Properties

Label 4608.2.a.s.1.3
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $4$
CM discriminant -24
Inner twists $4$

Related objects

Downloads

Learn more

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 1.3
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.449490 q^{5} -2.04989 q^{7} +O(q^{10})\) \(q+0.449490 q^{5} -2.04989 q^{7} +3.46410 q^{11} -4.79796 q^{25} -9.34847 q^{29} -3.60697 q^{31} -0.921404 q^{35} -2.79796 q^{49} -7.55051 q^{53} +1.55708 q^{55} +11.3137 q^{59} +9.79796 q^{73} -7.10102 q^{77} +17.4634 q^{79} -17.3205 q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) 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
\(5\) 0.449490 0.201018 0.100509 0.994936i \(-0.467953\pi\)
0.100509 + 0.994936i \(0.467953\pi\)
\(6\) 0 0
\(7\) −2.04989 −0.774785 −0.387392 0.921915i \(-0.626624\pi\)
−0.387392 + 0.921915i \(0.626624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 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
\(25\) −4.79796 −0.959592
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.34847 −1.73597 −0.867984 0.496593i \(-0.834584\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0 0
\(31\) −3.60697 −0.647830 −0.323915 0.946086i \(-0.604999\pi\)
−0.323915 + 0.946086i \(0.604999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.921404 −0.155746
\(36\) 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
\(49\) −2.79796 −0.399708
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.55051 −1.03714 −0.518571 0.855034i \(-0.673536\pi\)
−0.518571 + 0.855034i \(0.673536\pi\)
\(54\) 0 0
\(55\) 1.55708 0.209956
\(56\) 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 \(-0.305631\pi\)
0.573382 + 0.819288i \(0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.10102 −0.809236
\(78\) 0 0
\(79\) 17.4634 1.96478 0.982391 0.186834i \(-0.0598227\pi\)
0.982391 + 0.186834i \(0.0598227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −17.3205 −1.90117 −0.950586 0.310460i \(-0.899517\pi\)
−0.950586 + 0.310460i \(0.899517\pi\)
\(84\) 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
\(101\) −16.4495 −1.63679 −0.818393 0.574659i \(-0.805135\pi\)
−0.818393 + 0.574659i \(0.805135\pi\)
\(102\) 0 0
\(103\) 0.492810 0.0485580 0.0242790 0.999705i \(-0.492271\pi\)
0.0242790 + 0.999705i \(0.492271\pi\)
\(104\) 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
\(125\) −4.40408 −0.393913
\(126\) 0 0
\(127\) −19.0205 −1.68779 −0.843896 0.536507i \(-0.819744\pi\)
−0.843896 + 0.536507i \(0.819744\pi\)
\(128\) 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
\(145\) −4.20204 −0.348961
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.1464 −1.56854 −0.784268 0.620422i \(-0.786961\pi\)
−0.784268 + 0.620422i \(0.786961\pi\)
\(150\) 0 0
\(151\) 15.9063 1.29444 0.647218 0.762305i \(-0.275932\pi\)
0.647218 + 0.762305i \(0.275932\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.62129 −0.130225
\(156\) 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
\(173\) −26.2474 −1.99556 −0.997778 0.0666220i \(-0.978778\pi\)
−0.997778 + 0.0666220i \(0.978778\pi\)
\(174\) 0 0
\(175\) 9.83528 0.743477
\(176\) 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 \(-0.385285\pi\)
0.352636 + 0.935760i \(0.385285\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −27.1464 −1.93410 −0.967051 0.254581i \(-0.918062\pi\)
−0.967051 + 0.254581i \(0.918062\pi\)
\(198\) 0 0
\(199\) −27.2200 −1.92957 −0.964787 0.263031i \(-0.915278\pi\)
−0.964787 + 0.263031i \(0.915278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 19.1633 1.34500
\(204\) 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
\(217\) 7.39388 0.501929
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −28.7771 −1.92706 −0.963528 0.267608i \(-0.913767\pi\)
−0.963528 + 0.267608i \(0.913767\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.3923 −0.689761 −0.344881 0.938647i \(-0.612081\pi\)
−0.344881 + 0.938647i \(0.612081\pi\)
\(228\) 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 \(-0.104381\pi\)
0.946713 + 0.322078i \(0.104381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.25765 −0.0803486
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 31.1769 1.96787 0.983935 0.178529i \(-0.0571337\pi\)
0.983935 + 0.178529i \(0.0571337\pi\)
\(252\) 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
\(265\) −3.39388 −0.208484
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.6515 −0.893320 −0.446660 0.894704i \(-0.647387\pi\)
−0.446660 + 0.894704i \(0.647387\pi\)
\(270\) 0 0
\(271\) 32.8769 1.99713 0.998563 0.0535825i \(-0.0170640\pi\)
0.998563 + 0.0535825i \(0.0170640\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.6206 −1.00226
\(276\) 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
\(293\) 12.0454 0.703700 0.351850 0.936056i \(-0.385553\pi\)
0.351850 + 0.936056i \(0.385553\pi\)
\(294\) 0 0
\(295\) 5.08540 0.296083
\(296\) 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 \(-0.410691\pi\)
0.276907 + 0.960897i \(0.410691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −34.2474 −1.92353 −0.961764 0.273879i \(-0.911693\pi\)
−0.961764 + 0.273879i \(0.911693\pi\)
\(318\) 0 0
\(319\) −32.3840 −1.81316
\(320\) 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 \(-0.795482\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.4949 −0.676637
\(342\) 0 0
\(343\) 20.0847 1.08447
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.2487 1.30174 0.650870 0.759190i \(-0.274404\pi\)
0.650870 + 0.759190i \(0.274404\pi\)
\(348\) 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
\(365\) 4.40408 0.230520
\(366\) 0 0
\(367\) 30.3342 1.58343 0.791715 0.610890i \(-0.209188\pi\)
0.791715 + 0.610890i \(0.209188\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 15.4777 0.803562
\(372\) 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
\(385\) −3.19184 −0.162671
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.85357 −0.246086 −0.123043 0.992401i \(-0.539265\pi\)
−0.123043 + 0.992401i \(0.539265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.84961 0.394957
\(396\) 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
\(413\) −23.1918 −1.14120
\(414\) 0 0
\(415\) −7.78539 −0.382170
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.3923 −0.507697 −0.253849 0.967244i \(-0.581697\pi\)
−0.253849 + 0.967244i \(0.581697\pi\)
\(420\) 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
\(439\) 6.72112 0.320782 0.160391 0.987054i \(-0.448725\pi\)
0.160391 + 0.987054i \(0.448725\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.1769 −1.48126 −0.740630 0.671913i \(-0.765473\pi\)
−0.740630 + 0.671913i \(0.765473\pi\)
\(444\) 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
\(461\) 12.9444 0.602880 0.301440 0.953485i \(-0.402533\pi\)
0.301440 + 0.953485i \(0.402533\pi\)
\(462\) 0 0
\(463\) 42.6335 1.98135 0.990673 0.136260i \(-0.0435083\pi\)
0.990673 + 0.136260i \(0.0435083\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.3205 0.801498 0.400749 0.916188i \(-0.368750\pi\)
0.400749 + 0.916188i \(0.368750\pi\)
\(468\) 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
\(485\) −0.898979 −0.0408206
\(486\) 0 0
\(487\) −29.7627 −1.34868 −0.674338 0.738422i \(-0.735571\pi\)
−0.674338 + 0.738422i \(0.735571\pi\)
\(488\) 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
\(505\) −7.39388 −0.329023
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.8434 1.32278 0.661392 0.750040i \(-0.269966\pi\)
0.661392 + 0.750040i \(0.269966\pi\)
\(510\) 0 0
\(511\) −20.0847 −0.888496
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.221513 0.00976103
\(516\) 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
\(535\) 5.08540 0.219861
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.69241 −0.417482
\(540\) 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
\(553\) −35.7980 −1.52228
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.8434 0.925533 0.462767 0.886480i \(-0.346857\pi\)
0.462767 + 0.886480i \(0.346857\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −38.1051 −1.60594 −0.802970 0.596020i \(-0.796748\pi\)
−0.802970 + 0.596020i \(0.796748\pi\)
\(564\) 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 \(-0.290429\pi\)
0.611842 + 0.790980i \(0.290429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 35.5051 1.47300
\(582\) 0 0
\(583\) −26.1557 −1.08326
\(584\) 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 \(-0.987012\pi\)
−0.999168 + 0.0407909i \(0.987012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.449490 0.0182744
\(606\) 0 0
\(607\) −46.7333 −1.89684 −0.948422 0.317010i \(-0.897321\pi\)
−0.948422 + 0.317010i \(0.897321\pi\)
\(608\) 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
\(625\) 22.0102 0.880408
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 31.8912 1.26957 0.634785 0.772689i \(-0.281089\pi\)
0.634785 + 0.772689i \(0.281089\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.54950 −0.339276
\(636\) 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
\(653\) 21.7526 0.851243 0.425622 0.904901i \(-0.360055\pi\)
0.425622 + 0.904901i \(0.360055\pi\)
\(654\) 0 0
\(655\) −10.1708 −0.397406
\(656\) 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
\(677\) −38.7423 −1.48899 −0.744495 0.667628i \(-0.767310\pi\)
−0.744495 + 0.667628i \(0.767310\pi\)
\(678\) 0 0
\(679\) 4.09978 0.157335
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −51.9615 −1.98825 −0.994126 0.108227i \(-0.965483\pi\)
−0.994126 + 0.108227i \(0.965483\pi\)
\(684\) 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
\(701\) 52.9444 1.99968 0.999841 0.0178345i \(-0.00567720\pi\)
0.999841 + 0.0178345i \(0.00567720\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33.7196 1.26816
\(708\) 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
\(721\) −1.01021 −0.0376220
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 44.8536 1.66582
\(726\) 0 0
\(727\) −39.5193 −1.46569 −0.732845 0.680395i \(-0.761808\pi\)
−0.732845 + 0.680395i \(0.761808\pi\)
\(728\) 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
\(745\) −8.60612 −0.315304
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −23.1918 −0.847411
\(750\) 0 0
\(751\) 45.1762 1.64850 0.824251 0.566224i \(-0.191596\pi\)
0.824251 + 0.566224i \(0.191596\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.14972 0.260205
\(756\) 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 \(-0.155311\pi\)
0.883309 + 0.468792i \(0.155311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.7423 0.817985 0.408993 0.912538i \(-0.365880\pi\)
0.408993 + 0.912538i \(0.365880\pi\)
\(774\) 0 0
\(775\) 17.3061 0.621653
\(776\) 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
\(797\) 50.2474 1.77986 0.889928 0.456101i \(-0.150754\pi\)
0.889928 + 0.456101i \(0.150754\pi\)
\(798\) 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
\(821\) 11.9546 0.417218 0.208609 0.977999i \(-0.433106\pi\)
0.208609 + 0.977999i \(0.433106\pi\)
\(822\) 0 0
\(823\) −8.27820 −0.288560 −0.144280 0.989537i \(-0.546087\pi\)
−0.144280 + 0.989537i \(0.546087\pi\)
\(824\) 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
\(841\) 58.3939 2.01358
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.84337 −0.201018
\(846\) 0 0
\(847\) −2.04989 −0.0704350
\(848\) 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
\(865\) −11.7980 −0.401143
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 60.4949 2.05215
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.02788 0.305198
\(876\) 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
\(889\) 38.9898 1.30768
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −5.08540 −0.169986
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 33.7196 1.12461
\(900\) 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
\(917\) 46.3837 1.53172
\(918\) 0 0
\(919\) 59.6041 1.96616 0.983078 0.183187i \(-0.0586414\pi\)
0.983078 + 0.183187i \(0.0586414\pi\)
\(920\) 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
\(941\) 60.9444 1.98673 0.993365 0.115003i \(-0.0366878\pi\)
0.993365 + 0.115003i \(0.0366878\pi\)
\(942\) 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
\(961\) −17.9898 −0.580316
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.40408 0.141772
\(966\) 0 0
\(967\) 47.3047 1.52122 0.760609 0.649211i \(-0.224901\pi\)
0.760609 + 0.649211i \(0.224901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.46410 0.111168 0.0555842 0.998454i \(-0.482298\pi\)
0.0555842 + 0.998454i \(0.482298\pi\)
\(972\) 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
\(985\) −12.2020 −0.388789
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 23.6917 0.752591 0.376296 0.926500i \(-0.377198\pi\)
0.376296 + 0.926500i \(0.377198\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.2351 −0.387879
\(996\) 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.1.3 4
3.2 odd 2 4608.2.a.bc.1.1 yes 4
4.3 odd 2 inner 4608.2.a.s.1.4 yes 4
8.3 odd 2 4608.2.a.bc.1.2 yes 4
8.5 even 2 4608.2.a.bc.1.1 yes 4
12.11 even 2 4608.2.a.bc.1.2 yes 4
16.3 odd 4 4608.2.d.q.2305.3 8
16.5 even 4 4608.2.d.q.2305.6 8
16.11 odd 4 4608.2.d.q.2305.5 8
16.13 even 4 4608.2.d.q.2305.4 8
24.5 odd 2 CM 4608.2.a.s.1.3 4
24.11 even 2 inner 4608.2.a.s.1.4 yes 4
48.5 odd 4 4608.2.d.q.2305.4 8
48.11 even 4 4608.2.d.q.2305.3 8
48.29 odd 4 4608.2.d.q.2305.6 8
48.35 even 4 4608.2.d.q.2305.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.s.1.3 4 1.1 even 1 trivial
4608.2.a.s.1.3 4 24.5 odd 2 CM
4608.2.a.s.1.4 yes 4 4.3 odd 2 inner
4608.2.a.s.1.4 yes 4 24.11 even 2 inner
4608.2.a.bc.1.1 yes 4 3.2 odd 2
4608.2.a.bc.1.1 yes 4 8.5 even 2
4608.2.a.bc.1.2 yes 4 8.3 odd 2
4608.2.a.bc.1.2 yes 4 12.11 even 2
4608.2.d.q.2305.3 8 16.3 odd 4
4608.2.d.q.2305.3 8 48.11 even 4
4608.2.d.q.2305.4 8 16.13 even 4
4608.2.d.q.2305.4 8 48.5 odd 4
4608.2.d.q.2305.5 8 16.11 odd 4
4608.2.d.q.2305.5 8 48.35 even 4
4608.2.d.q.2305.6 8 16.5 even 4
4608.2.d.q.2305.6 8 48.29 odd 4