Properties

Label 300.3.g.a.101.1
Level $300$
Weight $3$
Character 300.101
Self dual yes
Analytic conductor $8.174$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 101.1
Character \(\chi\) \(=\) 300.101

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -13.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -13.0000 q^{7} +9.00000 q^{9} +23.0000 q^{13} +11.0000 q^{19} +39.0000 q^{21} -27.0000 q^{27} +59.0000 q^{31} +26.0000 q^{37} -69.0000 q^{39} +83.0000 q^{43} +120.000 q^{49} -33.0000 q^{57} -121.000 q^{61} -117.000 q^{63} -13.0000 q^{67} -46.0000 q^{73} -142.000 q^{79} +81.0000 q^{81} -299.000 q^{91} -177.000 q^{93} +167.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −3.00000 −1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −13.0000 −1.85714 −0.928571 0.371154i \(-0.878962\pi\)
−0.928571 + 0.371154i \(0.878962\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 23.0000 1.76923 0.884615 0.466321i \(-0.154421\pi\)
0.884615 + 0.466321i \(0.154421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 11.0000 0.578947 0.289474 0.957186i \(-0.406520\pi\)
0.289474 + 0.957186i \(0.406520\pi\)
\(20\) 0 0
\(21\) 39.0000 1.85714
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 59.0000 1.90323 0.951613 0.307299i \(-0.0994253\pi\)
0.951613 + 0.307299i \(0.0994253\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) −69.0000 −1.76923
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 83.0000 1.93023 0.965116 0.261822i \(-0.0843232\pi\)
0.965116 + 0.261822i \(0.0843232\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 120.000 2.44898
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −33.0000 −0.578947
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −121.000 −1.98361 −0.991803 0.127774i \(-0.959217\pi\)
−0.991803 + 0.127774i \(0.959217\pi\)
\(62\) 0 0
\(63\) −117.000 −1.85714
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −0.194030 −0.0970149 0.995283i \(-0.530929\pi\)
−0.0970149 + 0.995283i \(0.530929\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −46.0000 −0.630137 −0.315068 0.949069i \(-0.602027\pi\)
−0.315068 + 0.949069i \(0.602027\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −142.000 −1.79747 −0.898734 0.438494i \(-0.855512\pi\)
−0.898734 + 0.438494i \(0.855512\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −299.000 −3.28571
\(92\) 0 0
\(93\) −177.000 −1.90323
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 167.000 1.72165 0.860825 0.508902i \(-0.169948\pi\)
0.860825 + 0.508902i \(0.169948\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 194.000 1.88350 0.941748 0.336321i \(-0.109183\pi\)
0.941748 + 0.336321i \(0.109183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 71.0000 0.651376 0.325688 0.945477i \(-0.394404\pi\)
0.325688 + 0.945477i \(0.394404\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.702703
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 207.000 1.76923
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 146.000 1.14961 0.574803 0.818292i \(-0.305079\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(128\) 0 0
\(129\) −249.000 −1.93023
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −143.000 −1.07519
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −22.0000 −0.158273 −0.0791367 0.996864i \(-0.525216\pi\)
−0.0791367 + 0.996864i \(0.525216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −360.000 −2.44898
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 59.0000 0.390728 0.195364 0.980731i \(-0.437411\pi\)
0.195364 + 0.980731i \(0.437411\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −193.000 −1.22930 −0.614650 0.788800i \(-0.710703\pi\)
−0.614650 + 0.788800i \(0.710703\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −37.0000 −0.226994 −0.113497 0.993538i \(-0.536205\pi\)
−0.113497 + 0.993538i \(0.536205\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 360.000 2.13018
\(170\) 0 0
\(171\) 99.0000 0.578947
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.00552486 −0.00276243 0.999996i \(-0.500879\pi\)
−0.00276243 + 0.999996i \(0.500879\pi\)
\(182\) 0 0
\(183\) 363.000 1.98361
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 351.000 1.85714
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 143.000 0.740933 0.370466 0.928846i \(-0.379198\pi\)
0.370466 + 0.928846i \(0.379198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −109.000 −0.547739 −0.273869 0.961767i \(-0.588304\pi\)
−0.273869 + 0.961767i \(0.588304\pi\)
\(200\) 0 0
\(201\) 39.0000 0.194030
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 419.000 1.98578 0.992891 0.119027i \(-0.0379776\pi\)
0.992891 + 0.119027i \(0.0379776\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −767.000 −3.53456
\(218\) 0 0
\(219\) 138.000 0.630137
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 83.0000 0.372197 0.186099 0.982531i \(-0.440416\pi\)
0.186099 + 0.982531i \(0.440416\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −409.000 −1.78603 −0.893013 0.450031i \(-0.851413\pi\)
−0.893013 + 0.450031i \(0.851413\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 426.000 1.79747
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 479.000 1.98755 0.993776 0.111397i \(-0.0355327\pi\)
0.993776 + 0.111397i \(0.0355327\pi\)
\(242\) 0 0
\(243\) −243.000 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 253.000 1.02429
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −338.000 −1.30502
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 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
\(271\) 242.000 0.892989 0.446494 0.894786i \(-0.352672\pi\)
0.446494 + 0.894786i \(0.352672\pi\)
\(272\) 0 0
\(273\) 897.000 3.28571
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 407.000 1.46931 0.734657 0.678439i \(-0.237343\pi\)
0.734657 + 0.678439i \(0.237343\pi\)
\(278\) 0 0
\(279\) 531.000 1.90323
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −517.000 −1.82686 −0.913428 0.407001i \(-0.866574\pi\)
−0.913428 + 0.407001i \(0.866574\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) −501.000 −1.72165
\(292\) 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
\(301\) −1079.00 −3.58472
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −253.000 −0.824104 −0.412052 0.911160i \(-0.635188\pi\)
−0.412052 + 0.911160i \(0.635188\pi\)
\(308\) 0 0
\(309\) −582.000 −1.88350
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −457.000 −1.46006 −0.730032 0.683413i \(-0.760495\pi\)
−0.730032 + 0.683413i \(0.760495\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 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
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −213.000 −0.651376
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 362.000 1.09366 0.546828 0.837245i \(-0.315835\pi\)
0.546828 + 0.837245i \(0.315835\pi\)
\(332\) 0 0
\(333\) 234.000 0.702703
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 167.000 0.495549 0.247774 0.968818i \(-0.420301\pi\)
0.247774 + 0.968818i \(0.420301\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −923.000 −2.69096
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −502.000 −1.43840 −0.719198 0.694805i \(-0.755490\pi\)
−0.719198 + 0.694805i \(0.755490\pi\)
\(350\) 0 0
\(351\) −621.000 −1.76923
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −240.000 −0.664820
\(362\) 0 0
\(363\) −363.000 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 227.000 0.618529 0.309264 0.950976i \(-0.399917\pi\)
0.309264 + 0.950976i \(0.399917\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −577.000 −1.54692 −0.773458 0.633847i \(-0.781475\pi\)
−0.773458 + 0.633847i \(0.781475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 611.000 1.61214 0.806069 0.591822i \(-0.201591\pi\)
0.806069 + 0.591822i \(0.201591\pi\)
\(380\) 0 0
\(381\) −438.000 −1.14961
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 747.000 1.93023
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −793.000 −1.99748 −0.998741 0.0501728i \(-0.984023\pi\)
−0.998741 + 0.0501728i \(0.984023\pi\)
\(398\) 0 0
\(399\) 429.000 1.07519
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1357.00 3.36725
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −769.000 −1.88020 −0.940098 0.340905i \(-0.889267\pi\)
−0.940098 + 0.340905i \(0.889267\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 66.0000 0.158273
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −358.000 −0.850356 −0.425178 0.905110i \(-0.639789\pi\)
−0.425178 + 0.905110i \(0.639789\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1573.00 3.68384
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 503.000 1.16166 0.580831 0.814024i \(-0.302728\pi\)
0.580831 + 0.814024i \(0.302728\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −709.000 −1.61503 −0.807517 0.589844i \(-0.799189\pi\)
−0.807517 + 0.589844i \(0.799189\pi\)
\(440\) 0 0
\(441\) 1080.00 2.44898
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 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
\(453\) −177.000 −0.390728
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −814.000 −1.78118 −0.890591 0.454805i \(-0.849709\pi\)
−0.890591 + 0.454805i \(0.849709\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −526.000 −1.13607 −0.568035 0.823005i \(-0.692296\pi\)
−0.568035 + 0.823005i \(0.692296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 169.000 0.360341
\(470\) 0 0
\(471\) 579.000 1.22930
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 598.000 1.24324
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −613.000 −1.25873 −0.629363 0.777111i \(-0.716684\pi\)
−0.629363 + 0.777111i \(0.716684\pi\)
\(488\) 0 0
\(489\) 111.000 0.226994
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 851.000 1.70541 0.852705 0.522392i \(-0.174960\pi\)
0.852705 + 0.522392i \(0.174960\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1080.00 −2.13018
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 598.000 1.17025
\(512\) 0 0
\(513\) −297.000 −0.578947
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 803.000 1.53537 0.767686 0.640826i \(-0.221408\pi\)
0.767686 + 0.640826i \(0.221408\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 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
\(539\) 0 0
\(540\) 0 0
\(541\) −241.000 −0.445471 −0.222736 0.974879i \(-0.571499\pi\)
−0.222736 + 0.974879i \(0.571499\pi\)
\(542\) 0 0
\(543\) 3.00000 0.00552486
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) −1089.00 −1.98361
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1846.00 3.33816
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1909.00 3.41503
\(560\) 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
\(567\) −1053.00 −1.85714
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −181.000 −0.316988 −0.158494 0.987360i \(-0.550664\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1033.00 −1.79029 −0.895147 0.445770i \(-0.852930\pi\)
−0.895147 + 0.445770i \(0.852930\pi\)
\(578\) 0 0
\(579\) −429.000 −0.740933
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 649.000 1.10187
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 327.000 0.547739
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1199.00 1.99501 0.997504 0.0706077i \(-0.0224939\pi\)
0.997504 + 0.0706077i \(0.0224939\pi\)
\(602\) 0 0
\(603\) −117.000 −0.194030
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −814.000 −1.34102 −0.670511 0.741900i \(-0.733925\pi\)
−0.670511 + 0.741900i \(0.733925\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1126.00 −1.83687 −0.918434 0.395574i \(-0.870546\pi\)
−0.918434 + 0.395574i \(0.870546\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −949.000 −1.53312 −0.766559 0.642174i \(-0.778033\pi\)
−0.766559 + 0.642174i \(0.778033\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1261.00 −1.99842 −0.999208 0.0398015i \(-0.987327\pi\)
−0.999208 + 0.0398015i \(0.987327\pi\)
\(632\) 0 0
\(633\) −1257.00 −1.98578
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2760.00 4.33281
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 314.000 0.488336 0.244168 0.969733i \(-0.421485\pi\)
0.244168 + 0.969733i \(0.421485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2301.00 3.53456
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −414.000 −0.630137
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 122.000 0.184569 0.0922844 0.995733i \(-0.470583\pi\)
0.0922844 + 0.995733i \(0.470583\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −249.000 −0.372197
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1154.00 1.71471 0.857355 0.514725i \(-0.172106\pi\)
0.857355 + 0.514725i \(0.172106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2171.00 −3.19735
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1227.00 1.78603
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1318.00 −1.90738 −0.953690 0.300790i \(-0.902750\pi\)
−0.953690 + 0.300790i \(0.902750\pi\)
\(692\) 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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 286.000 0.406828
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1391.00 1.96192 0.980959 0.194214i \(-0.0622158\pi\)
0.980959 + 0.194214i \(0.0622158\pi\)
\(710\) 0 0
\(711\) −1278.00 −1.79747
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −2522.00 −3.49792
\(722\) 0 0
\(723\) −1437.00 −1.98755
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 947.000 1.30261 0.651307 0.758815i \(-0.274221\pi\)
0.651307 + 0.758815i \(0.274221\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1034.00 1.41064 0.705321 0.708888i \(-0.250803\pi\)
0.705321 + 0.708888i \(0.250803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1222.00 −1.65359 −0.826793 0.562506i \(-0.809837\pi\)
−0.826793 + 0.562506i \(0.809837\pi\)
\(740\) 0 0
\(741\) −759.000 −1.02429
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1202.00 1.60053 0.800266 0.599645i \(-0.204691\pi\)
0.800266 + 0.599645i \(0.204691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −673.000 −0.889036 −0.444518 0.895770i \(-0.646625\pi\)
−0.444518 + 0.895770i \(0.646625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −923.000 −1.20970
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 671.000 0.872562 0.436281 0.899811i \(-0.356295\pi\)
0.436281 + 0.899811i \(0.356295\pi\)
\(770\) 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
\(777\) 1014.00 1.30502
\(778\) 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
\(787\) −613.000 −0.778907 −0.389454 0.921046i \(-0.627336\pi\)
−0.389454 + 0.921046i \(0.627336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2783.00 −3.50946
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1261.00 −1.55487 −0.777435 0.628963i \(-0.783480\pi\)
−0.777435 + 0.628963i \(0.783480\pi\)
\(812\) 0 0
\(813\) −726.000 −0.892989
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 913.000 1.11750
\(818\) 0 0
\(819\) −2691.00 −3.28571
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 563.000 0.684083 0.342041 0.939685i \(-0.388882\pi\)
0.342041 + 0.939685i \(0.388882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 458.000 0.552473 0.276236 0.961090i \(-0.410913\pi\)
0.276236 + 0.961090i \(0.410913\pi\)
\(830\) 0 0
\(831\) −1221.00 −1.46931
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1593.00 −1.90323
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1573.00 −1.85714
\(848\) 0 0
\(849\) 1551.00 1.82686
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1177.00 −1.37984 −0.689918 0.723888i \(-0.742353\pi\)
−0.689918 + 0.723888i \(0.742353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1418.00 1.65076 0.825378 0.564580i \(-0.190962\pi\)
0.825378 + 0.564580i \(0.190962\pi\)
\(860\) 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
\(867\) −867.000 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −299.000 −0.343284
\(872\) 0 0
\(873\) 1503.00 1.72165
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1727.00 1.96921 0.984607 0.174785i \(-0.0559231\pi\)
0.984607 + 0.174785i \(0.0559231\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 443.000 0.501699 0.250849 0.968026i \(-0.419290\pi\)
0.250849 + 0.968026i \(0.419290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1898.00 −2.13498
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 3237.00 3.58472
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −214.000 −0.235943 −0.117971 0.993017i \(-0.537639\pi\)
−0.117971 + 0.993017i \(0.537639\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 971.000 1.05658 0.528292 0.849063i \(-0.322833\pi\)
0.528292 + 0.849063i \(0.322833\pi\)
\(920\) 0 0
\(921\) 759.000 0.824104
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1746.00 1.88350
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1320.00 1.41783
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1847.00 1.97118 0.985592 0.169138i \(-0.0540985\pi\)
0.985592 + 0.169138i \(0.0540985\pi\)
\(938\) 0 0
\(939\) 1371.00 1.46006
\(940\) 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
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1058.00 −1.11486
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2520.00 2.62227
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1534.00 −1.58635 −0.793175 0.608994i \(-0.791573\pi\)
−0.793175 + 0.608994i \(0.791573\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 286.000 0.293936
\(974\) 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
\(981\) 639.000 0.651376
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1739.00 1.75479 0.877397 0.479766i \(-0.159278\pi\)
0.877397 + 0.479766i \(0.159278\pi\)
\(992\) 0 0
\(993\) −1086.00 −1.09366
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1894.00 −1.89970 −0.949850 0.312707i \(-0.898764\pi\)
−0.949850 + 0.312707i \(0.898764\pi\)
\(998\) 0 0
\(999\) −702.000 −0.702703
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 300.3.g.a.101.1 1
3.2 odd 2 CM 300.3.g.a.101.1 1
4.3 odd 2 1200.3.l.e.401.1 1
5.2 odd 4 300.3.b.b.149.2 2
5.3 odd 4 300.3.b.b.149.1 2
5.4 even 2 300.3.g.c.101.1 yes 1
12.11 even 2 1200.3.l.e.401.1 1
15.2 even 4 300.3.b.b.149.2 2
15.8 even 4 300.3.b.b.149.1 2
15.14 odd 2 300.3.g.c.101.1 yes 1
20.3 even 4 1200.3.c.b.449.2 2
20.7 even 4 1200.3.c.b.449.1 2
20.19 odd 2 1200.3.l.a.401.1 1
60.23 odd 4 1200.3.c.b.449.2 2
60.47 odd 4 1200.3.c.b.449.1 2
60.59 even 2 1200.3.l.a.401.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.b.b.149.1 2 5.3 odd 4
300.3.b.b.149.1 2 15.8 even 4
300.3.b.b.149.2 2 5.2 odd 4
300.3.b.b.149.2 2 15.2 even 4
300.3.g.a.101.1 1 1.1 even 1 trivial
300.3.g.a.101.1 1 3.2 odd 2 CM
300.3.g.c.101.1 yes 1 5.4 even 2
300.3.g.c.101.1 yes 1 15.14 odd 2
1200.3.c.b.449.1 2 20.7 even 4
1200.3.c.b.449.1 2 60.47 odd 4
1200.3.c.b.449.2 2 20.3 even 4
1200.3.c.b.449.2 2 60.23 odd 4
1200.3.l.a.401.1 1 20.19 odd 2
1200.3.l.a.401.1 1 60.59 even 2
1200.3.l.e.401.1 1 4.3 odd 2
1200.3.l.e.401.1 1 12.11 even 2