Properties

Label 1764.2.e.e.1079.1
Level $1764$
Weight $2$
Character 1764.1079
Analytic conductor $14.086$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1079.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1079
Dual form 1764.2.e.e.1079.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.00000i q^{5} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.00000i q^{5} +2.82843i q^{8} -2.82843 q^{10} +7.07107 q^{13} +4.00000 q^{16} +2.00000i q^{17} +4.00000i q^{20} +1.00000 q^{25} -10.0000i q^{26} -9.89949i q^{29} -5.65685i q^{32} +2.82843 q^{34} +12.0000 q^{37} +5.65685 q^{40} +8.00000i q^{41} -1.41421i q^{50} -14.1421 q^{52} -12.7279i q^{53} -14.0000 q^{58} -15.5563 q^{61} -8.00000 q^{64} -14.1421i q^{65} -4.00000i q^{68} -7.07107 q^{73} -16.9706i q^{74} -8.00000i q^{80} +11.3137 q^{82} +4.00000 q^{85} +16.0000i q^{89} +18.3848 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 4q^{25} + 48q^{37} - 56q^{58} - 32q^{64} + 16q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) −2.82843 −0.894427
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 7.07107 1.96116 0.980581 0.196116i \(-0.0628330\pi\)
0.980581 + 0.196116i \(0.0628330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.00000i 0.894427i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) − 10.0000i − 1.96116i
\(27\) 0 0
\(28\) 0 0
\(29\) − 9.89949i − 1.83829i −0.393919 0.919145i \(-0.628881\pi\)
0.393919 0.919145i \(-0.371119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0000 1.97279 0.986394 0.164399i \(-0.0525685\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 5.65685 0.894427
\(41\) 8.00000i 1.24939i 0.780869 + 0.624695i \(0.214777\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 1.41421i − 0.200000i
\(51\) 0 0
\(52\) −14.1421 −1.96116
\(53\) − 12.7279i − 1.74831i −0.485643 0.874157i \(-0.661414\pi\)
0.485643 0.874157i \(-0.338586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −14.0000 −1.83829
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.5563 −1.99179 −0.995893 0.0905357i \(-0.971142\pi\)
−0.995893 + 0.0905357i \(0.971142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 14.1421i − 1.75412i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.07107 −0.827606 −0.413803 0.910366i \(-0.635800\pi\)
−0.413803 + 0.910366i \(0.635800\pi\)
\(74\) − 16.9706i − 1.97279i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 8.00000i − 0.894427i
\(81\) 0 0
\(82\) 11.3137 1.24939
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000i 1.69600i 0.529999 + 0.847998i \(0.322192\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.3848 1.86669 0.933346 0.358979i \(-0.116875\pi\)
0.933346 + 0.358979i \(0.116875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) − 20.0000i − 1.99007i −0.0995037 0.995037i \(-0.531726\pi\)
0.0995037 0.995037i \(-0.468274\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 20.0000i 1.96116i
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 21.2132i − 1.99557i −0.0665190 0.997785i \(-0.521189\pi\)
0.0665190 0.997785i \(-0.478811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 19.7990i 1.83829i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 22.0000i 1.99179i
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.0000i − 1.07331i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) −20.0000 −1.75412
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.65685 −0.485071
\(137\) − 9.89949i − 0.845771i −0.906183 0.422885i \(-0.861017\pi\)
0.906183 0.422885i \(-0.138983\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −19.7990 −1.64422
\(146\) 10.0000i 0.827606i
\(147\) 0 0
\(148\) −24.0000 −1.97279
\(149\) 4.24264i 0.347571i 0.984784 + 0.173785i \(0.0555999\pi\)
−0.984784 + 0.173785i \(0.944400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 24.0416 1.91873 0.959366 0.282166i \(-0.0910530\pi\)
0.959366 + 0.282166i \(0.0910530\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −11.3137 −0.894427
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) − 16.0000i − 1.24939i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 37.0000 2.84615
\(170\) − 5.65685i − 0.433861i
\(171\) 0 0
\(172\) 0 0
\(173\) − 4.00000i − 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 22.6274 1.69600
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.41421 0.105118 0.0525588 0.998618i \(-0.483262\pi\)
0.0525588 + 0.998618i \(0.483262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 24.0000i − 1.76452i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) − 26.0000i − 1.86669i
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i 0.654931 + 0.755689i \(0.272698\pi\)
−0.654931 + 0.755689i \(0.727302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 2.82843i 0.200000i
\(201\) 0 0
\(202\) −28.2843 −1.99007
\(203\) 0 0
\(204\) 0 0
\(205\) 16.0000 1.11749
\(206\) 0 0
\(207\) 0 0
\(208\) 28.2843 1.96116
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 25.4558i 1.74831i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 8.48528i 0.574696i
\(219\) 0 0
\(220\) 0 0
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −30.0000 −1.99557
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −24.0416 −1.58872 −0.794358 0.607450i \(-0.792192\pi\)
−0.794358 + 0.607450i \(0.792192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 28.0000 1.83829
\(233\) − 7.07107i − 0.463241i −0.972806 0.231621i \(-0.925597\pi\)
0.972806 0.231621i \(-0.0744028\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 15.5563 1.00207 0.501036 0.865426i \(-0.332952\pi\)
0.501036 + 0.865426i \(0.332952\pi\)
\(242\) 15.5563i 1.00000i
\(243\) 0 0
\(244\) 31.1127 1.99179
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −16.9706 −1.07331
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 32.0000i 1.99611i 0.0623783 + 0.998053i \(0.480131\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 28.2843i 1.75412i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −25.4558 −1.56374
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000i 1.58525i 0.609711 + 0.792624i \(0.291286\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 8.00000i 0.485071i
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 15.5563i − 0.928014i −0.885832 0.464007i \(-0.846411\pi\)
0.885832 0.464007i \(-0.153589\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 28.0000i 1.64422i
\(291\) 0 0
\(292\) 14.1421 0.827606
\(293\) 4.00000i 0.233682i 0.993151 + 0.116841i \(0.0372769\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 33.9411i 1.97279i
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 31.1127i 1.78151i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −1.41421 −0.0799361 −0.0399680 0.999201i \(-0.512726\pi\)
−0.0399680 + 0.999201i \(0.512726\pi\)
\(314\) − 34.0000i − 1.91873i
\(315\) 0 0
\(316\) 0 0
\(317\) 4.24264i 0.238290i 0.992877 + 0.119145i \(0.0380154\pi\)
−0.992877 + 0.119145i \(0.961985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 16.0000i 0.894427i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 7.07107 0.392232
\(326\) 0 0
\(327\) 0 0
\(328\) −22.6274 −1.24939
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) − 52.3259i − 2.84615i
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −5.65685 −0.304114
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 32.5269 1.74113 0.870563 0.492057i \(-0.163755\pi\)
0.870563 + 0.492057i \(0.163755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.0000i 1.80964i 0.425797 + 0.904819i \(0.359994\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 32.0000i − 1.69600i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) − 2.00000i − 0.105118i
\(363\) 0 0
\(364\) 0 0
\(365\) 14.1421i 0.740233i
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −33.9411 −1.76452
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 70.0000i − 3.60518i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.7990i 1.00774i
\(387\) 0 0
\(388\) −36.7696 −1.86669
\(389\) − 9.89949i − 0.501924i −0.967997 0.250962i \(-0.919253\pi\)
0.967997 0.250962i \(-0.0807470\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 30.0000 1.51138
\(395\) 0 0
\(396\) 0 0
\(397\) 35.3553 1.77443 0.887217 0.461353i \(-0.152636\pi\)
0.887217 + 0.461353i \(0.152636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 26.8701i 1.34183i 0.741536 + 0.670913i \(0.234098\pi\)
−0.741536 + 0.670913i \(0.765902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 40.0000i 1.99007i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −32.5269 −1.60835 −0.804176 0.594391i \(-0.797393\pi\)
−0.804176 + 0.594391i \(0.797393\pi\)
\(410\) − 22.6274i − 1.11749i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) − 40.0000i − 1.96116i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 36.0000 1.74831
\(425\) 2.00000i 0.0970143i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −41.0122 −1.97092 −0.985460 0.169907i \(-0.945653\pi\)
−0.985460 + 0.169907i \(0.945653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.0000 0.951303
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 32.0000 1.51695
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 38.1838i − 1.80200i −0.433816 0.901002i \(-0.642833\pi\)
0.433816 0.901002i \(-0.357167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 42.4264i 1.99557i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 34.0000i 1.58872i
\(459\) 0 0
\(460\) 0 0
\(461\) 20.0000i 0.931493i 0.884918 + 0.465746i \(0.154214\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) − 39.5980i − 1.83829i
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 84.8528 3.86896
\(482\) − 22.0000i − 1.00207i
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) − 36.7696i − 1.66962i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) − 44.0000i − 1.99179i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 19.7990 0.891702
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 24.0000i 1.07331i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −40.0000 −1.77998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 10.0000i − 0.443242i −0.975133 0.221621i \(-0.928865\pi\)
0.975133 0.221621i \(-0.0711348\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 1.00000i
\(513\) 0 0
\(514\) 45.2548 1.99611
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 40.0000 1.75412
\(521\) 22.0000i 0.963837i 0.876216 + 0.481919i \(0.160060\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 36.0000i 1.56374i
\(531\) 0 0
\(532\) 0 0
\(533\) 56.5685i 2.45026i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 36.7696 1.58525
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 11.3137 0.485071
\(545\) 12.0000i 0.514024i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 19.7990i 0.845771i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) − 39.5980i − 1.68236i
\(555\) 0 0
\(556\) 0 0
\(557\) 46.6690i 1.97743i 0.149805 + 0.988716i \(0.452135\pi\)
−0.149805 + 0.988716i \(0.547865\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −42.4264 −1.78489
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 9.89949i − 0.415008i −0.978234 0.207504i \(-0.933466\pi\)
0.978234 0.207504i \(-0.0665341\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −32.5269 −1.35411 −0.677057 0.735931i \(-0.736745\pi\)
−0.677057 + 0.735931i \(0.736745\pi\)
\(578\) − 18.3848i − 0.764706i
\(579\) 0 0
\(580\) 39.5980 1.64422
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) − 20.0000i − 0.827606i
\(585\) 0 0
\(586\) 5.65685 0.233682
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 48.0000 1.97279
\(593\) − 46.0000i − 1.88899i −0.328521 0.944497i \(-0.606550\pi\)
0.328521 0.944497i \(-0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 8.48528i − 0.347571i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −26.8701 −1.09605 −0.548026 0.836461i \(-0.684621\pi\)
−0.548026 + 0.836461i \(0.684621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 44.0000 1.78151
\(611\) 0 0
\(612\) 0 0
\(613\) −36.0000 −1.45403 −0.727013 0.686624i \(-0.759092\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.4975i 1.99269i 0.0854011 + 0.996347i \(0.472783\pi\)
−0.0854011 + 0.996347i \(0.527217\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 2.00000i 0.0799361i
\(627\) 0 0
\(628\) −48.0833 −1.91873
\(629\) 24.0000i 0.956943i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 22.6274 0.894427
\(641\) − 41.0122i − 1.61988i −0.586510 0.809942i \(-0.699498\pi\)
0.586510 0.809942i \(-0.300502\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 10.0000i − 0.392232i
\(651\) 0 0
\(652\) 0 0
\(653\) 49.4975i 1.93699i 0.249041 + 0.968493i \(0.419885\pi\)
−0.249041 + 0.968493i \(0.580115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 32.0000i 1.24939i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −43.8406 −1.70520 −0.852601 0.522562i \(-0.824976\pi\)
−0.852601 + 0.522562i \(0.824976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) −74.0000 −2.84615
\(677\) − 52.0000i − 1.99852i −0.0384331 0.999261i \(-0.512237\pi\)
0.0384331 0.999261i \(-0.487763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 11.3137i 0.433861i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −19.7990 −0.756481
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 90.0000i − 3.42873i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 8.00000i 0.304114i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) − 46.0000i − 1.74113i
\(699\) 0 0
\(700\) 0 0
\(701\) 43.8406i 1.65584i 0.560848 + 0.827919i \(0.310475\pi\)
−0.560848 + 0.827919i \(0.689525\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 48.0833 1.80964
\(707\) 0 0
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −45.2548 −1.69600
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 26.8701i − 1.00000i
\(723\) 0 0
\(724\) −2.82843 −0.105118
\(725\) − 9.89949i − 0.367658i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 20.0000 0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) 35.3553 1.30588 0.652940 0.757410i \(-0.273536\pi\)
0.652940 + 0.757410i \(0.273536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 48.0000i 1.76452i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 8.48528 0.310877
\(746\) 19.7990i 0.724893i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −98.9949 −3.60518
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000i 1.45000i 0.688749 + 0.724999i \(0.258160\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 18.3848 0.662972 0.331486 0.943460i \(-0.392450\pi\)
0.331486 + 0.943460i \(0.392450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 28.0000 1.00774
\(773\) 44.0000i 1.58257i 0.611448 + 0.791285i \(0.290588\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 52.0000i 1.86669i
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 48.0833i − 1.71617i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) − 42.4264i − 1.51138i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −110.000 −3.90621
\(794\) − 50.0000i − 1.77443i
\(795\) 0 0
\(796\) 0 0
\(797\) 52.0000i 1.84193i 0.389640 + 0.920967i \(0.372599\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) − 5.65685i − 0.200000i
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 56.5685 1.99007
\(809\) − 46.6690i − 1.64080i −0.571793 0.820398i \(-0.693752\pi\)
0.571793 0.820398i \(-0.306248\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 46.0000i 1.60835i
\(819\) 0 0
\(820\) −32.0000 −1.11749
\(821\) 55.1543i 1.92490i 0.271460 + 0.962450i \(0.412493\pi\)
−0.271460 + 0.962450i \(0.587507\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 24.0416 0.835000 0.417500 0.908677i \(-0.362906\pi\)
0.417500 + 0.908677i \(0.362906\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −56.5685 −1.96116
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −69.0000 −2.37931
\(842\) − 39.5980i − 1.36464i
\(843\) 0 0
\(844\) 0 0
\(845\) − 74.0000i − 2.54568i
\(846\) 0 0
\(847\) 0 0
\(848\) − 50.9117i − 1.74831i
\(849\) 0 0
\(850\) 2.82843 0.0970143
\(851\) 0 0
\(852\) 0 0
\(853\) 7.07107 0.242109 0.121054 0.992646i \(-0.461372\pi\)
0.121054 + 0.992646i \(0.461372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 8.00000i − 0.273275i −0.990621 0.136637i \(-0.956370\pi\)
0.990621 0.136637i \(-0.0436295\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 58.0000i 1.97092i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) − 16.9706i − 0.574696i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 50.0000i 1.68454i 0.539054 + 0.842271i \(0.318782\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) − 28.2843i − 0.951303i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 45.2548i − 1.51695i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −54.0000 −1.80200
\(899\) 0 0
\(900\) 0 0
\(901\) 25.4558 0.848057
\(902\) 0 0
\(903\) 0 0
\(904\) 60.0000 1.99557
\(905\) − 2.82843i − 0.0940201i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 11.3137i − 0.374224i
\(915\) 0 0
\(916\) 48.0833 1.58872
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 28.2843 0.931493
\(923\) 0 0
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 0 0
\(927\) 0 0
\(928\) −56.0000 −1.83829
\(929\) 40.0000i 1.31236i 0.754606 + 0.656179i \(0.227828\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.1421i 0.463241i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.07107 0.231002 0.115501 0.993307i \(-0.463153\pi\)
0.115501 + 0.993307i \(0.463153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 58.0000i − 1.89075i −0.325991 0.945373i \(-0.605698\pi\)
0.325991 0.945373i \(-0.394302\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −50.0000 −1.62307
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 21.2132i − 0.687163i −0.939123 0.343582i \(-0.888360\pi\)
0.939123 0.343582i \(-0.111640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) − 120.000i − 3.86896i
\(963\) 0 0
\(964\) −31.1127 −1.00207
\(965\) 28.0000i 0.901352i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) − 31.1127i − 1.00000i
\(969\) 0 0
\(970\) −52.0000 −1.66962
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −62.2254 −1.99179
\(977\) 49.4975i 1.58356i 0.610803 + 0.791782i \(0.290847\pi\)
−0.610803 + 0.791782i \(0.709153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 42.4264 1.35182
\(986\) − 28.0000i − 0.891702i
\(987\) 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
\(997\) −35.3553 −1.11971 −0.559857 0.828589i \(-0.689144\pi\)
−0.559857 + 0.828589i \(0.689144\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1764.2.e.e.1079.1 4
3.2 odd 2 inner 1764.2.e.e.1079.4 yes 4
4.3 odd 2 CM 1764.2.e.e.1079.1 4
7.6 odd 2 inner 1764.2.e.e.1079.2 yes 4
12.11 even 2 inner 1764.2.e.e.1079.4 yes 4
21.20 even 2 inner 1764.2.e.e.1079.3 yes 4
28.27 even 2 inner 1764.2.e.e.1079.2 yes 4
84.83 odd 2 inner 1764.2.e.e.1079.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.e.e.1079.1 4 1.1 even 1 trivial
1764.2.e.e.1079.1 4 4.3 odd 2 CM
1764.2.e.e.1079.2 yes 4 7.6 odd 2 inner
1764.2.e.e.1079.2 yes 4 28.27 even 2 inner
1764.2.e.e.1079.3 yes 4 21.20 even 2 inner
1764.2.e.e.1079.3 yes 4 84.83 odd 2 inner
1764.2.e.e.1079.4 yes 4 3.2 odd 2 inner
1764.2.e.e.1079.4 yes 4 12.11 even 2 inner