Properties

Label 304.5.e.b.113.1
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 113.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 304.113

$q$-expansion

\(f(q)\) \(=\) \(q-49.4743 q^{5} -93.1238 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-49.4743 q^{5} -93.1238 q^{7} +81.0000 q^{9} -173.124 q^{11} -219.866 q^{17} -361.000 q^{19} +158.000 q^{23} +1822.70 q^{25} +4607.23 q^{35} +800.896 q^{43} -4007.41 q^{45} -3077.04 q^{47} +6271.03 q^{49} +8565.17 q^{55} -7415.75 q^{61} -7543.02 q^{63} +1902.19 q^{73} +16121.9 q^{77} +6561.00 q^{81} +5678.00 q^{83} +10877.7 q^{85} +17860.2 q^{95} -14023.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 31q^{5} - 73q^{7} + 162q^{9} + O(q^{10}) \) \( 2q - 31q^{5} - 73q^{7} + 162q^{9} - 233q^{11} + 353q^{17} - 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} + 3527q^{43} - 2511q^{45} + 1207q^{47} + 4275q^{49} + 7459q^{55} - 3167q^{61} - 5913q^{63} + 10033q^{73} + 14917q^{77} + 13122q^{81} + 11356q^{83} + 21461q^{85} + 11191q^{95} - 18873q^{99} + O(q^{100}) \)

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
\(5\) −49.4743 −1.97897 −0.989485 0.144635i \(-0.953799\pi\)
−0.989485 + 0.144635i \(0.953799\pi\)
\(6\) 0 0
\(7\) −93.1238 −1.90048 −0.950242 0.311511i \(-0.899165\pi\)
−0.950242 + 0.311511i \(0.899165\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) −173.124 −1.43077 −0.715387 0.698728i \(-0.753750\pi\)
−0.715387 + 0.698728i \(0.753750\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −219.866 −0.760783 −0.380392 0.924826i \(-0.624211\pi\)
−0.380392 + 0.924826i \(0.624211\pi\)
\(18\) 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
\(25\) 1822.70 2.91632
\(26\) 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
\(35\) 4607.23 3.76100
\(36\) 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
\(43\) 800.896 0.433151 0.216575 0.976266i \(-0.430511\pi\)
0.216575 + 0.976266i \(0.430511\pi\)
\(44\) 0 0
\(45\) −4007.41 −1.97897
\(46\) 0 0
\(47\) −3077.04 −1.39296 −0.696479 0.717577i \(-0.745251\pi\)
−0.696479 + 0.717577i \(0.745251\pi\)
\(48\) 0 0
\(49\) 6271.03 2.61184
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 8565.17 2.83146
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −7415.75 −1.99294 −0.996472 0.0839219i \(-0.973255\pi\)
−0.996472 + 0.0839219i \(0.973255\pi\)
\(62\) 0 0
\(63\) −7543.02 −1.90048
\(64\) 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
\(73\) 1902.19 0.356951 0.178476 0.983944i \(-0.442883\pi\)
0.178476 + 0.983944i \(0.442883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16121.9 2.71917
\(78\) 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
\(85\) 10877.7 1.50557
\(86\) 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
\(95\) 17860.2 1.97897
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −14023.0 −1.43077
\(100\) 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
\(115\) −7816.93 −0.591072
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20474.8 1.44586
\(120\) 0 0
\(121\) 15330.8 1.04712
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −59255.4 −3.79235
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −22618.5 −1.31802 −0.659009 0.752135i \(-0.729024\pi\)
−0.659009 + 0.752135i \(0.729024\pi\)
\(132\) 0 0
\(133\) 33617.7 1.90048
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −35883.4 −1.91184 −0.955922 0.293622i \(-0.905139\pi\)
−0.955922 + 0.293622i \(0.905139\pi\)
\(138\) 0 0
\(139\) 33548.1 1.73636 0.868178 0.496253i \(-0.165291\pi\)
0.868178 + 0.496253i \(0.165291\pi\)
\(140\) 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
\(149\) 43285.3 1.94970 0.974851 0.222859i \(-0.0715390\pi\)
0.974851 + 0.222859i \(0.0715390\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −17809.2 −0.760783
\(154\) 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
\(161\) −14713.6 −0.567631
\(162\) 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
\(175\) −169737. −5.54243
\(176\) 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
\(187\) 38064.1 1.08851
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2315.78 −0.0634791 −0.0317395 0.999496i \(-0.510105\pi\)
−0.0317395 + 0.999496i \(0.510105\pi\)
\(192\) 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
\(199\) 50431.5 1.27349 0.636745 0.771075i \(-0.280280\pi\)
0.636745 + 0.771075i \(0.280280\pi\)
\(200\) 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
\(209\) 62497.7 1.43077
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −39623.7 −0.857193
\(216\) 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
\(225\) 147639. 2.91632
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 45558.3 0.868753 0.434376 0.900731i \(-0.356969\pi\)
0.434376 + 0.900731i \(0.356969\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −41536.1 −0.765093 −0.382547 0.923936i \(-0.624953\pi\)
−0.382547 + 0.923936i \(0.624953\pi\)
\(234\) 0 0
\(235\) 152234. 2.75662
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14367.8 −0.251533 −0.125766 0.992060i \(-0.540139\pi\)
−0.125766 + 0.992060i \(0.540139\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −310255. −5.16876
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −50915.4 −0.808168 −0.404084 0.914722i \(-0.632410\pi\)
−0.404084 + 0.914722i \(0.632410\pi\)
\(252\) 0 0
\(253\) −27353.6 −0.427339
\(254\) 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
\(263\) 130564. 1.88761 0.943806 0.330499i \(-0.107217\pi\)
0.943806 + 0.330499i \(0.107217\pi\)
\(264\) 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
\(275\) −315553. −4.17260
\(276\) 0 0
\(277\) 262.663 0.00342326 0.00171163 0.999999i \(-0.499455\pi\)
0.00171163 + 0.999999i \(0.499455\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −140748. −1.75740 −0.878698 0.477378i \(-0.841587\pi\)
−0.878698 + 0.477378i \(0.841587\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −35179.8 −0.421209
\(290\) 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
\(301\) −74582.5 −0.823197
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 366889. 3.94398
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 164753. 1.70339 0.851693 0.524042i \(-0.175576\pi\)
0.851693 + 0.524042i \(0.175576\pi\)
\(312\) 0 0
\(313\) 152162. 1.55316 0.776582 0.630016i \(-0.216952\pi\)
0.776582 + 0.630016i \(0.216952\pi\)
\(314\) 0 0
\(315\) 373186. 3.76100
\(316\) 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
\(323\) 79371.7 0.760783
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 286546. 2.64730
\(330\) 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
\(343\) −360392. −3.06328
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13124.9 0.109003 0.0545015 0.998514i \(-0.482643\pi\)
0.0545015 + 0.998514i \(0.482643\pi\)
\(348\) 0 0
\(349\) 191822. 1.57488 0.787438 0.616393i \(-0.211407\pi\)
0.787438 + 0.616393i \(0.211407\pi\)
\(350\) 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
\(359\) 42825.8 0.332289 0.166145 0.986101i \(-0.446868\pi\)
0.166145 + 0.986101i \(0.446868\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −94109.6 −0.706396
\(366\) 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
\(385\) −797621. −5.38115
\(386\) 0 0
\(387\) 64872.6 0.433151
\(388\) 0 0
\(389\) 240690. 1.59059 0.795296 0.606222i \(-0.207316\pi\)
0.795296 + 0.606222i \(0.207316\pi\)
\(390\) 0 0
\(391\) −34738.9 −0.227228
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 57272.2 0.363382 0.181691 0.983356i \(-0.441843\pi\)
0.181691 + 0.983356i \(0.441843\pi\)
\(398\) 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
\(405\) −324601. −1.97897
\(406\) 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
\(415\) −280915. −1.63109
\(416\) 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
\(423\) −249241. −1.39296
\(424\) 0 0
\(425\) −400751. −2.21869
\(426\) 0 0
\(427\) 690582. 3.78756
\(428\) 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
\(441\) 507954. 2.61184
\(442\) 0 0
\(443\) −229720. −1.17056 −0.585278 0.810833i \(-0.699015\pi\)
−0.585278 + 0.810833i \(0.699015\pi\)
\(444\) 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
\(457\) −347440. −1.66359 −0.831797 0.555080i \(-0.812688\pi\)
−0.831797 + 0.555080i \(0.812688\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 424783. 1.99878 0.999391 0.0348858i \(-0.0111067\pi\)
0.999391 + 0.0348858i \(0.0111067\pi\)
\(462\) 0 0
\(463\) 421201. 1.96484 0.982421 0.186679i \(-0.0597725\pi\)
0.982421 + 0.186679i \(0.0597725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 349048. 1.60048 0.800241 0.599678i \(-0.204705\pi\)
0.800241 + 0.599678i \(0.204705\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −138654. −0.619741
\(474\) 0 0
\(475\) −657995. −2.91632
\(476\) 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
\(495\) 693779. 2.83146
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 272748. 1.09537 0.547685 0.836684i \(-0.315509\pi\)
0.547685 + 0.836684i \(0.315509\pi\)
\(500\) 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
\(505\) 494644. 1.93959
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −177139. −0.678380
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 532709. 1.99301
\(518\) 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
\(539\) −1.08567e6 −3.73696
\(540\) 0 0
\(541\) −199899. −0.682994 −0.341497 0.939883i \(-0.610934\pi\)
−0.341497 + 0.939883i \(0.610934\pi\)
\(542\) 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
\(549\) −600676. −1.99294
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −483540. −1.55855 −0.779277 0.626680i \(-0.784413\pi\)
−0.779277 + 0.626680i \(0.784413\pi\)
\(558\) 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
\(567\) −610985. −1.90048
\(568\) 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
\(575\) 287987. 0.871038
\(576\) 0 0
\(577\) −465219. −1.39735 −0.698676 0.715439i \(-0.746227\pi\)
−0.698676 + 0.715439i \(0.746227\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −528757. −1.56640
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 615246. 1.78555 0.892776 0.450501i \(-0.148755\pi\)
0.892776 + 0.450501i \(0.148755\pi\)
\(588\) 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
\(595\) −1.01297e6 −2.86131
\(596\) 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
\(605\) −758482. −2.07221
\(606\) 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
\(613\) 636269. 1.69325 0.846623 0.532193i \(-0.178632\pi\)
0.846623 + 0.532193i \(0.178632\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 388478. 1.02046 0.510230 0.860038i \(-0.329560\pi\)
0.510230 + 0.860038i \(0.329560\pi\)
\(618\) 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
\(625\) 1.79243e6 4.58862
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 785431. 1.97265 0.986323 0.164824i \(-0.0527056\pi\)
0.986323 + 0.164824i \(0.0527056\pi\)
\(632\) 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
\(643\) 826891. 1.99998 0.999992 0.00402358i \(-0.00128075\pi\)
0.999992 + 0.00402358i \(0.00128075\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −623036. −1.48835 −0.744174 0.667986i \(-0.767156\pi\)
−0.744174 + 0.667986i \(0.767156\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −50179.0 −0.117678 −0.0588390 0.998267i \(-0.518740\pi\)
−0.0588390 + 0.998267i \(0.518740\pi\)
\(654\) 0 0
\(655\) 1.11903e6 2.60832
\(656\) 0 0
\(657\) 154078. 0.356951
\(658\) 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
\(665\) −1.66321e6 −3.76100
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.28384e6 2.85146
\(672\) 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
\(685\) 1.77530e6 3.78348
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −651845. −1.36517 −0.682587 0.730804i \(-0.739145\pi\)
−0.682587 + 0.730804i \(0.739145\pi\)
\(692\) 0 0
\(693\) 1.30588e6 2.71917
\(694\) 0 0
\(695\) −1.65977e6 −3.43620
\(696\) 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
\(707\) 931051. 1.86267
\(708\) 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
\(719\) −943912. −1.82589 −0.912943 0.408088i \(-0.866196\pi\)
−0.912943 + 0.408088i \(0.866196\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −418067. −0.791002 −0.395501 0.918466i \(-0.629429\pi\)
−0.395501 + 0.918466i \(0.629429\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) −176090. −0.329534
\(732\) 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
\(739\) 253921. 0.464953 0.232477 0.972602i \(-0.425317\pi\)
0.232477 + 0.972602i \(0.425317\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −2.14151e6 −3.85840
\(746\) 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
\(757\) −615167. −1.07350 −0.536749 0.843742i \(-0.680348\pi\)
−0.536749 + 0.843742i \(0.680348\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −862437. −1.48922 −0.744609 0.667501i \(-0.767364\pi\)
−0.744609 + 0.667501i \(0.767364\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 881095. 1.50557
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −996871. −1.68572 −0.842862 0.538130i \(-0.819131\pi\)
−0.842862 + 0.538130i \(0.819131\pi\)
\(770\) 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
\(785\) 2.43403e6 3.94991
\(786\) 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
\(799\) 676538. 1.05974
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −329315. −0.510717
\(804\) 0 0
\(805\) 727942. 1.12332
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.00522e6 −1.53590 −0.767950 0.640510i \(-0.778723\pi\)
−0.767950 + 0.640510i \(0.778723\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 463178. 0.697321
\(816\) 0 0
\(817\) −289123. −0.433151
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.16051e6 1.72172 0.860859 0.508844i \(-0.169927\pi\)
0.860859 + 0.508844i \(0.169927\pi\)
\(822\) 0 0
\(823\) 1.23849e6 1.82849 0.914245 0.405161i \(-0.132785\pi\)
0.914245 + 0.405161i \(0.132785\pi\)
\(824\) 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
\(833\) −1.37879e6 −1.98705
\(834\) 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
\(845\) −1.41303e6 −1.97897
\(846\) 0 0
\(847\) −1.42767e6 −1.99003
\(848\) 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
\(855\) 1.44668e6 1.97897
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −722367. −0.978975 −0.489487 0.872010i \(-0.662816\pi\)
−0.489487 + 0.872010i \(0.662816\pi\)
\(860\) 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
\(875\) 5.51809e6 7.20730
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −148469. −0.191286 −0.0956430 0.995416i \(-0.530491\pi\)
−0.0956430 + 0.995416i \(0.530491\pi\)
\(882\) 0 0
\(883\) −1.55636e6 −1.99613 −0.998065 0.0621808i \(-0.980194\pi\)
−0.998065 + 0.0621808i \(0.980194\pi\)
\(884\) 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
\(891\) −1.13586e6 −1.43077
\(892\) 0 0
\(893\) 1.11081e6 1.39296
\(894\) 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
\(913\) −982997. −1.17926
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.10632e6 2.50487
\(918\) 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
\(931\) −2.26384e6 −2.61184
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.88319e6 −2.15413
\(936\) 0 0
\(937\) 286932. 0.326813 0.163407 0.986559i \(-0.447752\pi\)
0.163407 + 0.986559i \(0.447752\pi\)
\(938\) 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
\(955\) 114571. 0.125623
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.34160e6 3.63343
\(960\) 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
\(973\) −3.12413e6 −3.29992
\(974\) 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
\(985\) 3.43935e6 3.54490
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 126542. 0.129372
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.49506e6 −2.52020
\(996\) 0 0
\(997\) −1.42600e6 −1.43459 −0.717296 0.696768i \(-0.754621\pi\)
−0.717296 + 0.696768i \(0.754621\pi\)
\(998\) 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.113.1 2
4.3 odd 2 76.5.c.a.37.1 2
12.11 even 2 684.5.h.b.37.2 2
19.18 odd 2 CM 304.5.e.b.113.1 2
76.75 even 2 76.5.c.a.37.1 2
228.227 odd 2 684.5.h.b.37.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.c.a.37.1 2 4.3 odd 2
76.5.c.a.37.1 2 76.75 even 2
304.5.e.b.113.1 2 1.1 even 1 trivial
304.5.e.b.113.1 2 19.18 odd 2 CM
684.5.h.b.37.2 2 12.11 even 2
684.5.h.b.37.2 2 228.227 odd 2