Properties

Label 1323.2.c.b.1322.3
Level $1323$
Weight $2$
Character 1323.1322
Analytic conductor $10.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1322.3
Root \(1.93649 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.b.1322.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{8} +O(q^{10})\) \(q+2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{8} -4.47214i q^{11} -1.73205i q^{13} -1.00000 q^{16} +7.74597 q^{17} -3.46410i q^{19} +10.0000 q^{22} -4.47214i q^{23} -5.00000 q^{25} +3.87298 q^{26} -4.47214i q^{29} +1.73205i q^{31} -6.70820i q^{32} +17.3205i q^{34} +5.00000 q^{37} +7.74597 q^{38} +7.74597 q^{41} -7.00000 q^{43} +13.4164i q^{44} +10.0000 q^{46} +7.74597 q^{47} -11.1803i q^{50} +5.19615i q^{52} -4.47214i q^{53} +10.0000 q^{58} -7.74597 q^{59} +8.66025i q^{61} -3.87298 q^{62} +13.0000 q^{64} -1.00000 q^{67} -23.2379 q^{68} +8.94427i q^{71} -6.92820i q^{73} +11.1803i q^{74} +10.3923i q^{76} +11.0000 q^{79} +17.3205i q^{82} -7.74597 q^{83} -15.6525i q^{86} -10.0000 q^{88} +15.4919 q^{89} +13.4164i q^{92} +17.3205i q^{94} +1.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{4} + O(q^{10}) \) \( 4q - 12q^{4} - 4q^{16} + 40q^{22} - 20q^{25} + 20q^{37} - 28q^{43} + 40q^{46} + 40q^{58} + 52q^{64} - 4q^{67} + 44q^{79} - 40q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i 0.612372 + 0.790569i \(0.290215\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) −3.00000 −1.50000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.23607i − 0.790569i
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.47214i − 1.34840i −0.738549 0.674200i \(-0.764489\pi\)
0.738549 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) − 1.73205i − 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 7.74597 1.87867 0.939336 0.342997i \(-0.111442\pi\)
0.939336 + 0.342997i \(0.111442\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.0000 2.13201
\(23\) − 4.47214i − 0.932505i −0.884652 0.466252i \(-0.845604\pi\)
0.884652 0.466252i \(-0.154396\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 3.87298 0.759555
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.47214i − 0.830455i −0.909718 0.415227i \(-0.863702\pi\)
0.909718 0.415227i \(-0.136298\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) − 6.70820i − 1.18585i
\(33\) 0 0
\(34\) 17.3205i 2.97044i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 7.74597 1.25656
\(39\) 0 0
\(40\) 0 0
\(41\) 7.74597 1.20972 0.604858 0.796333i \(-0.293230\pi\)
0.604858 + 0.796333i \(0.293230\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 13.4164i 2.02260i
\(45\) 0 0
\(46\) 10.0000 1.47442
\(47\) 7.74597 1.12987 0.564933 0.825137i \(-0.308902\pi\)
0.564933 + 0.825137i \(0.308902\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 11.1803i − 1.58114i
\(51\) 0 0
\(52\) 5.19615i 0.720577i
\(53\) − 4.47214i − 0.614295i −0.951662 0.307148i \(-0.900625\pi\)
0.951662 0.307148i \(-0.0993745\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −7.74597 −1.00844 −0.504219 0.863576i \(-0.668220\pi\)
−0.504219 + 0.863576i \(0.668220\pi\)
\(60\) 0 0
\(61\) 8.66025i 1.10883i 0.832240 + 0.554416i \(0.187058\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) −3.87298 −0.491869
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) −23.2379 −2.81801
\(69\) 0 0
\(70\) 0 0
\(71\) 8.94427i 1.06149i 0.847532 + 0.530745i \(0.178088\pi\)
−0.847532 + 0.530745i \(0.821912\pi\)
\(72\) 0 0
\(73\) − 6.92820i − 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 11.1803i 1.29969i
\(75\) 0 0
\(76\) 10.3923i 1.19208i
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 17.3205i 1.91273i
\(83\) −7.74597 −0.850230 −0.425115 0.905139i \(-0.639766\pi\)
−0.425115 + 0.905139i \(0.639766\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 15.6525i − 1.68785i
\(87\) 0 0
\(88\) −10.0000 −1.06600
\(89\) 15.4919 1.64214 0.821071 0.570826i \(-0.193377\pi\)
0.821071 + 0.570826i \(0.193377\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 13.4164i 1.39876i
\(93\) 0 0
\(94\) 17.3205i 1.78647i
\(95\) 0 0
\(96\) 0 0
\(97\) 1.73205i 0.175863i 0.996127 + 0.0879316i \(0.0280257\pi\)
−0.996127 + 0.0879316i \(0.971974\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 15.0000 1.50000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 5.19615i − 0.511992i −0.966678 0.255996i \(-0.917597\pi\)
0.966678 0.255996i \(-0.0824034\pi\)
\(104\) −3.87298 −0.379777
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) − 4.47214i − 0.432338i −0.976356 0.216169i \(-0.930644\pi\)
0.976356 0.216169i \(-0.0693562\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.94427i 0.841406i 0.907198 + 0.420703i \(0.138217\pi\)
−0.907198 + 0.420703i \(0.861783\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 13.4164i 1.24568i
\(117\) 0 0
\(118\) − 17.3205i − 1.59448i
\(119\) 0 0
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) −19.3649 −1.75322
\(123\) 0 0
\(124\) − 5.19615i − 0.466628i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 15.6525i 1.38350i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 2.23607i − 0.193167i
\(135\) 0 0
\(136\) − 17.3205i − 1.48522i
\(137\) − 4.47214i − 0.382080i −0.981582 0.191040i \(-0.938814\pi\)
0.981582 0.191040i \(-0.0611861\pi\)
\(138\) 0 0
\(139\) 8.66025i 0.734553i 0.930112 + 0.367277i \(0.119710\pi\)
−0.930112 + 0.367277i \(0.880290\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −20.0000 −1.67836
\(143\) −7.74597 −0.647750
\(144\) 0 0
\(145\) 0 0
\(146\) 15.4919 1.28212
\(147\) 0 0
\(148\) −15.0000 −1.23299
\(149\) − 17.8885i − 1.46549i −0.680505 0.732743i \(-0.738240\pi\)
0.680505 0.732743i \(-0.261760\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) −7.74597 −0.628281
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.8564i − 1.10586i −0.833227 0.552931i \(-0.813509\pi\)
0.833227 0.552931i \(-0.186491\pi\)
\(158\) 24.5967i 1.95681i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) −23.2379 −1.81458
\(165\) 0 0
\(166\) − 17.3205i − 1.34433i
\(167\) 7.74597 0.599401 0.299700 0.954033i \(-0.403113\pi\)
0.299700 + 0.954033i \(0.403113\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 0 0
\(172\) 21.0000 1.60123
\(173\) −15.4919 −1.17783 −0.588915 0.808195i \(-0.700445\pi\)
−0.588915 + 0.808195i \(0.700445\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.47214i 0.337100i
\(177\) 0 0
\(178\) 34.6410i 2.59645i
\(179\) 8.94427i 0.668526i 0.942480 + 0.334263i \(0.108487\pi\)
−0.942480 + 0.334263i \(0.891513\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −10.0000 −0.737210
\(185\) 0 0
\(186\) 0 0
\(187\) − 34.6410i − 2.53320i
\(188\) −23.2379 −1.69480
\(189\) 0 0
\(190\) 0 0
\(191\) − 17.8885i − 1.29437i −0.762333 0.647185i \(-0.775946\pi\)
0.762333 0.647185i \(-0.224054\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) −3.87298 −0.278064
\(195\) 0 0
\(196\) 0 0
\(197\) 8.94427i 0.637253i 0.947880 + 0.318626i \(0.103222\pi\)
−0.947880 + 0.318626i \(0.896778\pi\)
\(198\) 0 0
\(199\) − 22.5167i − 1.59616i −0.602549 0.798082i \(-0.705848\pi\)
0.602549 0.798082i \(-0.294152\pi\)
\(200\) 11.1803i 0.790569i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 11.6190 0.809531
\(207\) 0 0
\(208\) 1.73205i 0.120096i
\(209\) −15.4919 −1.07160
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) 13.4164i 0.921443i
\(213\) 0 0
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 2.23607i − 0.151446i
\(219\) 0 0
\(220\) 0 0
\(221\) − 13.4164i − 0.902485i
\(222\) 0 0
\(223\) − 24.2487i − 1.62381i −0.583787 0.811907i \(-0.698430\pi\)
0.583787 0.811907i \(-0.301570\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −20.0000 −1.33038
\(227\) 23.2379 1.54235 0.771177 0.636621i \(-0.219668\pi\)
0.771177 + 0.636621i \(0.219668\pi\)
\(228\) 0 0
\(229\) 25.9808i 1.71686i 0.512933 + 0.858429i \(0.328559\pi\)
−0.512933 + 0.858429i \(0.671441\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) 8.94427i 0.585959i 0.956119 + 0.292979i \(0.0946467\pi\)
−0.956119 + 0.292979i \(0.905353\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 23.2379 1.51266
\(237\) 0 0
\(238\) 0 0
\(239\) 8.94427i 0.578557i 0.957245 + 0.289278i \(0.0934153\pi\)
−0.957245 + 0.289278i \(0.906585\pi\)
\(240\) 0 0
\(241\) − 15.5885i − 1.00414i −0.864827 0.502070i \(-0.832572\pi\)
0.864827 0.502070i \(-0.167428\pi\)
\(242\) − 20.1246i − 1.29366i
\(243\) 0 0
\(244\) − 25.9808i − 1.66325i
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 3.87298 0.245935
\(249\) 0 0
\(250\) 0 0
\(251\) 23.2379 1.46676 0.733382 0.679817i \(-0.237941\pi\)
0.733382 + 0.679817i \(0.237941\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) − 2.23607i − 0.140303i
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) 23.2379 1.44954 0.724770 0.688991i \(-0.241946\pi\)
0.724770 + 0.688991i \(0.241946\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 4.47214i − 0.275764i −0.990449 0.137882i \(-0.955971\pi\)
0.990449 0.137882i \(-0.0440294\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 3.00000 0.183254
\(269\) −15.4919 −0.944560 −0.472280 0.881449i \(-0.656569\pi\)
−0.472280 + 0.881449i \(0.656569\pi\)
\(270\) 0 0
\(271\) 1.73205i 0.105215i 0.998615 + 0.0526073i \(0.0167532\pi\)
−0.998615 + 0.0526073i \(0.983247\pi\)
\(272\) −7.74597 −0.469668
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 22.3607i 1.34840i
\(276\) 0 0
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) −19.3649 −1.16143
\(279\) 0 0
\(280\) 0 0
\(281\) − 17.8885i − 1.06714i −0.845756 0.533571i \(-0.820850\pi\)
0.845756 0.533571i \(-0.179150\pi\)
\(282\) 0 0
\(283\) 12.1244i 0.720718i 0.932814 + 0.360359i \(0.117346\pi\)
−0.932814 + 0.360359i \(0.882654\pi\)
\(284\) − 26.8328i − 1.59223i
\(285\) 0 0
\(286\) − 17.3205i − 1.02418i
\(287\) 0 0
\(288\) 0 0
\(289\) 43.0000 2.52941
\(290\) 0 0
\(291\) 0 0
\(292\) 20.7846i 1.21633i
\(293\) 7.74597 0.452524 0.226262 0.974066i \(-0.427349\pi\)
0.226262 + 0.974066i \(0.427349\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 11.1803i − 0.649844i
\(297\) 0 0
\(298\) 40.0000 2.31714
\(299\) −7.74597 −0.447961
\(300\) 0 0
\(301\) 0 0
\(302\) − 29.0689i − 1.67273i
\(303\) 0 0
\(304\) 3.46410i 0.198680i
\(305\) 0 0
\(306\) 0 0
\(307\) 5.19615i 0.296560i 0.988945 + 0.148280i \(0.0473737\pi\)
−0.988945 + 0.148280i \(0.952626\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.4919 0.878467 0.439233 0.898373i \(-0.355250\pi\)
0.439233 + 0.898373i \(0.355250\pi\)
\(312\) 0 0
\(313\) 13.8564i 0.783210i 0.920133 + 0.391605i \(0.128080\pi\)
−0.920133 + 0.391605i \(0.871920\pi\)
\(314\) 30.9839 1.74852
\(315\) 0 0
\(316\) −33.0000 −1.85640
\(317\) − 4.47214i − 0.251180i −0.992082 0.125590i \(-0.959918\pi\)
0.992082 0.125590i \(-0.0400824\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 26.8328i − 1.49302i
\(324\) 0 0
\(325\) 8.66025i 0.480384i
\(326\) − 29.0689i − 1.60998i
\(327\) 0 0
\(328\) − 17.3205i − 0.956365i
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 23.2379 1.27535
\(333\) 0 0
\(334\) 17.3205i 0.947736i
\(335\) 0 0
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 22.3607i 1.21626i
\(339\) 0 0
\(340\) 0 0
\(341\) 7.74597 0.419468
\(342\) 0 0
\(343\) 0 0
\(344\) 15.6525i 0.843925i
\(345\) 0 0
\(346\) − 34.6410i − 1.86231i
\(347\) 8.94427i 0.480154i 0.970754 + 0.240077i \(0.0771726\pi\)
−0.970754 + 0.240077i \(0.922827\pi\)
\(348\) 0 0
\(349\) 22.5167i 1.20529i 0.798010 + 0.602645i \(0.205886\pi\)
−0.798010 + 0.602645i \(0.794114\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −30.0000 −1.59901
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −46.4758 −2.46321
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) − 4.47214i − 0.236030i −0.993012 0.118015i \(-0.962347\pi\)
0.993012 0.118015i \(-0.0376531\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.3923i 0.542474i 0.962513 + 0.271237i \(0.0874327\pi\)
−0.962513 + 0.271237i \(0.912567\pi\)
\(368\) 4.47214i 0.233126i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 77.4597 4.00534
\(375\) 0 0
\(376\) − 17.3205i − 0.893237i
\(377\) −7.74597 −0.398938
\(378\) 0 0
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 40.0000 2.04658
\(383\) 23.2379 1.18740 0.593701 0.804686i \(-0.297666\pi\)
0.593701 + 0.804686i \(0.297666\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 15.6525i − 0.796690i
\(387\) 0 0
\(388\) − 5.19615i − 0.263795i
\(389\) 8.94427i 0.453493i 0.973954 + 0.226746i \(0.0728088\pi\)
−0.973954 + 0.226746i \(0.927191\pi\)
\(390\) 0 0
\(391\) − 34.6410i − 1.75187i
\(392\) 0 0
\(393\) 0 0
\(394\) −20.0000 −1.00759
\(395\) 0 0
\(396\) 0 0
\(397\) − 29.4449i − 1.47780i −0.673818 0.738898i \(-0.735347\pi\)
0.673818 0.738898i \(-0.264653\pi\)
\(398\) 50.3488 2.52376
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 22.3607i 1.11664i 0.829626 + 0.558320i \(0.188554\pi\)
−0.829626 + 0.558320i \(0.811446\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 22.3607i − 1.10838i
\(408\) 0 0
\(409\) 22.5167i 1.11338i 0.830721 + 0.556689i \(0.187928\pi\)
−0.830721 + 0.556689i \(0.812072\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 15.5885i 0.767988i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −11.6190 −0.569666
\(417\) 0 0
\(418\) − 34.6410i − 1.69435i
\(419\) −15.4919 −0.756830 −0.378415 0.925636i \(-0.623531\pi\)
−0.378415 + 0.925636i \(0.623531\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) − 42.4853i − 2.06815i
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −38.7298 −1.87867
\(426\) 0 0
\(427\) 0 0
\(428\) 13.4164i 0.648507i
\(429\) 0 0
\(430\) 0 0
\(431\) − 31.3050i − 1.50791i −0.656928 0.753953i \(-0.728145\pi\)
0.656928 0.753953i \(-0.271855\pi\)
\(432\) 0 0
\(433\) − 15.5885i − 0.749133i −0.927200 0.374567i \(-0.877791\pi\)
0.927200 0.374567i \(-0.122209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.00000 0.143674
\(437\) −15.4919 −0.741080
\(438\) 0 0
\(439\) 3.46410i 0.165333i 0.996577 + 0.0826663i \(0.0263436\pi\)
−0.996577 + 0.0826663i \(0.973656\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 30.0000 1.42695
\(443\) 35.7771i 1.69982i 0.526927 + 0.849910i \(0.323344\pi\)
−0.526927 + 0.849910i \(0.676656\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 54.2218 2.56748
\(447\) 0 0
\(448\) 0 0
\(449\) 8.94427i 0.422106i 0.977475 + 0.211053i \(0.0676893\pi\)
−0.977475 + 0.211053i \(0.932311\pi\)
\(450\) 0 0
\(451\) − 34.6410i − 1.63118i
\(452\) − 26.8328i − 1.26211i
\(453\) 0 0
\(454\) 51.9615i 2.43868i
\(455\) 0 0
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) −58.0948 −2.71459
\(459\) 0 0
\(460\) 0 0
\(461\) −7.74597 −0.360766 −0.180383 0.983596i \(-0.557734\pi\)
−0.180383 + 0.983596i \(0.557734\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 4.47214i 0.207614i
\(465\) 0 0
\(466\) −20.0000 −0.926482
\(467\) −30.9839 −1.43376 −0.716881 0.697195i \(-0.754431\pi\)
−0.716881 + 0.697195i \(0.754431\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 17.3205i 0.797241i
\(473\) 31.3050i 1.43940i
\(474\) 0 0
\(475\) 17.3205i 0.794719i
\(476\) 0 0
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) − 8.66025i − 0.394874i
\(482\) 34.8569 1.58769
\(483\) 0 0
\(484\) 27.0000 1.22727
\(485\) 0 0
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 19.3649 0.876609
\(489\) 0 0
\(490\) 0 0
\(491\) 22.3607i 1.00912i 0.863376 + 0.504562i \(0.168346\pi\)
−0.863376 + 0.504562i \(0.831654\pi\)
\(492\) 0 0
\(493\) − 34.6410i − 1.56015i
\(494\) − 13.4164i − 0.603633i
\(495\) 0 0
\(496\) − 1.73205i − 0.0777714i
\(497\) 0 0
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 51.9615i 2.31916i
\(503\) −23.2379 −1.03613 −0.518063 0.855342i \(-0.673347\pi\)
−0.518063 + 0.855342i \(0.673347\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 44.7214i − 1.98811i
\(507\) 0 0
\(508\) 3.00000 0.133103
\(509\) −23.2379 −1.03000 −0.515001 0.857190i \(-0.672208\pi\)
−0.515001 + 0.857190i \(0.672208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 51.9615i 2.29192i
\(515\) 0 0
\(516\) 0 0
\(517\) − 34.6410i − 1.52351i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.74597 0.339357 0.169678 0.985499i \(-0.445727\pi\)
0.169678 + 0.985499i \(0.445727\pi\)
\(522\) 0 0
\(523\) − 19.0526i − 0.833110i −0.909110 0.416555i \(-0.863237\pi\)
0.909110 0.416555i \(-0.136763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 10.0000 0.436021
\(527\) 13.4164i 0.584428i
\(528\) 0 0
\(529\) 3.00000 0.130435
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 13.4164i − 0.581129i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.23607i 0.0965834i
\(537\) 0 0
\(538\) − 34.6410i − 1.49348i
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −3.87298 −0.166359
\(543\) 0 0
\(544\) − 51.9615i − 2.22783i
\(545\) 0 0
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 13.4164i 0.573121i
\(549\) 0 0
\(550\) −50.0000 −2.13201
\(551\) −15.4919 −0.659979
\(552\) 0 0
\(553\) 0 0
\(554\) 11.1803i 0.475007i
\(555\) 0 0
\(556\) − 25.9808i − 1.10183i
\(557\) 35.7771i 1.51592i 0.652299 + 0.757962i \(0.273805\pi\)
−0.652299 + 0.757962i \(0.726195\pi\)
\(558\) 0 0
\(559\) 12.1244i 0.512806i
\(560\) 0 0
\(561\) 0 0
\(562\) 40.0000 1.68730
\(563\) −7.74597 −0.326454 −0.163227 0.986589i \(-0.552190\pi\)
−0.163227 + 0.986589i \(0.552190\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −27.1109 −1.13956
\(567\) 0 0
\(568\) 20.0000 0.839181
\(569\) − 17.8885i − 0.749927i −0.927040 0.374963i \(-0.877655\pi\)
0.927040 0.374963i \(-0.122345\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 23.2379 0.971625
\(573\) 0 0
\(574\) 0 0
\(575\) 22.3607i 0.932505i
\(576\) 0 0
\(577\) 39.8372i 1.65844i 0.558920 + 0.829222i \(0.311216\pi\)
−0.558920 + 0.829222i \(0.688784\pi\)
\(578\) 96.1509i 3.99935i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) −15.4919 −0.641061
\(585\) 0 0
\(586\) 17.3205i 0.715504i
\(587\) −30.9839 −1.27884 −0.639421 0.768857i \(-0.720826\pi\)
−0.639421 + 0.768857i \(0.720826\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) −5.00000 −0.205499
\(593\) −30.9839 −1.27235 −0.636177 0.771543i \(-0.719485\pi\)
−0.636177 + 0.771543i \(0.719485\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 53.6656i 2.19823i
\(597\) 0 0
\(598\) − 17.3205i − 0.708288i
\(599\) 8.94427i 0.365453i 0.983164 + 0.182727i \(0.0584923\pi\)
−0.983164 + 0.182727i \(0.941508\pi\)
\(600\) 0 0
\(601\) 43.3013i 1.76630i 0.469095 + 0.883148i \(0.344580\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 39.0000 1.58689
\(605\) 0 0
\(606\) 0 0
\(607\) 31.1769i 1.26543i 0.774384 + 0.632716i \(0.218060\pi\)
−0.774384 + 0.632716i \(0.781940\pi\)
\(608\) −23.2379 −0.942421
\(609\) 0 0
\(610\) 0 0
\(611\) − 13.4164i − 0.542770i
\(612\) 0 0
\(613\) 23.0000 0.928961 0.464481 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) −11.6190 −0.468903
\(615\) 0 0
\(616\) 0 0
\(617\) 22.3607i 0.900207i 0.892976 + 0.450104i \(0.148613\pi\)
−0.892976 + 0.450104i \(0.851387\pi\)
\(618\) 0 0
\(619\) − 5.19615i − 0.208851i −0.994533 0.104425i \(-0.966700\pi\)
0.994533 0.104425i \(-0.0333004\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 34.6410i 1.38898i
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −30.9839 −1.23836
\(627\) 0 0
\(628\) 41.5692i 1.65879i
\(629\) 38.7298 1.54426
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) − 24.5967i − 0.978406i
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) − 44.7214i − 1.77054i
\(639\) 0 0
\(640\) 0 0
\(641\) − 4.47214i − 0.176639i −0.996092 0.0883194i \(-0.971850\pi\)
0.996092 0.0883194i \(-0.0281496\pi\)
\(642\) 0 0
\(643\) 19.0526i 0.751360i 0.926750 + 0.375680i \(0.122591\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 60.0000 2.36067
\(647\) −15.4919 −0.609051 −0.304525 0.952504i \(-0.598498\pi\)
−0.304525 + 0.952504i \(0.598498\pi\)
\(648\) 0 0
\(649\) 34.6410i 1.35978i
\(650\) −19.3649 −0.759555
\(651\) 0 0
\(652\) 39.0000 1.52736
\(653\) 22.3607i 0.875041i 0.899208 + 0.437521i \(0.144143\pi\)
−0.899208 + 0.437521i \(0.855857\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.74597 −0.302429
\(657\) 0 0
\(658\) 0 0
\(659\) − 17.8885i − 0.696839i −0.937339 0.348419i \(-0.886719\pi\)
0.937339 0.348419i \(-0.113281\pi\)
\(660\) 0 0
\(661\) 27.7128i 1.07790i 0.842337 + 0.538952i \(0.181179\pi\)
−0.842337 + 0.538952i \(0.818821\pi\)
\(662\) − 8.94427i − 0.347629i
\(663\) 0 0
\(664\) 17.3205i 0.672166i
\(665\) 0 0
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) −23.2379 −0.899101
\(669\) 0 0
\(670\) 0 0
\(671\) 38.7298 1.49515
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) − 22.3607i − 0.861301i
\(675\) 0 0
\(676\) −30.0000 −1.15385
\(677\) 30.9839 1.19081 0.595403 0.803427i \(-0.296992\pi\)
0.595403 + 0.803427i \(0.296992\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 17.3205i 0.663237i
\(683\) 35.7771i 1.36897i 0.729026 + 0.684486i \(0.239973\pi\)
−0.729026 + 0.684486i \(0.760027\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 7.00000 0.266872
\(689\) −7.74597 −0.295098
\(690\) 0 0
\(691\) − 32.9090i − 1.25192i −0.779857 0.625958i \(-0.784708\pi\)
0.779857 0.625958i \(-0.215292\pi\)
\(692\) 46.4758 1.76674
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) −50.3488 −1.90573
\(699\) 0 0
\(700\) 0 0
\(701\) 8.94427i 0.337820i 0.985631 + 0.168910i \(0.0540248\pi\)
−0.985631 + 0.168910i \(0.945975\pi\)
\(702\) 0 0
\(703\) − 17.3205i − 0.653255i
\(704\) − 58.1378i − 2.19115i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 11.0000 0.413114 0.206557 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 34.6410i − 1.29823i
\(713\) 7.74597 0.290089
\(714\) 0 0
\(715\) 0 0
\(716\) − 26.8328i − 1.00279i
\(717\) 0 0
\(718\) 10.0000 0.373197
\(719\) 15.4919 0.577752 0.288876 0.957367i \(-0.406719\pi\)
0.288876 + 0.957367i \(0.406719\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.6525i 0.582525i
\(723\) 0 0
\(724\) 0 0
\(725\) 22.3607i 0.830455i
\(726\) 0 0
\(727\) − 19.0526i − 0.706620i −0.935506 0.353310i \(-0.885056\pi\)
0.935506 0.353310i \(-0.114944\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −54.2218 −2.00546
\(732\) 0 0
\(733\) 36.3731i 1.34347i 0.740792 + 0.671735i \(0.234451\pi\)
−0.740792 + 0.671735i \(0.765549\pi\)
\(734\) −23.2379 −0.857727
\(735\) 0 0
\(736\) −30.0000 −1.10581
\(737\) 4.47214i 0.164733i
\(738\) 0 0
\(739\) 23.0000 0.846069 0.423034 0.906114i \(-0.360965\pi\)
0.423034 + 0.906114i \(0.360965\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 31.3050i − 1.14847i −0.818691 0.574234i \(-0.805300\pi\)
0.818691 0.574234i \(-0.194700\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.47214i 0.163737i
\(747\) 0 0
\(748\) 103.923i 3.79980i
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −7.74597 −0.282466
\(753\) 0 0
\(754\) − 17.3205i − 0.630776i
\(755\) 0 0
\(756\) 0 0
\(757\) −25.0000 −0.908640 −0.454320 0.890838i \(-0.650118\pi\)
−0.454320 + 0.890838i \(0.650118\pi\)
\(758\) 38.0132i 1.38070i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 53.6656i 1.94155i
\(765\) 0 0
\(766\) 51.9615i 1.87745i
\(767\) 13.4164i 0.484438i
\(768\) 0 0
\(769\) 34.6410i 1.24919i 0.780950 + 0.624593i \(0.214735\pi\)
−0.780950 + 0.624593i \(0.785265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 21.0000 0.755807
\(773\) 30.9839 1.11441 0.557206 0.830374i \(-0.311873\pi\)
0.557206 + 0.830374i \(0.311873\pi\)
\(774\) 0 0
\(775\) − 8.66025i − 0.311086i
\(776\) 3.87298 0.139032
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) − 26.8328i − 0.961385i
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 77.4597 2.76995
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 19.0526i − 0.679150i −0.940579 0.339575i \(-0.889717\pi\)
0.940579 0.339575i \(-0.110283\pi\)
\(788\) − 26.8328i − 0.955879i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 15.0000 0.532666
\(794\) 65.8407 2.33660
\(795\) 0 0
\(796\) 67.5500i 2.39425i
\(797\) −38.7298 −1.37188 −0.685941 0.727658i \(-0.740609\pi\)
−0.685941 + 0.727658i \(0.740609\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) 33.5410i 1.18585i
\(801\) 0 0
\(802\) −50.0000 −1.76556
\(803\) −30.9839 −1.09340
\(804\) 0 0
\(805\) 0 0
\(806\) 6.70820i 0.236286i
\(807\) 0 0
\(808\) 0 0
\(809\) − 4.47214i − 0.157232i −0.996905 0.0786160i \(-0.974950\pi\)
0.996905 0.0786160i \(-0.0250501\pi\)
\(810\) 0 0
\(811\) 10.3923i 0.364923i 0.983213 + 0.182462i \(0.0584065\pi\)
−0.983213 + 0.182462i \(0.941593\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 50.0000 1.75250
\(815\) 0 0
\(816\) 0 0
\(817\) 24.2487i 0.848355i
\(818\) −50.3488 −1.76040
\(819\) 0 0
\(820\) 0 0
\(821\) 22.3607i 0.780393i 0.920732 + 0.390197i \(0.127593\pi\)
−0.920732 + 0.390197i \(0.872407\pi\)
\(822\) 0 0
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) −11.6190 −0.404765
\(825\) 0 0
\(826\) 0 0
\(827\) 8.94427i 0.311023i 0.987834 + 0.155511i \(0.0497025\pi\)
−0.987834 + 0.155511i \(0.950297\pi\)
\(828\) 0 0
\(829\) 13.8564i 0.481253i 0.970618 + 0.240626i \(0.0773529\pi\)
−0.970618 + 0.240626i \(0.922647\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 22.5167i − 0.780625i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 46.4758 1.60740
\(837\) 0 0
\(838\) − 34.6410i − 1.19665i
\(839\) −7.74597 −0.267420 −0.133710 0.991020i \(-0.542689\pi\)
−0.133710 + 0.991020i \(0.542689\pi\)
\(840\) 0 0
\(841\) 9.00000 0.310345
\(842\) − 76.0263i − 2.62004i
\(843\) 0 0
\(844\) 57.0000 1.96202
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 4.47214i 0.153574i
\(849\) 0 0
\(850\) − 86.6025i − 2.97044i
\(851\) − 22.3607i − 0.766514i
\(852\) 0 0
\(853\) − 13.8564i − 0.474434i −0.971457 0.237217i \(-0.923765\pi\)
0.971457 0.237217i \(-0.0762353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 5.19615i 0.177290i 0.996063 + 0.0886452i \(0.0282537\pi\)
−0.996063 + 0.0886452i \(0.971746\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 70.0000 2.38421
\(863\) 35.7771i 1.21787i 0.793222 + 0.608933i \(0.208402\pi\)
−0.793222 + 0.608933i \(0.791598\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 34.8569 1.18448
\(867\) 0 0
\(868\) 0 0
\(869\) − 49.1935i − 1.66878i
\(870\) 0 0
\(871\) 1.73205i 0.0586883i
\(872\) 2.23607i 0.0757228i
\(873\) 0 0
\(874\) − 34.6410i − 1.17175i
\(875\) 0 0
\(876\) 0 0
\(877\) −25.0000 −0.844190 −0.422095 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) −7.74597 −0.261414
\(879\) 0 0
\(880\) 0 0
\(881\) −23.2379 −0.782905 −0.391452 0.920198i \(-0.628027\pi\)
−0.391452 + 0.920198i \(0.628027\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 40.2492i 1.35373i
\(885\) 0 0
\(886\) −80.0000 −2.68765
\(887\) −23.2379 −0.780252 −0.390126 0.920761i \(-0.627569\pi\)
−0.390126 + 0.920761i \(0.627569\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 72.7461i 2.43572i
\(893\) − 26.8328i − 0.897926i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 7.74597 0.258342
\(900\) 0 0
\(901\) − 34.6410i − 1.15406i
\(902\) 77.4597 2.57912
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) 0 0
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) −69.7137 −2.31353
\(909\) 0 0
\(910\) 0 0
\(911\) − 4.47214i − 0.148168i −0.997252 0.0740842i \(-0.976397\pi\)
0.997252 0.0740842i \(-0.0236034\pi\)
\(912\) 0 0
\(913\) 34.6410i 1.14645i
\(914\) 24.5967i 0.813588i
\(915\) 0 0
\(916\) − 77.9423i − 2.57529i
\(917\) 0 0
\(918\) 0 0
\(919\) −13.0000 −0.428830 −0.214415 0.976743i \(-0.568785\pi\)
−0.214415 + 0.976743i \(0.568785\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 17.3205i − 0.570421i
\(923\) 15.4919 0.509923
\(924\) 0 0
\(925\) −25.0000 −0.821995
\(926\) 17.8885i 0.587854i
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 30.9839 1.01655 0.508274 0.861196i \(-0.330284\pi\)
0.508274 + 0.861196i \(0.330284\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 26.8328i − 0.878938i
\(933\) 0 0
\(934\) − 69.2820i − 2.26698i
\(935\) 0 0
\(936\) 0 0
\(937\) 46.7654i 1.52776i 0.645359 + 0.763879i \(0.276708\pi\)
−0.645359 + 0.763879i \(0.723292\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.7298 1.26256 0.631278 0.775556i \(-0.282531\pi\)
0.631278 + 0.775556i \(0.282531\pi\)
\(942\) 0 0
\(943\) − 34.6410i − 1.12807i
\(944\) 7.74597 0.252110
\(945\) 0 0
\(946\) −70.0000 −2.27590
\(947\) − 4.47214i − 0.145325i −0.997357 0.0726624i \(-0.976850\pi\)
0.997357 0.0726624i \(-0.0231496\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −38.7298 −1.25656
\(951\) 0 0
\(952\) 0 0
\(953\) − 58.1378i − 1.88327i −0.336640 0.941634i \(-0.609290\pi\)
0.336640 0.941634i \(-0.390710\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 26.8328i − 0.867835i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 28.0000 0.903226
\(962\) 19.3649 0.624350
\(963\) 0 0
\(964\) 46.7654i 1.50621i
\(965\) 0 0
\(966\) 0 0
\(967\) −25.0000 −0.803946 −0.401973 0.915652i \(-0.631675\pi\)
−0.401973 + 0.915652i \(0.631675\pi\)
\(968\) 20.1246i 0.646830i
\(969\) 0 0
\(970\) 0 0
\(971\) 61.9677 1.98864 0.994320 0.106436i \(-0.0339438\pi\)
0.994320 + 0.106436i \(0.0339438\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 89.4427i − 2.86593i
\(975\) 0 0
\(976\) − 8.66025i − 0.277208i
\(977\) − 58.1378i − 1.85999i −0.367570 0.929996i \(-0.619810\pi\)
0.367570 0.929996i \(-0.380190\pi\)
\(978\) 0 0
\(979\) − 69.2820i − 2.21426i
\(980\) 0 0
\(981\) 0 0
\(982\) −50.0000 −1.59556
\(983\) 46.4758 1.48235 0.741174 0.671313i \(-0.234269\pi\)
0.741174 + 0.671313i \(0.234269\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 77.4597 2.46682
\(987\) 0 0
\(988\) 18.0000 0.572656
\(989\) 31.3050i 0.995440i
\(990\) 0 0
\(991\) −49.0000 −1.55654 −0.778268 0.627932i \(-0.783902\pi\)
−0.778268 + 0.627932i \(0.783902\pi\)
\(992\) 11.6190 0.368902
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.7654i 1.48107i 0.672015 + 0.740537i \(0.265429\pi\)
−0.672015 + 0.740537i \(0.734571\pi\)
\(998\) − 69.3181i − 2.19423i
\(999\) 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 1323.2.c.b.1322.3 4
3.2 odd 2 inner 1323.2.c.b.1322.1 4
7.2 even 3 189.2.p.c.80.2 yes 4
7.3 odd 6 189.2.p.c.26.1 4
7.6 odd 2 inner 1323.2.c.b.1322.4 4
21.2 odd 6 189.2.p.c.80.1 yes 4
21.17 even 6 189.2.p.c.26.2 yes 4
21.20 even 2 inner 1323.2.c.b.1322.2 4
63.2 odd 6 567.2.s.e.458.2 4
63.16 even 3 567.2.s.e.458.1 4
63.23 odd 6 567.2.i.c.269.1 4
63.31 odd 6 567.2.s.e.26.2 4
63.38 even 6 567.2.i.c.215.1 4
63.52 odd 6 567.2.i.c.215.2 4
63.58 even 3 567.2.i.c.269.2 4
63.59 even 6 567.2.s.e.26.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.c.26.1 4 7.3 odd 6
189.2.p.c.26.2 yes 4 21.17 even 6
189.2.p.c.80.1 yes 4 21.2 odd 6
189.2.p.c.80.2 yes 4 7.2 even 3
567.2.i.c.215.1 4 63.38 even 6
567.2.i.c.215.2 4 63.52 odd 6
567.2.i.c.269.1 4 63.23 odd 6
567.2.i.c.269.2 4 63.58 even 3
567.2.s.e.26.1 4 63.59 even 6
567.2.s.e.26.2 4 63.31 odd 6
567.2.s.e.458.1 4 63.16 even 3
567.2.s.e.458.2 4 63.2 odd 6
1323.2.c.b.1322.1 4 3.2 odd 2 inner
1323.2.c.b.1322.2 4 21.20 even 2 inner
1323.2.c.b.1322.3 4 1.1 even 1 trivial
1323.2.c.b.1322.4 4 7.6 odd 2 inner