Properties

Label 304.5.e.b.113.2
Level $304$
Weight $5$
Character 304.113
Self dual yes
Analytic conductor $31.424$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(31.4244687775\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 113.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 304.113

$q$-expansion

\(f(q)\) \(=\) \(q+18.4743 q^{5} +20.1238 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+18.4743 q^{5} +20.1238 q^{7} +81.0000 q^{9} -59.8762 q^{11} +572.866 q^{17} -361.000 q^{19} +158.000 q^{23} -283.702 q^{25} +371.771 q^{35} +2726.10 q^{43} +1496.41 q^{45} +4284.04 q^{47} -1996.03 q^{49} -1106.17 q^{55} +4248.75 q^{61} +1630.02 q^{63} +8130.81 q^{73} -1204.94 q^{77} +6561.00 q^{81} +5678.00 q^{83} +10583.3 q^{85} -6669.21 q^{95} -4849.98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 31q^{5} - 73q^{7} + 162q^{9} + O(q^{10}) \) \( 2q - 31q^{5} - 73q^{7} + 162q^{9} - 233q^{11} + 353q^{17} - 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} + 3527q^{43} - 2511q^{45} + 1207q^{47} + 4275q^{49} + 7459q^{55} - 3167q^{61} - 5913q^{63} + 10033q^{73} + 14917q^{77} + 13122q^{81} + 11356q^{83} + 21461q^{85} + 11191q^{95} - 18873q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 18.4743 0.738970 0.369485 0.929237i \(-0.379534\pi\)
0.369485 + 0.929237i \(0.379534\pi\)
\(6\) 0 0
\(7\) 20.1238 0.410689 0.205344 0.978690i \(-0.434169\pi\)
0.205344 + 0.978690i \(0.434169\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) −59.8762 −0.494845 −0.247422 0.968908i \(-0.579584\pi\)
−0.247422 + 0.968908i \(0.579584\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 572.866 1.98224 0.991118 0.132984i \(-0.0424559\pi\)
0.991118 + 0.132984i \(0.0424559\pi\)
\(18\) 0 0
\(19\) −361.000 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 158.000 0.298677 0.149338 0.988786i \(-0.452286\pi\)
0.149338 + 0.988786i \(0.452286\pi\)
\(24\) 0 0
\(25\) −283.702 −0.453923
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 371.771 0.303487
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2726.10 1.47437 0.737183 0.675693i \(-0.236155\pi\)
0.737183 + 0.675693i \(0.236155\pi\)
\(44\) 0 0
\(45\) 1496.41 0.738970
\(46\) 0 0
\(47\) 4284.04 1.93936 0.969680 0.244380i \(-0.0785844\pi\)
0.969680 + 0.244380i \(0.0785844\pi\)
\(48\) 0 0
\(49\) −1996.03 −0.831335
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −1106.17 −0.365676
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 4248.75 1.14183 0.570915 0.821009i \(-0.306589\pi\)
0.570915 + 0.821009i \(0.306589\pi\)
\(62\) 0 0
\(63\) 1630.02 0.410689
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 8130.81 1.52577 0.762883 0.646537i \(-0.223783\pi\)
0.762883 + 0.646537i \(0.223783\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1204.94 −0.203227
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 5678.00 0.824213 0.412106 0.911136i \(-0.364793\pi\)
0.412106 + 0.911136i \(0.364793\pi\)
\(84\) 0 0
\(85\) 10583.3 1.46481
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6669.21 −0.738970
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −4849.98 −0.494845
\(100\) 0 0
\(101\) −9998.00 −0.980100 −0.490050 0.871694i \(-0.663021\pi\)
−0.490050 + 0.871694i \(0.663021\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 2918.93 0.220713
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11528.2 0.814083
\(120\) 0 0
\(121\) −11055.8 −0.755128
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −16787.6 −1.07441
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 33665.5 1.96175 0.980873 0.194651i \(-0.0623574\pi\)
0.980873 + 0.194651i \(0.0623574\pi\)
\(132\) 0 0
\(133\) −7264.68 −0.410689
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8396.39 0.447354 0.223677 0.974663i \(-0.428194\pi\)
0.223677 + 0.974663i \(0.428194\pi\)
\(138\) 0 0
\(139\) −33381.1 −1.72771 −0.863856 0.503738i \(-0.831958\pi\)
−0.863856 + 0.503738i \(0.831958\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −30212.3 −1.36085 −0.680427 0.732816i \(-0.738206\pi\)
−0.680427 + 0.732816i \(0.738206\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 46402.2 1.98224
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −49198.0 −1.99594 −0.997972 0.0636620i \(-0.979722\pi\)
−0.997972 + 0.0636620i \(0.979722\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3179.55 0.122663
\(162\) 0 0
\(163\) −9362.00 −0.352366 −0.176183 0.984357i \(-0.556375\pi\)
−0.176183 + 0.984357i \(0.556375\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) −29241.0 −1.00000
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −5709.15 −0.186421
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −34301.1 −0.980900
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −61997.2 −1.69944 −0.849719 0.527235i \(-0.823229\pi\)
−0.849719 + 0.527235i \(0.823229\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −69518.0 −1.79129 −0.895643 0.444774i \(-0.853284\pi\)
−0.895643 + 0.444774i \(0.853284\pi\)
\(198\) 0 0
\(199\) −78104.5 −1.97229 −0.986143 0.165900i \(-0.946947\pi\)
−0.986143 + 0.165900i \(0.946947\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12798.0 0.298677
\(208\) 0 0
\(209\) 21615.3 0.494845
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 50362.7 1.08951
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −22979.9 −0.453923
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 59034.7 1.12574 0.562868 0.826547i \(-0.309698\pi\)
0.562868 + 0.826547i \(0.309698\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −66110.9 −1.21776 −0.608879 0.793263i \(-0.708380\pi\)
−0.608879 + 0.793263i \(0.708380\pi\)
\(234\) 0 0
\(235\) 79144.5 1.43313
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 105335. 1.84406 0.922032 0.387113i \(-0.126528\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −36875.2 −0.614331
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −74357.6 −1.18026 −0.590130 0.807308i \(-0.700924\pi\)
−0.590130 + 0.807308i \(0.700924\pi\)
\(252\) 0 0
\(253\) −9460.45 −0.147799
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −104877. −1.51625 −0.758124 0.652111i \(-0.773884\pi\)
−0.758124 + 0.652111i \(0.773884\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 126718. 1.72544 0.862720 0.505682i \(-0.168759\pi\)
0.862720 + 0.505682i \(0.168759\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16987.0 0.224622
\(276\) 0 0
\(277\) −133030. −1.73376 −0.866880 0.498517i \(-0.833878\pi\)
−0.866880 + 0.498517i \(0.833878\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 136595. 1.70554 0.852771 0.522286i \(-0.174921\pi\)
0.852771 + 0.522286i \(0.174921\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 244655. 2.92926
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 54859.5 0.605506
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 78492.4 0.843778
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5413.87 0.0559741 0.0279871 0.999608i \(-0.491090\pi\)
0.0279871 + 0.999608i \(0.491090\pi\)
\(312\) 0 0
\(313\) 152162. 1.55316 0.776582 0.630016i \(-0.216952\pi\)
0.776582 + 0.630016i \(0.216952\pi\)
\(314\) 0 0
\(315\) 30113.5 0.303487
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −206805. −1.98224
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 86211.1 0.796473
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −88484.9 −0.752109
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 201682. 1.67497 0.837487 0.546457i \(-0.184024\pi\)
0.837487 + 0.546457i \(0.184024\pi\)
\(348\) 0 0
\(349\) −225949. −1.85506 −0.927532 0.373745i \(-0.878074\pi\)
−0.927532 + 0.373745i \(0.878074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10882.0 0.0873292 0.0436646 0.999046i \(-0.486097\pi\)
0.0436646 + 0.999046i \(0.486097\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −241539. −1.87412 −0.937061 0.349165i \(-0.886465\pi\)
−0.937061 + 0.349165i \(0.886465\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 150211. 1.12750
\(366\) 0 0
\(367\) 266878. 1.98144 0.990719 0.135923i \(-0.0434001\pi\)
0.990719 + 0.135923i \(0.0434001\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −22260.3 −0.150179
\(386\) 0 0
\(387\) 220814. 1.47437
\(388\) 0 0
\(389\) 38543.1 0.254711 0.127355 0.991857i \(-0.459351\pi\)
0.127355 + 0.991857i \(0.459351\pi\)
\(390\) 0 0
\(391\) 90512.9 0.592048
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −297079. −1.88491 −0.942456 0.334329i \(-0.891490\pi\)
−0.942456 + 0.334329i \(0.891490\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 121210. 0.738970
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 104897. 0.609068
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −229522. −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 347008. 1.93936
\(424\) 0 0
\(425\) −162523. −0.899783
\(426\) 0 0
\(427\) 85500.8 0.468937
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −57038.0 −0.298677
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −161679. −0.831335
\(442\) 0 0
\(443\) −160753. −0.819126 −0.409563 0.912282i \(-0.634319\pi\)
−0.409563 + 0.912282i \(0.634319\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 374513. 1.79322 0.896612 0.442817i \(-0.146021\pi\)
0.896612 + 0.442817i \(0.146021\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −199550. −0.938967 −0.469484 0.882941i \(-0.655560\pi\)
−0.469484 + 0.882941i \(0.655560\pi\)
\(462\) 0 0
\(463\) −279914. −1.30576 −0.652879 0.757462i \(-0.726439\pi\)
−0.652879 + 0.757462i \(0.726439\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 51999.4 0.238432 0.119216 0.992868i \(-0.461962\pi\)
0.119216 + 0.992868i \(0.461962\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −163229. −0.729583
\(474\) 0 0
\(475\) 102416. 0.453923
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −428482. −1.86750 −0.933752 0.357921i \(-0.883486\pi\)
−0.933752 + 0.357921i \(0.883486\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −360562. −1.49561 −0.747803 0.663921i \(-0.768891\pi\)
−0.747803 + 0.663921i \(0.768891\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −89599.7 −0.365676
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −497221. −1.99687 −0.998433 0.0559673i \(-0.982176\pi\)
−0.998433 + 0.0559673i \(0.982176\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −358882. −1.41846 −0.709228 0.704979i \(-0.750956\pi\)
−0.709228 + 0.704979i \(0.750956\pi\)
\(504\) 0 0
\(505\) −184706. −0.724265
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 163622. 0.626615
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −256512. −0.959682
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −254877. −0.910792
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 119515. 0.411382
\(540\) 0 0
\(541\) 576412. 1.96942 0.984711 0.174196i \(-0.0557327\pi\)
0.984711 + 0.174196i \(0.0557327\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 344149. 1.14183
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −94987.4 −0.306165 −0.153083 0.988213i \(-0.548920\pi\)
−0.153083 + 0.988213i \(0.548920\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 132032. 0.410689
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 442318. 1.35663 0.678317 0.734770i \(-0.262710\pi\)
0.678317 + 0.734770i \(0.262710\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −44824.9 −0.135576
\(576\) 0 0
\(577\) −179948. −0.540500 −0.270250 0.962790i \(-0.587106\pi\)
−0.270250 + 0.962790i \(0.587106\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 114263. 0.338495
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −38758.7 −0.112485 −0.0562423 0.998417i \(-0.517912\pi\)
−0.0562423 + 0.998417i \(0.517912\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −702398. −1.99744 −0.998720 0.0505740i \(-0.983895\pi\)
−0.998720 + 0.0505740i \(0.983895\pi\)
\(594\) 0 0
\(595\) 212975. 0.601583
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −204248. −0.558017
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 28243.5 0.0751620 0.0375810 0.999294i \(-0.488035\pi\)
0.0375810 + 0.999294i \(0.488035\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −761325. −1.99986 −0.999930 0.0118526i \(-0.996227\pi\)
−0.999930 + 0.0118526i \(0.996227\pi\)
\(618\) 0 0
\(619\) 328078. 0.856241 0.428120 0.903722i \(-0.359176\pi\)
0.428120 + 0.903722i \(0.359176\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −132825. −0.340031
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −506384. −1.27181 −0.635903 0.771769i \(-0.719372\pi\)
−0.635903 + 0.771769i \(0.719372\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −410564. −0.993023 −0.496511 0.868030i \(-0.665386\pi\)
−0.496511 + 0.868030i \(0.665386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 795843. 1.90116 0.950580 0.310480i \(-0.100490\pi\)
0.950580 + 0.310480i \(0.100490\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 762372. 1.78789 0.893945 0.448178i \(-0.147927\pi\)
0.893945 + 0.448178i \(0.147927\pi\)
\(654\) 0 0
\(655\) 621945. 1.44967
\(656\) 0 0
\(657\) 658595. 1.52577
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −134209. −0.303487
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −254399. −0.565029
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 155117. 0.330581
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −278468. −0.583202 −0.291601 0.956540i \(-0.594188\pi\)
−0.291601 + 0.956540i \(0.594188\pi\)
\(692\) 0 0
\(693\) −97599.7 −0.203227
\(694\) 0 0
\(695\) −616692. −1.27673
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 222802. 0.453402 0.226701 0.973964i \(-0.427206\pi\)
0.226701 + 0.973964i \(0.427206\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −201197. −0.402516
\(708\) 0 0
\(709\) 731762. 1.45572 0.727859 0.685727i \(-0.240515\pi\)
0.727859 + 0.685727i \(0.240515\pi\)
\(710\) 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
\(719\) 837359. 1.61977 0.809886 0.586588i \(-0.199529\pi\)
0.809886 + 0.586588i \(0.199529\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −631766. −1.19533 −0.597664 0.801747i \(-0.703904\pi\)
−0.597664 + 0.801747i \(0.703904\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 1.56169e6 2.92254
\(732\) 0 0
\(733\) 538322. 1.00192 0.500961 0.865470i \(-0.332980\pi\)
0.500961 + 0.865470i \(0.332980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.04695e6 −1.91707 −0.958536 0.284970i \(-0.908016\pi\)
−0.958536 + 0.284970i \(0.908016\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −558150. −1.00563
\(746\) 0 0
\(747\) 459918. 0.824213
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.14504e6 1.99815 0.999076 0.0429669i \(-0.0136810\pi\)
0.999076 + 0.0429669i \(0.0136810\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −238330. −0.411537 −0.205769 0.978601i \(-0.565969\pi\)
−0.205769 + 0.978601i \(0.565969\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 857246. 1.46481
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.04962e6 1.77493 0.887465 0.460875i \(-0.152464\pi\)
0.887465 + 0.460875i \(0.152464\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −908896. −1.47494
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 2.45418e6 3.84427
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −486842. −0.755018
\(804\) 0 0
\(805\) 58739.9 0.0906445
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −223470. −0.341446 −0.170723 0.985319i \(-0.554610\pi\)
−0.170723 + 0.985319i \(0.554610\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −172956. −0.260388
\(816\) 0 0
\(817\) −984124. −1.47437
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.17432e6 −1.74220 −0.871101 0.491104i \(-0.836594\pi\)
−0.871101 + 0.491104i \(0.836594\pi\)
\(822\) 0 0
\(823\) −143923. −0.212486 −0.106243 0.994340i \(-0.533882\pi\)
−0.106243 + 0.994340i \(0.533882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.14346e6 −1.64790
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 527643. 0.738970
\(846\) 0 0
\(847\) −222485. −0.310123
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −394318. −0.541937 −0.270968 0.962588i \(-0.587344\pi\)
−0.270968 + 0.962588i \(0.587344\pi\)
\(854\) 0 0
\(855\) −540206. −0.738970
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1.47565e6 1.99985 0.999927 0.0120966i \(-0.00385055\pi\)
0.999927 + 0.0120966i \(0.00385055\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −337829. −0.441247
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.41242e6 1.81975 0.909877 0.414879i \(-0.136176\pi\)
0.909877 + 0.414879i \(0.136176\pi\)
\(882\) 0 0
\(883\) 694207. 0.890365 0.445182 0.895440i \(-0.353139\pi\)
0.445182 + 0.895440i \(0.353139\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −392848. −0.494845
\(892\) 0 0
\(893\) −1.54654e6 −1.93936
\(894\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −809838. −0.980100
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −339977. −0.407857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 677477. 0.805667
\(918\) 0 0
\(919\) −1.41552e6 −1.67604 −0.838022 0.545636i \(-0.816288\pi\)
−0.838022 + 0.545636i \(0.816288\pi\)
\(920\) 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
\(929\) −1.31392e6 −1.52243 −0.761214 0.648501i \(-0.775396\pi\)
−0.761214 + 0.648501i \(0.775396\pi\)
\(930\) 0 0
\(931\) 720568. 0.831335
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −633687. −0.724856
\(936\) 0 0
\(937\) 1.35678e6 1.54536 0.772682 0.634794i \(-0.218915\pi\)
0.772682 + 0.634794i \(0.218915\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.55528e6 −1.73424 −0.867120 0.498099i \(-0.834031\pi\)
−0.867120 + 0.498099i \(0.834031\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.14535e6 −1.25583
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 168967. 0.183723
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.33392e6 −1.42652 −0.713259 0.700900i \(-0.752782\pi\)
−0.713259 + 0.700900i \(0.752782\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −671754. −0.709553
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1.28429e6 −1.32371
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 430724. 0.440359
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.44292e6 −1.45746
\(996\) 0 0
\(997\) −486609. −0.489542 −0.244771 0.969581i \(-0.578713\pi\)
−0.244771 + 0.969581i \(0.578713\pi\)
\(998\) 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 304.5.e.b.113.2 2
4.3 odd 2 76.5.c.a.37.2 2
12.11 even 2 684.5.h.b.37.1 2
19.18 odd 2 CM 304.5.e.b.113.2 2
76.75 even 2 76.5.c.a.37.2 2
228.227 odd 2 684.5.h.b.37.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.c.a.37.2 2 4.3 odd 2
76.5.c.a.37.2 2 76.75 even 2
304.5.e.b.113.2 2 1.1 even 1 trivial
304.5.e.b.113.2 2 19.18 odd 2 CM
684.5.h.b.37.1 2 12.11 even 2
684.5.h.b.37.1 2 228.227 odd 2