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Properties

Label 720.3.j.b.559.1
Level $720$
Weight $3$
Character 720.559
Analytic conductor $19.619$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,3,Mod(559,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.559");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 720.j (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.6185790339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 559.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 720.559
Dual form 720.3.j.b.559.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.00000 - 3.00000i) q^{5} +O(q^{10})\) \(q+(-4.00000 - 3.00000i) q^{5} -24.0000i q^{13} +30.0000i q^{17} +(7.00000 + 24.0000i) q^{25} -40.0000 q^{29} +24.0000i q^{37} -80.0000 q^{41} -49.0000 q^{49} +90.0000i q^{53} +22.0000 q^{61} +(-72.0000 + 96.0000i) q^{65} +96.0000i q^{73} +(90.0000 - 120.000i) q^{85} -160.000 q^{89} -144.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{5} + 14 q^{25} - 80 q^{29} - 160 q^{41} - 98 q^{49} + 44 q^{61} - 144 q^{65} + 180 q^{85} - 320 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 24.0000i − 1.84615i −0.384615 − 0.923077i \(-0.625666\pi\)
0.384615 − 0.923077i \(-0.374334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.0000i 1.76471i 0.470588 + 0.882353i \(0.344042\pi\)
−0.470588 + 0.882353i \(0.655958\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −40.0000 −1.37931 −0.689655 − 0.724138i \(-0.742238\pi\)
−0.689655 + 0.724138i \(0.742238\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 24.0000i 0.648649i 0.945946 + 0.324324i \(0.105137\pi\)
−0.945946 + 0.324324i \(0.894863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −80.0000 −1.95122 −0.975610 − 0.219512i \(-0.929553\pi\)
−0.975610 + 0.219512i \(0.929553\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 90.0000i 1.69811i 0.528302 + 0.849057i \(0.322829\pi\)
−0.528302 + 0.849057i \(0.677171\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 22.0000 0.360656 0.180328 − 0.983607i \(-0.442284\pi\)
0.180328 + 0.983607i \(0.442284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −72.0000 + 96.0000i −1.10769 + 1.47692i
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 96.0000i 1.31507i 0.753425 + 0.657534i \(0.228401\pi\)
−0.753425 + 0.657534i \(0.771599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 90.0000 − 120.000i 1.05882 − 1.41176i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −160.000 −1.79775 −0.898876 − 0.438202i \(-0.855615\pi\)
−0.898876 + 0.438202i \(0.855615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 144.000i − 1.48454i −0.670103 − 0.742268i \(-0.733750\pi\)
0.670103 − 0.742268i \(-0.266250\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −40.0000 −0.396040 −0.198020 − 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −182.000 −1.66972 −0.834862 − 0.550459i \(-0.814453\pi\)
−0.834862 + 0.550459i \(0.814453\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 30.0000i 0.265487i 0.991150 + 0.132743i \(0.0423786\pi\)
−0.991150 + 0.132743i \(0.957621\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 44.0000 − 117.000i 0.352000 − 0.936000i
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 210.000i 1.53285i 0.642336 + 0.766423i \(0.277965\pi\)
−0.642336 + 0.766423i \(0.722035\pi\)
\(138\) 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
\(145\) 160.000 + 120.000i 1.10345 + 0.827586i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 280.000 1.87919 0.939597 − 0.342282i \(-0.111200\pi\)
0.939597 + 0.342282i \(0.111200\pi\)
\(150\) 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
\(157\) 264.000i 1.68153i 0.541401 + 0.840764i \(0.317894\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −407.000 −2.40828
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 330.000i − 1.90751i −0.300578 − 0.953757i \(-0.597180\pi\)
0.300578 − 0.953757i \(-0.402820\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 38.0000 0.209945 0.104972 − 0.994475i \(-0.466525\pi\)
0.104972 + 0.994475i \(0.466525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 72.0000 − 96.0000i 0.389189 − 0.518919i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) − 336.000i − 1.74093i −0.492228 − 0.870466i \(-0.663817\pi\)
0.492228 − 0.870466i \(-0.336183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 390.000i − 1.97970i −0.142132 − 0.989848i \(-0.545396\pi\)
0.142132 − 0.989848i \(-0.454604\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 320.000 + 240.000i 1.56098 + 1.17073i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 720.000 3.25792
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −442.000 −1.93013 −0.965066 − 0.262009i \(-0.915615\pi\)
−0.965066 + 0.262009i \(0.915615\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 210.000i 0.901288i 0.892704 + 0.450644i \(0.148806\pi\)
−0.892704 + 0.450644i \(0.851194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 418.000 1.73444 0.867220 − 0.497925i \(-0.165905\pi\)
0.867220 + 0.497925i \(0.165905\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 196.000 + 147.000i 0.800000 + 0.600000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 510.000i − 1.98444i −0.124514 − 0.992218i \(-0.539737\pi\)
0.124514 − 0.992218i \(-0.460263\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 270.000 − 360.000i 1.01887 − 1.35849i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 520.000 1.93309 0.966543 − 0.256506i \(-0.0825712\pi\)
0.966543 + 0.256506i \(0.0825712\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 504.000i − 1.81949i −0.415162 − 0.909747i \(-0.636275\pi\)
0.415162 − 0.909747i \(-0.363725\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −320.000 −1.13879 −0.569395 − 0.822064i \(-0.692822\pi\)
−0.569395 + 0.822064i \(0.692822\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −611.000 −2.11419
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 570.000i − 1.94539i −0.232082 − 0.972696i \(-0.574554\pi\)
0.232082 − 0.972696i \(-0.425446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −88.0000 − 66.0000i −0.288525 − 0.216393i
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 624.000i 1.99361i 0.0798722 + 0.996805i \(0.474549\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 150.000i 0.473186i 0.971609 + 0.236593i \(0.0760308\pi\)
−0.971609 + 0.236593i \(0.923969\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 576.000 − 168.000i 1.77231 − 0.516923i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 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
\(337\) 576.000i 1.70920i 0.519288 + 0.854599i \(0.326197\pi\)
−0.519288 + 0.854599i \(0.673803\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −598.000 −1.71347 −0.856734 − 0.515759i \(-0.827510\pi\)
−0.856734 + 0.515759i \(0.827510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 450.000i 1.27479i 0.770538 + 0.637394i \(0.219988\pi\)
−0.770538 + 0.637394i \(0.780012\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 288.000 − 384.000i 0.789041 − 1.05205i
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 504.000i 1.35121i 0.737265 + 0.675603i \(0.236117\pi\)
−0.737265 + 0.675603i \(0.763883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 960.000i 2.54642i
\(378\) 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
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −680.000 −1.74807 −0.874036 − 0.485861i \(-0.838506\pi\)
−0.874036 + 0.485861i \(0.838506\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 456.000i − 1.14861i −0.818640 − 0.574307i \(-0.805271\pi\)
0.818640 − 0.574307i \(-0.194729\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 80.0000 0.199501 0.0997506 − 0.995012i \(-0.468195\pi\)
0.0997506 + 0.995012i \(0.468195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −782.000 −1.91198 −0.955990 − 0.293399i \(-0.905214\pi\)
−0.955990 + 0.293399i \(0.905214\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 58.0000 0.137767 0.0688836 − 0.997625i \(-0.478056\pi\)
0.0688836 + 0.997625i \(0.478056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −720.000 + 210.000i −1.69412 + 0.494118i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) − 816.000i − 1.88453i −0.334873 − 0.942263i \(-0.608693\pi\)
0.334873 − 0.942263i \(-0.391307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 640.000 + 480.000i 1.43820 + 1.07865i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −560.000 −1.24722 −0.623608 − 0.781737i \(-0.714334\pi\)
−0.623608 + 0.781737i \(0.714334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 336.000i 0.735230i 0.929978 + 0.367615i \(0.119826\pi\)
−0.929978 + 0.367615i \(0.880174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −760.000 −1.64859 −0.824295 − 0.566161i \(-0.808428\pi\)
−0.824295 + 0.566161i \(0.808428\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 576.000 1.19751
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −432.000 + 576.000i −0.890722 + 1.18763i
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) − 1200.00i − 2.43408i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 160.000 + 120.000i 0.316832 + 0.237624i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 440.000 0.864440 0.432220 − 0.901768i \(-0.357730\pi\)
0.432220 + 0.901768i \(0.357730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) −880.000 −1.68906 −0.844530 − 0.535509i \(-0.820120\pi\)
−0.844530 + 0.535509i \(0.820120\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −529.000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1920.00i 3.60225i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −682.000 −1.26063 −0.630314 − 0.776340i \(-0.717074\pi\)
−0.630314 + 0.776340i \(0.717074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 728.000 + 546.000i 1.33578 + 1.00183i
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 330.000i − 0.592460i −0.955117 − 0.296230i \(-0.904271\pi\)
0.955117 − 0.296230i \(-0.0957294\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 90.0000 − 120.000i 0.159292 − 0.212389i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1040.00 −1.82777 −0.913884 − 0.405975i \(-0.866932\pi\)
−0.913884 + 0.405975i \(0.866932\pi\)
\(570\) 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
\(577\) − 96.0000i − 0.166378i −0.996534 − 0.0831889i \(-0.973490\pi\)
0.996534 − 0.0831889i \(-0.0265105\pi\)
\(578\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 930.000i − 1.56830i −0.620573 − 0.784148i \(-0.713100\pi\)
0.620573 − 0.784148i \(-0.286900\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1102.00 1.83361 0.916805 − 0.399334i \(-0.130759\pi\)
0.916805 + 0.399334i \(0.130759\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −484.000 − 363.000i −0.800000 − 0.600000i
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1224.00i 1.99674i 0.0570962 + 0.998369i \(0.481816\pi\)
−0.0570962 + 0.998369i \(0.518184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 210.000i − 0.340357i −0.985413 − 0.170178i \(-0.945566\pi\)
0.985413 − 0.170178i \(-0.0544344\pi\)
\(618\) 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
\(625\) −527.000 + 336.000i −0.843200 + 0.537600i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −720.000 −1.14467
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1176.00i 1.84615i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 400.000 0.624025 0.312012 − 0.950078i \(-0.398997\pi\)
0.312012 + 0.950078i \(0.398997\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 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
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 630.000i − 0.964778i −0.875957 − 0.482389i \(-0.839769\pi\)
0.875957 − 0.482389i \(-0.160231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1178.00 −1.78215 −0.891074 − 0.453858i \(-0.850047\pi\)
−0.891074 + 0.453858i \(0.850047\pi\)
\(662\) 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
\(673\) 1104.00i 1.64042i 0.572065 + 0.820208i \(0.306142\pi\)
−0.572065 + 0.820208i \(0.693858\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1350.00i 1.99409i 0.0768095 + 0.997046i \(0.475527\pi\)
−0.0768095 + 0.997046i \(0.524473\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 630.000 − 840.000i 0.919708 − 1.22628i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2160.00 3.13498
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 2400.00i − 3.44333i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 520.000 0.741797 0.370899 − 0.928673i \(-0.379050\pi\)
0.370899 + 0.928673i \(0.379050\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 518.000 0.730606 0.365303 − 0.930889i \(-0.380965\pi\)
0.365303 + 0.930889i \(0.380965\pi\)
\(710\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −280.000 − 960.000i −0.386207 − 1.32414i
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 216.000i 0.294679i 0.989086 + 0.147340i \(0.0470711\pi\)
−0.989086 + 0.147340i \(0.952929\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −1120.00 − 840.000i −1.50336 − 1.12752i
\(746\) 0 0
\(747\) 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
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 936.000i 1.23646i 0.785997 + 0.618230i \(0.212150\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1520.00 1.99737 0.998686 − 0.0512484i \(-0.0163200\pi\)
0.998686 + 0.0512484i \(0.0163200\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 − 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 390.000i − 0.504528i −0.967658 − 0.252264i \(-0.918825\pi\)
0.967658 − 0.252264i \(-0.0811751\pi\)
\(774\) 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
\(784\) 0 0
\(785\) 792.000 − 1056.00i 1.00892 − 1.34522i
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 528.000i − 0.665826i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1110.00i − 1.39272i −0.717691 − 0.696361i \(-0.754801\pi\)
0.717691 − 0.696361i \(-0.245199\pi\)
\(798\) 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
\(809\) 560.000 0.692213 0.346106 − 0.938195i \(-0.387504\pi\)
0.346106 + 0.938195i \(0.387504\pi\)
\(810\) 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
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1400.00 −1.70524 −0.852619 − 0.522533i \(-0.824987\pi\)
−0.852619 + 0.522533i \(0.824987\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1258.00 1.51749 0.758745 − 0.651387i \(-0.225813\pi\)
0.758745 + 0.651387i \(0.225813\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1470.00i − 1.76471i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 759.000 0.902497
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1628.00 + 1221.00i 1.92663 + 1.44497i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 1656.00i − 1.94138i −0.240328 − 0.970692i \(-0.577255\pi\)
0.240328 − 0.970692i \(-0.422745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1650.00i − 1.92532i −0.270712 − 0.962660i \(-0.587259\pi\)
0.270712 − 0.962660i \(-0.412741\pi\)
\(858\) 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
\(865\) −990.000 + 1320.00i −1.14451 + 1.52601i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 696.000i − 0.793615i −0.917902 − 0.396807i \(-0.870118\pi\)
0.917902 − 0.396807i \(-0.129882\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1600.00 1.81612 0.908059 − 0.418842i \(-0.137564\pi\)
0.908059 + 0.418842i \(0.137564\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2700.00 −2.99667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −152.000 − 114.000i −0.167956 − 0.125967i
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −576.000 + 168.000i −0.622703 + 0.181622i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1840.00 1.98062 0.990312 − 0.138859i \(-0.0443435\pi\)
0.990312 + 0.138859i \(0.0443435\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1824.00i 1.94664i 0.229456 + 0.973319i \(0.426305\pi\)
−0.229456 + 0.973319i \(0.573695\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1160.00 1.23273 0.616366 − 0.787460i \(-0.288604\pi\)
0.616366 + 0.787460i \(0.288604\pi\)
\(942\) 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
\(949\) 2304.00 2.42782
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1230.00i 1.29066i 0.763903 + 0.645331i \(0.223280\pi\)
−0.763903 + 0.645331i \(0.776720\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1008.00 + 1344.00i −1.04456 + 1.39275i
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1890.00i − 1.93449i −0.253838 − 0.967247i \(-0.581693\pi\)
0.253838 − 0.967247i \(-0.418307\pi\)
\(978\) 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
\(985\) −1170.00 + 1560.00i −1.18782 + 1.58376i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 744.000i 0.746239i 0.927783 + 0.373119i \(0.121712\pi\)
−0.927783 + 0.373119i \(0.878288\pi\)
\(998\) 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 720.3.j.b.559.1 ✓ 2
3.2 odd 2 720.3.j.c.559.2 yes 2
4.3 odd 2 CM 720.3.j.b.559.1 ✓ 2
5.2 odd 4 3600.3.e.a.3151.1 1
5.3 odd 4 3600.3.e.e.3151.1 1
5.4 even 2 inner 720.3.j.b.559.2 yes 2
12.11 even 2 720.3.j.c.559.2 yes 2
15.2 even 4 3600.3.e.b.3151.1 1
15.8 even 4 3600.3.e.d.3151.1 1
15.14 odd 2 720.3.j.c.559.1 yes 2
20.3 even 4 3600.3.e.e.3151.1 1
20.7 even 4 3600.3.e.a.3151.1 1
20.19 odd 2 inner 720.3.j.b.559.2 yes 2
60.23 odd 4 3600.3.e.d.3151.1 1
60.47 odd 4 3600.3.e.b.3151.1 1
60.59 even 2 720.3.j.c.559.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.3.j.b.559.1 ✓ 2 1.1 even 1 trivial
720.3.j.b.559.1 ✓ 2 4.3 odd 2 CM
720.3.j.b.559.2 yes 2 5.4 even 2 inner
720.3.j.b.559.2 yes 2 20.19 odd 2 inner
720.3.j.c.559.1 yes 2 15.14 odd 2
720.3.j.c.559.1 yes 2 60.59 even 2
720.3.j.c.559.2 yes 2 3.2 odd 2
720.3.j.c.559.2 yes 2 12.11 even 2
3600.3.e.a.3151.1 1 5.2 odd 4
3600.3.e.a.3151.1 1 20.7 even 4
3600.3.e.b.3151.1 1 15.2 even 4
3600.3.e.b.3151.1 1 60.47 odd 4
3600.3.e.d.3151.1 1 15.8 even 4
3600.3.e.d.3151.1 1 60.23 odd 4
3600.3.e.e.3151.1 1 5.3 odd 4
3600.3.e.e.3151.1 1 20.3 even 4