Properties

Label 4608.2.a.s.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $4$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
Root \(1.93185\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.44949 q^{5} +4.87832 q^{7} +O(q^{10})\) \(q-4.44949 q^{5} +4.87832 q^{7} -3.46410 q^{11} +14.7980 q^{25} +5.34847 q^{29} -10.5352 q^{31} -21.7060 q^{35} +16.7980 q^{49} -12.4495 q^{53} +15.4135 q^{55} +11.3137 q^{59} -9.79796 q^{73} -16.8990 q^{77} -3.32124 q^{79} +17.3205 q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 20 q^{25} - 8 q^{29} + 28 q^{49} - 40 q^{53} - 48 q^{77} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(4\) 0 0
\(5\) −4.44949 −1.98987 −0.994936 0.100509i \(-0.967953\pi\)
−0.994936 + 0.100509i \(0.967953\pi\)
\(6\) 0 0
\(7\) 4.87832 1.84383 0.921915 0.387392i \(-0.126624\pi\)
0.921915 + 0.387392i \(0.126624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(25\) 14.7980 2.95959
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.34847 0.993186 0.496593 0.867984i \(-0.334584\pi\)
0.496593 + 0.867984i \(0.334584\pi\)
\(30\) 0 0
\(31\) −10.5352 −1.89217 −0.946086 0.323915i \(-0.895001\pi\)
−0.946086 + 0.323915i \(0.895001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −21.7060 −3.66899
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 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
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 16.7980 2.39971
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.4495 −1.71007 −0.855034 0.518571i \(-0.826464\pi\)
−0.855034 + 0.518571i \(0.826464\pi\)
\(54\) 0 0
\(55\) 15.4135 2.07835
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 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
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −9.79796 −1.14676 −0.573382 0.819288i \(-0.694369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.8990 −1.92582
\(78\) 0 0
\(79\) −3.32124 −0.373668 −0.186834 0.982391i \(-0.559823\pi\)
−0.186834 + 0.982391i \(0.559823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.3205 1.90117 0.950586 0.310460i \(-0.100483\pi\)
0.950586 + 0.310460i \(0.100483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 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
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.5505 −1.14932 −0.574659 0.818393i \(-0.694865\pi\)
−0.574659 + 0.818393i \(0.694865\pi\)
\(102\) 0 0
\(103\) −20.2918 −1.99941 −0.999705 0.0242790i \(-0.992271\pi\)
−0.999705 + 0.0242790i \(0.992271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −43.5959 −3.89934
\(126\) 0 0
\(127\) −12.0922 −1.07301 −0.536507 0.843896i \(-0.680256\pi\)
−0.536507 + 0.843896i \(0.680256\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −22.6274 −1.97697 −0.988483 0.151330i \(-0.951644\pi\)
−0.988483 + 0.151330i \(0.951644\pi\)
\(132\) 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
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −23.7980 −1.97631
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.1464 1.24084 0.620422 0.784268i \(-0.286961\pi\)
0.620422 + 0.784268i \(0.286961\pi\)
\(150\) 0 0
\(151\) −18.7347 −1.52461 −0.762305 0.647218i \(-0.775932\pi\)
−0.762305 + 0.647218i \(0.775932\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 46.8761 3.76518
\(156\) 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
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 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
\(171\) 0 0
\(172\) 0 0
\(173\) −1.75255 −0.133244 −0.0666220 0.997778i \(-0.521222\pi\)
−0.0666220 + 0.997778i \(0.521222\pi\)
\(174\) 0 0
\(175\) 72.1891 5.45698
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 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
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −9.79796 −0.705273 −0.352636 0.935760i \(-0.614715\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.14643 0.509162 0.254581 0.967051i \(-0.418062\pi\)
0.254581 + 0.967051i \(0.418062\pi\)
\(198\) 0 0
\(199\) 7.42101 0.526062 0.263031 0.964787i \(-0.415278\pi\)
0.263031 + 0.964787i \(0.415278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 26.0915 1.83127
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −51.3939 −3.48884
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.99247 −0.535215 −0.267608 0.963528i \(-0.586233\pi\)
−0.267608 + 0.963528i \(0.586233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 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
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −29.3939 −1.89343 −0.946713 0.322078i \(-0.895619\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −74.7423 −4.77511
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −31.1769 −1.96787 −0.983935 0.178529i \(-0.942866\pi\)
−0.983935 + 0.178529i \(0.942866\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 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
\(265\) 55.3939 3.40282
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −29.3485 −1.78941 −0.894704 0.446660i \(-0.852613\pi\)
−0.894704 + 0.446660i \(0.852613\pi\)
\(270\) 0 0
\(271\) −1.76416 −0.107165 −0.0535825 0.998563i \(-0.517064\pi\)
−0.0535825 + 0.998563i \(0.517064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −51.2616 −3.09119
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.0454 −1.87211 −0.936056 0.351850i \(-0.885553\pi\)
−0.936056 + 0.351850i \(0.885553\pi\)
\(294\) 0 0
\(295\) −50.3402 −2.93092
\(296\) 0 0
\(297\) 0 0
\(298\) 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
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −9.79796 −0.553813 −0.276907 0.960897i \(-0.589309\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.75255 −0.547758 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(318\) 0 0
\(319\) −18.5276 −1.03735
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.3939 1.60119 0.800593 0.599208i \(-0.204518\pi\)
0.800593 + 0.599208i \(0.204518\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 36.4949 1.97631
\(342\) 0 0
\(343\) 47.7975 2.58082
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.2487 −1.30174 −0.650870 0.759190i \(-0.725596\pi\)
−0.650870 + 0.759190i \(0.725596\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 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
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 43.5959 2.28191
\(366\) 0 0
\(367\) 23.4060 1.22178 0.610890 0.791715i \(-0.290812\pi\)
0.610890 + 0.791715i \(0.290812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −60.7325 −3.15308
\(372\) 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
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 75.1918 3.83213
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −39.1464 −1.98480 −0.992401 0.123043i \(-0.960735\pi\)
−0.992401 + 0.123043i \(0.960735\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.7778 0.743552
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 55.1918 2.71581
\(414\) 0 0
\(415\) −77.0674 −3.78309
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923 0.507697 0.253849 0.967244i \(-0.418303\pi\)
0.253849 + 0.967244i \(0.418303\pi\)
\(420\) 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
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 41.3621 1.97411 0.987054 0.160391i \(-0.0512755\pi\)
0.987054 + 0.160391i \(0.0512755\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.1769 1.48126 0.740630 0.671913i \(-0.234527\pi\)
0.740630 + 0.671913i \(0.234527\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −40.9444 −1.90697 −0.953485 0.301440i \(-0.902533\pi\)
−0.953485 + 0.301440i \(0.902533\pi\)
\(462\) 0 0
\(463\) −5.86393 −0.272520 −0.136260 0.990673i \(-0.543508\pi\)
−0.136260 + 0.990673i \(0.543508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.3205 −0.801498 −0.400749 0.916188i \(-0.631250\pi\)
−0.400749 + 0.916188i \(0.631250\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 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
\(485\) 8.89898 0.404082
\(486\) 0 0
\(487\) 32.5911 1.47684 0.738422 0.674338i \(-0.235571\pi\)
0.738422 + 0.674338i \(0.235571\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.6274 1.02116 0.510581 0.859830i \(-0.329431\pi\)
0.510581 + 0.859830i \(0.329431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 51.3939 2.28700
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −33.8434 −1.50008 −0.750040 0.661392i \(-0.769966\pi\)
−0.750040 + 0.661392i \(0.769966\pi\)
\(510\) 0 0
\(511\) −47.7975 −2.11444
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 90.2882 3.97857
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 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
\(535\) −50.3402 −2.17640
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −58.1898 −2.50641
\(540\) 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
\(547\) 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
\(553\) −16.2020 −0.688981
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −41.8434 −1.77296 −0.886480 0.462767i \(-0.846857\pi\)
−0.886480 + 0.462767i \(0.846857\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.1051 1.60594 0.802970 0.596020i \(-0.203252\pi\)
0.802970 + 0.596020i \(0.203252\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −29.3939 −1.22368 −0.611842 0.790980i \(-0.709571\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 84.4949 3.50544
\(582\) 0 0
\(583\) 43.1263 1.78611
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.2548 −1.86787 −0.933933 0.357447i \(-0.883647\pi\)
−0.933933 + 0.357447i \(0.883647\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 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
\(601\) 48.9898 1.99834 0.999168 0.0407909i \(-0.0129877\pi\)
0.999168 + 0.0407909i \(0.0129877\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.44949 −0.180897
\(606\) 0 0
\(607\) 15.6206 0.634019 0.317010 0.948422i \(-0.397321\pi\)
0.317010 + 0.948422i \(0.397321\pi\)
\(608\) 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
\(617\) 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
\(625\) 119.990 4.79959
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 38.8194 1.54538 0.772689 0.634785i \(-0.218911\pi\)
0.772689 + 0.634785i \(0.218911\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 53.8043 2.13516
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −39.1918 −1.53841
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.2474 1.80980 0.904901 0.425622i \(-0.139945\pi\)
0.904901 + 0.425622i \(0.139945\pi\)
\(654\) 0 0
\(655\) 100.680 3.93391
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 45.2548 1.76288 0.881439 0.472298i \(-0.156575\pi\)
0.881439 + 0.472298i \(0.156575\pi\)
\(660\) 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
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.7423 1.33526 0.667628 0.744495i \(-0.267310\pi\)
0.667628 + 0.744495i \(0.267310\pi\)
\(678\) 0 0
\(679\) −9.75663 −0.374425
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 51.9615 1.98825 0.994126 0.108227i \(-0.0345173\pi\)
0.994126 + 0.108227i \(0.0345173\pi\)
\(684\) 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
\(701\) −0.944387 −0.0356690 −0.0178345 0.999841i \(-0.505677\pi\)
−0.0178345 + 0.999841i \(0.505677\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −56.3470 −2.11915
\(708\) 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
\(721\) −98.9898 −3.68657
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 79.1464 2.93942
\(726\) 0 0
\(727\) 36.6909 1.36079 0.680395 0.732845i \(-0.261808\pi\)
0.680395 + 0.732845i \(0.261808\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −67.3939 −2.46912
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 55.1918 2.01667
\(750\) 0 0
\(751\) −31.0340 −1.13245 −0.566224 0.824251i \(-0.691596\pi\)
−0.566224 + 0.824251i \(0.691596\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 83.3600 3.03378
\(756\) 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
\(769\) −48.9898 −1.76662 −0.883309 0.468792i \(-0.844689\pi\)
−0.883309 + 0.468792i \(0.844689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −50.7423 −1.82508 −0.912538 0.408993i \(-0.865880\pi\)
−0.912538 + 0.408993i \(0.865880\pi\)
\(774\) 0 0
\(775\) −155.899 −5.60006
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(797\) 25.7526 0.912202 0.456101 0.889928i \(-0.349246\pi\)
0.456101 + 0.889928i \(0.349246\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.9411 1.19776
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 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
\(821\) 56.0454 1.95600 0.977999 0.208609i \(-0.0668936\pi\)
0.977999 + 0.208609i \(0.0668936\pi\)
\(822\) 0 0
\(823\) −56.7756 −1.97907 −0.989537 0.144280i \(-0.953913\pi\)
−0.989537 + 0.144280i \(0.953913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −56.5685 −1.96708 −0.983540 0.180688i \(-0.942168\pi\)
−0.983540 + 0.180688i \(0.942168\pi\)
\(828\) 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
\(841\) −0.393877 −0.0135820
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 57.8434 1.98987
\(846\) 0 0
\(847\) 4.87832 0.167621
\(848\) 0 0
\(849\) 0 0
\(850\) 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
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 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
\(865\) 7.79796 0.265139
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.5051 0.390284
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −212.675 −7.18971
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 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
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −58.9898 −1.97845
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 50.3402 1.68269
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −56.3470 −1.87928
\(900\) 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
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −110.384 −3.64519
\(918\) 0 0
\(919\) 11.1066 0.366374 0.183187 0.983078i \(-0.441359\pi\)
0.183187 + 0.983078i \(0.441359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 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
\(937\) −58.0000 −1.89478 −0.947389 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.05561 0.230006 0.115003 0.993365i \(-0.463312\pi\)
0.115003 + 0.993365i \(0.463312\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.5685 1.83823 0.919115 0.393989i \(-0.128905\pi\)
0.919115 + 0.393989i \(0.128905\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 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
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 79.9898 2.58032
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 43.5959 1.40340
\(966\) 0 0
\(967\) 40.3765 1.29842 0.649211 0.760609i \(-0.275099\pi\)
0.649211 + 0.760609i \(0.275099\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.46410 −0.111168 −0.0555842 0.998454i \(-0.517702\pi\)
−0.0555842 + 0.998454i \(0.517702\pi\)
\(972\) 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
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −31.7980 −1.01317
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 58.3327 1.85300 0.926500 0.376296i \(-0.122802\pi\)
0.926500 + 0.376296i \(0.122802\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −33.0197 −1.04680
\(996\) 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 4608.2.a.s.1.2 yes 4
3.2 odd 2 4608.2.a.bc.1.4 yes 4
4.3 odd 2 inner 4608.2.a.s.1.1 4
8.3 odd 2 4608.2.a.bc.1.3 yes 4
8.5 even 2 4608.2.a.bc.1.4 yes 4
12.11 even 2 4608.2.a.bc.1.3 yes 4
16.3 odd 4 4608.2.d.q.2305.8 8
16.5 even 4 4608.2.d.q.2305.1 8
16.11 odd 4 4608.2.d.q.2305.2 8
16.13 even 4 4608.2.d.q.2305.7 8
24.5 odd 2 CM 4608.2.a.s.1.2 yes 4
24.11 even 2 inner 4608.2.a.s.1.1 4
48.5 odd 4 4608.2.d.q.2305.7 8
48.11 even 4 4608.2.d.q.2305.8 8
48.29 odd 4 4608.2.d.q.2305.1 8
48.35 even 4 4608.2.d.q.2305.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.s.1.1 4 4.3 odd 2 inner
4608.2.a.s.1.1 4 24.11 even 2 inner
4608.2.a.s.1.2 yes 4 1.1 even 1 trivial
4608.2.a.s.1.2 yes 4 24.5 odd 2 CM
4608.2.a.bc.1.3 yes 4 8.3 odd 2
4608.2.a.bc.1.3 yes 4 12.11 even 2
4608.2.a.bc.1.4 yes 4 3.2 odd 2
4608.2.a.bc.1.4 yes 4 8.5 even 2
4608.2.d.q.2305.1 8 16.5 even 4
4608.2.d.q.2305.1 8 48.29 odd 4
4608.2.d.q.2305.2 8 16.11 odd 4
4608.2.d.q.2305.2 8 48.35 even 4
4608.2.d.q.2305.7 8 16.13 even 4
4608.2.d.q.2305.7 8 48.5 odd 4
4608.2.d.q.2305.8 8 16.3 odd 4
4608.2.d.q.2305.8 8 48.11 even 4