Properties

Label 3200.2.f.s.449.6
Level $3200$
Weight $2$
Character 3200.449
Analytic conductor $25.552$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.6
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.s.449.5

$q$-expansion

\(f(q)\) \(=\) \(q+0.317837 q^{3} -2.89898 q^{9} +O(q^{10})\) \(q+0.317837 q^{3} -2.89898 q^{9} +3.78194i q^{11} +1.89898i q^{17} -5.97469i q^{19} -1.87492 q^{27} +1.20204i q^{33} -6.79796 q^{41} -8.48528 q^{43} +7.00000 q^{49} +0.603566i q^{51} -1.89898i q^{57} -14.1421i q^{59} -16.3670 q^{67} -15.6969i q^{73} +8.10102 q^{81} -17.0027 q^{83} -4.10102 q^{89} -10.0000i q^{97} -10.9638i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{41} + 56 q^{49} + 104 q^{81} - 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0.317837 0.183503 0.0917517 0.995782i \(-0.470753\pi\)
0.0917517 + 0.995782i \(0.470753\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.89898 −0.966326
\(10\) 0 0
\(11\) 3.78194i 1.14030i 0.821541 + 0.570149i \(0.193114\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.89898i 0.460570i 0.973123 + 0.230285i \(0.0739659\pi\)
−0.973123 + 0.230285i \(0.926034\pi\)
\(18\) 0 0
\(19\) − 5.97469i − 1.37069i −0.728219 0.685344i \(-0.759652\pi\)
0.728219 0.685344i \(-0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.87492 −0.360828
\(28\) 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
\(33\) 1.20204i 0.209248i
\(34\) 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
\(41\) −6.79796 −1.06166 −0.530831 0.847477i \(-0.678120\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0.603566i 0.0845162i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.89898i − 0.251526i
\(58\) 0 0
\(59\) − 14.1421i − 1.84115i −0.390567 0.920575i \(-0.627721\pi\)
0.390567 0.920575i \(-0.372279\pi\)
\(60\) 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
\(67\) −16.3670 −1.99955 −0.999773 0.0212861i \(-0.993224\pi\)
−0.999773 + 0.0212861i \(0.993224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 15.6969i − 1.83719i −0.395203 0.918594i \(-0.629326\pi\)
0.395203 0.918594i \(-0.370674\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 8.10102 0.900113
\(82\) 0 0
\(83\) −17.0027 −1.86629 −0.933143 0.359506i \(-0.882945\pi\)
−0.933143 + 0.359506i \(0.882945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.10102 −0.434707 −0.217354 0.976093i \(-0.569742\pi\)
−0.217354 + 0.976093i \(0.569742\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) − 10.9638i − 1.10190i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.0956 −1.45935 −0.729676 0.683793i \(-0.760329\pi\)
−0.729676 + 0.683793i \(0.760329\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.7980i 1.76836i 0.467143 + 0.884182i \(0.345283\pi\)
−0.467143 + 0.884182i \(0.654717\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.30306 −0.300278
\(122\) 0 0
\(123\) −2.16064 −0.194819
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −2.69694 −0.237452
\(130\) 0 0
\(131\) 14.1421i 1.23560i 0.786334 + 0.617802i \(0.211977\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.5959i 1.93050i 0.261329 + 0.965250i \(0.415839\pi\)
−0.261329 + 0.965250i \(0.584161\pi\)
\(138\) 0 0
\(139\) − 23.2952i − 1.97587i −0.154859 0.987937i \(-0.549492\pi\)
0.154859 0.987937i \(-0.450508\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.22486 0.183503
\(148\) 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
\(153\) − 5.50510i − 0.445061i
\(154\) 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
\(163\) −14.4600 −1.13259 −0.566296 0.824202i \(-0.691624\pi\)
−0.566296 + 0.824202i \(0.691624\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 17.3205i 1.32453i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.49490i − 0.337857i
\(178\) 0 0
\(179\) − 25.4880i − 1.90506i −0.304446 0.952529i \(-0.598471\pi\)
0.304446 0.952529i \(-0.401529\pi\)
\(180\) 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
\(187\) −7.18182 −0.525187
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 25.6969i 1.84971i 0.380325 + 0.924853i \(0.375812\pi\)
−0.380325 + 0.924853i \(0.624188\pi\)
\(194\) 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
\(201\) −5.20204 −0.366924
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.5959 1.56299
\(210\) 0 0
\(211\) − 0.603566i − 0.0415512i −0.999784 0.0207756i \(-0.993386\pi\)
0.999784 0.0207756i \(-0.00661356\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 4.98907i − 0.337130i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.82843 −0.187729 −0.0938647 0.995585i \(-0.529922\pi\)
−0.0938647 + 0.995585i \(0.529922\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 30.0000i − 1.96537i −0.185296 0.982683i \(-0.559325\pi\)
0.185296 0.982683i \(-0.440675\pi\)
\(234\) 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
\(241\) 27.6969 1.78412 0.892058 0.451920i \(-0.149261\pi\)
0.892058 + 0.451920i \(0.149261\pi\)
\(242\) 0 0
\(243\) 8.19955 0.526002
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.40408 −0.342470
\(250\) 0 0
\(251\) 10.3602i 0.653930i 0.945036 + 0.326965i \(0.106026\pi\)
−0.945036 + 0.326965i \(0.893974\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 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
\(267\) −1.30346 −0.0797703
\(268\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −31.7805 −1.88915 −0.944577 0.328291i \(-0.893527\pi\)
−0.944577 + 0.328291i \(0.893527\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.3939 0.787875
\(290\) 0 0
\(291\) − 3.17837i − 0.186319i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 7.09082i − 0.411451i
\(298\) 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
\(307\) −33.6875 −1.92265 −0.961324 0.275421i \(-0.911183\pi\)
−0.961324 + 0.275421i \(0.911183\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 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
\(321\) −4.79796 −0.267796
\(322\) 0 0
\(323\) 11.3458 0.631298
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 35.2446i − 1.93722i −0.248590 0.968609i \(-0.579967\pi\)
0.248590 0.968609i \(-0.420033\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 36.3939i 1.98250i 0.131995 + 0.991250i \(0.457862\pi\)
−0.131995 + 0.991250i \(0.542138\pi\)
\(338\) 0 0
\(339\) 5.97469i 0.324501i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.1886 0.708002 0.354001 0.935245i \(-0.384821\pi\)
0.354001 + 0.935245i \(0.384821\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 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
\(361\) −16.6969 −0.878786
\(362\) 0 0
\(363\) −1.04984 −0.0551021
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 19.7071 1.02591
\(370\) 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
\(379\) 28.6663i 1.47249i 0.676715 + 0.736245i \(0.263403\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(380\) 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
\(387\) 24.5987 1.25042
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 4.49490i 0.226738i
\(394\) 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
\(401\) 31.2929 1.56269 0.781345 0.624099i \(-0.214534\pi\)
0.781345 + 0.624099i \(0.214534\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −40.3939 −1.99735 −0.998674 0.0514740i \(-0.983608\pi\)
−0.998674 + 0.0514740i \(0.983608\pi\)
\(410\) 0 0
\(411\) 7.18182i 0.354253i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 7.40408i − 0.362579i
\(418\) 0 0
\(419\) 2.79632i 0.136609i 0.997665 + 0.0683046i \(0.0217590\pi\)
−0.997665 + 0.0683046i \(0.978241\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 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
\(433\) 4.30306i 0.206792i 0.994640 + 0.103396i \(0.0329709\pi\)
−0.994640 + 0.103396i \(0.967029\pi\)
\(434\) 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
\(441\) −20.2929 −0.966326
\(442\) 0 0
\(443\) 34.9589 1.66095 0.830473 0.557059i \(-0.188070\pi\)
0.830473 + 0.557059i \(0.188070\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.8990 −1.22225 −0.611124 0.791535i \(-0.709282\pi\)
−0.611124 + 0.791535i \(0.709282\pi\)
\(450\) 0 0
\(451\) − 25.7095i − 1.21061i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3939i 0.766873i 0.923567 + 0.383437i \(0.125260\pi\)
−0.923567 + 0.383437i \(0.874740\pi\)
\(458\) 0 0
\(459\) − 3.56043i − 0.166186i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.1127 1.43972 0.719862 0.694117i \(-0.244205\pi\)
0.719862 + 0.694117i \(0.244205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 32.0908i − 1.47554i
\(474\) 0 0
\(475\) 0 0
\(476\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −4.59592 −0.207835
\(490\) 0 0
\(491\) 14.1421i 0.638226i 0.947717 + 0.319113i \(0.103385\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 42.4264i 1.89927i 0.313363 + 0.949633i \(0.398544\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 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
\(507\) −4.13188 −0.183503
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11.2020i 0.494582i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.1918 1.84846 0.924229 0.381839i \(-0.124709\pi\)
0.924229 + 0.381839i \(0.124709\pi\)
\(522\) 0 0
\(523\) 20.1810 0.882455 0.441228 0.897395i \(-0.354543\pi\)
0.441228 + 0.897395i \(0.354543\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 40.9978i 1.77915i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 8.10102i − 0.349585i
\(538\) 0 0
\(539\) 26.4736i 1.14030i
\(540\) 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
\(547\) 35.5945 1.52191 0.760956 0.648803i \(-0.224730\pi\)
0.760956 + 0.648803i \(0.224730\pi\)
\(548\) 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
\(561\) −2.28265 −0.0963736
\(562\) 0 0
\(563\) 36.7696 1.54965 0.774826 0.632175i \(-0.217837\pi\)
0.774826 + 0.632175i \(0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.40408 0.0588622 0.0294311 0.999567i \(-0.490630\pi\)
0.0294311 + 0.999567i \(0.490630\pi\)
\(570\) 0 0
\(571\) − 42.4264i − 1.77549i −0.460336 0.887745i \(-0.652271\pi\)
0.460336 0.887745i \(-0.347729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 46.3939i − 1.93140i −0.259656 0.965701i \(-0.583609\pi\)
0.259656 0.965701i \(-0.416391\pi\)
\(578\) 0 0
\(579\) 8.16744i 0.339427i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.2378 1.20677 0.603386 0.797449i \(-0.293818\pi\)
0.603386 + 0.797449i \(0.293818\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 30.1918i − 1.23983i −0.784669 0.619915i \(-0.787167\pi\)
0.784669 0.619915i \(-0.212833\pi\)
\(594\) 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
\(601\) −37.6969 −1.53769 −0.768845 0.639435i \(-0.779168\pi\)
−0.768845 + 0.639435i \(0.779168\pi\)
\(602\) 0 0
\(603\) 47.4476 1.93222
\(604\) 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
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 42.4264i 1.70526i 0.522514 + 0.852631i \(0.324994\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.18182 0.286814
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) − 0.191836i − 0.00762479i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) −8.48528 −0.334627 −0.167313 0.985904i \(-0.553509\pi\)
−0.167313 + 0.985904i \(0.553509\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 53.4847 2.09946
\(650\) 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
\(657\) 45.5051i 1.77532i
\(658\) 0 0
\(659\) 39.6301i 1.54377i 0.635763 + 0.771885i \(0.280686\pi\)
−0.635763 + 0.771885i \(0.719314\pi\)
\(660\) 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
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.898979 −0.0344490
\(682\) 0 0
\(683\) −20.8167 −0.796530 −0.398265 0.917270i \(-0.630387\pi\)
−0.398265 + 0.917270i \(0.630387\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 52.5651i − 1.99967i −0.0181572 0.999835i \(-0.505780\pi\)
0.0181572 0.999835i \(-0.494220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.9092i − 0.488970i
\(698\) 0 0
\(699\) − 9.53512i − 0.360651i
\(700\) 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
\(723\) 8.80312 0.327392
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −21.6969 −0.803590
\(730\) 0 0
\(731\) − 16.1134i − 0.595975i
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 61.8990i − 2.28008i
\(738\) 0 0
\(739\) 42.4264i 1.56068i 0.625355 + 0.780340i \(0.284954\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 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
\(747\) 49.2904 1.80344
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 3.29286i 0.119998i
\(754\) 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
\(761\) 36.7980 1.33392 0.666962 0.745091i \(-0.267594\pi\)
0.666962 + 0.745091i \(0.267594\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −33.0908 −1.19329 −0.596643 0.802507i \(-0.703499\pi\)
−0.596643 + 0.802507i \(0.703499\pi\)
\(770\) 0 0
\(771\) − 9.53512i − 0.343399i
\(772\) 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
\(779\) 40.6157i 1.45521i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −25.4558 −0.907403 −0.453701 0.891154i \(-0.649897\pi\)
−0.453701 + 0.891154i \(0.649897\pi\)
\(788\) 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
\(801\) 11.8888 0.420069
\(802\) 0 0
\(803\) 59.3649 2.09494
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) − 42.4264i − 1.48979i −0.667180 0.744896i \(-0.732499\pi\)
0.667180 0.744896i \(-0.267501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 50.6969i 1.77366i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.8659 1.28195 0.640976 0.767561i \(-0.278530\pi\)
0.640976 + 0.767561i \(0.278530\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 13.2929i 0.460570i
\(834\) 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
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) −5.72107 −0.197044
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.1010 −0.346666
\(850\) 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
\(857\) 7.40408i 0.252919i 0.991972 + 0.126459i \(0.0403613\pi\)
−0.991972 + 0.126459i \(0.959639\pi\)
\(858\) 0 0
\(859\) 45.9868i 1.56905i 0.620097 + 0.784525i \(0.287093\pi\)
−0.620097 + 0.784525i \(0.712907\pi\)
\(860\) 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
\(867\) 4.25707 0.144578
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 28.9898i 0.981156i
\(874\) 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
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −27.9664 −0.941145 −0.470573 0.882361i \(-0.655953\pi\)
−0.470573 + 0.882361i \(0.655953\pi\)
\(884\) 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
\(891\) 30.6376i 1.02640i
\(892\) 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
\(907\) −59.3970 −1.97224 −0.986122 0.166022i \(-0.946908\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) − 64.3031i − 2.12812i
\(914\) 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
\(921\) −10.7071 −0.352812
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) − 41.8228i − 1.37069i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 27.0908i − 0.885018i −0.896764 0.442509i \(-0.854088\pi\)
0.896764 0.442509i \(-0.145912\pi\)
\(938\) 0 0
\(939\) − 3.17837i − 0.103722i
\(940\) 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
\(947\) 53.7401 1.74632 0.873160 0.487435i \(-0.162067\pi\)
0.873160 + 0.487435i \(0.162067\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 60.1918i 1.94980i 0.222633 + 0.974902i \(0.428535\pi\)
−0.222633 + 0.974902i \(0.571465\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 43.7620 1.41021
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 3.60612 0.115845
\(970\) 0 0
\(971\) 62.3217i 2.00000i 0.000892350 1.00000i \(0.499716\pi\)
−0.000892350 1.00000i \(0.500284\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.8888i 1.62808i 0.580812 + 0.814038i \(0.302735\pi\)
−0.580812 + 0.814038i \(0.697265\pi\)
\(978\) 0 0
\(979\) − 15.5098i − 0.495696i
\(980\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) − 11.2020i − 0.355486i
\(994\) 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 3200.2.f.s.449.6 8
4.3 odd 2 inner 3200.2.f.s.449.3 8
5.2 odd 4 3200.2.d.q.1601.2 yes 4
5.3 odd 4 3200.2.d.n.1601.3 yes 4
5.4 even 2 inner 3200.2.f.s.449.4 8
8.3 odd 2 CM 3200.2.f.s.449.6 8
8.5 even 2 inner 3200.2.f.s.449.3 8
20.3 even 4 3200.2.d.n.1601.2 4
20.7 even 4 3200.2.d.q.1601.3 yes 4
20.19 odd 2 inner 3200.2.f.s.449.5 8
40.3 even 4 3200.2.d.n.1601.3 yes 4
40.13 odd 4 3200.2.d.n.1601.2 4
40.19 odd 2 inner 3200.2.f.s.449.4 8
40.27 even 4 3200.2.d.q.1601.2 yes 4
40.29 even 2 inner 3200.2.f.s.449.5 8
40.37 odd 4 3200.2.d.q.1601.3 yes 4
80.3 even 4 6400.2.a.cq.1.3 4
80.13 odd 4 6400.2.a.cq.1.2 4
80.27 even 4 6400.2.a.cr.1.3 4
80.37 odd 4 6400.2.a.cr.1.2 4
80.43 even 4 6400.2.a.cq.1.2 4
80.53 odd 4 6400.2.a.cq.1.3 4
80.67 even 4 6400.2.a.cr.1.2 4
80.77 odd 4 6400.2.a.cr.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.n.1601.2 4 20.3 even 4
3200.2.d.n.1601.2 4 40.13 odd 4
3200.2.d.n.1601.3 yes 4 5.3 odd 4
3200.2.d.n.1601.3 yes 4 40.3 even 4
3200.2.d.q.1601.2 yes 4 5.2 odd 4
3200.2.d.q.1601.2 yes 4 40.27 even 4
3200.2.d.q.1601.3 yes 4 20.7 even 4
3200.2.d.q.1601.3 yes 4 40.37 odd 4
3200.2.f.s.449.3 8 4.3 odd 2 inner
3200.2.f.s.449.3 8 8.5 even 2 inner
3200.2.f.s.449.4 8 5.4 even 2 inner
3200.2.f.s.449.4 8 40.19 odd 2 inner
3200.2.f.s.449.5 8 20.19 odd 2 inner
3200.2.f.s.449.5 8 40.29 even 2 inner
3200.2.f.s.449.6 8 1.1 even 1 trivial
3200.2.f.s.449.6 8 8.3 odd 2 CM
6400.2.a.cq.1.2 4 80.13 odd 4
6400.2.a.cq.1.2 4 80.43 even 4
6400.2.a.cq.1.3 4 80.3 even 4
6400.2.a.cq.1.3 4 80.53 odd 4
6400.2.a.cr.1.2 4 80.37 odd 4
6400.2.a.cr.1.2 4 80.67 even 4
6400.2.a.cr.1.3 4 80.27 even 4
6400.2.a.cr.1.3 4 80.77 odd 4