Properties

Label 128.4.b.b.65.2
Level $128$
Weight $4$
Character 128.65
Analytic conductor $7.552$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,4,Mod(65,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.b (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: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 128.65
Dual form 128.4.b.b.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{3} +19.0000 q^{9} +O(q^{10})\) \(q+2.82843i q^{3} +19.0000 q^{9} +70.7107i q^{11} -90.0000 q^{17} +127.279i q^{19} +125.000 q^{25} +130.108i q^{27} -200.000 q^{33} +522.000 q^{41} -483.661i q^{43} -343.000 q^{49} -254.558i q^{51} -360.000 q^{57} -325.269i q^{59} -1094.60i q^{67} +430.000 q^{73} +353.553i q^{75} +145.000 q^{81} +681.651i q^{83} -1026.00 q^{89} +1910.00 q^{97} +1343.50i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 38 q^{9} - 180 q^{17} + 250 q^{25} - 400 q^{33} + 1044 q^{41} - 686 q^{49} - 720 q^{57} + 860 q^{73} + 290 q^{81} - 2052 q^{89} + 3820 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 2.82843i 0.544331i 0.962250 + 0.272166i \(0.0877398\pi\)
−0.962250 + 0.272166i \(0.912260\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 19.0000 0.703704
\(10\) 0 0
\(11\) 70.7107i 1.93819i 0.246691 + 0.969094i \(0.420657\pi\)
−0.246691 + 0.969094i \(0.579343\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −90.0000 −1.28401 −0.642006 0.766700i \(-0.721898\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(18\) 0 0
\(19\) 127.279i 1.53683i 0.639949 + 0.768417i \(0.278955\pi\)
−0.639949 + 0.768417i \(0.721045\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 125.000 1.00000
\(26\) 0 0
\(27\) 130.108i 0.927379i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −200.000 −1.05502
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 522.000 1.98836 0.994179 0.107738i \(-0.0343608\pi\)
0.994179 + 0.107738i \(0.0343608\pi\)
\(42\) 0 0
\(43\) − 483.661i − 1.71529i −0.514239 0.857647i \(-0.671926\pi\)
0.514239 0.857647i \(-0.328074\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) − 254.558i − 0.698928i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −360.000 −0.836547
\(58\) 0 0
\(59\) − 325.269i − 0.717736i −0.933388 0.358868i \(-0.883163\pi\)
0.933388 0.358868i \(-0.116837\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1094.60i − 1.99592i −0.0638199 0.997961i \(-0.520328\pi\)
0.0638199 0.997961i \(-0.479672\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 430.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) 0 0
\(75\) 353.553i 0.544331i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 145.000 0.198903
\(82\) 0 0
\(83\) 681.651i 0.901457i 0.892661 + 0.450728i \(0.148836\pi\)
−0.892661 + 0.450728i \(0.851164\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1026.00 −1.22198 −0.610988 0.791640i \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1910.00 1.99929 0.999645 0.0266459i \(-0.00848265\pi\)
0.999645 + 0.0266459i \(0.00848265\pi\)
\(98\) 0 0
\(99\) 1343.50i 1.36391i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1405.73i 1.27006i 0.772486 + 0.635032i \(0.219013\pi\)
−0.772486 + 0.635032i \(0.780987\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −270.000 −0.224774 −0.112387 0.993665i \(-0.535850\pi\)
−0.112387 + 0.993665i \(0.535850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3669.00 −2.75657
\(122\) 0 0
\(123\) 1476.44i 1.08233i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 1368.00 0.933687
\(130\) 0 0
\(131\) 2729.43i 1.82039i 0.414176 + 0.910197i \(0.364070\pi\)
−0.414176 + 0.910197i \(0.635930\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2250.00 1.40314 0.701571 0.712599i \(-0.252482\pi\)
0.701571 + 0.712599i \(0.252482\pi\)
\(138\) 0 0
\(139\) − 2927.42i − 1.78634i −0.449723 0.893168i \(-0.648477\pi\)
0.449723 0.893168i \(-0.351523\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 970.151i − 0.544331i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −1710.00 −0.903564
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 4047.48i − 1.94493i −0.233056 0.972463i \(-0.574873\pi\)
0.233056 0.972463i \(-0.425127\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) 2418.31i 1.08148i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 920.000 0.390686
\(178\) 0 0
\(179\) − 2870.85i − 1.19876i −0.800465 0.599379i \(-0.795414\pi\)
0.800465 0.599379i \(-0.204586\pi\)
\(180\) 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
\(187\) − 6363.96i − 2.48866i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −2090.00 −0.779490 −0.389745 0.920923i \(-0.627437\pi\)
−0.389745 + 0.920923i \(0.627437\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 3096.00 1.08644
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9000.00 −2.97867
\(210\) 0 0
\(211\) − 381.838i − 0.124582i −0.998058 0.0622910i \(-0.980159\pi\)
0.998058 0.0622910i \(-0.0198407\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1216.22i 0.375273i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 2375.00 0.703704
\(226\) 0 0
\(227\) 1903.53i 0.556572i 0.960498 + 0.278286i \(0.0897663\pi\)
−0.960498 + 0.278286i \(0.910234\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6030.00 1.69544 0.847722 0.530441i \(-0.177974\pi\)
0.847722 + 0.530441i \(0.177974\pi\)
\(234\) 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
\(241\) 1222.00 0.326622 0.163311 0.986575i \(-0.447783\pi\)
0.163311 + 0.986575i \(0.447783\pi\)
\(242\) 0 0
\(243\) 3923.03i 1.03565i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1928.00 −0.490691
\(250\) 0 0
\(251\) 6689.23i 1.68215i 0.540916 + 0.841077i \(0.318078\pi\)
−0.540916 + 0.841077i \(0.681922\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3870.00 −0.939315 −0.469658 0.882849i \(-0.655623\pi\)
−0.469658 + 0.882849i \(0.655623\pi\)
\(258\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) − 2901.97i − 0.665159i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8838.83i 1.93819i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9342.00 1.98326 0.991632 0.129099i \(-0.0412086\pi\)
0.991632 + 0.129099i \(0.0412086\pi\)
\(282\) 0 0
\(283\) 5116.62i 1.07474i 0.843346 + 0.537371i \(0.180582\pi\)
−0.843346 + 0.537371i \(0.819418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) 5402.30i 1.08828i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −9200.00 −1.79743
\(298\) 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
\(307\) − 7204.00i − 1.33926i −0.742693 0.669632i \(-0.766452\pi\)
0.742693 0.669632i \(-0.233548\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −8390.00 −1.51511 −0.757557 0.652769i \(-0.773607\pi\)
−0.757557 + 0.652769i \(0.773607\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3976.00 −0.691335
\(322\) 0 0
\(323\) − 11455.1i − 1.97331i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8782.27i 1.45836i 0.684322 + 0.729180i \(0.260098\pi\)
−0.684322 + 0.729180i \(0.739902\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11410.0 1.84434 0.922170 0.386786i \(-0.126415\pi\)
0.922170 + 0.386786i \(0.126415\pi\)
\(338\) 0 0
\(339\) − 763.675i − 0.122351i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 11435.3i − 1.76911i −0.466437 0.884554i \(-0.654463\pi\)
0.466437 0.884554i \(-0.345537\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4770.00 0.719211 0.359605 0.933104i \(-0.382911\pi\)
0.359605 + 0.933104i \(0.382911\pi\)
\(354\) 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
\(361\) −9341.00 −1.36186
\(362\) 0 0
\(363\) − 10377.5i − 1.50049i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 9918.00 1.39922
\(370\) 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
\(377\) 0 0
\(378\) 0 0
\(379\) − 9036.82i − 1.22478i −0.790557 0.612389i \(-0.790209\pi\)
0.790557 0.612389i \(-0.209791\pi\)
\(380\) 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
\(387\) − 9189.56i − 1.20706i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −7720.00 −0.990897
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7002.00 −0.871978 −0.435989 0.899952i \(-0.643601\pi\)
−0.435989 + 0.899952i \(0.643601\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 16346.0 1.97618 0.988090 0.153877i \(-0.0491758\pi\)
0.988090 + 0.153877i \(0.0491758\pi\)
\(410\) 0 0
\(411\) 6363.96i 0.763774i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8280.00 0.972358
\(418\) 0 0
\(419\) − 3493.11i − 0.407278i −0.979046 0.203639i \(-0.934723\pi\)
0.979046 0.203639i \(-0.0652769\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11250.0 −1.28401
\(426\) 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
\(433\) 5510.00 0.611533 0.305766 0.952107i \(-0.401087\pi\)
0.305766 + 0.952107i \(0.401087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −6517.00 −0.703704
\(442\) 0 0
\(443\) 3736.35i 0.400721i 0.979722 + 0.200361i \(0.0642114\pi\)
−0.979722 + 0.200361i \(0.935789\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17514.0 −1.84084 −0.920420 0.390932i \(-0.872153\pi\)
−0.920420 + 0.390932i \(0.872153\pi\)
\(450\) 0 0
\(451\) 36911.0i 3.85381i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18070.0 −1.84963 −0.924813 0.380422i \(-0.875779\pi\)
−0.924813 + 0.380422i \(0.875779\pi\)
\(458\) 0 0
\(459\) − 11709.7i − 1.19077i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 13471.8i − 1.33490i −0.744653 0.667452i \(-0.767385\pi\)
0.744653 0.667452i \(-0.232615\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 34200.0 3.32456
\(474\) 0 0
\(475\) 15909.9i 1.53683i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 11448.0 1.05868
\(490\) 0 0
\(491\) 18002.9i 1.65471i 0.561681 + 0.827354i \(0.310155\pi\)
−0.561681 + 0.827354i \(0.689845\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12855.2i 1.15326i 0.817005 + 0.576631i \(0.195633\pi\)
−0.817005 + 0.576631i \(0.804367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6214.05i 0.544331i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −16560.0 −1.42523
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9162.00 0.770431 0.385215 0.922827i \(-0.374127\pi\)
0.385215 + 0.922827i \(0.374127\pi\)
\(522\) 0 0
\(523\) 23444.8i 1.96017i 0.198569 + 0.980087i \(0.436371\pi\)
−0.198569 + 0.980087i \(0.563629\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) − 6180.11i − 0.505074i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8120.00 0.652521
\(538\) 0 0
\(539\) − 24253.8i − 1.93819i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 13313.4i − 1.04066i −0.853966 0.520329i \(-0.825809\pi\)
0.853966 0.520329i \(-0.174191\pi\)
\(548\) 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
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18000.0 1.35465
\(562\) 0 0
\(563\) 12391.3i 0.927589i 0.885943 + 0.463795i \(0.153512\pi\)
−0.885943 + 0.463795i \(0.846488\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2394.00 0.176383 0.0881913 0.996104i \(-0.471891\pi\)
0.0881913 + 0.996104i \(0.471891\pi\)
\(570\) 0 0
\(571\) 3691.10i 0.270521i 0.990810 + 0.135261i \(0.0431872\pi\)
−0.990810 + 0.135261i \(0.956813\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19550.0 −1.41053 −0.705266 0.708943i \(-0.749173\pi\)
−0.705266 + 0.708943i \(0.749173\pi\)
\(578\) 0 0
\(579\) − 5911.41i − 0.424300i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 26493.9i − 1.86289i −0.363876 0.931447i \(-0.618547\pi\)
0.363876 0.931447i \(-0.381453\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26190.0 −1.81365 −0.906825 0.421507i \(-0.861501\pi\)
−0.906825 + 0.421507i \(0.861501\pi\)
\(594\) 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
\(601\) 14398.0 0.977216 0.488608 0.872503i \(-0.337505\pi\)
0.488608 + 0.872503i \(0.337505\pi\)
\(602\) 0 0
\(603\) − 20797.4i − 1.40454i
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28530.0 −1.86155 −0.930774 0.365596i \(-0.880865\pi\)
−0.930774 + 0.365596i \(0.880865\pi\)
\(618\) 0 0
\(619\) − 2418.31i − 0.157027i −0.996913 0.0785136i \(-0.974983\pi\)
0.996913 0.0785136i \(-0.0250174\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) − 25455.8i − 1.62139i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 1080.00 0.0678138
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6678.00 0.411490 0.205745 0.978606i \(-0.434038\pi\)
0.205745 + 0.978606i \(0.434038\pi\)
\(642\) 0 0
\(643\) − 15757.2i − 0.966411i −0.875507 0.483205i \(-0.839472\pi\)
0.875507 0.483205i \(-0.160528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 23000.0 1.39111
\(650\) 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
\(657\) 8170.00 0.485148
\(658\) 0 0
\(659\) − 16107.9i − 0.952161i −0.879402 0.476081i \(-0.842057\pi\)
0.879402 0.476081i \(-0.157943\pi\)
\(660\) 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
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19190.0 1.09914 0.549569 0.835448i \(-0.314792\pi\)
0.549569 + 0.835448i \(0.314792\pi\)
\(674\) 0 0
\(675\) 16263.5i 0.927379i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −5384.00 −0.302959
\(682\) 0 0
\(683\) − 33632.8i − 1.88422i −0.335300 0.942112i \(-0.608838\pi\)
0.335300 0.942112i \(-0.391162\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 36274.6i 1.99703i 0.0544477 + 0.998517i \(0.482660\pi\)
−0.0544477 + 0.998517i \(0.517340\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −46980.0 −2.55308
\(698\) 0 0
\(699\) 17055.4i 0.922883i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 3456.34i 0.177791i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −7181.00 −0.364833
\(730\) 0 0
\(731\) 43529.5i 2.20246i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 77400.0 3.86847
\(738\) 0 0
\(739\) − 17691.8i − 0.880655i −0.897837 0.440327i \(-0.854862\pi\)
0.897837 0.440327i \(-0.145138\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12951.4i 0.634358i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −18920.0 −0.915648
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34182.0 −1.62825 −0.814124 0.580691i \(-0.802782\pi\)
−0.814124 + 0.580691i \(0.802782\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −40106.0 −1.88070 −0.940351 0.340207i \(-0.889503\pi\)
−0.940351 + 0.340207i \(0.889503\pi\)
\(770\) 0 0
\(771\) − 10946.0i − 0.511298i
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 66439.8i 3.05578i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 43605.9i 1.97507i 0.157396 + 0.987536i \(0.449690\pi\)
−0.157396 + 0.987536i \(0.550310\pi\)
\(788\) 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
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −19494.0 −0.859908
\(802\) 0 0
\(803\) 30405.6i 1.33623i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14346.0 0.623459 0.311730 0.950171i \(-0.399092\pi\)
0.311730 + 0.950171i \(0.399092\pi\)
\(810\) 0 0
\(811\) − 26855.9i − 1.16281i −0.813614 0.581405i \(-0.802503\pi\)
0.813614 0.581405i \(-0.197497\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 61560.0 2.63612
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) −25000.0 −1.05502
\(826\) 0 0
\(827\) − 41360.1i − 1.73909i −0.493850 0.869547i \(-0.664411\pi\)
0.493850 0.869547i \(-0.335589\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 30870.0 1.28401
\(834\) 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
\(843\) 26423.2i 1.07955i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14472.0 −0.585015
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18630.0 −0.742577 −0.371289 0.928518i \(-0.621084\pi\)
−0.371289 + 0.928518i \(0.621084\pi\)
\(858\) 0 0
\(859\) − 21255.6i − 0.844276i −0.906532 0.422138i \(-0.861280\pi\)
0.906532 0.422138i \(-0.138720\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9014.20i 0.353101i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 36290.0 1.40691
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −41742.0 −1.59628 −0.798141 0.602471i \(-0.794183\pi\)
−0.798141 + 0.602471i \(0.794183\pi\)
\(882\) 0 0
\(883\) − 52209.9i − 1.98981i −0.100806 0.994906i \(-0.532142\pi\)
0.100806 0.994906i \(-0.467858\pi\)
\(884\) 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
\(891\) 10253.0i 0.385511i
\(892\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 47933.4i − 1.75480i −0.479762 0.877399i \(-0.659277\pi\)
0.479762 0.877399i \(-0.340723\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −48200.0 −1.74719
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 20376.0 0.729003
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6966.00 0.246014 0.123007 0.992406i \(-0.460746\pi\)
0.123007 + 0.992406i \(0.460746\pi\)
\(930\) 0 0
\(931\) − 43656.8i − 1.53683i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 56270.0 1.96186 0.980929 0.194367i \(-0.0622652\pi\)
0.980929 + 0.194367i \(0.0622652\pi\)
\(938\) 0 0
\(939\) − 23730.5i − 0.824724i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2525.79i 0.0866705i 0.999061 + 0.0433353i \(0.0137984\pi\)
−0.999061 + 0.0433353i \(0.986202\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45990.0 −1.56323 −0.781617 0.623759i \(-0.785605\pi\)
−0.781617 + 0.623759i \(0.785605\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 26708.8i 0.893749i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 32400.0 1.07414
\(970\) 0 0
\(971\) − 60514.2i − 1.99999i −0.00267705 0.999996i \(-0.500852\pi\)
0.00267705 0.999996i \(-0.499148\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17370.0 −0.568798 −0.284399 0.958706i \(-0.591794\pi\)
−0.284399 + 0.958706i \(0.591794\pi\)
\(978\) 0 0
\(979\) − 72549.2i − 2.36842i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −24840.0 −0.793830
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 128.4.b.b.65.2 yes 2
3.2 odd 2 1152.4.d.e.577.1 2
4.3 odd 2 inner 128.4.b.b.65.1 2
8.3 odd 2 CM 128.4.b.b.65.2 yes 2
8.5 even 2 inner 128.4.b.b.65.1 2
12.11 even 2 1152.4.d.e.577.2 2
16.3 odd 4 256.4.a.k.1.2 2
16.5 even 4 256.4.a.k.1.2 2
16.11 odd 4 256.4.a.k.1.1 2
16.13 even 4 256.4.a.k.1.1 2
24.5 odd 2 1152.4.d.e.577.2 2
24.11 even 2 1152.4.d.e.577.1 2
48.5 odd 4 2304.4.a.bf.1.2 2
48.11 even 4 2304.4.a.bf.1.1 2
48.29 odd 4 2304.4.a.bf.1.1 2
48.35 even 4 2304.4.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.b.65.1 2 4.3 odd 2 inner
128.4.b.b.65.1 2 8.5 even 2 inner
128.4.b.b.65.2 yes 2 1.1 even 1 trivial
128.4.b.b.65.2 yes 2 8.3 odd 2 CM
256.4.a.k.1.1 2 16.11 odd 4
256.4.a.k.1.1 2 16.13 even 4
256.4.a.k.1.2 2 16.3 odd 4
256.4.a.k.1.2 2 16.5 even 4
1152.4.d.e.577.1 2 3.2 odd 2
1152.4.d.e.577.1 2 24.11 even 2
1152.4.d.e.577.2 2 12.11 even 2
1152.4.d.e.577.2 2 24.5 odd 2
2304.4.a.bf.1.1 2 48.11 even 4
2304.4.a.bf.1.1 2 48.29 odd 4
2304.4.a.bf.1.2 2 48.5 odd 4
2304.4.a.bf.1.2 2 48.35 even 4