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Properties

Label 315.3.e.a.244.1
Level $315$
Weight $3$
Character 315.244
Self dual yes
Analytic conductor $8.583$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,3,Mod(244,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.244");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.e (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: yes
Analytic conductor: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 244.1
Character \(\chi\) \(=\) 315.244

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} -5.00000 q^{5} -7.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{4} -5.00000 q^{5} -7.00000 q^{7} +13.0000 q^{11} +19.0000 q^{13} +16.0000 q^{16} +29.0000 q^{17} -20.0000 q^{20} +25.0000 q^{25} -28.0000 q^{28} -23.0000 q^{29} +35.0000 q^{35} +52.0000 q^{44} -31.0000 q^{47} +49.0000 q^{49} +76.0000 q^{52} -65.0000 q^{55} +64.0000 q^{64} -95.0000 q^{65} +116.000 q^{68} -2.00000 q^{71} +34.0000 q^{73} -91.0000 q^{77} -157.000 q^{79} -80.0000 q^{80} +86.0000 q^{83} -145.000 q^{85} -133.000 q^{91} -149.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) −5.00000 −1.00000
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.0000 1.18182 0.590909 − 0.806738i \(-0.298769\pi\)
0.590909 + 0.806738i \(0.298769\pi\)
\(12\) 0 0
\(13\) 19.0000 1.46154 0.730769 − 0.682625i \(-0.239162\pi\)
0.730769 + 0.682625i \(0.239162\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 29.0000 1.70588 0.852941 − 0.522007i \(-0.174817\pi\)
0.852941 + 0.522007i \(0.174817\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −20.0000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −28.0000 −1.00000
\(29\) −23.0000 −0.793103 −0.396552 − 0.918012i \(-0.629793\pi\)
−0.396552 + 0.918012i \(0.629793\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 35.0000 1.00000
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 52.0000 1.18182
\(45\) 0 0
\(46\) 0 0
\(47\) −31.0000 −0.659574 −0.329787 − 0.944055i \(-0.606977\pi\)
−0.329787 + 0.944055i \(0.606977\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 76.0000 1.46154
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −65.0000 −1.18182
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) −95.0000 −1.46154
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 116.000 1.70588
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.0281690 −0.0140845 − 0.999901i \(-0.504483\pi\)
−0.0140845 + 0.999901i \(0.504483\pi\)
\(72\) 0 0
\(73\) 34.0000 0.465753 0.232877 − 0.972506i \(-0.425186\pi\)
0.232877 + 0.972506i \(0.425186\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −91.0000 −1.18182
\(78\) 0 0
\(79\) −157.000 −1.98734 −0.993671 − 0.112331i \(-0.964168\pi\)
−0.993671 + 0.112331i \(0.964168\pi\)
\(80\) −80.0000 −1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) 86.0000 1.03614 0.518072 − 0.855337i \(-0.326650\pi\)
0.518072 + 0.855337i \(0.326650\pi\)
\(84\) 0 0
\(85\) −145.000 −1.70588
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −133.000 −1.46154
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −149.000 −1.53608 −0.768041 − 0.640400i \(-0.778768\pi\)
−0.768041 + 0.640400i \(0.778768\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 100.000 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 199.000 1.93204 0.966019 − 0.258470i \(-0.0832182\pi\)
0.966019 + 0.258470i \(0.0832182\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −97.0000 −0.889908 −0.444954 − 0.895553i \(-0.646780\pi\)
−0.444954 + 0.895553i \(0.646780\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −112.000 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −92.0000 −0.793103
\(117\) 0 0
\(118\) 0 0
\(119\) −203.000 −1.70588
\(120\) 0 0
\(121\) 48.0000 0.396694
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 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
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 140.000 1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 247.000 1.72727
\(144\) 0 0
\(145\) 115.000 0.793103
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 262.000 1.75839 0.879195 − 0.476463i \(-0.158081\pi\)
0.879195 + 0.476463i \(0.158081\pi\)
\(150\) 0 0
\(151\) −13.0000 −0.0860927 −0.0430464 − 0.999073i \(-0.513706\pi\)
−0.0430464 + 0.999073i \(0.513706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −134.000 −0.853503 −0.426752 − 0.904369i \(-0.640342\pi\)
−0.426752 + 0.904369i \(0.640342\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −271.000 −1.62275 −0.811377 − 0.584523i \(-0.801282\pi\)
−0.811377 + 0.584523i \(0.801282\pi\)
\(168\) 0 0
\(169\) 192.000 1.13609
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 221.000 1.27746 0.638728 − 0.769432i \(-0.279461\pi\)
0.638728 + 0.769432i \(0.279461\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 208.000 1.18182
\(177\) 0 0
\(178\) 0 0
\(179\) −218.000 −1.21788 −0.608939 − 0.793217i \(-0.708404\pi\)
−0.608939 + 0.793217i \(0.708404\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 377.000 2.01604
\(188\) −124.000 −0.659574
\(189\) 0 0
\(190\) 0 0
\(191\) −347.000 −1.81675 −0.908377 − 0.418152i \(-0.862678\pi\)
−0.908377 + 0.418152i \(0.862678\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 161.000 0.793103
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 304.000 1.46154
\(209\) 0 0
\(210\) 0 0
\(211\) 107.000 0.507109 0.253555 − 0.967321i \(-0.418400\pi\)
0.253555 + 0.967321i \(0.418400\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −260.000 −1.18182
\(221\) 551.000 2.49321
\(222\) 0 0
\(223\) −401.000 −1.79821 −0.899103 − 0.437737i \(-0.855780\pi\)
−0.899103 + 0.437737i \(0.855780\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −391.000 −1.72247 −0.861233 − 0.508209i \(-0.830308\pi\)
−0.861233 + 0.508209i \(0.830308\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 155.000 0.659574
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 397.000 1.66109 0.830544 − 0.556953i \(-0.188030\pi\)
0.830544 + 0.556953i \(0.188030\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −245.000 −1.00000
\(246\) 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
\(256\) 256.000 1.00000
\(257\) 494.000 1.92218 0.961089 − 0.276237i \(-0.0890875\pi\)
0.961089 + 0.276237i \(0.0890875\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −380.000 −1.46154
\(261\) 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
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 464.000 1.70588
\(273\) 0 0
\(274\) 0 0
\(275\) 325.000 1.18182
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −527.000 −1.87544 −0.937722 − 0.347385i \(-0.887070\pi\)
−0.937722 + 0.347385i \(0.887070\pi\)
\(282\) 0 0
\(283\) 559.000 1.97527 0.987633 − 0.156787i \(-0.0501135\pi\)
0.987633 + 0.156787i \(0.0501135\pi\)
\(284\) −8.00000 −0.0281690
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 552.000 1.91003
\(290\) 0 0
\(291\) 0 0
\(292\) 136.000 0.465753
\(293\) −19.0000 −0.0648464 −0.0324232 − 0.999474i \(-0.510322\pi\)
−0.0324232 + 0.999474i \(0.510322\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −569.000 −1.85342 −0.926710 − 0.375777i \(-0.877376\pi\)
−0.926710 + 0.375777i \(0.877376\pi\)
\(308\) −364.000 −1.18182
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −221.000 −0.706070 −0.353035 − 0.935610i \(-0.614850\pi\)
−0.353035 + 0.935610i \(0.614850\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −628.000 −1.98734
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −299.000 −0.937304
\(320\) −320.000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 475.000 1.46154
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 217.000 0.659574
\(330\) 0 0
\(331\) −598.000 −1.80665 −0.903323 − 0.428960i \(-0.858880\pi\)
−0.903323 + 0.428960i \(0.858880\pi\)
\(332\) 344.000 1.03614
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −580.000 −1.70588
\(341\) 0 0
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −139.000 −0.393768 −0.196884 − 0.980427i \(-0.563082\pi\)
−0.196884 + 0.980427i \(0.563082\pi\)
\(354\) 0 0
\(355\) 10.0000 0.0281690
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −578.000 −1.61003 −0.805014 − 0.593256i \(-0.797842\pi\)
−0.805014 + 0.593256i \(0.797842\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) −532.000 −1.46154
\(365\) −170.000 −0.465753
\(366\) 0 0
\(367\) 391.000 1.06540 0.532698 − 0.846306i \(-0.321178\pi\)
0.532698 + 0.846306i \(0.321178\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −437.000 −1.15915
\(378\) 0 0
\(379\) −502.000 −1.32454 −0.662269 − 0.749266i \(-0.730406\pi\)
−0.662269 + 0.749266i \(0.730406\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −514.000 −1.34204 −0.671018 − 0.741441i \(-0.734143\pi\)
−0.671018 + 0.741441i \(0.734143\pi\)
\(384\) 0 0
\(385\) 455.000 1.18182
\(386\) 0 0
\(387\) 0 0
\(388\) −596.000 −1.53608
\(389\) −743.000 −1.91003 −0.955013 − 0.296564i \(-0.904159\pi\)
−0.955013 + 0.296564i \(0.904159\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 785.000 1.98734
\(396\) 0 0
\(397\) −389.000 −0.979849 −0.489924 − 0.871765i \(-0.662976\pi\)
−0.489924 + 0.871765i \(0.662976\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 73.0000 0.182045 0.0910224 − 0.995849i \(-0.470986\pi\)
0.0910224 + 0.995849i \(0.470986\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 796.000 1.93204
\(413\) 0 0
\(414\) 0 0
\(415\) −430.000 −1.03614
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 527.000 1.25178 0.625891 − 0.779911i \(-0.284736\pi\)
0.625891 + 0.779911i \(0.284736\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 725.000 1.70588
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 853.000 1.97912 0.989559 − 0.144127i \(-0.0460375\pi\)
0.989559 + 0.144127i \(0.0460375\pi\)
\(432\) 0 0
\(433\) 754.000 1.74134 0.870670 − 0.491868i \(-0.163686\pi\)
0.870670 + 0.491868i \(0.163686\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −388.000 −0.889908
\(437\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −448.000 −1.00000
\(449\) 817.000 1.81960 0.909800 − 0.415048i \(-0.136235\pi\)
0.909800 + 0.415048i \(0.136235\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 665.000 1.46154
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −368.000 −0.793103
\(465\) 0 0
\(466\) 0 0
\(467\) −871.000 −1.86510 −0.932548 − 0.361046i \(-0.882420\pi\)
−0.932548 + 0.361046i \(0.882420\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −812.000 −1.70588
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 192.000 0.396694
\(485\) 745.000 1.53608
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −107.000 −0.217923 −0.108961 − 0.994046i \(-0.534752\pi\)
−0.108961 + 0.994046i \(0.534752\pi\)
\(492\) 0 0
\(493\) −667.000 −1.35294
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.0000 0.0281690
\(498\) 0 0
\(499\) 683.000 1.36874 0.684369 − 0.729136i \(-0.260078\pi\)
0.684369 + 0.729136i \(0.260078\pi\)
\(500\) −500.000 −1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) −439.000 −0.872763 −0.436382 − 0.899762i \(-0.643740\pi\)
−0.436382 + 0.899762i \(0.643740\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −238.000 −0.465753
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −995.000 −1.93204
\(516\) 0 0
\(517\) −403.000 −0.779497
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −326.000 −0.623327 −0.311663 − 0.950193i \(-0.600886\pi\)
−0.311663 + 0.950193i \(0.600886\pi\)
\(524\) 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
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 637.000 1.18182
\(540\) 0 0
\(541\) 767.000 1.41774 0.708872 − 0.705337i \(-0.249204\pi\)
0.708872 + 0.705337i \(0.249204\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 485.000 0.889908
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1099.00 1.98734
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 560.000 1.00000
\(561\) 0 0
\(562\) 0 0
\(563\) −874.000 −1.55240 −0.776199 − 0.630488i \(-0.782855\pi\)
−0.776199 + 0.630488i \(0.782855\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1102.00 1.93673 0.968366 − 0.249536i \(-0.0802781\pi\)
0.968366 + 0.249536i \(0.0802781\pi\)
\(570\) 0 0
\(571\) −118.000 −0.206655 −0.103327 − 0.994647i \(-0.532949\pi\)
−0.103327 + 0.994647i \(0.532949\pi\)
\(572\) 988.000 1.72727
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −29.0000 −0.0502600 −0.0251300 − 0.999684i \(-0.508000\pi\)
−0.0251300 + 0.999684i \(0.508000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 460.000 0.793103
\(581\) −602.000 −1.03614
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1094.00 1.86371 0.931857 − 0.362826i \(-0.118188\pi\)
0.931857 + 0.362826i \(0.118188\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −619.000 −1.04384 −0.521922 − 0.852993i \(-0.674785\pi\)
−0.521922 + 0.852993i \(0.674785\pi\)
\(594\) 0 0
\(595\) 1015.00 1.70588
\(596\) 1048.00 1.75839
\(597\) 0 0
\(598\) 0 0
\(599\) −323.000 −0.539232 −0.269616 − 0.962968i \(-0.586897\pi\)
−0.269616 + 0.962968i \(0.586897\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −52.0000 −0.0860927
\(605\) −240.000 −0.396694
\(606\) 0 0
\(607\) −809.000 −1.33278 −0.666392 − 0.745601i \(-0.732162\pi\)
−0.666392 + 0.745601i \(0.732162\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −589.000 −0.963993
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 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
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −536.000 −0.853503
\(629\) 0 0
\(630\) 0 0
\(631\) 947.000 1.50079 0.750396 − 0.660988i \(-0.229863\pi\)
0.750396 + 0.660988i \(0.229863\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 931.000 1.46154
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 958.000 1.49454 0.747270 − 0.664521i \(-0.231364\pi\)
0.747270 + 0.664521i \(0.231364\pi\)
\(642\) 0 0
\(643\) −1241.00 −1.93002 −0.965008 − 0.262221i \(-0.915545\pi\)
−0.965008 + 0.262221i \(0.915545\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 974.000 1.50541 0.752705 − 0.658358i \(-0.228749\pi\)
0.752705 + 0.658358i \(0.228749\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1283.00 −1.94689 −0.973445 − 0.228923i \(-0.926480\pi\)
−0.973445 + 0.228923i \(0.926480\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1084.00 −1.62275
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 768.000 1.13609
\(677\) −1291.00 −1.90694 −0.953471 − 0.301484i \(-0.902518\pi\)
−0.953471 + 0.301484i \(0.902518\pi\)
\(678\) 0 0
\(679\) 1043.00 1.53608
\(680\) 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
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 884.000 1.27746
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −700.000 −1.00000
\(701\) 313.000 0.446505 0.223252 − 0.974761i \(-0.428333\pi\)
0.223252 + 0.974761i \(0.428333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 832.000 1.18182
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1417.00 −1.99859 −0.999295 − 0.0375492i \(-0.988045\pi\)
−0.999295 + 0.0375492i \(0.988045\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1235.00 −1.72727
\(716\) −872.000 −1.21788
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1393.00 −1.93204
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −575.000 −0.793103
\(726\) 0 0
\(727\) 1426.00 1.96149 0.980743 − 0.195304i \(-0.0625693\pi\)
0.980743 + 0.195304i \(0.0625693\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1061.00 −1.44748 −0.723738 − 0.690075i \(-0.757578\pi\)
−0.723738 + 0.690075i \(0.757578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1357.00 −1.83627 −0.918133 − 0.396273i \(-0.870303\pi\)
−0.918133 + 0.396273i \(0.870303\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −1310.00 −1.75839
\(746\) 0 0
\(747\) 0 0
\(748\) 1508.00 2.01604
\(749\) 0 0
\(750\) 0 0
\(751\) −1333.00 −1.77497 −0.887483 − 0.460840i \(-0.847548\pi\)
−0.887483 + 0.460840i \(0.847548\pi\)
\(752\) −496.000 −0.659574
\(753\) 0 0
\(754\) 0 0
\(755\) 65.0000 0.0860927
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 679.000 0.889908
\(764\) −1388.00 −1.81675
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1541.00 1.99353 0.996766 − 0.0803607i \(-0.0256072\pi\)
0.996766 + 0.0803607i \(0.0256072\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −26.0000 −0.0332907
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 670.000 0.853503
\(786\) 0 0
\(787\) −449.000 −0.570521 −0.285260 − 0.958450i \(-0.592080\pi\)
−0.285260 + 0.958450i \(0.592080\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1531.00 −1.92095 −0.960477 − 0.278360i \(-0.910209\pi\)
−0.960477 + 0.278360i \(0.910209\pi\)
\(798\) 0 0
\(799\) −899.000 −1.12516
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 442.000 0.550436
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 97.0000 0.119901 0.0599506 − 0.998201i \(-0.480906\pi\)
0.0599506 + 0.998201i \(0.480906\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 644.000 0.793103
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1607.00 −1.95737 −0.978685 − 0.205369i \(-0.934160\pi\)
−0.978685 + 0.205369i \(0.934160\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1216.00 1.46154
\(833\) 1421.00 1.70588
\(834\) 0 0
\(835\) 1355.00 1.62275
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −312.000 −0.370987
\(842\) 0 0
\(843\) 0 0
\(844\) 428.000 0.507109
\(845\) −960.000 −1.13609
\(846\) 0 0
\(847\) −336.000 −0.396694
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −86.0000 −0.100821 −0.0504103 − 0.998729i \(-0.516053\pi\)
−0.0504103 + 0.998729i \(0.516053\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −706.000 −0.823804 −0.411902 − 0.911228i \(-0.635135\pi\)
−0.411902 + 0.911228i \(0.635135\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1105.00 −1.27746
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2041.00 −2.34868
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 875.000 1.00000
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1040.00 −1.18182
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 2204.00 2.49321
\(885\) 0 0
\(886\) 0 0
\(887\) 494.000 0.556933 0.278467 − 0.960446i \(-0.410174\pi\)
0.278467 + 0.960446i \(0.410174\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −1604.00 −1.79821
\(893\) 0 0
\(894\) 0 0
\(895\) 1090.00 1.21788
\(896\) 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
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −1564.00 −1.72247
\(909\) 0 0
\(910\) 0 0
\(911\) 1678.00 1.84193 0.920966 − 0.389643i \(-0.127402\pi\)
0.920966 + 0.389643i \(0.127402\pi\)
\(912\) 0 0
\(913\) 1118.00 1.22453
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −997.000 −1.08487 −0.542437 − 0.840096i \(-0.682498\pi\)
−0.542437 + 0.840096i \(0.682498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −38.0000 −0.0411701
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1885.00 −2.01604
\(936\) 0 0
\(937\) −1829.00 −1.95197 −0.975987 − 0.217828i \(-0.930103\pi\)
−0.975987 + 0.217828i \(0.930103\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 620.000 0.659574
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 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
\(949\) 646.000 0.680717
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1735.00 1.81675
\(956\) 1588.00 1.66109
\(957\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 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
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −980.000 −1.00000
\(981\) 0 0
\(982\) 0 0
\(983\) 1121.00 1.14039 0.570193 − 0.821511i \(-0.306868\pi\)
0.570193 + 0.821511i \(0.306868\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 722.000 0.728557 0.364279 − 0.931290i \(-0.381316\pi\)
0.364279 + 0.931290i \(0.381316\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1651.00 1.65597 0.827984 − 0.560752i \(-0.189488\pi\)
0.827984 + 0.560752i \(0.189488\pi\)
\(998\) 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 315.3.e.a.244.1 1
3.2 odd 2 35.3.c.a.34.1 ✓ 1
5.4 even 2 315.3.e.b.244.1 1
7.6 odd 2 315.3.e.b.244.1 1
12.11 even 2 560.3.p.b.209.1 1
15.2 even 4 175.3.d.e.76.2 2
15.8 even 4 175.3.d.e.76.1 2
15.14 odd 2 35.3.c.b.34.1 yes 1
21.2 odd 6 245.3.i.b.129.1 2
21.5 even 6 245.3.i.a.129.1 2
21.11 odd 6 245.3.i.b.19.1 2
21.17 even 6 245.3.i.a.19.1 2
21.20 even 2 35.3.c.b.34.1 yes 1
35.34 odd 2 CM 315.3.e.a.244.1 1
60.59 even 2 560.3.p.a.209.1 1
84.83 odd 2 560.3.p.a.209.1 1
105.44 odd 6 245.3.i.a.129.1 2
105.59 even 6 245.3.i.b.19.1 2
105.62 odd 4 175.3.d.e.76.1 2
105.74 odd 6 245.3.i.a.19.1 2
105.83 odd 4 175.3.d.e.76.2 2
105.89 even 6 245.3.i.b.129.1 2
105.104 even 2 35.3.c.a.34.1 ✓ 1
420.419 odd 2 560.3.p.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.a.34.1 ✓ 1 3.2 odd 2
35.3.c.a.34.1 ✓ 1 105.104 even 2
35.3.c.b.34.1 yes 1 15.14 odd 2
35.3.c.b.34.1 yes 1 21.20 even 2
175.3.d.e.76.1 2 15.8 even 4
175.3.d.e.76.1 2 105.62 odd 4
175.3.d.e.76.2 2 15.2 even 4
175.3.d.e.76.2 2 105.83 odd 4
245.3.i.a.19.1 2 21.17 even 6
245.3.i.a.19.1 2 105.74 odd 6
245.3.i.a.129.1 2 21.5 even 6
245.3.i.a.129.1 2 105.44 odd 6
245.3.i.b.19.1 2 21.11 odd 6
245.3.i.b.19.1 2 105.59 even 6
245.3.i.b.129.1 2 21.2 odd 6
245.3.i.b.129.1 2 105.89 even 6
315.3.e.a.244.1 1 1.1 even 1 trivial
315.3.e.a.244.1 1 35.34 odd 2 CM
315.3.e.b.244.1 1 5.4 even 2
315.3.e.b.244.1 1 7.6 odd 2
560.3.p.a.209.1 1 60.59 even 2
560.3.p.a.209.1 1 84.83 odd 2
560.3.p.b.209.1 1 12.11 even 2
560.3.p.b.209.1 1 420.419 odd 2