Properties

Label 900.3.c.c.451.1
Level $900$
Weight $3$
Character 900.451
Self dual yes
Analytic conductor $24.523$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 451.1
Character \(\chi\) \(=\) 900.451

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +8.00000 q^{8} +10.0000 q^{13} +16.0000 q^{16} +16.0000 q^{17} +20.0000 q^{26} -40.0000 q^{29} +32.0000 q^{32} +32.0000 q^{34} +70.0000 q^{37} +80.0000 q^{41} +49.0000 q^{49} +40.0000 q^{52} -56.0000 q^{53} -80.0000 q^{58} -22.0000 q^{61} +64.0000 q^{64} +64.0000 q^{68} -110.000 q^{73} +140.000 q^{74} +160.000 q^{82} -160.000 q^{89} +130.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 10.0000 0.769231 0.384615 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 16.0000 0.941176 0.470588 0.882353i \(-0.344042\pi\)
0.470588 + 0.882353i \(0.344042\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 20.0000 0.769231
\(27\) 0 0
\(28\) 0 0
\(29\) −40.0000 −1.37931 −0.689655 0.724138i \(-0.742238\pi\)
−0.689655 + 0.724138i \(0.742238\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 32.0000 0.941176
\(35\) 0 0
\(36\) 0 0
\(37\) 70.0000 1.89189 0.945946 0.324324i \(-0.105137\pi\)
0.945946 + 0.324324i \(0.105137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 80.0000 1.95122 0.975610 0.219512i \(-0.0704466\pi\)
0.975610 + 0.219512i \(0.0704466\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 40.0000 0.769231
\(53\) −56.0000 −1.05660 −0.528302 0.849057i \(-0.677171\pi\)
−0.528302 + 0.849057i \(0.677171\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −80.0000 −1.37931
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −22.0000 −0.360656 −0.180328 0.983607i \(-0.557716\pi\)
−0.180328 + 0.983607i \(0.557716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 64.0000 0.941176
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −110.000 −1.50685 −0.753425 0.657534i \(-0.771599\pi\)
−0.753425 + 0.657534i \(0.771599\pi\)
\(74\) 140.000 1.89189
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 160.000 1.95122
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −160.000 −1.79775 −0.898876 0.438202i \(-0.855615\pi\)
−0.898876 + 0.438202i \(0.855615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 130.000 1.34021 0.670103 0.742268i \(-0.266250\pi\)
0.670103 + 0.742268i \(0.266250\pi\)
\(98\) 98.0000 1.00000
\(99\) 0 0
\(100\) 0 0
\(101\) −40.0000 −0.396040 −0.198020 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 80.0000 0.769231
\(105\) 0 0
\(106\) −112.000 −1.05660
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 182.000 1.66972 0.834862 0.550459i \(-0.185547\pi\)
0.834862 + 0.550459i \(0.185547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −224.000 −1.98230 −0.991150 0.132743i \(-0.957621\pi\)
−0.991150 + 0.132743i \(0.957621\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −160.000 −1.37931
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) −44.0000 −0.360656
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 128.000 0.941176
\(137\) −176.000 −1.28467 −0.642336 0.766423i \(-0.722035\pi\)
−0.642336 + 0.766423i \(0.722035\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −220.000 −1.50685
\(147\) 0 0
\(148\) 280.000 1.89189
\(149\) −280.000 −1.87919 −0.939597 0.342282i \(-0.888800\pi\)
−0.939597 + 0.342282i \(0.888800\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −170.000 −1.08280 −0.541401 0.840764i \(-0.682106\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 320.000 1.95122
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −69.0000 −0.408284
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −104.000 −0.601156 −0.300578 0.953757i \(-0.597180\pi\)
−0.300578 + 0.953757i \(0.597180\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −320.000 −1.79775
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 38.0000 0.209945 0.104972 0.994475i \(-0.466525\pi\)
0.104972 + 0.994475i \(0.466525\pi\)
\(182\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 190.000 0.984456 0.492228 0.870466i \(-0.336183\pi\)
0.492228 + 0.870466i \(0.336183\pi\)
\(194\) 260.000 1.34021
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) −56.0000 −0.284264 −0.142132 0.989848i \(-0.545396\pi\)
−0.142132 + 0.989848i \(0.545396\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −80.0000 −0.396040
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 160.000 0.769231
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −224.000 −1.05660
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 364.000 1.66972
\(219\) 0 0
\(220\) 0 0
\(221\) 160.000 0.723982
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −448.000 −1.98230
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −442.000 −1.93013 −0.965066 0.262009i \(-0.915615\pi\)
−0.965066 + 0.262009i \(0.915615\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −320.000 −1.37931
\(233\) −416.000 −1.78541 −0.892704 0.450644i \(-0.851194\pi\)
−0.892704 + 0.450644i \(0.851194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −418.000 −1.73444 −0.867220 0.497925i \(-0.834095\pi\)
−0.867220 + 0.497925i \(0.834095\pi\)
\(242\) 242.000 1.00000
\(243\) 0 0
\(244\) −88.0000 −0.360656
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 256.000 1.00000
\(257\) 64.0000 0.249027 0.124514 0.992218i \(-0.460263\pi\)
0.124514 + 0.992218i \(0.460263\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 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
\(269\) −520.000 −1.93309 −0.966543 0.256506i \(-0.917429\pi\)
−0.966543 + 0.256506i \(0.917429\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 256.000 0.941176
\(273\) 0 0
\(274\) −352.000 −1.28467
\(275\) 0 0
\(276\) 0 0
\(277\) −230.000 −0.830325 −0.415162 0.909747i \(-0.636275\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 320.000 1.13879 0.569395 0.822064i \(-0.307178\pi\)
0.569395 + 0.822064i \(0.307178\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −33.0000 −0.114187
\(290\) 0 0
\(291\) 0 0
\(292\) −440.000 −1.50685
\(293\) 136.000 0.464164 0.232082 0.972696i \(-0.425446\pi\)
0.232082 + 0.972696i \(0.425446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 560.000 1.89189
\(297\) 0 0
\(298\) −560.000 −1.87919
\(299\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −50.0000 −0.159744 −0.0798722 0.996805i \(-0.525451\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) −340.000 −1.08280
\(315\) 0 0
\(316\) 0 0
\(317\) 616.000 1.94322 0.971609 0.236593i \(-0.0760308\pi\)
0.971609 + 0.236593i \(0.0760308\pi\)
\(318\) 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
\(327\) 0 0
\(328\) 640.000 1.95122
\(329\) 0 0
\(330\) 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
\(337\) −350.000 −1.03858 −0.519288 0.854599i \(-0.673803\pi\)
−0.519288 + 0.854599i \(0.673803\pi\)
\(338\) −138.000 −0.408284
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −208.000 −0.601156
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −598.000 −1.71347 −0.856734 0.515759i \(-0.827510\pi\)
−0.856734 + 0.515759i \(0.827510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 544.000 1.54108 0.770538 0.637394i \(-0.219988\pi\)
0.770538 + 0.637394i \(0.219988\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −640.000 −1.79775
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 76.0000 0.209945
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 550.000 1.47453 0.737265 0.675603i \(-0.236117\pi\)
0.737265 + 0.675603i \(0.236117\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −400.000 −1.06101
\(378\) 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
\(385\) 0 0
\(386\) 380.000 0.984456
\(387\) 0 0
\(388\) 520.000 1.34021
\(389\) 680.000 1.74807 0.874036 0.485861i \(-0.161494\pi\)
0.874036 + 0.485861i \(0.161494\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 392.000 1.00000
\(393\) 0 0
\(394\) −112.000 −0.284264
\(395\) 0 0
\(396\) 0 0
\(397\) −650.000 −1.63728 −0.818640 0.574307i \(-0.805271\pi\)
−0.818640 + 0.574307i \(0.805271\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 80.0000 0.199501 0.0997506 0.995012i \(-0.468195\pi\)
0.0997506 + 0.995012i \(0.468195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −160.000 −0.396040
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 782.000 1.91198 0.955990 0.293399i \(-0.0947863\pi\)
0.955990 + 0.293399i \(0.0947863\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 320.000 0.769231
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −58.0000 −0.137767 −0.0688836 0.997625i \(-0.521944\pi\)
−0.0688836 + 0.997625i \(0.521944\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −448.000 −1.05660
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −290.000 −0.669746 −0.334873 0.942263i \(-0.608693\pi\)
−0.334873 + 0.942263i \(0.608693\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 728.000 1.66972
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 320.000 0.723982
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 560.000 1.24722 0.623608 0.781737i \(-0.285666\pi\)
0.623608 + 0.781737i \(0.285666\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −896.000 −1.98230
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 850.000 1.85996 0.929978 0.367615i \(-0.119826\pi\)
0.929978 + 0.367615i \(0.119826\pi\)
\(458\) −884.000 −1.93013
\(459\) 0 0
\(460\) 0 0
\(461\) −760.000 −1.64859 −0.824295 0.566161i \(-0.808428\pi\)
−0.824295 + 0.566161i \(0.808428\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −640.000 −1.37931
\(465\) 0 0
\(466\) −832.000 −1.78541
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 700.000 1.45530
\(482\) −836.000 −1.73444
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −176.000 −0.360656
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −640.000 −1.29817
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 0 0
\(509\) 440.000 0.864440 0.432220 0.901768i \(-0.357730\pi\)
0.432220 + 0.901768i \(0.357730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) 128.000 0.249027
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −880.000 −1.68906 −0.844530 0.535509i \(-0.820120\pi\)
−0.844530 + 0.535509i \(0.820120\pi\)
\(522\) 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
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 800.000 1.50094
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1040.00 −1.93309
\(539\) 0 0
\(540\) 0 0
\(541\) −682.000 −1.26063 −0.630314 0.776340i \(-0.717074\pi\)
−0.630314 + 0.776340i \(0.717074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 512.000 0.941176
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −704.000 −1.28467
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −460.000 −0.830325
\(555\) 0 0
\(556\) 0 0
\(557\) −1064.00 −1.91023 −0.955117 0.296230i \(-0.904271\pi\)
−0.955117 + 0.296230i \(0.904271\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 640.000 1.13879
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1040.00 1.82777 0.913884 0.405975i \(-0.133068\pi\)
0.913884 + 0.405975i \(0.133068\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1150.00 1.99307 0.996534 0.0831889i \(-0.0265105\pi\)
0.996534 + 0.0831889i \(0.0265105\pi\)
\(578\) −66.0000 −0.114187
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −880.000 −1.50685
\(585\) 0 0
\(586\) 272.000 0.464164
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1120.00 1.89189
\(593\) 736.000 1.24115 0.620573 0.784148i \(-0.286900\pi\)
0.620573 + 0.784148i \(0.286900\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1120.00 −1.87919
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1102.00 −1.83361 −0.916805 0.399334i \(-0.869241\pi\)
−0.916805 + 0.399334i \(0.869241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(613\) 70.0000 0.114192 0.0570962 0.998369i \(-0.481816\pi\)
0.0570962 + 0.998369i \(0.481816\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1216.00 1.97083 0.985413 0.170178i \(-0.0544344\pi\)
0.985413 + 0.170178i \(0.0544344\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −100.000 −0.159744
\(627\) 0 0
\(628\) −680.000 −1.08280
\(629\) 1120.00 1.78060
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1232.00 1.94322
\(635\) 0 0
\(636\) 0 0
\(637\) 490.000 0.769231
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −400.000 −0.624025 −0.312012 0.950078i \(-0.601003\pi\)
−0.312012 + 0.950078i \(0.601003\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 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
\(651\) 0 0
\(652\) 0 0
\(653\) 1144.00 1.75191 0.875957 0.482389i \(-0.160231\pi\)
0.875957 + 0.482389i \(0.160231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1280.00 1.95122
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1178.00 1.78215 0.891074 0.453858i \(-0.149953\pi\)
0.891074 + 0.453858i \(0.149953\pi\)
\(662\) 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
\(673\) −770.000 −1.14413 −0.572065 0.820208i \(-0.693858\pi\)
−0.572065 + 0.820208i \(0.693858\pi\)
\(674\) −700.000 −1.03858
\(675\) 0 0
\(676\) −276.000 −0.408284
\(677\) −104.000 −0.153619 −0.0768095 0.997046i \(-0.524473\pi\)
−0.0768095 + 0.997046i \(0.524473\pi\)
\(678\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −560.000 −0.812772
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −416.000 −0.601156
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1280.00 1.83644
\(698\) −1196.00 −1.71347
\(699\) 0 0
\(700\) 0 0
\(701\) −520.000 −0.741797 −0.370899 0.928673i \(-0.620950\pi\)
−0.370899 + 0.928673i \(0.620950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1088.00 1.54108
\(707\) 0 0
\(708\) 0 0
\(709\) 518.000 0.730606 0.365303 0.930889i \(-0.380965\pi\)
0.365303 + 0.930889i \(0.380965\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1280.00 −1.79775
\(713\) 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
\(721\) 0 0
\(722\) 722.000 1.00000
\(723\) 0 0
\(724\) 152.000 0.209945
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1450.00 1.97817 0.989086 0.147340i \(-0.0470711\pi\)
0.989086 + 0.147340i \(0.0470711\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1100.00 1.47453
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −800.000 −1.06101
\(755\) 0 0
\(756\) 0 0
\(757\) −1190.00 −1.57199 −0.785997 0.618230i \(-0.787850\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1520.00 1.99737 0.998686 0.0512484i \(-0.0163200\pi\)
0.998686 + 0.0512484i \(0.0163200\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 760.000 0.984456
\(773\) −1496.00 −1.93532 −0.967658 0.252264i \(-0.918825\pi\)
−0.967658 + 0.252264i \(0.918825\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1040.00 1.34021
\(777\) 0 0
\(778\) 1360.00 1.74807
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −224.000 −0.284264
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −220.000 −0.277427
\(794\) −1300.00 −1.63728
\(795\) 0 0
\(796\) 0 0
\(797\) 1144.00 1.43538 0.717691 0.696361i \(-0.245199\pi\)
0.717691 + 0.696361i \(0.245199\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 160.000 0.199501
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −320.000 −0.396040
\(809\) 560.000 0.692213 0.346106 0.938195i \(-0.387504\pi\)
0.346106 + 0.938195i \(0.387504\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1564.00 1.91198
\(819\) 0 0
\(820\) 0 0
\(821\) 1400.00 1.70524 0.852619 0.522533i \(-0.175013\pi\)
0.852619 + 0.522533i \(0.175013\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1258.00 −1.51749 −0.758745 0.651387i \(-0.774187\pi\)
−0.758745 + 0.651387i \(0.774187\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 640.000 0.769231
\(833\) 784.000 0.941176
\(834\) 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
\(841\) 759.000 0.902497
\(842\) −116.000 −0.137767
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −896.000 −1.05660
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −410.000 −0.480657 −0.240328 0.970692i \(-0.577255\pi\)
−0.240328 + 0.970692i \(0.577255\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −464.000 −0.541424 −0.270712 0.962660i \(-0.587259\pi\)
−0.270712 + 0.962660i \(0.587259\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −580.000 −0.669746
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1456.00 1.66972
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1610.00 −1.83580 −0.917902 0.396807i \(-0.870118\pi\)
−0.917902 + 0.396807i \(0.870118\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1600.00 −1.81612 −0.908059 0.418842i \(-0.862436\pi\)
−0.908059 + 0.418842i \(0.862436\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 640.000 0.723982
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1120.00 1.24722
\(899\) 0 0
\(900\) 0 0
\(901\) −896.000 −0.994451
\(902\) 0 0
\(903\) 0 0
\(904\) −1792.00 −1.98230
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1700.00 1.85996
\(915\) 0 0
\(916\) −1768.00 −1.93013
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1520.00 −1.64859
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) −1280.00 −1.37931
\(929\) −1840.00 −1.98062 −0.990312 0.138859i \(-0.955657\pi\)
−0.990312 + 0.138859i \(0.955657\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1664.00 −1.78541
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 430.000 0.458911 0.229456 0.973319i \(-0.426305\pi\)
0.229456 + 0.973319i \(0.426305\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1160.00 1.23273 0.616366 0.787460i \(-0.288604\pi\)
0.616366 + 0.787460i \(0.288604\pi\)
\(942\) 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
\(949\) −1100.00 −1.15911
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1456.00 1.52781 0.763903 0.645331i \(-0.223280\pi\)
0.763903 + 0.645331i \(0.223280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 1400.00 1.45530
\(963\) 0 0
\(964\) −1672.00 −1.73444
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 968.000 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −352.000 −0.360656
\(977\) 496.000 0.507677 0.253838 0.967247i \(-0.418307\pi\)
0.253838 + 0.967247i \(0.418307\pi\)
\(978\) 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
\(985\) 0 0
\(986\) −1280.00 −1.29817
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1850.00 −1.85557 −0.927783 0.373119i \(-0.878288\pi\)
−0.927783 + 0.373119i \(0.878288\pi\)
\(998\) 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 900.3.c.c.451.1 1
3.2 odd 2 900.3.c.b.451.1 1
4.3 odd 2 CM 900.3.c.c.451.1 1
5.2 odd 4 900.3.f.b.199.2 2
5.3 odd 4 900.3.f.b.199.1 2
5.4 even 2 36.3.d.a.19.1 1
12.11 even 2 900.3.c.b.451.1 1
15.2 even 4 900.3.f.a.199.1 2
15.8 even 4 900.3.f.a.199.2 2
15.14 odd 2 36.3.d.b.19.1 yes 1
20.3 even 4 900.3.f.b.199.1 2
20.7 even 4 900.3.f.b.199.2 2
20.19 odd 2 36.3.d.a.19.1 1
40.19 odd 2 576.3.g.a.127.1 1
40.29 even 2 576.3.g.a.127.1 1
45.4 even 6 324.3.f.f.55.1 2
45.14 odd 6 324.3.f.e.55.1 2
45.29 odd 6 324.3.f.e.271.1 2
45.34 even 6 324.3.f.f.271.1 2
60.23 odd 4 900.3.f.a.199.2 2
60.47 odd 4 900.3.f.a.199.1 2
60.59 even 2 36.3.d.b.19.1 yes 1
80.19 odd 4 2304.3.b.d.127.1 2
80.29 even 4 2304.3.b.d.127.1 2
80.59 odd 4 2304.3.b.d.127.2 2
80.69 even 4 2304.3.b.d.127.2 2
120.29 odd 2 576.3.g.c.127.1 1
120.59 even 2 576.3.g.c.127.1 1
180.59 even 6 324.3.f.e.55.1 2
180.79 odd 6 324.3.f.f.271.1 2
180.119 even 6 324.3.f.e.271.1 2
180.139 odd 6 324.3.f.f.55.1 2
240.29 odd 4 2304.3.b.e.127.2 2
240.59 even 4 2304.3.b.e.127.1 2
240.149 odd 4 2304.3.b.e.127.1 2
240.179 even 4 2304.3.b.e.127.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.d.a.19.1 1 5.4 even 2
36.3.d.a.19.1 1 20.19 odd 2
36.3.d.b.19.1 yes 1 15.14 odd 2
36.3.d.b.19.1 yes 1 60.59 even 2
324.3.f.e.55.1 2 45.14 odd 6
324.3.f.e.55.1 2 180.59 even 6
324.3.f.e.271.1 2 45.29 odd 6
324.3.f.e.271.1 2 180.119 even 6
324.3.f.f.55.1 2 45.4 even 6
324.3.f.f.55.1 2 180.139 odd 6
324.3.f.f.271.1 2 45.34 even 6
324.3.f.f.271.1 2 180.79 odd 6
576.3.g.a.127.1 1 40.19 odd 2
576.3.g.a.127.1 1 40.29 even 2
576.3.g.c.127.1 1 120.29 odd 2
576.3.g.c.127.1 1 120.59 even 2
900.3.c.b.451.1 1 3.2 odd 2
900.3.c.b.451.1 1 12.11 even 2
900.3.c.c.451.1 1 1.1 even 1 trivial
900.3.c.c.451.1 1 4.3 odd 2 CM
900.3.f.a.199.1 2 15.2 even 4
900.3.f.a.199.1 2 60.47 odd 4
900.3.f.a.199.2 2 15.8 even 4
900.3.f.a.199.2 2 60.23 odd 4
900.3.f.b.199.1 2 5.3 odd 4
900.3.f.b.199.1 2 20.3 even 4
900.3.f.b.199.2 2 5.2 odd 4
900.3.f.b.199.2 2 20.7 even 4
2304.3.b.d.127.1 2 80.19 odd 4
2304.3.b.d.127.1 2 80.29 even 4
2304.3.b.d.127.2 2 80.59 odd 4
2304.3.b.d.127.2 2 80.69 even 4
2304.3.b.e.127.1 2 240.59 even 4
2304.3.b.e.127.1 2 240.149 odd 4
2304.3.b.e.127.2 2 240.29 odd 4
2304.3.b.e.127.2 2 240.179 even 4