.248708
\(777\) 0 0
\(778\) −21.0000 −0.752886
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 20.7846 0.743256
\(783\) 0 0
\(784\) 0 0
\(785\) − 18.0000i − 0.642448i
\(786\) 0 0
\(787\) 3.46410i 0.123482i 0.998092 + 0.0617409i \(0.0196653\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) − 3.00000i − 0.106871i
\(789\) 0 0
\(790\) 19.0526i 0.677860i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 27.7128 0.983491
\(795\) 0 0
\(796\) − 8.66025i − 0.306955i
\(797\) 41.5692 1.47246 0.736229 0.676733i \(-0.236605\pi\)
0.736229 + 0.676733i \(0.236605\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 2.00000i 0.0707107i
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) − 5.19615i − 0.182800i
\(809\) − 24.0000i − 0.843795i −0.906644 0.421898i \(-0.861364\pi\)
0.906644 0.421898i \(-0.138636\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.46410 −0.121342
\(816\) 0 0
\(817\) 55.4256i 1.93910i
\(818\) −15.5885 −0.545038
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 9.00000i 0.314102i 0.987590 + 0.157051i \(0.0501987\pi\)
−0.987590 + 0.157051i \(0.949801\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 10.3923 0.362033
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.00000i − 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) 0 0
\(829\) − 34.6410i − 1.20313i −0.798823 0.601566i \(-0.794544\pi\)
0.798823 0.601566i \(-0.205456\pi\)
\(830\) 30.0000i 1.04132i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 30.0000 1.03819
\(836\) 0 0
\(837\) 0 0
\(838\) − 3.46410i − 0.119665i
\(839\) −6.92820 −0.239188 −0.119594 0.992823i \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(840\) 0 0
\(841\) −52.0000 −1.79310
\(842\) − 32.0000i − 1.10279i
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) −22.5167 −0.774597
\(846\) 0 0
\(847\) 0 0
\(848\) − 3.00000i − 0.103020i
\(849\) 0 0
\(850\) 6.92820i 0.237635i
\(851\) 24.0000i 0.822709i
\(852\) 0 0
\(853\) 6.92820i 0.237217i 0.992941 + 0.118609i \(0.0378434\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) −6.92820 −0.236663 −0.118331 0.992974i \(-0.537755\pi\)
−0.118331 + 0.992974i \(0.537755\pi\)
\(858\) 0 0
\(859\) 24.2487i 0.827355i 0.910423 + 0.413678i \(0.135756\pi\)
−0.910423 + 0.413678i \(0.864244\pi\)
\(860\) −13.8564 −0.472500
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) 18.0000i 0.612727i 0.951915 + 0.306364i \(0.0991123\pi\)
−0.951915 + 0.306364i \(0.900888\pi\)
\(864\) 0 0
\(865\) −9.00000 −0.306009
\(866\) 1.73205 0.0588575
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 16.0000i 0.541828i
\(873\) 0 0
\(874\) − 41.5692i − 1.40610i
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 31.1769 1.05217
\(879\) 0 0
\(880\) 0 0
\(881\) −31.1769 −1.05038 −0.525188 0.850986i \(-0.676005\pi\)
−0.525188 + 0.850986i \(0.676005\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9.00000 −0.302361
\(887\) −17.3205 −0.581566 −0.290783 0.956789i \(-0.593916\pi\)
−0.290783 + 0.956789i \(0.593916\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 18.0000i 0.603361i
\(891\) 0 0
\(892\) 15.5885i 0.521940i
\(893\) − 24.0000i − 0.803129i
\(894\) 0 0
\(895\) − 25.9808i − 0.868441i
\(896\) 0 0
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −31.1769 −1.03981
\(900\) 0 0
\(901\) − 10.3923i − 0.346218i
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) − 30.0000i − 0.997234i
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −25.9808 −0.862202
\(909\) 0 0
\(910\) 0 0
\(911\) − 30.0000i − 0.993944i −0.867766 0.496972i \(-0.834445\pi\)
0.867766 0.496972i \(-0.165555\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 41.0000i 1.35616i
\(915\) 0 0
\(916\) − 3.46410i − 0.114457i
\(917\) 0 0
\(918\) 0 0
\(919\) 7.00000 0.230909 0.115454 0.993313i \(-0.463168\pi\)
0.115454 + 0.993313i \(0.463168\pi\)
\(920\) 10.3923 0.342624
\(921\) 0 0
\(922\) − 22.5167i − 0.741547i
\(923\) 0 0
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) − 23.0000i − 0.755827i
\(927\) 0 0
\(928\) 9.00000 0.295439
\(929\) −38.1051 −1.25019 −0.625094 0.780549i \(-0.714939\pi\)
−0.625094 + 0.780549i \(0.714939\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 30.0000i 0.982683i
\(933\) 0 0
\(934\) − 5.19615i − 0.170023i
\(935\) 0 0
\(936\) 0 0
\(937\) 57.1577i 1.86726i 0.358239 + 0.933630i \(0.383377\pi\)
−0.358239 + 0.933630i \(0.616623\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) −55.4256 −1.80682 −0.903412 0.428774i \(-0.858946\pi\)
−0.903412 + 0.428774i \(0.858946\pi\)
\(942\) 0 0
\(943\) − 20.7846i − 0.676840i
\(944\) 12.1244 0.394614
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000i 0.877382i 0.898638 + 0.438691i \(0.144558\pi\)
−0.898638 + 0.438691i \(0.855442\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 13.8564 0.449561
\(951\) 0 0
\(952\) 0 0
\(953\) − 12.0000i − 0.388718i −0.980930 0.194359i \(-0.937737\pi\)
0.980930 0.194359i \(-0.0622627\pi\)
\(954\) 0 0
\(955\) − 10.3923i − 0.336287i
\(956\) − 24.0000i − 0.776215i
\(957\) 0 0
\(958\) 13.8564i 0.447680i
\(959\) 0 0
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) − 1.73205i − 0.0557856i
\(965\) −17.3205 −0.557567
\(966\) 0 0
\(967\) 23.0000 0.739630 0.369815 0.929105i \(-0.379421\pi\)
0.369815 + 0.929105i \(0.379421\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 0 0
\(970\) 12.0000 0.385297
\(971\) 1.73205 0.0555842 0.0277921 0.999614i \(-0.491152\pi\)
0.0277921 + 0.999614i \(0.491152\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 1.00000i − 0.0320421i
\(975\) 0 0
\(976\) − 3.46410i − 0.110883i
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −3.00000 −0.0957338
\(983\) 27.7128 0.883901 0.441951 0.897039i \(-0.354287\pi\)
0.441951 + 0.897039i \(0.354287\pi\)
\(984\) 0 0
\(985\) − 5.19615i − 0.165563i
\(986\) 31.1769 0.992875
\(987\) 0 0
\(988\) 0 0
\(989\) − 48.0000i − 1.52631i
\(990\) 0 0
\(991\) −5.00000 −0.158830 −0.0794151 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) 3.46410 0.109985
\(993\) 0 0
\(994\) 0 0
\(995\) − 15.0000i − 0.475532i
\(996\) 0 0
\(997\) 48.4974i 1.53593i 0.640493 + 0.767964i \(0.278730\pi\)
−0.640493 + 0.767964i \(0.721270\pi\)
\(998\) 14.0000i 0.443162i
\(999\) 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 2646.2.d.a.2645.1 4
3.2 odd 2 inner 2646.2.d.a.2645.4 4
7.4 even 3 378.2.k.c.215.2 yes 4
7.5 odd 6 378.2.k.c.269.1 yes 4
7.6 odd 2 inner 2646.2.d.a.2645.2 4
21.5 even 6 378.2.k.c.269.2 yes 4
21.11 odd 6 378.2.k.c.215.1 4
21.20 even 2 inner 2646.2.d.a.2645.3 4
63.4 even 3 1134.2.t.c.593.1 4
63.5 even 6 1134.2.l.b.269.2 4
63.11 odd 6 1134.2.l.b.215.2 4
63.25 even 3 1134.2.l.b.215.1 4
63.32 odd 6 1134.2.t.c.593.2 4
63.40 odd 6 1134.2.l.b.269.1 4
63.47 even 6 1134.2.t.c.1025.1 4
63.61 odd 6 1134.2.t.c.1025.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.c.215.1 4 21.11 odd 6
378.2.k.c.215.2 yes 4 7.4 even 3
378.2.k.c.269.1 yes 4 7.5 odd 6
378.2.k.c.269.2 yes 4 21.5 even 6
1134.2.l.b.215.1 4 63.25 even 3
1134.2.l.b.215.2 4 63.11 odd 6
1134.2.l.b.269.1 4 63.40 odd 6
1134.2.l.b.269.2 4 63.5 even 6
1134.2.t.c.593.1 4 63.4 even 3
1134.2.t.c.593.2 4 63.32 odd 6
1134.2.t.c.1025.1 4 63.47 even 6
1134.2.t.c.1025.2 4 63.61 odd 6
2646.2.d.a.2645.1 4 1.1 even 1 trivial
2646.2.d.a.2645.2 4 7.6 odd 2 inner
2646.2.d.a.2645.3 4 21.20 even 2 inner
2646.2.d.a.2645.4 4 3.2 odd 2 inner