Properties

Label 256.7.c.d.255.1
Level $256$
Weight $7$
Character 256.255
Analytic conductor $58.894$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 256.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.8938454067\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 255.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.255
Dual form 256.7.c.d.255.2

$q$-expansion

\(f(q)\) \(=\) \(q-46.0000i q^{3} -1387.00 q^{9} +O(q^{10})\) \(q-46.0000i q^{3} -1387.00 q^{9} -2338.00i q^{11} -1726.00 q^{17} +2482.00i q^{19} -15625.0 q^{25} +30268.0i q^{27} -107548. q^{33} -134642. q^{41} -74914.0i q^{43} +117649. q^{49} +79396.0i q^{51} +114172. q^{57} +304958. i q^{59} +596626. i q^{67} +593134. q^{73} +718750. i q^{75} +381205. q^{81} -678926. i q^{83} +357262. q^{89} +1.82275e6 q^{97} +3.24281e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2774 q^{9} - 3452 q^{17} - 31250 q^{25} - 215096 q^{33} - 269284 q^{41} + 235298 q^{49} + 228344 q^{57} + 1186268 q^{73} + 762410 q^{81} + 714524 q^{89} + 3645508 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 46.0000i − 1.70370i −0.523783 0.851852i \(-0.675480\pi\)
0.523783 0.851852i \(-0.324520\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1387.00 −1.90261
\(10\) 0 0
\(11\) − 2338.00i − 1.75657i −0.478134 0.878287i \(-0.658687\pi\)
0.478134 0.878287i \(-0.341313\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1726.00 −0.351313 −0.175656 0.984452i \(-0.556205\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(18\) 0 0
\(19\) 2482.00i 0.361860i 0.983496 + 0.180930i \(0.0579108\pi\)
−0.983496 + 0.180930i \(0.942089\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −15625.0 −1.00000
\(26\) 0 0
\(27\) 30268.0i 1.53777i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −107548. −2.99268
\(34\) 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
\(41\) −134642. −1.95357 −0.976785 0.214222i \(-0.931278\pi\)
−0.976785 + 0.214222i \(0.931278\pi\)
\(42\) 0 0
\(43\) − 74914.0i − 0.942232i −0.882071 0.471116i \(-0.843851\pi\)
0.882071 0.471116i \(-0.156149\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 79396.0i 0.598533i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 114172. 0.616503
\(58\) 0 0
\(59\) 304958.i 1.48485i 0.669927 + 0.742427i \(0.266326\pi\)
−0.669927 + 0.742427i \(0.733674\pi\)
\(60\) 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
\(67\) 596626.i 1.98371i 0.127380 + 0.991854i \(0.459343\pi\)
−0.127380 + 0.991854i \(0.540657\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 593134. 1.52470 0.762350 0.647165i \(-0.224046\pi\)
0.762350 + 0.647165i \(0.224046\pi\)
\(74\) 0 0
\(75\) 718750.i 1.70370i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 381205. 0.717304
\(82\) 0 0
\(83\) − 678926.i − 1.18738i −0.804695 0.593688i \(-0.797671\pi\)
0.804695 0.593688i \(-0.202329\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 357262. 0.506777 0.253388 0.967365i \(-0.418455\pi\)
0.253388 + 0.967365i \(0.418455\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.82275e6 1.99716 0.998580 0.0532728i \(-0.0169653\pi\)
0.998580 + 0.0532728i \(0.0169653\pi\)
\(98\) 0 0
\(99\) 3.24281e6i 3.34207i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 474014.i 0.386937i 0.981107 + 0.193468i \(0.0619737\pi\)
−0.981107 + 0.193468i \(0.938026\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.81289e6 −1.94948 −0.974738 0.223350i \(-0.928301\pi\)
−0.974738 + 0.223350i \(0.928301\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.69468e6 −2.08555
\(122\) 0 0
\(123\) 6.19353e6i 3.32830i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −3.44604e6 −1.60528
\(130\) 0 0
\(131\) 2.95362e6i 1.31383i 0.753963 + 0.656917i \(0.228140\pi\)
−0.753963 + 0.656917i \(0.771860\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 80206.0 0.0311921 0.0155961 0.999878i \(-0.495035\pi\)
0.0155961 + 0.999878i \(0.495035\pi\)
\(138\) 0 0
\(139\) − 3.19856e6i − 1.19100i −0.803357 0.595498i \(-0.796955\pi\)
0.803357 0.595498i \(-0.203045\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 5.41185e6i − 1.70370i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 2.39396e6 0.668410
\(154\) 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
\(163\) 7.72059e6i 1.78274i 0.453277 + 0.891370i \(0.350255\pi\)
−0.453277 + 0.891370i \(0.649745\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −4.82681e6 −1.00000
\(170\) 0 0
\(171\) − 3.44253e6i − 0.688478i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.40281e7 2.52975
\(178\) 0 0
\(179\) − 3.22888e6i − 0.562979i −0.959564 0.281490i \(-0.909171\pi\)
0.959564 0.281490i \(-0.0908285\pi\)
\(180\) 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
\(187\) 4.03539e6i 0.617107i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.00100e7 −1.39240 −0.696198 0.717850i \(-0.745126\pi\)
−0.696198 + 0.717850i \(0.745126\pi\)
\(194\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 2.74448e7 3.37965
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.80292e6 0.635634
\(210\) 0 0
\(211\) − 1.86421e7i − 1.98448i −0.124340 0.992240i \(-0.539681\pi\)
0.124340 0.992240i \(-0.460319\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 2.72842e7i − 2.59764i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.16719e7 1.90261
\(226\) 0 0
\(227\) − 1.97707e7i − 1.69023i −0.534586 0.845114i \(-0.679533\pi\)
0.534586 0.845114i \(-0.320467\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.10622e7 −0.874530 −0.437265 0.899333i \(-0.644053\pi\)
−0.437265 + 0.899333i \(0.644053\pi\)
\(234\) 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
\(241\) −2.65018e7 −1.89332 −0.946659 0.322237i \(-0.895565\pi\)
−0.946659 + 0.322237i \(0.895565\pi\)
\(242\) 0 0
\(243\) 4.52994e6i 0.315699i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3.12306e7 −2.02294
\(250\) 0 0
\(251\) − 1.31193e7i − 0.829640i −0.909904 0.414820i \(-0.863845\pi\)
0.909904 0.414820i \(-0.136155\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.89723e7 −1.11769 −0.558844 0.829273i \(-0.688755\pi\)
−0.558844 + 0.829273i \(0.688755\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.64341e7i − 0.863398i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.65312e7i 1.75657i
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.28969e7 −1.93333 −0.966667 0.256038i \(-0.917583\pi\)
−0.966667 + 0.256038i \(0.917583\pi\)
\(282\) 0 0
\(283\) 1.91505e7i 0.844931i 0.906379 + 0.422466i \(0.138835\pi\)
−0.906379 + 0.422466i \(0.861165\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.11585e7 −0.876579
\(290\) 0 0
\(291\) − 8.38467e7i − 3.40257i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7.07666e7 2.70121
\(298\) 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
\(307\) − 5.97121e6i − 0.206370i −0.994662 0.103185i \(-0.967097\pi\)
0.994662 0.103185i \(-0.0329034\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −9.06351e6 −0.295572 −0.147786 0.989019i \(-0.547215\pi\)
−0.147786 + 0.989019i \(0.547215\pi\)
\(314\) 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
\(321\) 2.18046e7 0.659225
\(322\) 0 0
\(323\) − 4.28393e6i − 0.127126i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 4.59882e6i − 0.126813i −0.997988 0.0634063i \(-0.979804\pi\)
0.997988 0.0634063i \(-0.0201964\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.36426e7 1.40159 0.700794 0.713364i \(-0.252829\pi\)
0.700794 + 0.713364i \(0.252829\pi\)
\(338\) 0 0
\(339\) 1.29393e8i 3.32133i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 4.72029e7i − 1.12975i −0.825178 0.564873i \(-0.808925\pi\)
0.825178 0.564873i \(-0.191075\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.52211e7 −1.48274 −0.741368 0.671099i \(-0.765823\pi\)
−0.741368 + 0.671099i \(0.765823\pi\)
\(354\) 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
\(361\) 4.08856e7 0.869057
\(362\) 0 0
\(363\) 1.69955e8i 3.55316i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 1.86748e8 3.71687
\(370\) 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
\(379\) 2.72157e7i 0.499921i 0.968256 + 0.249961i \(0.0804177\pi\)
−0.968256 + 0.249961i \(0.919582\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.03906e8i 1.79270i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1.35866e8 2.23838
\(394\) 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
\(401\) −7.99344e7 −1.23965 −0.619827 0.784738i \(-0.712797\pi\)
−0.619827 + 0.784738i \(0.712797\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.30356e8 −1.90529 −0.952644 0.304088i \(-0.901648\pi\)
−0.952644 + 0.304088i \(0.901648\pi\)
\(410\) 0 0
\(411\) − 3.68948e6i − 0.0531422i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.47134e8 −2.02910
\(418\) 0 0
\(419\) − 1.34918e8i − 1.83412i −0.398744 0.917062i \(-0.630554\pi\)
0.398744 0.917062i \(-0.369446\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.69688e7 0.351313
\(426\) 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
\(433\) −1.32005e8 −1.62603 −0.813014 0.582244i \(-0.802175\pi\)
−0.813014 + 0.582244i \(0.802175\pi\)
\(434\) 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
\(441\) −1.63179e8 −1.90261
\(442\) 0 0
\(443\) 1.59916e8i 1.83942i 0.392595 + 0.919711i \(0.371577\pi\)
−0.392595 + 0.919711i \(0.628423\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.25702e8 1.38869 0.694344 0.719643i \(-0.255694\pi\)
0.694344 + 0.719643i \(0.255694\pi\)
\(450\) 0 0
\(451\) 3.14793e8i 3.43159i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.35637e8 −1.42112 −0.710558 0.703639i \(-0.751557\pi\)
−0.710558 + 0.703639i \(0.751557\pi\)
\(458\) 0 0
\(459\) − 5.22426e7i − 0.540240i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 2.22058e7i − 0.218030i −0.994040 0.109015i \(-0.965230\pi\)
0.994040 0.109015i \(-0.0347696\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.75149e8 −1.65510
\(474\) 0 0
\(475\) − 3.87812e7i − 0.361860i
\(476\) 0 0
\(477\) 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
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 3.55147e8 3.03726
\(490\) 0 0
\(491\) − 8.73643e7i − 0.738056i −0.929418 0.369028i \(-0.879691\pi\)
0.929418 0.369028i \(-0.120309\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 8.32468e7i − 0.669986i −0.942221 0.334993i \(-0.891266\pi\)
0.942221 0.334993i \(-0.108734\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.22033e8i 1.70370i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7.51252e7 −0.556459
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.98899e8 1.40644 0.703218 0.710974i \(-0.251746\pi\)
0.703218 + 0.710974i \(0.251746\pi\)
\(522\) 0 0
\(523\) − 2.63549e8i − 1.84228i −0.389229 0.921141i \(-0.627259\pi\)
0.389229 0.921141i \(-0.372741\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) − 4.22977e8i − 2.82509i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.48528e8 −0.959150
\(538\) 0 0
\(539\) − 2.75063e8i − 1.75657i
\(540\) 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
\(547\) − 1.50088e8i − 0.917030i −0.888687 0.458515i \(-0.848382\pi\)
0.888687 0.458515i \(-0.151618\pi\)
\(548\) 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
\(561\) 1.85628e8 1.05137
\(562\) 0 0
\(563\) − 2.03362e8i − 1.13958i −0.821791 0.569789i \(-0.807025\pi\)
0.821791 0.569789i \(-0.192975\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.62709e8 1.96889 0.984445 0.175695i \(-0.0562174\pi\)
0.984445 + 0.175695i \(0.0562174\pi\)
\(570\) 0 0
\(571\) 3.58715e8i 1.92682i 0.268035 + 0.963409i \(0.413626\pi\)
−0.268035 + 0.963409i \(0.586374\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.99757e6 −0.0103986 −0.00519929 0.999986i \(-0.501655\pi\)
−0.00519929 + 0.999986i \(0.501655\pi\)
\(578\) 0 0
\(579\) 4.60461e8i 2.37223i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 2.97402e8i − 1.47038i −0.677862 0.735189i \(-0.737094\pi\)
0.677862 0.735189i \(-0.262906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.68860e8 1.28933 0.644663 0.764467i \(-0.276998\pi\)
0.644663 + 0.764467i \(0.276998\pi\)
\(594\) 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
\(601\) 2.26861e8 1.04505 0.522525 0.852624i \(-0.324990\pi\)
0.522525 + 0.852624i \(0.324990\pi\)
\(602\) 0 0
\(603\) − 8.27520e8i − 3.77422i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 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
\(617\) −3.44191e8 −1.46536 −0.732679 0.680575i \(-0.761730\pi\)
−0.732679 + 0.680575i \(0.761730\pi\)
\(618\) 0 0
\(619\) 4.68505e8i 1.97534i 0.156543 + 0.987671i \(0.449965\pi\)
−0.156543 + 0.987671i \(0.550035\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) − 2.66934e8i − 1.08293i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −8.57535e8 −3.38096
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.82154e8 −1.83068 −0.915338 0.402687i \(-0.868076\pi\)
−0.915338 + 0.402687i \(0.868076\pi\)
\(642\) 0 0
\(643\) − 2.83407e8i − 1.06605i −0.846099 0.533025i \(-0.821055\pi\)
0.846099 0.533025i \(-0.178945\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 7.12992e8 2.60826
\(650\) 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
\(657\) −8.22677e8 −2.90090
\(658\) 0 0
\(659\) − 3.12918e8i − 1.09339i −0.837332 0.546694i \(-0.815886\pi\)
0.837332 0.546694i \(-0.184114\pi\)
\(660\) 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
\(673\) −2.41386e8 −0.791895 −0.395947 0.918273i \(-0.629584\pi\)
−0.395947 + 0.918273i \(0.629584\pi\)
\(674\) 0 0
\(675\) − 4.72938e8i − 1.53777i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.09454e8 −2.87965
\(682\) 0 0
\(683\) − 4.93943e8i − 1.55030i −0.631779 0.775148i \(-0.717675\pi\)
0.631779 0.775148i \(-0.282325\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.55966e8i 1.98814i 0.108734 + 0.994071i \(0.465320\pi\)
−0.108734 + 0.994071i \(0.534680\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.32392e8 0.686314
\(698\) 0 0
\(699\) 5.08862e8i 1.48994i
\(700\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.21908e9i 3.22565i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 4.86276e8 1.25516
\(730\) 0 0
\(731\) 1.29302e8i 0.331018i
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.39491e9 3.48453
\(738\) 0 0
\(739\) − 4.94167e8i − 1.22445i −0.790685 0.612224i \(-0.790275\pi\)
0.790685 0.612224i \(-0.209725\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9.41670e8i 2.25911i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −6.03488e8 −1.41346
\(754\) 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
\(761\) −2.86987e8 −0.651191 −0.325595 0.945509i \(-0.605565\pi\)
−0.325595 + 0.945509i \(0.605565\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.98978e8 1.53704 0.768519 0.639827i \(-0.220994\pi\)
0.768519 + 0.639827i \(0.220994\pi\)
\(770\) 0 0
\(771\) 8.72725e8i 1.90421i
\(772\) 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
\(779\) − 3.34181e8i − 0.706919i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.26584e8i 1.90091i 0.310868 + 0.950453i \(0.399380\pi\)
−0.310868 + 0.950453i \(0.600620\pi\)
\(788\) 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
\(801\) −4.95522e8 −0.964197
\(802\) 0 0
\(803\) − 1.38675e9i − 2.67825i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.53143e8 1.61130 0.805649 0.592393i \(-0.201817\pi\)
0.805649 + 0.592393i \(0.201817\pi\)
\(810\) 0 0
\(811\) 3.45583e8i 0.647873i 0.946079 + 0.323937i \(0.105006\pi\)
−0.946079 + 0.323937i \(0.894994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.85937e8 0.340956
\(818\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 1.68044e9 2.99268
\(826\) 0 0
\(827\) − 5.79438e8i − 1.02445i −0.858851 0.512225i \(-0.828821\pi\)
0.858851 0.512225i \(-0.171179\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.03062e8 −0.351313
\(834\) 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
\(841\) −5.94823e8 −1.00000
\(842\) 0 0
\(843\) 1.97326e9i 3.29383i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.80924e8 1.43951
\(850\) 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
\(857\) 9.11769e8 1.44858 0.724290 0.689496i \(-0.242168\pi\)
0.724290 + 0.689496i \(0.242168\pi\)
\(858\) 0 0
\(859\) 8.15878e8i 1.28720i 0.765363 + 0.643599i \(0.222560\pi\)
−0.765363 + 0.643599i \(0.777440\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.73291e8i 1.49343i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.52816e9 −3.79981
\(874\) 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
\(881\) 3.74799e8 0.548114 0.274057 0.961714i \(-0.411634\pi\)
0.274057 + 0.961714i \(0.411634\pi\)
\(882\) 0 0
\(883\) 1.34895e9i 1.95935i 0.200584 + 0.979676i \(0.435716\pi\)
−0.200584 + 0.979676i \(0.564284\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 8.91257e8i − 1.26000i
\(892\) 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
\(907\) − 8.05321e8i − 1.07931i −0.841885 0.539656i \(-0.818554\pi\)
0.841885 0.539656i \(-0.181446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1.58733e9 −2.08571
\(914\) 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
\(921\) −2.74676e8 −0.351594
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.55501e9 −1.93948 −0.969739 0.244146i \(-0.921492\pi\)
−0.969739 + 0.244146i \(0.921492\pi\)
\(930\) 0 0
\(931\) 2.92005e8i 0.361860i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.52100e9 −1.84889 −0.924443 0.381320i \(-0.875470\pi\)
−0.924443 + 0.381320i \(0.875470\pi\)
\(938\) 0 0
\(939\) 4.16921e8i 0.503567i
\(940\) 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
\(947\) − 1.69218e9i − 1.99249i −0.0865891 0.996244i \(-0.527597\pi\)
0.0865891 0.996244i \(-0.472403\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.84034e8 −0.443701 −0.221851 0.975081i \(-0.571210\pi\)
−0.221851 + 0.975081i \(0.571210\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) − 6.57457e8i − 0.736188i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) −1.97061e8 −0.216585
\(970\) 0 0
\(971\) − 1.83097e9i − 1.99997i −0.00535408 0.999986i \(-0.501704\pi\)
0.00535408 0.999986i \(-0.498296\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.56343e9 −1.67647 −0.838234 0.545310i \(-0.816412\pi\)
−0.838234 + 0.545310i \(0.816412\pi\)
\(978\) 0 0
\(979\) − 8.35279e8i − 0.890191i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −2.11546e8 −0.216051
\(994\) 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 256.7.c.d.255.1 2
4.3 odd 2 inner 256.7.c.d.255.2 2
8.3 odd 2 CM 256.7.c.d.255.1 2
8.5 even 2 inner 256.7.c.d.255.2 2
16.3 odd 4 32.7.d.a.15.1 1
16.5 even 4 32.7.d.a.15.1 1
16.11 odd 4 8.7.d.a.3.1 1
16.13 even 4 8.7.d.a.3.1 1
48.5 odd 4 288.7.b.a.271.1 1
48.11 even 4 72.7.b.a.19.1 1
48.29 odd 4 72.7.b.a.19.1 1
48.35 even 4 288.7.b.a.271.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.7.d.a.3.1 1 16.11 odd 4
8.7.d.a.3.1 1 16.13 even 4
32.7.d.a.15.1 1 16.3 odd 4
32.7.d.a.15.1 1 16.5 even 4
72.7.b.a.19.1 1 48.11 even 4
72.7.b.a.19.1 1 48.29 odd 4
256.7.c.d.255.1 2 1.1 even 1 trivial
256.7.c.d.255.1 2 8.3 odd 2 CM
256.7.c.d.255.2 2 4.3 odd 2 inner
256.7.c.d.255.2 2 8.5 even 2 inner
288.7.b.a.271.1 1 48.5 odd 4
288.7.b.a.271.1 1 48.35 even 4