Properties

Label 224.3.h.c.209.2
Level 224
Weight 3
Character 224.209
Self dual yes
Analytic conductor 6.104
Analytic rank 0
Dimension 2
CM discriminant -56
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.2
Root \(1.41421\) of \(x^{2} - 2\)
Character \(\chi\) \(=\) 224.209

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843 q^{3} +8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.82843 q^{3} +8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} -25.4558 q^{13} +24.0000 q^{15} -8.48528 q^{19} +19.7990 q^{21} +10.0000 q^{23} +47.0000 q^{25} -28.2843 q^{27} +59.3970 q^{35} -72.0000 q^{39} -8.48528 q^{45} +49.0000 q^{49} -24.0000 q^{57} -76.3675 q^{59} +8.48528 q^{61} -7.00000 q^{63} -216.000 q^{65} +28.2843 q^{69} +110.000 q^{71} +132.936 q^{75} -130.000 q^{79} -71.0000 q^{81} +25.4558 q^{83} -178.191 q^{91} -72.0000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 14q^{7} - 2q^{9} + 48q^{15} + 20q^{23} + 94q^{25} - 144q^{39} + 98q^{49} - 48q^{57} - 14q^{63} - 432q^{65} + 220q^{71} - 260q^{79} - 142q^{81} - 144q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 2.82843 0.942809 0.471405 0.881917i \(-0.343747\pi\)
0.471405 + 0.881917i \(0.343747\pi\)
\(4\) 0 0
\(5\) 8.48528 1.69706 0.848528 0.529150i \(-0.177489\pi\)
0.848528 + 0.529150i \(0.177489\pi\)
\(6\) 0 0
\(7\) 7.00000 1.00000
\(8\) 0 0
\(9\) −1.00000 −0.111111
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −25.4558 −1.95814 −0.979071 0.203519i \(-0.934762\pi\)
−0.979071 + 0.203519i \(0.934762\pi\)
\(14\) 0 0
\(15\) 24.0000 1.60000
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −8.48528 −0.446594 −0.223297 0.974750i \(-0.571682\pi\)
−0.223297 + 0.974750i \(0.571682\pi\)
\(20\) 0 0
\(21\) 19.7990 0.942809
\(22\) 0 0
\(23\) 10.0000 0.434783 0.217391 0.976085i \(-0.430245\pi\)
0.217391 + 0.976085i \(0.430245\pi\)
\(24\) 0 0
\(25\) 47.0000 1.88000
\(26\) 0 0
\(27\) −28.2843 −1.04757
\(28\) 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
\(35\) 59.3970 1.69706
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −72.0000 −1.84615
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −8.48528 −0.188562
\(46\) 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
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −24.0000 −0.421053
\(58\) 0 0
\(59\) −76.3675 −1.29436 −0.647182 0.762335i \(-0.724053\pi\)
−0.647182 + 0.762335i \(0.724053\pi\)
\(60\) 0 0
\(61\) 8.48528 0.139103 0.0695515 0.997578i \(-0.477843\pi\)
0.0695515 + 0.997578i \(0.477843\pi\)
\(62\) 0 0
\(63\) −7.00000 −0.111111
\(64\) 0 0
\(65\) −216.000 −3.32308
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 28.2843 0.409917
\(70\) 0 0
\(71\) 110.000 1.54930 0.774648 0.632393i \(-0.217927\pi\)
0.774648 + 0.632393i \(0.217927\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 132.936 1.77248
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −130.000 −1.64557 −0.822785 0.568353i \(-0.807581\pi\)
−0.822785 + 0.568353i \(0.807581\pi\)
\(80\) 0 0
\(81\) −71.0000 −0.876543
\(82\) 0 0
\(83\) 25.4558 0.306697 0.153348 0.988172i \(-0.450994\pi\)
0.153348 + 0.988172i \(0.450994\pi\)
\(84\) 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
\(91\) −178.191 −1.95814
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −72.0000 −0.757895
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −161.220 −1.59624 −0.798121 0.602498i \(-0.794172\pi\)
−0.798121 + 0.602498i \(0.794172\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 168.000 1.60000
\(106\) 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
\(113\) −26.0000 −0.230088 −0.115044 0.993360i \(-0.536701\pi\)
−0.115044 + 0.993360i \(0.536701\pi\)
\(114\) 0 0
\(115\) 84.8528 0.737851
\(116\) 0 0
\(117\) 25.4558 0.217571
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 186.676 1.49341
\(126\) 0 0
\(127\) 250.000 1.96850 0.984252 0.176771i \(-0.0565653\pi\)
0.984252 + 0.176771i \(0.0565653\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −246.073 −1.87842 −0.939211 0.343342i \(-0.888441\pi\)
−0.939211 + 0.343342i \(0.888441\pi\)
\(132\) 0 0
\(133\) −59.3970 −0.446594
\(134\) 0 0
\(135\) −240.000 −1.77778
\(136\) 0 0
\(137\) 50.0000 0.364964 0.182482 0.983209i \(-0.441587\pi\)
0.182482 + 0.983209i \(0.441587\pi\)
\(138\) 0 0
\(139\) 263.044 1.89240 0.946200 0.323581i \(-0.104887\pi\)
0.946200 + 0.323581i \(0.104887\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 138.593 0.942809
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 202.000 1.33775 0.668874 0.743376i \(-0.266776\pi\)
0.668874 + 0.743376i \(0.266776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 313.955 1.99972 0.999858 0.0168519i \(-0.00536439\pi\)
0.999858 + 0.0168519i \(0.00536439\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 70.0000 0.434783
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 479.000 2.83432
\(170\) 0 0
\(171\) 8.48528 0.0496215
\(172\) 0 0
\(173\) −229.103 −1.32429 −0.662146 0.749375i \(-0.730354\pi\)
−0.662146 + 0.749375i \(0.730354\pi\)
\(174\) 0 0
\(175\) 329.000 1.88000
\(176\) 0 0
\(177\) −216.000 −1.22034
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −263.044 −1.45328 −0.726640 0.687018i \(-0.758919\pi\)
−0.726640 + 0.687018i \(0.758919\pi\)
\(182\) 0 0
\(183\) 24.0000 0.131148
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −197.990 −1.04757
\(190\) 0 0
\(191\) −130.000 −0.680628 −0.340314 0.940312i \(-0.610533\pi\)
−0.340314 + 0.940312i \(0.610533\pi\)
\(192\) 0 0
\(193\) −314.000 −1.62694 −0.813472 0.581605i \(-0.802425\pi\)
−0.813472 + 0.581605i \(0.802425\pi\)
\(194\) 0 0
\(195\) −610.940 −3.13303
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −10.0000 −0.0483092
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 311.127 1.46069
\(214\) 0 0
\(215\) 0 0
\(216\) 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
\(225\) −47.0000 −0.208889
\(226\) 0 0
\(227\) 398.808 1.75686 0.878432 0.477867i \(-0.158590\pi\)
0.878432 + 0.477867i \(0.158590\pi\)
\(228\) 0 0
\(229\) 246.073 1.07456 0.537278 0.843405i \(-0.319453\pi\)
0.537278 + 0.843405i \(0.319453\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −430.000 −1.84549 −0.922747 0.385407i \(-0.874061\pi\)
−0.922747 + 0.385407i \(0.874061\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −367.696 −1.55146
\(238\) 0 0
\(239\) −422.000 −1.76569 −0.882845 0.469664i \(-0.844375\pi\)
−0.882845 + 0.469664i \(0.844375\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 53.7401 0.221153
\(244\) 0 0
\(245\) 415.779 1.69706
\(246\) 0 0
\(247\) 216.000 0.874494
\(248\) 0 0
\(249\) 72.0000 0.289157
\(250\) 0 0
\(251\) 500.632 1.99455 0.997274 0.0737859i \(-0.0235081\pi\)
0.997274 + 0.0737859i \(0.0235081\pi\)
\(252\) 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
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −274.000 −1.04183 −0.520913 0.853610i \(-0.674408\pi\)
−0.520913 + 0.853610i \(0.674408\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −161.220 −0.599332 −0.299666 0.954044i \(-0.596875\pi\)
−0.299666 + 0.954044i \(0.596875\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −504.000 −1.84615
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 338.000 1.20285 0.601423 0.798930i \(-0.294600\pi\)
0.601423 + 0.798930i \(0.294600\pi\)
\(282\) 0 0
\(283\) −313.955 −1.10938 −0.554692 0.832056i \(-0.687164\pi\)
−0.554692 + 0.832056i \(0.687164\pi\)
\(284\) 0 0
\(285\) −203.647 −0.714550
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 449.720 1.53488 0.767440 0.641121i \(-0.221530\pi\)
0.767440 + 0.641121i \(0.221530\pi\)
\(294\) 0 0
\(295\) −648.000 −2.19661
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −254.558 −0.851366
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −456.000 −1.50495
\(304\) 0 0
\(305\) 72.0000 0.236066
\(306\) 0 0
\(307\) −449.720 −1.46489 −0.732443 0.680829i \(-0.761620\pi\)
−0.732443 + 0.680829i \(0.761620\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −59.3970 −0.188562
\(316\) 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
\(325\) −1196.42 −3.68131
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 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
\(337\) −26.0000 −0.0771513 −0.0385757 0.999256i \(-0.512282\pi\)
−0.0385757 + 0.999256i \(0.512282\pi\)
\(338\) 0 0
\(339\) −73.5391 −0.216930
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 343.000 1.00000
\(344\) 0 0
\(345\) 240.000 0.695652
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 687.308 1.96936 0.984682 0.174362i \(-0.0557862\pi\)
0.984682 + 0.174362i \(0.0557862\pi\)
\(350\) 0 0
\(351\) 720.000 2.05128
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 933.381 2.62924
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 682.000 1.89972 0.949861 0.312673i \(-0.101225\pi\)
0.949861 + 0.312673i \(0.101225\pi\)
\(360\) 0 0
\(361\) −289.000 −0.800554
\(362\) 0 0
\(363\) 342.240 0.942809
\(364\) 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
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 528.000 1.40800
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 707.107 1.85592
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −696.000 −1.77099
\(394\) 0 0
\(395\) −1103.09 −2.79262
\(396\) 0 0
\(397\) 483.661 1.21829 0.609145 0.793059i \(-0.291513\pi\)
0.609145 + 0.793059i \(0.291513\pi\)
\(398\) 0 0
\(399\) −168.000 −0.421053
\(400\) 0 0
\(401\) 550.000 1.37157 0.685786 0.727804i \(-0.259459\pi\)
0.685786 + 0.727804i \(0.259459\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −602.455 −1.48754
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 141.421 0.344091
\(412\) 0 0
\(413\) −534.573 −1.29436
\(414\) 0 0
\(415\) 216.000 0.520482
\(416\) 0 0
\(417\) 744.000 1.78417
\(418\) 0 0
\(419\) 500.632 1.19482 0.597412 0.801934i \(-0.296196\pi\)
0.597412 + 0.801934i \(0.296196\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 59.3970 0.139103
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 538.000 1.24826 0.624130 0.781321i \(-0.285454\pi\)
0.624130 + 0.781321i \(0.285454\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −84.8528 −0.194171
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −49.0000 −0.111111
\(442\) 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
\(449\) 2.00000 0.00445434 0.00222717 0.999998i \(-0.499291\pi\)
0.00222717 + 0.999998i \(0.499291\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 571.342 1.26124
\(454\) 0 0
\(455\) −1512.00 −3.32308
\(456\) 0 0
\(457\) 886.000 1.93873 0.969365 0.245623i \(-0.0789925\pi\)
0.969365 + 0.245623i \(0.0789925\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −263.044 −0.570594 −0.285297 0.958439i \(-0.592092\pi\)
−0.285297 + 0.958439i \(0.592092\pi\)
\(462\) 0 0
\(463\) −226.000 −0.488121 −0.244060 0.969760i \(-0.578480\pi\)
−0.244060 + 0.969760i \(0.578480\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −483.661 −1.03568 −0.517838 0.855478i \(-0.673263\pi\)
−0.517838 + 0.855478i \(0.673263\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 888.000 1.88535
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −398.808 −0.839596
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 197.990 0.409917
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −470.000 −0.965092 −0.482546 0.875871i \(-0.660288\pi\)
−0.482546 + 0.875871i \(0.660288\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(497\) 770.000 1.54930
\(498\) 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
\(505\) −1368.00 −2.70891
\(506\) 0 0
\(507\) 1354.82 2.67222
\(508\) 0 0
\(509\) −941.866 −1.85042 −0.925212 0.379450i \(-0.876113\pi\)
−0.925212 + 0.379450i \(0.876113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 240.000 0.467836
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −648.000 −1.24855
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −958.837 −1.83334 −0.916670 0.399645i \(-0.869133\pi\)
−0.916670 + 0.399645i \(0.869133\pi\)
\(524\) 0 0
\(525\) 930.553 1.77248
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −429.000 −0.810964
\(530\) 0 0
\(531\) 76.3675 0.143818
\(532\) 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
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −744.000 −1.37017
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −8.48528 −0.0154559
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −910.000 −1.64557
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 398.808 0.708363 0.354181 0.935177i \(-0.384760\pi\)
0.354181 + 0.935177i \(0.384760\pi\)
\(564\) 0 0
\(565\) −220.617 −0.390473
\(566\) 0 0
\(567\) −497.000 −0.876543
\(568\) 0 0
\(569\) −1130.00 −1.98594 −0.992970 0.118365i \(-0.962235\pi\)
−0.992970 + 0.118365i \(0.962235\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −367.696 −0.641702
\(574\) 0 0
\(575\) 470.000 0.817391
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) −888.126 −1.53390
\(580\) 0 0
\(581\) 178.191 0.306697
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 216.000 0.369231
\(586\) 0 0
\(587\) −1162.48 −1.98038 −0.990190 0.139725i \(-0.955378\pi\)
−0.990190 + 0.139725i \(0.955378\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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
\(597\) 0 0
\(598\) 0 0
\(599\) 1070.00 1.78631 0.893155 0.449748i \(-0.148486\pi\)
0.893155 + 0.449748i \(0.148486\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1026.72 1.69706
\(606\) 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
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1034.00 −1.67585 −0.837925 0.545785i \(-0.816232\pi\)
−0.837925 + 0.545785i \(0.816232\pi\)
\(618\) 0 0
\(619\) 1111.57 1.79575 0.897877 0.440246i \(-0.145109\pi\)
0.897877 + 0.440246i \(0.145109\pi\)
\(620\) 0 0
\(621\) −282.843 −0.455463
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 409.000 0.654400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 110.000 0.174326 0.0871632 0.996194i \(-0.472220\pi\)
0.0871632 + 0.996194i \(0.472220\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2121.32 3.34066
\(636\) 0 0
\(637\) −1247.34 −1.95814
\(638\) 0 0
\(639\) −110.000 −0.172144
\(640\) 0 0
\(641\) 1030.00 1.60686 0.803432 0.595396i \(-0.203005\pi\)
0.803432 + 0.595396i \(0.203005\pi\)
\(642\) 0 0
\(643\) 738.219 1.14809 0.574043 0.818825i \(-0.305374\pi\)
0.574043 + 0.818825i \(0.305374\pi\)
\(644\) 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
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −2088.00 −3.18779
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1264.31 1.91272 0.956359 0.292193i \(-0.0943851\pi\)
0.956359 + 0.292193i \(0.0943851\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −504.000 −0.757895
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −670.000 −0.995542 −0.497771 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) 0 0
\(675\) −1329.36 −1.96942
\(676\) 0 0
\(677\) 483.661 0.714418 0.357209 0.934024i \(-0.383728\pi\)
0.357209 + 0.934024i \(0.383728\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1128.00 1.65639
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 424.264 0.619364
\(686\) 0 0
\(687\) 696.000 1.01310
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −246.073 −0.356112 −0.178056 0.984020i \(-0.556981\pi\)
−0.178056 + 0.984020i \(0.556981\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2232.00 3.21151
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1216.22 −1.73995
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1128.54 −1.59624
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 130.000 0.182841
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1193.60 −1.66471
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 791.000 1.08505
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1417.04 −1.93321 −0.966604 0.256273i \(-0.917505\pi\)
−0.966604 + 0.256273i \(0.917505\pi\)
\(734\) 0 0
\(735\) 1176.00 1.60000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 610.940 0.824481
\(742\) 0 0
\(743\) −1430.00 −1.92463 −0.962315 0.271937i \(-0.912336\pi\)
−0.962315 + 0.271937i \(0.912336\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −25.4558 −0.0340774
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −998.000 −1.32889 −0.664447 0.747335i \(-0.731333\pi\)
−0.664447 + 0.747335i \(0.731333\pi\)
\(752\) 0 0
\(753\) 1416.00 1.88048
\(754\) 0 0
\(755\) 1714.03 2.27023
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1944.00 2.53455
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −873.984 −1.13064 −0.565320 0.824872i \(-0.691247\pi\)
−0.565320 + 0.824872i \(0.691247\pi\)
\(774\) 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
\(785\) 2664.00 3.39363
\(786\) 0 0
\(787\) −1501.89 −1.90838 −0.954190 0.299202i \(-0.903280\pi\)
−0.954190 + 0.299202i \(0.903280\pi\)
\(788\) 0 0
\(789\) −774.989 −0.982242
\(790\) 0 0
\(791\) −182.000 −0.230088
\(792\) 0 0
\(793\) −216.000 −0.272383
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 449.720 0.564266 0.282133 0.959375i \(-0.408958\pi\)
0.282133 + 0.959375i \(0.408958\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 593.970 0.737851
\(806\) 0 0
\(807\) −456.000 −0.565056
\(808\) 0 0
\(809\) −650.000 −0.803461 −0.401731 0.915758i \(-0.631591\pi\)
−0.401731 + 0.915758i \(0.631591\pi\)
\(810\) 0 0
\(811\) 1450.98 1.78913 0.894564 0.446939i \(-0.147486\pi\)
0.894564 + 0.446939i \(0.147486\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 178.191 0.217571
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −946.000 −1.14945 −0.574727 0.818345i \(-0.694892\pi\)
−0.574727 + 0.818345i \(0.694892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 76.3675 0.0921201 0.0460600 0.998939i \(-0.485333\pi\)
0.0460600 + 0.998939i \(0.485333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 956.008 1.13406
\(844\) 0 0
\(845\) 4064.45 4.81000
\(846\) 0 0
\(847\) 847.000 1.00000
\(848\) 0 0
\(849\) −888.000 −1.04594
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 449.720 0.527221 0.263611 0.964629i \(-0.415087\pi\)
0.263611 + 0.964629i \(0.415087\pi\)
\(854\) 0 0
\(855\) 72.0000 0.0842105
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1196.42 −1.39281 −0.696406 0.717648i \(-0.745218\pi\)
−0.696406 + 0.717648i \(0.745218\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1474.00 −1.70800 −0.853998 0.520277i \(-0.825829\pi\)
−0.853998 + 0.520277i \(0.825829\pi\)
\(864\) 0 0
\(865\) −1944.00 −2.24740
\(866\) 0 0
\(867\) 817.415 0.942809
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1306.73 1.49341
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 1272.00 1.44710
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −1832.82 −2.07098
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1750.00 1.96850
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −720.000 −0.802676
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2232.00 −2.46630
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 161.220 0.177360
\(910\) 0 0
\(911\) 922.000 1.01207 0.506037 0.862512i \(-0.331110\pi\)
0.506037 + 0.862512i \(0.331110\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 203.647 0.222565
\(916\) 0 0
\(917\) −1722.51 −1.87842
\(918\) 0 0
\(919\) 1550.00 1.68662 0.843308 0.537431i \(-0.180605\pi\)
0.843308 + 0.537431i \(0.180605\pi\)
\(920\) 0 0
\(921\) −1272.00 −1.38111
\(922\) 0 0
\(923\) −2800.14 −3.03374
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −415.779 −0.446594
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1111.57 −1.18127 −0.590633 0.806940i \(-0.701122\pi\)
−0.590633 + 0.806940i \(0.701122\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1680.00 −1.77778
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1010.00 1.05981 0.529906 0.848057i \(-0.322227\pi\)
0.529906 + 0.848057i \(0.322227\pi\)
\(954\) 0 0
\(955\) −1103.09 −1.15506
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 350.000 0.364964
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2664.38 −2.76101
\(966\) 0 0
\(967\) −1430.00 −1.47880 −0.739400 0.673266i \(-0.764891\pi\)
−0.739400 + 0.673266i \(0.764891\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.48528 −0.00873870 −0.00436935 0.999990i \(-0.501391\pi\)
−0.00436935 + 0.999990i \(0.501391\pi\)
\(972\) 0 0
\(973\) 1841.31 1.89240
\(974\) 0 0
\(975\) −3384.00 −3.47077
\(976\) 0 0
\(977\) −1630.00 −1.66837 −0.834186 0.551483i \(-0.814062\pi\)
−0.834186 + 0.551483i \(0.814062\pi\)
\(978\) 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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −610.000 −0.615540 −0.307770 0.951461i \(-0.599583\pi\)
−0.307770 + 0.951461i \(0.599583\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1977.07 1.98302 0.991510 0.130032i \(-0.0415080\pi\)
0.991510 + 0.130032i \(0.0415080\pi\)
\(998\) 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 224.3.h.c.209.2 2
3.2 odd 2 2016.3.l.c.433.1 2
4.3 odd 2 56.3.h.c.13.1 2
7.6 odd 2 inner 224.3.h.c.209.1 2
8.3 odd 2 56.3.h.c.13.2 yes 2
8.5 even 2 inner 224.3.h.c.209.1 2
12.11 even 2 504.3.l.a.181.1 2
21.20 even 2 2016.3.l.c.433.2 2
24.5 odd 2 2016.3.l.c.433.2 2
24.11 even 2 504.3.l.a.181.2 2
28.3 even 6 392.3.j.a.117.1 4
28.11 odd 6 392.3.j.a.117.2 4
28.19 even 6 392.3.j.a.325.1 4
28.23 odd 6 392.3.j.a.325.2 4
28.27 even 2 56.3.h.c.13.2 yes 2
56.3 even 6 392.3.j.a.117.2 4
56.11 odd 6 392.3.j.a.117.1 4
56.13 odd 2 CM 224.3.h.c.209.2 2
56.19 even 6 392.3.j.a.325.2 4
56.27 even 2 56.3.h.c.13.1 2
56.51 odd 6 392.3.j.a.325.1 4
84.83 odd 2 504.3.l.a.181.2 2
168.83 odd 2 504.3.l.a.181.1 2
168.125 even 2 2016.3.l.c.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.h.c.13.1 2 4.3 odd 2
56.3.h.c.13.1 2 56.27 even 2
56.3.h.c.13.2 yes 2 8.3 odd 2
56.3.h.c.13.2 yes 2 28.27 even 2
224.3.h.c.209.1 2 7.6 odd 2 inner
224.3.h.c.209.1 2 8.5 even 2 inner
224.3.h.c.209.2 2 1.1 even 1 trivial
224.3.h.c.209.2 2 56.13 odd 2 CM
392.3.j.a.117.1 4 28.3 even 6
392.3.j.a.117.1 4 56.11 odd 6
392.3.j.a.117.2 4 28.11 odd 6
392.3.j.a.117.2 4 56.3 even 6
392.3.j.a.325.1 4 28.19 even 6
392.3.j.a.325.1 4 56.51 odd 6
392.3.j.a.325.2 4 28.23 odd 6
392.3.j.a.325.2 4 56.19 even 6
504.3.l.a.181.1 2 12.11 even 2
504.3.l.a.181.1 2 168.83 odd 2
504.3.l.a.181.2 2 24.11 even 2
504.3.l.a.181.2 2 84.83 odd 2
2016.3.l.c.433.1 2 3.2 odd 2
2016.3.l.c.433.1 2 168.125 even 2
2016.3.l.c.433.2 2 21.20 even 2
2016.3.l.c.433.2 2 24.5 odd 2