Properties

Label 256.4.a.h.1.1
Level $256$
Weight $4$
Character 256.1
Self dual yes
Analytic conductor $15.104$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,4,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

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: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 64)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 256.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0000 q^{3} +73.0000 q^{9} +O(q^{10})\) \(q+10.0000 q^{3} +73.0000 q^{9} -18.0000 q^{11} +90.0000 q^{17} +106.000 q^{19} -125.000 q^{25} +460.000 q^{27} -180.000 q^{33} -522.000 q^{41} -290.000 q^{43} -343.000 q^{49} +900.000 q^{51} +1060.00 q^{57} +846.000 q^{59} -70.0000 q^{67} +430.000 q^{73} -1250.00 q^{75} +2629.00 q^{81} -1350.00 q^{83} -1026.00 q^{89} -1910.00 q^{97} -1314.00 q^{99} +O(q^{100})\)

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
\(3\) 10.0000 1.92450 0.962250 0.272166i \(-0.0877398\pi\)
0.962250 + 0.272166i \(0.0877398\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 73.0000 2.70370
\(10\) 0 0
\(11\) −18.0000 −0.493382 −0.246691 0.969094i \(-0.579343\pi\)
−0.246691 + 0.969094i \(0.579343\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 90.0000 1.28401 0.642006 0.766700i \(-0.278102\pi\)
0.642006 + 0.766700i \(0.278102\pi\)
\(18\) 0 0
\(19\) 106.000 1.27990 0.639949 0.768417i \(-0.278955\pi\)
0.639949 + 0.768417i \(0.278955\pi\)
\(20\) 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
\(27\) 460.000 3.27878
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −180.000 −0.949514
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −522.000 −1.98836 −0.994179 0.107738i \(-0.965639\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) −290.000 −1.02848 −0.514239 0.857647i \(-0.671926\pi\)
−0.514239 + 0.857647i \(0.671926\pi\)
\(44\) 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
\(51\) 900.000 2.47108
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1060.00 2.46317
\(58\) 0 0
\(59\) 846.000 1.86678 0.933388 0.358868i \(-0.116837\pi\)
0.933388 + 0.358868i \(0.116837\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −70.0000 −0.127640 −0.0638199 0.997961i \(-0.520328\pi\)
−0.0638199 + 0.997961i \(0.520328\pi\)
\(68\) 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
\(75\) −1250.00 −1.92450
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 0 0
\(83\) −1350.00 −1.78532 −0.892661 0.450728i \(-0.851164\pi\)
−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 \(-0.991517\pi\)
−0.999645 + 0.0266459i \(0.991517\pi\)
\(98\) 0 0
\(99\) −1314.00 −1.33396
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1710.00 1.54497 0.772486 0.635032i \(-0.219013\pi\)
0.772486 + 0.635032i \(0.219013\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 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
\(121\) −1007.00 −0.756574
\(122\) 0 0
\(123\) −5220.00 −3.82660
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −2900.00 −1.97931
\(130\) 0 0
\(131\) 1242.00 0.828351 0.414176 0.910197i \(-0.364070\pi\)
0.414176 + 0.910197i \(0.364070\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2250.00 −1.40314 −0.701571 0.712599i \(-0.747518\pi\)
−0.701571 + 0.712599i \(0.747518\pi\)
\(138\) 0 0
\(139\) −1474.00 −0.899446 −0.449723 0.893168i \(-0.648477\pi\)
−0.449723 + 0.893168i \(0.648477\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3430.00 −1.92450
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 6570.00 3.47159
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 970.000 0.466112 0.233056 0.972463i \(-0.425127\pi\)
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
\(171\) 7738.00 3.46047
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8460.00 3.59261
\(178\) 0 0
\(179\) 3834.00 1.60093 0.800465 0.599379i \(-0.204586\pi\)
0.800465 + 0.599379i \(0.204586\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1620.00 −0.633509
\(188\) 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 \(-0.372563\pi\)
0.389745 + 0.920923i \(0.372563\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −700.000 −0.245643
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1908.00 −0.631479
\(210\) 0 0
\(211\) −6118.00 −1.99612 −0.998058 0.0622910i \(-0.980159\pi\)
−0.998058 + 0.0622910i \(0.980159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4300.00 1.32679
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −9125.00 −2.70370
\(226\) 0 0
\(227\) 6570.00 1.92100 0.960498 0.278286i \(-0.0897663\pi\)
0.960498 + 0.278286i \(0.0897663\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 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 \(-0.552217\pi\)
−0.163311 + 0.986575i \(0.552217\pi\)
\(242\) 0 0
\(243\) 13870.0 3.66157
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −13500.0 −3.43585
\(250\) 0 0
\(251\) 4302.00 1.08183 0.540916 0.841077i \(-0.318078\pi\)
0.540916 + 0.841077i \(0.318078\pi\)
\(252\) 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
\(267\) −10260.0 −2.35169
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2250.00 0.493382
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 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
\(283\) 8030.00 1.68669 0.843346 0.537371i \(-0.180582\pi\)
0.843346 + 0.537371i \(0.180582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) −19100.0 −3.84764
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8280.00 −1.61769
\(298\) 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
\(307\) −7990.00 −1.48539 −0.742693 0.669632i \(-0.766452\pi\)
−0.742693 + 0.669632i \(0.766452\pi\)
\(308\) 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 \(-0.226393\pi\)
0.757557 + 0.652769i \(0.226393\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 17100.0 2.97330
\(322\) 0 0
\(323\) 9540.00 1.64340
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8242.00 −1.36864 −0.684322 0.729180i \(-0.739902\pi\)
−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
\(339\) −2700.00 −0.432578
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6030.00 0.932874 0.466437 0.884554i \(-0.345537\pi\)
0.466437 + 0.884554i \(0.345537\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 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
\(361\) 4377.00 0.638140
\(362\) 0 0
\(363\) −10070.0 −1.45603
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −38106.0 −5.37593
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11666.0 −1.58111 −0.790557 0.612389i \(-0.790209\pi\)
−0.790557 + 0.612389i \(0.790209\pi\)
\(380\) 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
\(387\) −21170.0 −2.78070
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12420.0 1.59416
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7002.00 0.871978 0.435989 0.899952i \(-0.356399\pi\)
0.435989 + 0.899952i \(0.356399\pi\)
\(402\) 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 \(-0.950824\pi\)
−0.988090 + 0.153877i \(0.950824\pi\)
\(410\) 0 0
\(411\) −22500.0 −2.70035
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −14740.0 −1.73099
\(418\) 0 0
\(419\) 16794.0 1.95809 0.979046 0.203639i \(-0.0652769\pi\)
0.979046 + 0.203639i \(0.0652769\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 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 \(-0.598913\pi\)
−0.305766 + 0.952107i \(0.598913\pi\)
\(434\) 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
\(441\) −25039.0 −2.70370
\(442\) 0 0
\(443\) 18270.0 1.95944 0.979722 0.200361i \(-0.0642114\pi\)
0.979722 + 0.200361i \(0.0642114\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17514.0 1.84084 0.920420 0.390932i \(-0.127847\pi\)
0.920420 + 0.390932i \(0.127847\pi\)
\(450\) 0 0
\(451\) 9396.00 0.981021
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18070.0 1.84963 0.924813 0.380422i \(-0.124221\pi\)
0.924813 + 0.380422i \(0.124221\pi\)
\(458\) 0 0
\(459\) 41400.0 4.20999
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15030.0 −1.48931 −0.744653 0.667452i \(-0.767385\pi\)
−0.744653 + 0.667452i \(0.767385\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5220.00 0.507433
\(474\) 0 0
\(475\) −13250.0 −1.27990
\(476\) 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
\(489\) 9700.00 0.897033
\(490\) 0 0
\(491\) 12222.0 1.12336 0.561681 0.827354i \(-0.310155\pi\)
0.561681 + 0.827354i \(0.310155\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −18214.0 −1.63401 −0.817005 0.576631i \(-0.804367\pi\)
−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
\(507\) −21970.0 −1.92450
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 48760.0 4.19650
\(514\) 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 \(-0.625873\pi\)
−0.385215 + 0.922827i \(0.625873\pi\)
\(522\) 0 0
\(523\) 4750.00 0.397138 0.198569 0.980087i \(-0.436371\pi\)
0.198569 + 0.980087i \(0.436371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 61758.0 5.04721
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 38340.0 3.08099
\(538\) 0 0
\(539\) 6174.00 0.493382
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21850.0 1.70793 0.853966 0.520329i \(-0.174191\pi\)
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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −16200.0 −1.21919
\(562\) 0 0
\(563\) −23670.0 −1.77189 −0.885943 0.463795i \(-0.846488\pi\)
−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 \(-0.528109\pi\)
−0.0881913 + 0.996104i \(0.528109\pi\)
\(570\) 0 0
\(571\) 27038.0 1.98162 0.990810 0.135261i \(-0.0431872\pi\)
0.990810 + 0.135261i \(0.0431872\pi\)
\(572\) 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
\(579\) 20900.0 1.50013
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10350.0 0.727752 0.363876 0.931447i \(-0.381453\pi\)
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
\(603\) −5110.00 −0.345100
\(604\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 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
\(619\) −30706.0 −1.99383 −0.996913 0.0785136i \(-0.974983\pi\)
−0.996913 + 0.0785136i \(0.974983\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) −19080.0 −1.21528
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −61180.0 −3.84153
\(634\) 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 \(-0.565962\pi\)
−0.205745 + 0.978606i \(0.565962\pi\)
\(642\) 0 0
\(643\) −28550.0 −1.75101 −0.875507 0.483205i \(-0.839472\pi\)
−0.875507 + 0.483205i \(0.839472\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −15228.0 −0.921034
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 31390.0 1.86399
\(658\) 0 0
\(659\) 29754.0 1.75880 0.879402 0.476081i \(-0.157943\pi\)
0.879402 + 0.476081i \(0.157943\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 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 \(-0.685208\pi\)
−0.549569 + 0.835448i \(0.685208\pi\)
\(674\) 0 0
\(675\) −57500.0 −3.27878
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 65700.0 3.69696
\(682\) 0 0
\(683\) −11970.0 −0.670599 −0.335300 0.942112i \(-0.608838\pi\)
−0.335300 + 0.942112i \(0.608838\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1978.00 0.108895 0.0544477 0.998517i \(-0.482660\pi\)
0.0544477 + 0.998517i \(0.482660\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −46980.0 −2.55308
\(698\) 0 0
\(699\) 60300.0 3.26288
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 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
\(723\) −12220.0 −0.628585
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 67717.0 3.44038
\(730\) 0 0
\(731\) −26100.0 −1.32058
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1260.00 0.0629752
\(738\) 0 0
\(739\) 36074.0 1.79567 0.897837 0.440327i \(-0.145138\pi\)
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
\(747\) −98550.0 −4.82698
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 43020.0 2.08199
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34182.0 1.62825 0.814124 0.580691i \(-0.197218\pi\)
0.814124 + 0.580691i \(0.197218\pi\)
\(762\) 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 \(-0.110497\pi\)
0.940351 + 0.340207i \(0.110497\pi\)
\(770\) 0 0
\(771\) −38700.0 −1.80771
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −55332.0 −2.54490
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −6950.00 −0.314791 −0.157396 0.987536i \(-0.550310\pi\)
−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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −74898.0 −3.30386
\(802\) 0 0
\(803\) −7740.00 −0.340148
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −14346.0 −0.623459 −0.311730 0.950171i \(-0.600908\pi\)
−0.311730 + 0.950171i \(0.600908\pi\)
\(810\) 0 0
\(811\) 37582.0 1.62723 0.813614 0.581405i \(-0.197497\pi\)
0.813614 + 0.581405i \(0.197497\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −30740.0 −1.31635
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 22500.0 0.949514
\(826\) 0 0
\(827\) −23490.0 −0.987699 −0.493850 0.869547i \(-0.664411\pi\)
−0.493850 + 0.869547i \(0.664411\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 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
\(843\) 93420.0 3.81679
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 80300.0 3.24604
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18630.0 0.742577 0.371289 0.928518i \(-0.378916\pi\)
0.371289 + 0.928518i \(0.378916\pi\)
\(858\) 0 0
\(859\) 45646.0 1.81306 0.906532 0.422138i \(-0.138720\pi\)
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
\(867\) 31870.0 1.24840
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −139430. −5.40549
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 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
\(883\) 5290.00 0.201611 0.100806 0.994906i \(-0.467858\pi\)
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
\(891\) −47322.0 −1.77929
\(892\) 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
\(907\) −26210.0 −0.959525 −0.479762 0.877399i \(-0.659277\pi\)
−0.479762 + 0.877399i \(0.659277\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 24300.0 0.880846
\(914\) 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
\(921\) −79900.0 −2.85863
\(922\) 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 \(-0.539254\pi\)
−0.123007 + 0.992406i \(0.539254\pi\)
\(930\) 0 0
\(931\) −36358.0 −1.27990
\(932\) 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
\(939\) 83900.0 2.91584
\(940\) 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
\(947\) −58230.0 −1.99812 −0.999061 0.0433353i \(-0.986202\pi\)
−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 \(-0.214395\pi\)
0.781617 + 0.623759i \(0.214395\pi\)
\(954\) 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
\(963\) 124830. 4.17714
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 95400.0 3.16273
\(970\) 0 0
\(971\) −162.000 −0.00535410 −0.00267705 0.999996i \(-0.500852\pi\)
−0.00267705 + 0.999996i \(0.500852\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17370.0 0.568798 0.284399 0.958706i \(-0.408206\pi\)
0.284399 + 0.958706i \(0.408206\pi\)
\(978\) 0 0
\(979\) 18468.0 0.602901
\(980\) 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
\(993\) −82420.0 −2.63396
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 256.4.a.h.1.1 1
3.2 odd 2 2304.4.a.i.1.1 1
4.3 odd 2 256.4.a.a.1.1 1
8.3 odd 2 CM 256.4.a.h.1.1 1
8.5 even 2 256.4.a.a.1.1 1
12.11 even 2 2304.4.a.h.1.1 1
16.3 odd 4 64.4.b.a.33.1 2
16.5 even 4 64.4.b.a.33.1 2
16.11 odd 4 64.4.b.a.33.2 yes 2
16.13 even 4 64.4.b.a.33.2 yes 2
24.5 odd 2 2304.4.a.h.1.1 1
24.11 even 2 2304.4.a.i.1.1 1
48.5 odd 4 576.4.d.a.289.2 2
48.11 even 4 576.4.d.a.289.1 2
48.29 odd 4 576.4.d.a.289.1 2
48.35 even 4 576.4.d.a.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
64.4.b.a.33.1 2 16.3 odd 4
64.4.b.a.33.1 2 16.5 even 4
64.4.b.a.33.2 yes 2 16.11 odd 4
64.4.b.a.33.2 yes 2 16.13 even 4
256.4.a.a.1.1 1 4.3 odd 2
256.4.a.a.1.1 1 8.5 even 2
256.4.a.h.1.1 1 1.1 even 1 trivial
256.4.a.h.1.1 1 8.3 odd 2 CM
576.4.d.a.289.1 2 48.11 even 4
576.4.d.a.289.1 2 48.29 odd 4
576.4.d.a.289.2 2 48.5 odd 4
576.4.d.a.289.2 2 48.35 even 4
2304.4.a.h.1.1 1 12.11 even 2
2304.4.a.h.1.1 1 24.5 odd 2
2304.4.a.i.1.1 1 3.2 odd 2
2304.4.a.i.1.1 1 24.11 even 2