Properties

Label 4900.2.a.bj.1.2
Level $4900$
Weight $2$
Character 4900.1
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{21})\)
Defining polynomial: \(x^{4} - 13 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 980)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
Root \(-1.17325\) of defining polynomial
Character \(\chi\) \(=\) 4900.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.17325 q^{3} -1.62348 q^{9} +O(q^{10})\) \(q-1.17325 q^{3} -1.62348 q^{9} +6.62348 q^{11} -5.64539 q^{13} -7.99190 q^{17} +5.42451 q^{27} -0.623475 q^{29} -7.77102 q^{33} +6.62348 q^{39} +12.4640 q^{47} +9.37652 q^{51} -12.0000 q^{71} +13.4164 q^{73} +15.8704 q^{79} -1.49390 q^{81} -8.94427 q^{83} +0.731495 q^{87} -12.6849 q^{97} -10.7530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{9} + O(q^{10}) \) \( 4 q + 14 q^{9} + 6 q^{11} + 18 q^{29} + 6 q^{39} + 58 q^{51} - 48 q^{71} + 2 q^{79} + 76 q^{81} - 84 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.17325 −0.677378 −0.338689 0.940898i \(-0.609984\pi\)
−0.338689 + 0.940898i \(0.609984\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.62348 −0.541158
\(10\) 0 0
\(11\) 6.62348 1.99705 0.998526 0.0542666i \(-0.0172821\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) −5.64539 −1.56575 −0.782875 0.622179i \(-0.786247\pi\)
−0.782875 + 0.622179i \(0.786247\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.99190 −1.93832 −0.969160 0.246433i \(-0.920742\pi\)
−0.969160 + 0.246433i \(0.920742\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 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
\(27\) 5.42451 1.04395
\(28\) 0 0
\(29\) −0.623475 −0.115776 −0.0578882 0.998323i \(-0.518437\pi\)
−0.0578882 + 0.998323i \(0.518437\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −7.77102 −1.35276
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 6.62348 1.06060
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.4640 1.81807 0.909033 0.416724i \(-0.136822\pi\)
0.909033 + 0.416724i \(0.136822\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 9.37652 1.31298
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 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
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 13.4164 1.57027 0.785136 0.619324i \(-0.212593\pi\)
0.785136 + 0.619324i \(0.212593\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.8704 1.78556 0.892781 0.450490i \(-0.148751\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) 0 0
\(81\) −1.49390 −0.165989
\(82\) 0 0
\(83\) −8.94427 −0.981761 −0.490881 0.871227i \(-0.663325\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.731495 0.0784245
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.6849 −1.28796 −0.643979 0.765043i \(-0.722718\pi\)
−0.643979 + 0.765043i \(0.722718\pi\)
\(98\) 0 0
\(99\) −10.7530 −1.08072
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 7.77102 0.765701 0.382851 0.923810i \(-0.374942\pi\)
0.382851 + 0.923810i \(0.374942\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −9.87043 −0.945415 −0.472708 0.881219i \(-0.656723\pi\)
−0.472708 + 0.881219i \(0.656723\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.16515 0.847319
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 32.8704 2.98822
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −14.6235 −1.23152
\(142\) 0 0
\(143\) −37.3921 −3.12688
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 23.8704 1.94255 0.971274 0.237964i \(-0.0764802\pi\)
0.971274 + 0.237964i \(0.0764802\pi\)
\(152\) 0 0
\(153\) 12.9746 1.04894
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −13.4164 −1.07075 −0.535373 0.844616i \(-0.679829\pi\)
−0.535373 + 0.844616i \(0.679829\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.42451 −0.419761 −0.209881 0.977727i \(-0.567308\pi\)
−0.209881 + 0.977727i \(0.567308\pi\)
\(168\) 0 0
\(169\) 18.8704 1.45157
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0314 1.14282 0.571409 0.820666i \(-0.306397\pi\)
0.571409 + 0.820666i \(0.306397\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 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
\(187\) −52.9341 −3.87093
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.6235 1.34755 0.673774 0.738938i \(-0.264672\pi\)
0.673774 + 0.738938i \(0.264672\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 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
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 3.87043 0.266451 0.133226 0.991086i \(-0.457467\pi\)
0.133226 + 0.991086i \(0.457467\pi\)
\(212\) 0 0
\(213\) 14.0790 0.964680
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −15.7409 −1.06367
\(220\) 0 0
\(221\) 45.1174 3.03492
\(222\) 0 0
\(223\) 21.8501 1.46319 0.731594 0.681740i \(-0.238777\pi\)
0.731594 + 0.681740i \(0.238777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.4083 1.42092 0.710460 0.703738i \(-0.248487\pi\)
0.710460 + 0.703738i \(0.248487\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −18.6200 −1.20950
\(238\) 0 0
\(239\) −21.1174 −1.36597 −0.682985 0.730433i \(-0.739318\pi\)
−0.682985 + 0.730433i \(0.739318\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −14.5208 −0.931510
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.4939 0.665024
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.47214 0.278964 0.139482 0.990225i \(-0.455456\pi\)
0.139482 + 0.990225i \(0.455456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.01220 0.0626534
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 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
\(281\) 11.3765 0.678667 0.339333 0.940666i \(-0.389799\pi\)
0.339333 + 0.940666i \(0.389799\pi\)
\(282\) 0 0
\(283\) 19.0618 1.13311 0.566553 0.824025i \(-0.308277\pi\)
0.566553 + 0.824025i \(0.308277\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 46.8704 2.75708
\(290\) 0 0
\(291\) 14.8826 0.872435
\(292\) 0 0
\(293\) 32.9200 1.92320 0.961602 0.274446i \(-0.0884946\pi\)
0.961602 + 0.274446i \(0.0884946\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 35.9291 2.08482
\(298\) 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
\(307\) 33.1408 1.89145 0.945724 0.324971i \(-0.105355\pi\)
0.945724 + 0.324971i \(0.105355\pi\)
\(308\) 0 0
\(309\) −9.11738 −0.518669
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 35.2665 1.99338 0.996689 0.0813030i \(-0.0259081\pi\)
0.996689 + 0.0813030i \(0.0259081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) −4.12957 −0.231212
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.5805 0.640404
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −30.6235 −1.63456
\(352\) 0 0
\(353\) 6.08715 0.323986 0.161993 0.986792i \(-0.448208\pi\)
0.161993 + 0.986792i \(0.448208\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −38.5654 −2.02416
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.731495 −0.0381837 −0.0190919 0.999818i \(-0.506077\pi\)
−0.0190919 + 0.999818i \(0.506077\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 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
\(377\) 3.51976 0.181277
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −35.7771 −1.82812 −0.914062 0.405575i \(-0.867071\pi\)
−0.914062 + 0.405575i \(0.867071\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.6235 1.24846 0.624230 0.781241i \(-0.285413\pi\)
0.624230 + 0.781241i \(0.285413\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.7244 0.989941 0.494971 0.868910i \(-0.335179\pi\)
0.494971 + 0.868910i \(0.335179\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 39.1174 1.95343 0.976714 0.214544i \(-0.0688266\pi\)
0.976714 + 0.214544i \(0.0688266\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −33.8704 −1.65074 −0.825372 0.564590i \(-0.809034\pi\)
−0.825372 + 0.564590i \(0.809034\pi\)
\(422\) 0 0
\(423\) −20.2351 −0.983862
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 43.8704 2.11808
\(430\) 0 0
\(431\) −34.3643 −1.65527 −0.827636 0.561266i \(-0.810315\pi\)
−0.827636 + 0.561266i \(0.810315\pi\)
\(432\) 0 0
\(433\) −40.2492 −1.93425 −0.967127 0.254293i \(-0.918157\pi\)
−0.967127 + 0.254293i \(0.918157\pi\)
\(434\) 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
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7.03952 −0.332958
\(448\) 0 0
\(449\) 40.3643 1.90491 0.952455 0.304679i \(-0.0985491\pi\)
0.952455 + 0.304679i \(0.0985491\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −28.0061 −1.31584
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) −43.3521 −2.02350
\(460\) 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
\(467\) 14.3688 0.664908 0.332454 0.943119i \(-0.392123\pi\)
0.332454 + 0.943119i \(0.392123\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.7409 0.725300
\(472\) 0 0
\(473\) 0 0
\(474\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) −9.11738 −0.411461 −0.205731 0.978609i \(-0.565957\pi\)
−0.205731 + 0.978609i \(0.565957\pi\)
\(492\) 0 0
\(493\) 4.98275 0.224412
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −35.8704 −1.60578 −0.802890 0.596127i \(-0.796706\pi\)
−0.802890 + 0.596127i \(0.796706\pi\)
\(500\) 0 0
\(501\) 6.36433 0.284337
\(502\) 0 0
\(503\) 1.61501 0.0720099 0.0360049 0.999352i \(-0.488537\pi\)
0.0360049 + 0.999352i \(0.488537\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −22.1398 −0.983263
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 82.5552 3.63077
\(518\) 0 0
\(519\) −17.6357 −0.774120
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 26.8328 1.17332 0.586659 0.809834i \(-0.300443\pi\)
0.586659 + 0.809834i \(0.300443\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −28.1581 −1.21511
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.12957 −0.263531 −0.131765 0.991281i \(-0.542065\pi\)
−0.131765 + 0.991281i \(0.542065\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(561\) 62.1052 2.62208
\(562\) 0 0
\(563\) 44.7214 1.88478 0.942390 0.334515i \(-0.108573\pi\)
0.942390 + 0.334515i \(0.108573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) −21.8501 −0.912800
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −46.5573 −1.93820 −0.969102 0.246661i \(-0.920667\pi\)
−0.969102 + 0.246661i \(0.920667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.94427 0.369170 0.184585 0.982817i \(-0.440906\pi\)
0.184585 + 0.982817i \(0.440906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −41.8642 −1.71916 −0.859579 0.511003i \(-0.829274\pi\)
−0.859579 + 0.511003i \(0.829274\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 45.1174 1.84345 0.921723 0.387849i \(-0.126782\pi\)
0.921723 + 0.387849i \(0.126782\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.8896 1.17259 0.586296 0.810097i \(-0.300586\pi\)
0.586296 + 0.810097i \(0.300586\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −70.3643 −2.84664
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.12957 0.323633 0.161817 0.986821i \(-0.448265\pi\)
0.161817 + 0.986821i \(0.448265\pi\)
\(632\) 0 0
\(633\) −4.54099 −0.180488
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 19.4817 0.770684
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 40.1804 1.58456 0.792279 0.610158i \(-0.208894\pi\)
0.792279 + 0.610158i \(0.208894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8885 0.703271 0.351636 0.936137i \(-0.385626\pi\)
0.351636 + 0.936137i \(0.385626\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 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
\(657\) −21.7812 −0.849766
\(658\) 0 0
\(659\) 30.6235 1.19292 0.596461 0.802642i \(-0.296573\pi\)
0.596461 + 0.802642i \(0.296573\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) −52.9341 −2.05579
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −25.6357 −0.991132
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.8409 −0.724115 −0.362058 0.932156i \(-0.617926\pi\)
−0.362058 + 0.932156i \(0.617926\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −25.1174 −0.962500
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.3643 1.97777 0.988887 0.148671i \(-0.0474996\pi\)
0.988887 + 0.148671i \(0.0474996\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −45.6113 −1.71297 −0.856484 0.516174i \(-0.827356\pi\)
−0.856484 + 0.516174i \(0.827356\pi\)
\(710\) 0 0
\(711\) −25.7652 −0.966272
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.7760 0.925278
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −53.6656 −1.99035 −0.995174 0.0981255i \(-0.968715\pi\)
−0.995174 + 0.0981255i \(0.968715\pi\)
\(728\) 0 0
\(729\) 21.5183 0.796974
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.5968 1.97964 0.989821 0.142318i \(-0.0454555\pi\)
0.989821 + 0.142318i \(0.0454555\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 51.6113 1.89855 0.949276 0.314445i \(-0.101818\pi\)
0.949276 + 0.314445i \(0.101818\pi\)
\(740\) 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
\(747\) 14.5208 0.531288
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −39.6113 −1.44544 −0.722718 0.691143i \(-0.757107\pi\)
−0.722718 + 0.691143i \(0.757107\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 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
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −5.24695 −0.188964
\(772\) 0 0
\(773\) −46.9990 −1.69044 −0.845218 0.534421i \(-0.820530\pi\)
−0.845218 + 0.534421i \(0.820530\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −79.4817 −2.84408
\(782\) 0 0
\(783\) −3.38205 −0.120865
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −55.7224 −1.98629 −0.993145 0.116892i \(-0.962707\pi\)
−0.993145 + 0.116892i \(0.962707\pi\)
\(788\) 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
\(797\) −34.8247 −1.23355 −0.616777 0.787138i \(-0.711562\pi\)
−0.616777 + 0.787138i \(0.711562\pi\)
\(798\) 0 0
\(799\) −99.6113 −3.52399
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 88.8632 3.13592
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.3643 −0.575339 −0.287670 0.957730i \(-0.592880\pi\)
−0.287670 + 0.957730i \(0.592880\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.3765 −0.815846 −0.407923 0.913016i \(-0.633747\pi\)
−0.407923 + 0.913016i \(0.633747\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 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
\(841\) −28.6113 −0.986596
\(842\) 0 0
\(843\) −13.3476 −0.459714
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −22.3643 −0.767542
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −40.2492 −1.37811 −0.689054 0.724710i \(-0.741974\pi\)
−0.689054 + 0.724710i \(0.741974\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 49.1935 1.68042 0.840209 0.542263i \(-0.182432\pi\)
0.840209 + 0.542263i \(0.182432\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 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
\(867\) −54.9909 −1.86759
\(868\) 0 0
\(869\) 105.117 3.56586
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 20.5936 0.696989
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) −38.6235 −1.30274
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35.7771 1.20128 0.600639 0.799521i \(-0.294913\pi\)
0.600639 + 0.799521i \(0.294913\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.89482 −0.331489
\(892\) 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
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −59.2422 −1.96063
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −31.6113 −1.04276 −0.521380 0.853325i \(-0.674583\pi\)
−0.521380 + 0.853325i \(0.674583\pi\)
\(920\) 0 0
\(921\) −38.8826 −1.28123
\(922\) 0 0
\(923\) 67.7447 2.22984
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −12.6161 −0.414366
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −49.3455 −1.61205 −0.806024 0.591883i \(-0.798385\pi\)
−0.806024 + 0.591883i \(0.798385\pi\)
\(938\) 0 0
\(939\) −41.3765 −1.35027
\(940\) 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
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −75.7409 −2.45865
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.84504 0.156618
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 16.0244 0.511620
\(982\) 0 0
\(983\) 33.5826 1.07112 0.535559 0.844498i \(-0.320101\pi\)
0.535559 + 0.844498i \(0.320101\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) −9.38603 −0.297857
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.1479 0.448069 0.224034 0.974581i \(-0.428077\pi\)
0.224034 + 0.974581i \(0.428077\pi\)
\(998\) 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 4900.2.a.bj.1.2 4
5.2 odd 4 980.2.e.d.589.3 yes 4
5.3 odd 4 980.2.e.d.589.2 4
5.4 even 2 inner 4900.2.a.bj.1.3 4
7.6 odd 2 inner 4900.2.a.bj.1.3 4
35.2 odd 12 980.2.q.i.949.2 8
35.3 even 12 980.2.q.i.569.3 8
35.12 even 12 980.2.q.i.949.3 8
35.13 even 4 980.2.e.d.589.3 yes 4
35.17 even 12 980.2.q.i.569.2 8
35.18 odd 12 980.2.q.i.569.2 8
35.23 odd 12 980.2.q.i.949.3 8
35.27 even 4 980.2.e.d.589.2 4
35.32 odd 12 980.2.q.i.569.3 8
35.33 even 12 980.2.q.i.949.2 8
35.34 odd 2 CM 4900.2.a.bj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.e.d.589.2 4 5.3 odd 4
980.2.e.d.589.2 4 35.27 even 4
980.2.e.d.589.3 yes 4 5.2 odd 4
980.2.e.d.589.3 yes 4 35.13 even 4
980.2.q.i.569.2 8 35.17 even 12
980.2.q.i.569.2 8 35.18 odd 12
980.2.q.i.569.3 8 35.3 even 12
980.2.q.i.569.3 8 35.32 odd 12
980.2.q.i.949.2 8 35.2 odd 12
980.2.q.i.949.2 8 35.33 even 12
980.2.q.i.949.3 8 35.12 even 12
980.2.q.i.949.3 8 35.23 odd 12
4900.2.a.bj.1.2 4 1.1 even 1 trivial
4900.2.a.bj.1.2 4 35.34 odd 2 CM
4900.2.a.bj.1.3 4 5.4 even 2 inner
4900.2.a.bj.1.3 4 7.6 odd 2 inner