Properties

Label 3200.2.d.n.1601.2
Level $3200$
Weight $2$
Character 3200.1601
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1601.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3200.1601
Dual form 3200.2.d.n.1601.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.317837i q^{3} +2.89898 q^{9} +O(q^{10})\) \(q-0.317837i q^{3} +2.89898 q^{9} -3.78194i q^{11} +1.89898 q^{17} -5.97469i q^{19} -1.87492i q^{27} -1.20204 q^{33} -6.79796 q^{41} +8.48528i q^{43} -7.00000 q^{49} -0.603566i q^{51} -1.89898 q^{57} -14.1421i q^{59} -16.3670i q^{67} +15.6969 q^{73} +8.10102 q^{81} +17.0027i q^{83} +4.10102 q^{89} -10.0000 q^{97} -10.9638i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} - 12 q^{17} - 44 q^{33} + 12 q^{41} - 28 q^{49} + 12 q^{57} + 4 q^{73} + 52 q^{81} + 36 q^{89} - 40 q^{97}+O(q^{100}) \) 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
\(3\) − 0.317837i − 0.183503i −0.995782 0.0917517i \(-0.970753\pi\)
0.995782 0.0917517i \(-0.0292466\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.89898 0.966326
\(10\) 0 0
\(11\) − 3.78194i − 1.14030i −0.821541 0.570149i \(-0.806886\pi\)
0.821541 0.570149i \(-0.193114\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.89898 0.460570 0.230285 0.973123i \(-0.426034\pi\)
0.230285 + 0.973123i \(0.426034\pi\)
\(18\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.87492i − 0.360828i
\(28\) 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
\(33\) −1.20204 −0.209248
\(34\) 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
\(41\) −6.79796 −1.06166 −0.530831 0.847477i \(-0.678120\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) − 0.603566i − 0.0845162i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.89898 −0.251526
\(58\) 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
\(67\) − 16.3670i − 1.99955i −0.0212861 0.999773i \(-0.506776\pi\)
0.0212861 0.999773i \(-0.493224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 15.6969 1.83719 0.918594 0.395203i \(-0.129326\pi\)
0.918594 + 0.395203i \(0.129326\pi\)
\(74\) 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
\(83\) 17.0027i 1.86629i 0.359506 + 0.933143i \(0.382945\pi\)
−0.359506 + 0.933143i \(0.617055\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.10102 0.434707 0.217354 0.976093i \(-0.430258\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.0956i − 1.45935i −0.683793 0.729676i \(-0.739671\pi\)
0.683793 0.729676i \(-0.260329\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.7980 −1.76836 −0.884182 0.467143i \(-0.845283\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 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
\(123\) 2.16064i 0.194819i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 2.69694 0.237452
\(130\) 0 0
\(131\) − 14.1421i − 1.23560i −0.786334 0.617802i \(-0.788023\pi\)
0.786334 0.617802i \(-0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.5959 1.93050 0.965250 0.261329i \(-0.0841608\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 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
\(147\) 2.22486i 0.183503i
\(148\) 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
\(153\) 5.50510 0.445061
\(154\) 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
\(163\) 14.4600i 1.13259i 0.824202 + 0.566296i \(0.191624\pi\)
−0.824202 + 0.566296i \(0.808376\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 17.3205i − 1.32453i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.49490 −0.337857
\(178\) 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
\(187\) − 7.18182i − 0.525187i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −25.6969 −1.84971 −0.924853 0.380325i \(-0.875812\pi\)
−0.924853 + 0.380325i \(0.875812\pi\)
\(194\) 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
\(201\) −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 \(0.00661356\pi\)
−0.999784 + 0.0207756i \(0.993386\pi\)
\(212\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.82843i − 0.187729i −0.995585 0.0938647i \(-0.970078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 30.0000 1.96537 0.982683 0.185296i \(-0.0593245\pi\)
0.982683 + 0.185296i \(0.0593245\pi\)
\(234\) 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
\(243\) − 8.19955i − 0.526002i
\(244\) 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 \(-0.893974\pi\)
0.945036 0.326965i \(-0.106026\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 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
\(267\) − 1.30346i − 0.0797703i
\(268\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 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
\(283\) 31.7805i 1.88915i 0.328291 + 0.944577i \(0.393527\pi\)
−0.328291 + 0.944577i \(0.606473\pi\)
\(284\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.09082 −0.411451
\(298\) 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
\(307\) − 33.6875i − 1.92265i −0.275421 0.961324i \(-0.588817\pi\)
0.275421 0.961324i \(-0.411183\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 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
\(321\) −4.79796 −0.267796
\(322\) 0 0
\(323\) − 11.3458i − 0.631298i
\(324\) 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 \(0.420033\pi\)
−0.248590 + 0.968609i \(0.579967\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 36.3939 1.98250 0.991250 0.131995i \(-0.0421382\pi\)
0.991250 + 0.131995i \(0.0421382\pi\)
\(338\) 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
\(347\) 13.1886i 0.708002i 0.935245 + 0.354001i \(0.115179\pi\)
−0.935245 + 0.354001i \(0.884821\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 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
\(363\) 1.04984i 0.0551021i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −19.7071 −1.02591
\(370\) 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
\(379\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.5987i 1.25042i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.49490 −0.226738
\(394\) 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
\(401\) 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 \(-0.0163919\pi\)
0.998674 + 0.0514740i \(0.0163919\pi\)
\(410\) 0 0
\(411\) − 7.18182i − 0.354253i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.40408 −0.362579
\(418\) 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
\(433\) −4.30306 −0.206792 −0.103396 0.994640i \(-0.532971\pi\)
−0.103396 + 0.994640i \(0.532971\pi\)
\(434\) 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
\(443\) − 34.9589i − 1.66095i −0.557059 0.830473i \(-0.688070\pi\)
0.557059 0.830473i \(-0.311930\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 25.8990 1.22225 0.611124 0.791535i \(-0.290718\pi\)
0.611124 + 0.791535i \(0.290718\pi\)
\(450\) 0 0
\(451\) 25.7095i 1.21061i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3939 0.766873 0.383437 0.923567i \(-0.374740\pi\)
0.383437 + 0.923567i \(0.374740\pi\)
\(458\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.1127i 1.43972i 0.694117 + 0.719862i \(0.255795\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 32.0908 1.47554
\(474\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 4.59592 0.207835
\(490\) 0 0
\(491\) − 14.1421i − 0.638226i −0.947717 0.319113i \(-0.896615\pi\)
0.947717 0.319113i \(-0.103385\pi\)
\(492\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 4.13188i − 0.183503i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −11.2020 −0.494582
\(514\) 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
\(523\) − 20.1810i − 0.882455i −0.897395 0.441228i \(-0.854543\pi\)
0.897395 0.441228i \(-0.145457\pi\)
\(524\) 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
\(537\) −8.10102 −0.349585
\(538\) 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
\(547\) 35.5945i 1.52191i 0.648803 + 0.760956i \(0.275270\pi\)
−0.648803 + 0.760956i \(0.724730\pi\)
\(548\) 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
\(561\) −2.28265 −0.0963736
\(562\) 0 0
\(563\) − 36.7696i − 1.54965i −0.632175 0.774826i \(-0.717837\pi\)
0.632175 0.774826i \(-0.282163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.40408 −0.0588622 −0.0294311 0.999567i \(-0.509370\pi\)
−0.0294311 + 0.999567i \(0.509370\pi\)
\(570\) 0 0
\(571\) 42.4264i 1.77549i 0.460336 + 0.887745i \(0.347729\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −46.3939 −1.93140 −0.965701 0.259656i \(-0.916391\pi\)
−0.965701 + 0.259656i \(0.916391\pi\)
\(578\) 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
\(587\) 29.2378i 1.20677i 0.797449 + 0.603386i \(0.206182\pi\)
−0.797449 + 0.603386i \(0.793818\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.1918 1.23983 0.619915 0.784669i \(-0.287167\pi\)
0.619915 + 0.784669i \(0.287167\pi\)
\(594\) 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
\(603\) − 47.4476i − 1.93222i
\(604\) 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
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 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
\(627\) 7.18182i 0.286814i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0.191836 0.00762479
\(634\) 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
\(643\) 8.48528i 0.334627i 0.985904 + 0.167313i \(0.0535092\pi\)
−0.985904 + 0.167313i \(0.946491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −53.4847 −2.09946
\(650\) 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
\(657\) 45.5051 1.77532
\(658\) 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
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.898979 −0.0344490
\(682\) 0 0
\(683\) 20.8167i 0.796530i 0.917270 + 0.398265i \(0.130387\pi\)
−0.917270 + 0.398265i \(0.869613\pi\)
\(684\) 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 \(0.494220\pi\)
−0.0181572 + 0.999835i \(0.505780\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.9092 −0.488970
\(698\) 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
\(723\) − 8.80312i − 0.327392i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 21.6969 0.803590
\(730\) 0 0
\(731\) 16.1134i 0.595975i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −61.8990 −2.28008
\(738\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 49.2904i 1.80344i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −3.29286 −0.119998
\(754\) 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
\(761\) 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 \(-0.296501\pi\)
0.596643 + 0.802507i \(0.296501\pi\)
\(770\) 0 0
\(771\) 9.53512i 0.343399i
\(772\) 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
\(779\) 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
\(787\) − 25.4558i − 0.907403i −0.891154 0.453701i \(-0.850103\pi\)
0.891154 0.453701i \(-0.149897\pi\)
\(788\) 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
\(801\) 11.8888 0.420069
\(802\) 0 0
\(803\) − 59.3649i − 2.09494i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 42.4264i 1.48979i 0.667180 + 0.744896i \(0.267501\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 50.6969 1.77366
\(818\) 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
\(825\) 0 0
\(826\) 0 0
\(827\) 36.8659i 1.28195i 0.767561 + 0.640976i \(0.221470\pi\)
−0.767561 + 0.640976i \(0.778530\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −13.2929 −0.460570
\(834\) 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
\(843\) 5.72107i 0.197044i
\(844\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.40408 0.252919 0.126459 0.991972i \(-0.459639\pi\)
0.126459 + 0.991972i \(0.459639\pi\)
\(858\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.25707i 0.144578i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −28.9898 −0.981156
\(874\) 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
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 27.9664i 0.941145i 0.882361 + 0.470573i \(0.155953\pi\)
−0.882361 + 0.470573i \(0.844047\pi\)
\(884\) 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
\(891\) − 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
\(907\) − 59.3970i − 1.97224i −0.166022 0.986122i \(-0.553092\pi\)
0.166022 0.986122i \(-0.446908\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 64.3031 2.12812
\(914\) 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 \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) 41.8228i 1.37069i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.0908 −0.885018 −0.442509 0.896764i \(-0.645912\pi\)
−0.442509 + 0.896764i \(0.645912\pi\)
\(938\) 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
\(947\) 53.7401i 1.74632i 0.487435 + 0.873160i \(0.337933\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −60.1918 −1.94980 −0.974902 0.222633i \(-0.928535\pi\)
−0.974902 + 0.222633i \(0.928535\pi\)
\(954\) 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
\(963\) − 43.7620i − 1.41021i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) −3.60612 −0.115845
\(970\) 0 0
\(971\) − 62.3217i − 2.00000i −0.000892350 1.00000i \(-0.500284\pi\)
0.000892350 1.00000i \(-0.499716\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.8888 1.62808 0.814038 0.580812i \(-0.197265\pi\)
0.814038 + 0.580812i \(0.197265\pi\)
\(978\) 0 0
\(979\) − 15.5098i − 0.495696i
\(980\) 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
\(993\) 11.2020 0.355486
\(994\) 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 3200.2.d.n.1601.2 4
4.3 odd 2 inner 3200.2.d.n.1601.3 yes 4
5.2 odd 4 3200.2.f.s.449.3 8
5.3 odd 4 3200.2.f.s.449.5 8
5.4 even 2 3200.2.d.q.1601.3 yes 4
8.3 odd 2 CM 3200.2.d.n.1601.2 4
8.5 even 2 inner 3200.2.d.n.1601.3 yes 4
16.3 odd 4 6400.2.a.cq.1.2 4
16.5 even 4 6400.2.a.cq.1.2 4
16.11 odd 4 6400.2.a.cq.1.3 4
16.13 even 4 6400.2.a.cq.1.3 4
20.3 even 4 3200.2.f.s.449.4 8
20.7 even 4 3200.2.f.s.449.6 8
20.19 odd 2 3200.2.d.q.1601.2 yes 4
40.3 even 4 3200.2.f.s.449.5 8
40.13 odd 4 3200.2.f.s.449.4 8
40.19 odd 2 3200.2.d.q.1601.3 yes 4
40.27 even 4 3200.2.f.s.449.3 8
40.29 even 2 3200.2.d.q.1601.2 yes 4
40.37 odd 4 3200.2.f.s.449.6 8
80.19 odd 4 6400.2.a.cr.1.3 4
80.29 even 4 6400.2.a.cr.1.2 4
80.59 odd 4 6400.2.a.cr.1.2 4
80.69 even 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 1.1 even 1 trivial
3200.2.d.n.1601.2 4 8.3 odd 2 CM
3200.2.d.n.1601.3 yes 4 4.3 odd 2 inner
3200.2.d.n.1601.3 yes 4 8.5 even 2 inner
3200.2.d.q.1601.2 yes 4 20.19 odd 2
3200.2.d.q.1601.2 yes 4 40.29 even 2
3200.2.d.q.1601.3 yes 4 5.4 even 2
3200.2.d.q.1601.3 yes 4 40.19 odd 2
3200.2.f.s.449.3 8 5.2 odd 4
3200.2.f.s.449.3 8 40.27 even 4
3200.2.f.s.449.4 8 20.3 even 4
3200.2.f.s.449.4 8 40.13 odd 4
3200.2.f.s.449.5 8 5.3 odd 4
3200.2.f.s.449.5 8 40.3 even 4
3200.2.f.s.449.6 8 20.7 even 4
3200.2.f.s.449.6 8 40.37 odd 4
6400.2.a.cq.1.2 4 16.3 odd 4
6400.2.a.cq.1.2 4 16.5 even 4
6400.2.a.cq.1.3 4 16.11 odd 4
6400.2.a.cq.1.3 4 16.13 even 4
6400.2.a.cr.1.2 4 80.29 even 4
6400.2.a.cr.1.2 4 80.59 odd 4
6400.2.a.cr.1.3 4 80.19 odd 4
6400.2.a.cr.1.3 4 80.69 even 4