Properties

Label 6400.2.a.cr.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.14626 q^{3} +6.89898 q^{9} +O(q^{10})\) \(q-3.14626 q^{3} +6.89898 q^{9} -6.61037 q^{11} +7.89898 q^{17} -2.51059 q^{19} -12.2672 q^{27} +20.7980 q^{33} -12.7980 q^{41} +8.48528 q^{43} -7.00000 q^{49} -24.8523 q^{51} +7.89898 q^{57} +14.1421 q^{59} -7.88171 q^{67} -13.6969 q^{73} +17.8990 q^{81} +14.1742 q^{83} -13.8990 q^{89} +10.0000 q^{97} -45.6048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9} + O(q^{10}) \) \( 4 q + 8 q^{9} + 12 q^{17} + 44 q^{33} - 12 q^{41} - 28 q^{49} + 12 q^{57} + 4 q^{73} + 52 q^{81} - 36 q^{89} + 40 q^{97} + O(q^{100}) \)

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
\(3\) −3.14626 −1.81650 −0.908248 0.418432i \(-0.862580\pi\)
−0.908248 + 0.418432i \(0.862580\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 6.89898 2.29966
\(10\) 0 0
\(11\) −6.61037 −1.99310 −0.996550 0.0829925i \(-0.973552\pi\)
−0.996550 + 0.0829925i \(0.973552\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.89898 1.91578 0.957892 0.287129i \(-0.0927008\pi\)
0.957892 + 0.287129i \(0.0927008\pi\)
\(18\) 0 0
\(19\) −2.51059 −0.575969 −0.287984 0.957635i \(-0.592985\pi\)
−0.287984 + 0.957635i \(0.592985\pi\)
\(20\) 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
\(27\) −12.2672 −2.36083
\(28\) 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
\(33\) 20.7980 3.62046
\(34\) 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
\(41\) −12.7980 −1.99871 −0.999353 0.0359748i \(-0.988546\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 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
\(51\) −24.8523 −3.48001
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.89898 1.04625
\(58\) 0 0
\(59\) 14.1421 1.84115 0.920575 0.390567i \(-0.127721\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 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
\(67\) −7.88171 −0.962905 −0.481452 0.876472i \(-0.659891\pi\)
−0.481452 + 0.876472i \(0.659891\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −13.6969 −1.60311 −0.801553 0.597924i \(-0.795992\pi\)
−0.801553 + 0.597924i \(0.795992\pi\)
\(74\) 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
\(81\) 17.8990 1.98878
\(82\) 0 0
\(83\) 14.1742 1.55583 0.777913 0.628372i \(-0.216279\pi\)
0.777913 + 0.628372i \(0.216279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.8990 −1.47329 −0.736644 0.676280i \(-0.763591\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 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 \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −45.6048 −4.58345
\(100\) 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
\(107\) −4.70334 −0.454689 −0.227345 0.973814i \(-0.573004\pi\)
−0.227345 + 0.973814i \(0.573004\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.797959 −0.0750657 −0.0375328 0.999295i \(-0.511950\pi\)
−0.0375328 + 0.999295i \(0.511950\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 32.6969 2.97245
\(122\) 0 0
\(123\) 40.2658 3.63064
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −26.6969 −2.35053
\(130\) 0 0
\(131\) −14.1421 −1.23560 −0.617802 0.786334i \(-0.711977\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.5959 −1.41788 −0.708942 0.705266i \(-0.750827\pi\)
−0.708942 + 0.705266i \(0.750827\pi\)
\(138\) 0 0
\(139\) −14.8099 −1.25616 −0.628080 0.778148i \(-0.716159\pi\)
−0.628080 + 0.778148i \(0.716159\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 22.0239 1.81650
\(148\) 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
\(153\) 54.4949 4.40565
\(154\) 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
\(163\) −10.9959 −0.861263 −0.430632 0.902528i \(-0.641709\pi\)
−0.430632 + 0.902528i \(0.641709\pi\)
\(164\) 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
\(171\) −17.3205 −1.32453
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −44.4949 −3.34444
\(178\) 0 0
\(179\) 5.68896 0.425213 0.212607 0.977138i \(-0.431805\pi\)
0.212607 + 0.977138i \(0.431805\pi\)
\(180\) 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
\(187\) −52.2151 −3.81835
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −3.69694 −0.266111 −0.133056 0.991109i \(-0.542479\pi\)
−0.133056 + 0.991109i \(0.542479\pi\)
\(194\) 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
\(201\) 24.7980 1.74911
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.5959 1.14796
\(210\) 0 0
\(211\) 24.8523 1.71090 0.855451 0.517884i \(-0.173280\pi\)
0.855451 + 0.517884i \(0.173280\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 43.0942 2.91204
\(220\) 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
\(227\) 2.82843 0.187729 0.0938647 0.995585i \(-0.470078\pi\)
0.0938647 + 0.995585i \(0.470078\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 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
\(241\) −1.69694 −0.109309 −0.0546547 0.998505i \(-0.517406\pi\)
−0.0546547 + 0.998505i \(0.517406\pi\)
\(242\) 0 0
\(243\) −19.5133 −1.25178
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −44.5959 −2.82615
\(250\) 0 0
\(251\) 20.7525 1.30989 0.654943 0.755678i \(-0.272693\pi\)
0.654943 + 0.755678i \(0.272693\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 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
\(267\) 43.7299 2.67622
\(268\) 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
\(275\) 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 \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −6.32464 −0.375961 −0.187980 0.982173i \(-0.560194\pi\)
−0.187980 + 0.982173i \(0.560194\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 45.3939 2.67023
\(290\) 0 0
\(291\) −31.4626 −1.84437
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 81.0908 4.70537
\(298\) 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
\(307\) −25.2022 −1.43837 −0.719183 0.694820i \(-0.755484\pi\)
−0.719183 + 0.694820i \(0.755484\pi\)
\(308\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 14.7980 0.825942
\(322\) 0 0
\(323\) −19.8311 −1.10343
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.78874 0.538038 0.269019 0.963135i \(-0.413301\pi\)
0.269019 + 0.963135i \(0.413301\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.3939 1.21987 0.609936 0.792451i \(-0.291195\pi\)
0.609936 + 0.792451i \(0.291195\pi\)
\(338\) 0 0
\(339\) 2.51059 0.136357
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.5809 1.26589 0.632945 0.774197i \(-0.281846\pi\)
0.632945 + 0.774197i \(0.281846\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 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
\(361\) −12.6969 −0.668260
\(362\) 0 0
\(363\) −102.873 −5.39944
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −88.2929 −4.59634
\(370\) 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
\(379\) 37.1516 1.90835 0.954175 0.299249i \(-0.0967363\pi\)
0.954175 + 0.299249i \(0.0967363\pi\)
\(380\) 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
\(387\) 58.5398 2.97574
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 44.4949 2.24447
\(394\) 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
\(401\) −37.2929 −1.86232 −0.931158 0.364615i \(-0.881200\pi\)
−0.931158 + 0.364615i \(0.881200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 18.3939 0.909519 0.454759 0.890614i \(-0.349725\pi\)
0.454759 + 0.890614i \(0.349725\pi\)
\(410\) 0 0
\(411\) 52.2151 2.57558
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 46.5959 2.28181
\(418\) 0 0
\(419\) 33.9732 1.65970 0.829851 0.557986i \(-0.188426\pi\)
0.829851 + 0.557986i \(0.188426\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 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
\(433\) 33.6969 1.61937 0.809686 0.586864i \(-0.199638\pi\)
0.809686 + 0.586864i \(0.199638\pi\)
\(434\) 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
\(441\) −48.2929 −2.29966
\(442\) 0 0
\(443\) 37.7873 1.79533 0.897664 0.440681i \(-0.145263\pi\)
0.897664 + 0.440681i \(0.145263\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.1010 0.759854 0.379927 0.925016i \(-0.375949\pi\)
0.379927 + 0.925016i \(0.375949\pi\)
\(450\) 0 0
\(451\) 84.5992 3.98362
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −42.3939 −1.98310 −0.991551 0.129718i \(-0.958593\pi\)
−0.991551 + 0.129718i \(0.958593\pi\)
\(458\) 0 0
\(459\) −96.8985 −4.52284
\(460\) 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
\(467\) −31.1127 −1.43972 −0.719862 0.694117i \(-0.755795\pi\)
−0.719862 + 0.694117i \(0.755795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −56.0908 −2.57906
\(474\) 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
\(489\) 34.5959 1.56448
\(490\) 0 0
\(491\) 14.1421 0.638226 0.319113 0.947717i \(-0.396615\pi\)
0.319113 + 0.947717i \(0.396615\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 42.4264 1.89927 0.949633 0.313363i \(-0.101456\pi\)
0.949633 + 0.313363i \(0.101456\pi\)
\(500\) 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
\(507\) 40.9014 1.81650
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 30.7980 1.35976
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 36.1918 1.58559 0.792797 0.609486i \(-0.208624\pi\)
0.792797 + 0.609486i \(0.208624\pi\)
\(522\) 0 0
\(523\) 45.6369 1.99556 0.997781 0.0665832i \(-0.0212098\pi\)
0.997781 + 0.0665832i \(0.0212098\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 97.5663 4.23402
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −17.8990 −0.772398
\(538\) 0 0
\(539\) 46.2726 1.99310
\(540\) 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
\(547\) 44.0798 1.88472 0.942358 0.334606i \(-0.108603\pi\)
0.942358 + 0.334606i \(0.108603\pi\)
\(548\) 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
\(561\) 164.283 6.93602
\(562\) 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
\(569\) 40.5959 1.70187 0.850935 0.525271i \(-0.176036\pi\)
0.850935 + 0.525271i \(0.176036\pi\)
\(570\) 0 0
\(571\) −42.4264 −1.77549 −0.887745 0.460336i \(-0.847729\pi\)
−0.887745 + 0.460336i \(0.847729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −12.3939 −0.515964 −0.257982 0.966150i \(-0.583058\pi\)
−0.257982 + 0.966150i \(0.583058\pi\)
\(578\) 0 0
\(579\) 11.6315 0.483391
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.8455 0.777836 0.388918 0.921272i \(-0.372849\pi\)
0.388918 + 0.921272i \(0.372849\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 48.1918 1.97900 0.989501 0.144528i \(-0.0461663\pi\)
0.989501 + 0.144528i \(0.0461663\pi\)
\(594\) 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
\(601\) 8.30306 0.338689 0.169344 0.985557i \(-0.445835\pi\)
0.169344 + 0.985557i \(0.445835\pi\)
\(602\) 0 0
\(603\) −54.3758 −2.21435
\(604\) 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
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −42.4264 −1.70526 −0.852631 0.522514i \(-0.824994\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −52.2151 −2.08527
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −78.1918 −3.10785
\(634\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −93.4847 −3.66960
\(650\) 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
\(657\) −94.4949 −3.68660
\(658\) 0 0
\(659\) 8.45317 0.329289 0.164644 0.986353i \(-0.447352\pi\)
0.164644 + 0.986353i \(0.447352\pi\)
\(660\) 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
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 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
\(681\) −8.89898 −0.341010
\(682\) 0 0
\(683\) −51.9294 −1.98702 −0.993512 0.113728i \(-0.963721\pi\)
−0.993512 + 0.113728i \(0.963721\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −27.1092 −1.03128 −0.515642 0.856804i \(-0.672447\pi\)
−0.515642 + 0.856804i \(0.672447\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −101.091 −3.82909
\(698\) 0 0
\(699\) −94.3879 −3.57008
\(700\) 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
\(723\) 5.33902 0.198560
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 7.69694 0.285072
\(730\) 0 0
\(731\) 67.0251 2.47901
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.1010 1.91917
\(738\) 0 0
\(739\) 42.4264 1.56068 0.780340 0.625355i \(-0.215046\pi\)
0.780340 + 0.625355i \(0.215046\pi\)
\(740\) 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
\(747\) 97.7878 3.57787
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −65.2929 −2.37940
\(754\) 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
\(761\) −17.2020 −0.623574 −0.311787 0.950152i \(-0.600927\pi\)
−0.311787 + 0.950152i \(0.600927\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −55.0908 −1.98663 −0.993313 0.115454i \(-0.963168\pi\)
−0.993313 + 0.115454i \(0.963168\pi\)
\(770\) 0 0
\(771\) −94.3879 −3.39930
\(772\) 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
\(779\) 32.1304 1.15119
\(780\) 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 \(-0.350103\pi\)
0.453701 + 0.891154i \(0.350103\pi\)
\(788\) 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
\(801\) −95.8888 −3.38806
\(802\) 0 0
\(803\) 90.5418 3.19515
\(804\) 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
\(811\) −42.4264 −1.48979 −0.744896 0.667180i \(-0.767501\pi\)
−0.744896 + 0.667180i \(0.767501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −21.3031 −0.745300
\(818\) 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
\(825\) 0 0
\(826\) 0 0
\(827\) −56.6649 −1.97043 −0.985215 0.171321i \(-0.945196\pi\)
−0.985215 + 0.171321i \(0.945196\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −55.2929 −1.91578
\(834\) 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
\(843\) −56.6328 −1.95054
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 19.8990 0.682931
\(850\) 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
\(857\) 46.5959 1.59169 0.795843 0.605503i \(-0.207028\pi\)
0.795843 + 0.605503i \(0.207028\pi\)
\(858\) 0 0
\(859\) 54.4721 1.85856 0.929282 0.369370i \(-0.120427\pi\)
0.929282 + 0.369370i \(0.120427\pi\)
\(860\) 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
\(867\) −142.821 −4.85046
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 68.9898 2.33495
\(874\) 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
\(883\) −31.4305 −1.05772 −0.528861 0.848709i \(-0.677381\pi\)
−0.528861 + 0.848709i \(0.677381\pi\)
\(884\) 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
\(891\) −118.319 −3.96383
\(892\) 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
\(913\) −93.6969 −3.10092
\(914\) 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
\(921\) 79.2929 2.61279
\(922\) 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
\(931\) 17.5741 0.575969
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 61.0908 1.99575 0.997875 0.0651578i \(-0.0207551\pi\)
0.997875 + 0.0651578i \(0.0207551\pi\)
\(938\) 0 0
\(939\) −31.4626 −1.02674
\(940\) 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
\(947\) −53.7401 −1.74632 −0.873160 0.487435i \(-0.837933\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.1918 0.589291 0.294646 0.955607i \(-0.404798\pi\)
0.294646 + 0.955607i \(0.404798\pi\)
\(954\) 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
\(963\) −32.4483 −1.04563
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 62.3939 2.00438
\(970\) 0 0
\(971\) −31.2090 −1.00155 −0.500773 0.865579i \(-0.666951\pi\)
−0.500773 + 0.865579i \(0.666951\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.8888 1.82003 0.910017 0.414572i \(-0.136069\pi\)
0.910017 + 0.414572i \(0.136069\pi\)
\(978\) 0 0
\(979\) 91.8773 2.93641
\(980\) 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
\(993\) −30.7980 −0.977344
\(994\) 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 6400.2.a.cr.1.1 4
4.3 odd 2 inner 6400.2.a.cr.1.4 4
5.4 even 2 6400.2.a.cq.1.4 4
8.3 odd 2 CM 6400.2.a.cr.1.1 4
8.5 even 2 inner 6400.2.a.cr.1.4 4
16.3 odd 4 3200.2.d.q.1601.4 yes 4
16.5 even 4 3200.2.d.q.1601.4 yes 4
16.11 odd 4 3200.2.d.q.1601.1 yes 4
16.13 even 4 3200.2.d.q.1601.1 yes 4
20.19 odd 2 6400.2.a.cq.1.1 4
40.19 odd 2 6400.2.a.cq.1.4 4
40.29 even 2 6400.2.a.cq.1.1 4
80.3 even 4 3200.2.f.s.449.1 8
80.13 odd 4 3200.2.f.s.449.8 8
80.19 odd 4 3200.2.d.n.1601.1 4
80.27 even 4 3200.2.f.s.449.2 8
80.29 even 4 3200.2.d.n.1601.4 yes 4
80.37 odd 4 3200.2.f.s.449.7 8
80.43 even 4 3200.2.f.s.449.8 8
80.53 odd 4 3200.2.f.s.449.1 8
80.59 odd 4 3200.2.d.n.1601.4 yes 4
80.67 even 4 3200.2.f.s.449.7 8
80.69 even 4 3200.2.d.n.1601.1 4
80.77 odd 4 3200.2.f.s.449.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.n.1601.1 4 80.19 odd 4
3200.2.d.n.1601.1 4 80.69 even 4
3200.2.d.n.1601.4 yes 4 80.29 even 4
3200.2.d.n.1601.4 yes 4 80.59 odd 4
3200.2.d.q.1601.1 yes 4 16.11 odd 4
3200.2.d.q.1601.1 yes 4 16.13 even 4
3200.2.d.q.1601.4 yes 4 16.3 odd 4
3200.2.d.q.1601.4 yes 4 16.5 even 4
3200.2.f.s.449.1 8 80.3 even 4
3200.2.f.s.449.1 8 80.53 odd 4
3200.2.f.s.449.2 8 80.27 even 4
3200.2.f.s.449.2 8 80.77 odd 4
3200.2.f.s.449.7 8 80.37 odd 4
3200.2.f.s.449.7 8 80.67 even 4
3200.2.f.s.449.8 8 80.13 odd 4
3200.2.f.s.449.8 8 80.43 even 4
6400.2.a.cq.1.1 4 20.19 odd 2
6400.2.a.cq.1.1 4 40.29 even 2
6400.2.a.cq.1.4 4 5.4 even 2
6400.2.a.cq.1.4 4 40.19 odd 2
6400.2.a.cr.1.1 4 1.1 even 1 trivial
6400.2.a.cr.1.1 4 8.3 odd 2 CM
6400.2.a.cr.1.4 4 4.3 odd 2 inner
6400.2.a.cr.1.4 4 8.5 even 2 inner