Properties

Label 225.4.b.g.199.1
Level $225$
Weight $4$
Character 225.199
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.g.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{4} -20.0000i q^{7} +O(q^{10})\) \(q+8.00000 q^{4} -20.0000i q^{7} -70.0000i q^{13} +64.0000 q^{16} -56.0000 q^{19} -160.000i q^{28} +308.000 q^{31} -110.000i q^{37} -520.000i q^{43} -57.0000 q^{49} -560.000i q^{52} +182.000 q^{61} +512.000 q^{64} +880.000i q^{67} +1190.00i q^{73} -448.000 q^{76} -884.000 q^{79} -1400.00 q^{91} +1330.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{4} + O(q^{10}) \) \( 2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 8.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) − 20.0000i − 1.07990i −0.841698 0.539949i \(-0.818443\pi\)
0.841698 0.539949i \(-0.181557\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 70.0000i − 1.49342i −0.665148 0.746712i \(-0.731631\pi\)
0.665148 0.746712i \(-0.268369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −56.0000 −0.676173 −0.338086 0.941115i \(-0.609780\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) − 160.000i − 1.07990i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 308.000 1.78447 0.892233 0.451576i \(-0.149138\pi\)
0.892233 + 0.451576i \(0.149138\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 110.000i − 0.488754i −0.969680 0.244377i \(-0.921417\pi\)
0.969680 0.244377i \(-0.0785834\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 520.000i − 1.84417i −0.386989 0.922084i \(-0.626485\pi\)
0.386989 0.922084i \(-0.373515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) 0 0
\(52\) − 560.000i − 1.49342i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 182.000 0.382012 0.191006 0.981589i \(-0.438825\pi\)
0.191006 + 0.981589i \(0.438825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 880.000i 1.60461i 0.596912 + 0.802307i \(0.296394\pi\)
−0.596912 + 0.802307i \(0.703606\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1190.00i 1.90793i 0.299916 + 0.953966i \(0.403041\pi\)
−0.299916 + 0.953966i \(0.596959\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −448.000 −0.676173
\(77\) 0 0
\(78\) 0 0
\(79\) −884.000 −1.25896 −0.629480 0.777017i \(-0.716732\pi\)
−0.629480 + 0.777017i \(0.716732\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −1400.00 −1.61275
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1330.00i 1.39218i 0.717957 + 0.696088i \(0.245078\pi\)
−0.717957 + 0.696088i \(0.754922\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1820.00i 1.74107i 0.492109 + 0.870534i \(0.336226\pi\)
−0.492109 + 0.870534i \(0.663774\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 646.000 0.567666 0.283833 0.958874i \(-0.408394\pi\)
0.283833 + 0.958874i \(0.408394\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1280.00i − 1.07990i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 2464.00 1.78447
\(125\) 0 0
\(126\) 0 0
\(127\) − 380.000i − 0.265508i −0.991149 0.132754i \(-0.957618\pi\)
0.991149 0.132754i \(-0.0423821\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 1120.00i 0.730198i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −2576.00 −1.57190 −0.785948 0.618293i \(-0.787825\pi\)
−0.785948 + 0.618293i \(0.787825\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) − 880.000i − 0.488754i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 1748.00 0.942054 0.471027 0.882119i \(-0.343883\pi\)
0.471027 + 0.882119i \(0.343883\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3850.00i 1.95709i 0.206028 + 0.978546i \(0.433946\pi\)
−0.206028 + 0.978546i \(0.566054\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 3400.00i − 1.63379i −0.576783 0.816897i \(-0.695692\pi\)
0.576783 0.816897i \(-0.304308\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −2703.00 −1.23031
\(170\) 0 0
\(171\) 0 0
\(172\) − 4160.00i − 1.84417i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 3458.00 1.42006 0.710031 0.704171i \(-0.248681\pi\)
0.710031 + 0.704171i \(0.248681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 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
\(193\) − 1150.00i − 0.428906i −0.976734 0.214453i \(-0.931203\pi\)
0.976734 0.214453i \(-0.0687968\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −456.000 −0.166181
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 5236.00 1.86518 0.932588 0.360942i \(-0.117545\pi\)
0.932588 + 0.360942i \(0.117545\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) − 4480.00i − 1.49342i
\(209\) 0 0
\(210\) 0 0
\(211\) 6032.00 1.96806 0.984028 0.178011i \(-0.0569664\pi\)
0.984028 + 0.178011i \(0.0569664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6160.00i − 1.92704i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 3220.00i − 0.966938i −0.875362 0.483469i \(-0.839377\pi\)
0.875362 0.483469i \(-0.160623\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −4466.00 −1.28874 −0.644370 0.764714i \(-0.722880\pi\)
−0.644370 + 0.764714i \(0.722880\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 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
\(241\) −7378.00 −1.97203 −0.986014 0.166662i \(-0.946701\pi\)
−0.986014 + 0.166662i \(0.946701\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1456.00 0.382012
\(245\) 0 0
\(246\) 0 0
\(247\) 3920.00i 1.00981i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −2200.00 −0.527804
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 7040.00i 1.60461i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 812.000 0.182013 0.0910064 0.995850i \(-0.470992\pi\)
0.0910064 + 0.995850i \(0.470992\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4030.00i 0.874149i 0.899425 + 0.437074i \(0.143985\pi\)
−0.899425 + 0.437074i \(0.856015\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 5600.00i 1.17627i 0.808761 + 0.588137i \(0.200138\pi\)
−0.808761 + 0.588137i \(0.799862\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4913.00 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 9520.00i 1.90793i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −10400.0 −1.99152
\(302\) 0 0
\(303\) 0 0
\(304\) −3584.00 −0.676173
\(305\) 0 0
\(306\) 0 0
\(307\) − 10640.0i − 1.97804i −0.147797 0.989018i \(-0.547218\pi\)
0.147797 0.989018i \(-0.452782\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10010.0i 1.80766i 0.427888 + 0.903832i \(0.359258\pi\)
−0.427888 + 0.903832i \(0.640742\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −7072.00 −1.25896
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 992.000 0.164729 0.0823644 0.996602i \(-0.473753\pi\)
0.0823644 + 0.996602i \(0.473753\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4930.00i 0.796897i 0.917191 + 0.398448i \(0.130451\pi\)
−0.917191 + 0.398448i \(0.869549\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 5720.00i − 0.900440i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 11914.0 1.82734 0.913670 0.406456i \(-0.133236\pi\)
0.913670 + 0.406456i \(0.133236\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3723.00 −0.542790
\(362\) 0 0
\(363\) 0 0
\(364\) −11200.0 −1.61275
\(365\) 0 0
\(366\) 0 0
\(367\) − 4340.00i − 0.617292i −0.951177 0.308646i \(-0.900124\pi\)
0.951177 0.308646i \(-0.0998758\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12350.0i 1.71437i 0.515011 + 0.857183i \(0.327788\pi\)
−0.515011 + 0.857183i \(0.672212\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8584.00 1.16340 0.581702 0.813402i \(-0.302387\pi\)
0.581702 + 0.813402i \(0.302387\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 10640.0i 1.39218i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1190.00i − 0.150439i −0.997167 0.0752196i \(-0.976034\pi\)
0.997167 0.0752196i \(-0.0239658\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) − 21560.0i − 2.66496i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −8246.00 −0.996916 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14560.0i 1.74107i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 17138.0 1.98398 0.991989 0.126322i \(-0.0403172\pi\)
0.991989 + 0.126322i \(0.0403172\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3640.00i − 0.412534i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 2590.00i − 0.287454i −0.989617 0.143727i \(-0.954091\pi\)
0.989617 0.143727i \(-0.0459087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5168.00 0.567666
\(437\) 0 0
\(438\) 0 0
\(439\) −14924.0 −1.62251 −0.811257 0.584690i \(-0.801216\pi\)
−0.811257 + 0.584690i \(0.801216\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) − 10240.0i − 1.07990i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 12710.0i − 1.30098i −0.759514 0.650491i \(-0.774563\pi\)
0.759514 0.650491i \(-0.225437\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) − 19780.0i − 1.98543i −0.120482 0.992716i \(-0.538444\pi\)
0.120482 0.992716i \(-0.461556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 17600.0 1.73282
\(470\) 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
\(481\) −7700.00 −0.729916
\(482\) 0 0
\(483\) 0 0
\(484\) −10648.0 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) − 20900.0i − 1.94470i −0.233526 0.972351i \(-0.575026\pi\)
0.233526 0.972351i \(-0.424974\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 19712.0 1.78447
\(497\) 0 0
\(498\) 0 0
\(499\) 15136.0 1.35788 0.678938 0.734195i \(-0.262440\pi\)
0.678938 + 0.734195i \(0.262440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 3040.00i − 0.265508i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 23800.0 2.06037
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) − 12040.0i − 1.00664i −0.864100 0.503320i \(-0.832112\pi\)
0.864100 0.503320i \(-0.167888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12167.0 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 8960.00i 0.730198i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22678.0 −1.80222 −0.901112 0.433586i \(-0.857248\pi\)
−0.901112 + 0.433586i \(0.857248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 1640.00i − 0.128193i −0.997944 0.0640963i \(-0.979584\pi\)
0.997944 0.0640963i \(-0.0204165\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 17680.0i 1.35955i
\(554\) 0 0
\(555\) 0 0
\(556\) −20608.0 −1.57190
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −36400.0 −2.75413
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 23312.0 1.70854 0.854270 0.519829i \(-0.174004\pi\)
0.854270 + 0.519829i \(0.174004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17710.0i 1.27778i 0.769300 + 0.638888i \(0.220605\pi\)
−0.769300 + 0.638888i \(0.779395\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −17248.0 −1.20661
\(590\) 0 0
\(591\) 0 0
\(592\) − 7040.00i − 0.488754i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −29302.0 −1.98877 −0.994387 0.105801i \(-0.966259\pi\)
−0.994387 + 0.105801i \(0.966259\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 13984.0 0.942054
\(605\) 0 0
\(606\) 0 0
\(607\) 28420.0i 1.90038i 0.311667 + 0.950191i \(0.399113\pi\)
−0.311667 + 0.950191i \(0.600887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17390.0i 1.14580i 0.819625 + 0.572900i \(0.194182\pi\)
−0.819625 + 0.572900i \(0.805818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 26656.0 1.73085 0.865424 0.501040i \(-0.167049\pi\)
0.865424 + 0.501040i \(0.167049\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 30800.0i 1.95709i
\(629\) 0 0
\(630\) 0 0
\(631\) 1892.00 0.119365 0.0596825 0.998217i \(-0.480991\pi\)
0.0596825 + 0.998217i \(0.480991\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3990.00i 0.248178i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 13160.0i 0.807122i 0.914953 + 0.403561i \(0.132228\pi\)
−0.914953 + 0.403561i \(0.867772\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 27200.0i − 1.63379i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −20482.0 −1.20523 −0.602615 0.798032i \(-0.705875\pi\)
−0.602615 + 0.798032i \(0.705875\pi\)
\(662\) 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
\(673\) 24050.0i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −21624.0 −1.23031
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 26600.0 1.50341
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) − 33280.0i − 1.84417i
\(689\) 0 0
\(690\) 0 0
\(691\) −16072.0 −0.884816 −0.442408 0.896814i \(-0.645876\pi\)
−0.442408 + 0.896814i \(0.645876\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 6160.00i 0.330482i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36146.0 −1.91466 −0.957328 0.289003i \(-0.906676\pi\)
−0.957328 + 0.289003i \(0.906676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 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
\(721\) 36400.0 1.88018
\(722\) 0 0
\(723\) 0 0
\(724\) 27664.0 1.42006
\(725\) 0 0
\(726\) 0 0
\(727\) 10780.0i 0.549942i 0.961452 + 0.274971i \(0.0886683\pi\)
−0.961452 + 0.274971i \(0.911332\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 15050.0i 0.758369i 0.925321 + 0.379184i \(0.123795\pi\)
−0.925321 + 0.379184i \(0.876205\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −31376.0 −1.56182 −0.780910 0.624644i \(-0.785244\pi\)
−0.780910 + 0.624644i \(0.785244\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23452.0 −1.13951 −0.569757 0.821813i \(-0.692963\pi\)
−0.569757 + 0.821813i \(0.692963\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41470.0i 1.99109i 0.0943039 + 0.995543i \(0.469937\pi\)
−0.0943039 + 0.995543i \(0.530063\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) − 12920.0i − 0.613022i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4606.00 0.215990 0.107995 0.994151i \(-0.465557\pi\)
0.107995 + 0.994151i \(0.465557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 9200.00i − 0.428906i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −3648.00 −0.166181
\(785\) 0 0
\(786\) 0 0
\(787\) − 43400.0i − 1.96575i −0.184281 0.982874i \(-0.558996\pi\)
0.184281 0.982874i \(-0.441004\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 12740.0i − 0.570505i
\(794\) 0 0
\(795\) 0 0
\(796\) 41888.0 1.86518
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 39368.0 1.70456 0.852280 0.523087i \(-0.175220\pi\)
0.852280 + 0.523087i \(0.175220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 29120.0i 1.24698i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) − 12220.0i − 0.517573i −0.965935 0.258786i \(-0.916677\pi\)
0.965935 0.258786i \(-0.0833226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −17066.0 −0.714990 −0.357495 0.933915i \(-0.616369\pi\)
−0.357495 + 0.933915i \(0.616369\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 35840.0i − 1.49342i
\(833\) 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
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 48256.0 1.96806
\(845\) 0 0
\(846\) 0 0
\(847\) 26620.0i 1.07990i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 46690.0i − 1.87413i −0.349151 0.937066i \(-0.613530\pi\)
0.349151 0.937066i \(-0.386470\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −31304.0 −1.24340 −0.621699 0.783256i \(-0.713557\pi\)
−0.621699 + 0.783256i \(0.713557\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) − 49280.0i − 1.92704i
\(869\) 0 0
\(870\) 0 0
\(871\) 61600.0 2.39637
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 50150.0i − 1.93095i −0.260491 0.965476i \(-0.583885\pi\)
0.260491 0.965476i \(-0.416115\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) − 20680.0i − 0.788151i −0.919078 0.394076i \(-0.871065\pi\)
0.919078 0.394076i \(-0.128935\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −7600.00 −0.286722
\(890\) 0 0
\(891\) 0 0
\(892\) − 25760.0i − 0.966938i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 44840.0i − 1.64155i −0.571250 0.820776i \(-0.693541\pi\)
0.571250 0.820776i \(-0.306459\pi\)
\(908\) 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
\(914\) 0 0
\(915\) 0 0
\(916\) −35728.0 −1.28874
\(917\) 0 0
\(918\) 0 0
\(919\) −2756.00 −0.0989250 −0.0494625 0.998776i \(-0.515751\pi\)
−0.0494625 + 0.998776i \(0.515751\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 3192.00 0.112367
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 55510.0i 1.93536i 0.252181 + 0.967680i \(0.418852\pi\)
−0.252181 + 0.967680i \(0.581148\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 83300.0 2.84935
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 65073.0 2.18432
\(962\) 0 0
\(963\) 0 0
\(964\) −59024.0 −1.97203
\(965\) 0 0
\(966\) 0 0
\(967\) 50020.0i 1.66343i 0.555204 + 0.831714i \(0.312640\pi\)
−0.555204 + 0.831714i \(0.687360\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 51520.0i 1.69749i
\(974\) 0 0
\(975\) 0 0
\(976\) 11648.0 0.382012
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 31360.0i 1.00981i
\(989\) 0 0
\(990\) 0 0
\(991\) −45628.0 −1.46258 −0.731292 0.682064i \(-0.761082\pi\)
−0.731292 + 0.682064i \(0.761082\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 28910.0i − 0.918344i −0.888347 0.459172i \(-0.848146\pi\)
0.888347 0.459172i \(-0.151854\pi\)
\(998\) 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 225.4.b.g.199.1 2
3.2 odd 2 CM 225.4.b.g.199.1 2
5.2 odd 4 9.4.a.a.1.1 1
5.3 odd 4 225.4.a.d.1.1 1
5.4 even 2 inner 225.4.b.g.199.2 2
15.2 even 4 9.4.a.a.1.1 1
15.8 even 4 225.4.a.d.1.1 1
15.14 odd 2 inner 225.4.b.g.199.2 2
20.7 even 4 144.4.a.d.1.1 1
35.2 odd 12 441.4.e.i.361.1 2
35.12 even 12 441.4.e.j.361.1 2
35.17 even 12 441.4.e.j.226.1 2
35.27 even 4 441.4.a.f.1.1 1
35.32 odd 12 441.4.e.i.226.1 2
40.27 even 4 576.4.a.l.1.1 1
40.37 odd 4 576.4.a.m.1.1 1
45.2 even 12 81.4.c.b.28.1 2
45.7 odd 12 81.4.c.b.28.1 2
45.22 odd 12 81.4.c.b.55.1 2
45.32 even 12 81.4.c.b.55.1 2
55.32 even 4 1089.4.a.g.1.1 1
60.47 odd 4 144.4.a.d.1.1 1
65.12 odd 4 1521.4.a.g.1.1 1
105.2 even 12 441.4.e.i.361.1 2
105.17 odd 12 441.4.e.j.226.1 2
105.32 even 12 441.4.e.i.226.1 2
105.47 odd 12 441.4.e.j.361.1 2
105.62 odd 4 441.4.a.f.1.1 1
120.77 even 4 576.4.a.m.1.1 1
120.107 odd 4 576.4.a.l.1.1 1
165.32 odd 4 1089.4.a.g.1.1 1
195.77 even 4 1521.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.4.a.a.1.1 1 5.2 odd 4
9.4.a.a.1.1 1 15.2 even 4
81.4.c.b.28.1 2 45.2 even 12
81.4.c.b.28.1 2 45.7 odd 12
81.4.c.b.55.1 2 45.22 odd 12
81.4.c.b.55.1 2 45.32 even 12
144.4.a.d.1.1 1 20.7 even 4
144.4.a.d.1.1 1 60.47 odd 4
225.4.a.d.1.1 1 5.3 odd 4
225.4.a.d.1.1 1 15.8 even 4
225.4.b.g.199.1 2 1.1 even 1 trivial
225.4.b.g.199.1 2 3.2 odd 2 CM
225.4.b.g.199.2 2 5.4 even 2 inner
225.4.b.g.199.2 2 15.14 odd 2 inner
441.4.a.f.1.1 1 35.27 even 4
441.4.a.f.1.1 1 105.62 odd 4
441.4.e.i.226.1 2 35.32 odd 12
441.4.e.i.226.1 2 105.32 even 12
441.4.e.i.361.1 2 35.2 odd 12
441.4.e.i.361.1 2 105.2 even 12
441.4.e.j.226.1 2 35.17 even 12
441.4.e.j.226.1 2 105.17 odd 12
441.4.e.j.361.1 2 35.12 even 12
441.4.e.j.361.1 2 105.47 odd 12
576.4.a.l.1.1 1 40.27 even 4
576.4.a.l.1.1 1 120.107 odd 4
576.4.a.m.1.1 1 40.37 odd 4
576.4.a.m.1.1 1 120.77 even 4
1089.4.a.g.1.1 1 55.32 even 4
1089.4.a.g.1.1 1 165.32 odd 4
1521.4.a.g.1.1 1 65.12 odd 4
1521.4.a.g.1.1 1 195.77 even 4