Properties

Label 1216.3.e.f.1025.2
Level $1216$
Weight $3$
Character 1216.1025
Self dual yes
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1025.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 1216.1025

$q$-expansion

\(f(q)\) \(=\) \(q-0.725083 q^{5} +13.8248 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-0.725083 q^{5} +13.8248 q^{7} +9.00000 q^{9} +20.3746 q^{11} +18.9244 q^{17} +19.0000 q^{19} -30.0000 q^{23} -24.4743 q^{25} -10.0241 q^{35} -53.8248 q^{43} -6.52575 q^{45} -86.5739 q^{47} +142.124 q^{49} -14.7733 q^{55} -5.12376 q^{61} +124.423 q^{63} -112.072 q^{73} +281.674 q^{77} +81.0000 q^{81} -90.0000 q^{83} -13.7218 q^{85} -13.7766 q^{95} +183.371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{5} + 5 q^{7} + 18 q^{9} + O(q^{10}) \) \( 2 q - 9 q^{5} + 5 q^{7} + 18 q^{9} + 3 q^{11} - 15 q^{17} + 38 q^{19} - 60 q^{23} + 19 q^{25} + 63 q^{35} - 85 q^{43} - 81 q^{45} - 75 q^{47} + 171 q^{49} + 129 q^{55} + 103 q^{61} + 45 q^{63} + 25 q^{73} + 435 q^{77} + 162 q^{81} - 180 q^{83} + 267 q^{85} - 171 q^{95} + 27 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) −0.725083 −0.145017 −0.0725083 0.997368i \(-0.523100\pi\)
−0.0725083 + 0.997368i \(0.523100\pi\)
\(6\) 0 0
\(7\) 13.8248 1.97496 0.987482 0.157730i \(-0.0504176\pi\)
0.987482 + 0.157730i \(0.0504176\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 20.3746 1.85224 0.926118 0.377235i \(-0.123125\pi\)
0.926118 + 0.377235i \(0.123125\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.9244 1.11320 0.556601 0.830780i \(-0.312105\pi\)
0.556601 + 0.830780i \(0.312105\pi\)
\(18\) 0 0
\(19\) 19.0000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −30.0000 −1.30435 −0.652174 0.758069i \(-0.726143\pi\)
−0.652174 + 0.758069i \(0.726143\pi\)
\(24\) 0 0
\(25\) −24.4743 −0.978970
\(26\) 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
\(35\) −10.0241 −0.286403
\(36\) 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
\(43\) −53.8248 −1.25174 −0.625869 0.779928i \(-0.715256\pi\)
−0.625869 + 0.779928i \(0.715256\pi\)
\(44\) 0 0
\(45\) −6.52575 −0.145017
\(46\) 0 0
\(47\) −86.5739 −1.84200 −0.920999 0.389564i \(-0.872626\pi\)
−0.920999 + 0.389564i \(0.872626\pi\)
\(48\) 0 0
\(49\) 142.124 2.90048
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −14.7733 −0.268605
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −5.12376 −0.0839960 −0.0419980 0.999118i \(-0.513372\pi\)
−0.0419980 + 0.999118i \(0.513372\pi\)
\(62\) 0 0
\(63\) 124.423 1.97496
\(64\) 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
\(73\) −112.072 −1.53524 −0.767618 0.640907i \(-0.778558\pi\)
−0.767618 + 0.640907i \(0.778558\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 281.674 3.65810
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −90.0000 −1.08434 −0.542169 0.840270i \(-0.682397\pi\)
−0.542169 + 0.840270i \(0.682397\pi\)
\(84\) 0 0
\(85\) −13.7218 −0.161433
\(86\) 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
\(95\) −13.7766 −0.145017
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 183.371 1.85224
\(100\) 0 0
\(101\) 102.000 1.00990 0.504950 0.863148i \(-0.331511\pi\)
0.504950 + 0.863148i \(0.331511\pi\)
\(102\) 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
\(115\) 21.7525 0.189152
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 261.625 2.19853
\(120\) 0 0
\(121\) 294.124 2.43077
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 35.8729 0.286983
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −238.622 −1.82154 −0.910771 0.412911i \(-0.864512\pi\)
−0.910771 + 0.412911i \(0.864512\pi\)
\(132\) 0 0
\(133\) 262.670 1.97496
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −40.6769 −0.296912 −0.148456 0.988919i \(-0.547430\pi\)
−0.148456 + 0.988919i \(0.547430\pi\)
\(138\) 0 0
\(139\) 71.3713 0.513462 0.256731 0.966483i \(-0.417354\pi\)
0.256731 + 0.966483i \(0.417354\pi\)
\(140\) 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
\(149\) −296.120 −1.98739 −0.993693 0.112137i \(-0.964231\pi\)
−0.993693 + 0.112137i \(0.964231\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 170.320 1.11320
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.0636943 −0.0318471 0.999493i \(-0.510139\pi\)
−0.0318471 + 0.999493i \(0.510139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −414.743 −2.57604
\(162\) 0 0
\(163\) −250.000 −1.53374 −0.766871 0.641801i \(-0.778187\pi\)
−0.766871 + 0.641801i \(0.778187\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 171.000 1.00000
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −338.350 −1.93343
\(176\) 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
\(187\) 385.577 2.06191
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −274.368 −1.43648 −0.718241 0.695795i \(-0.755052\pi\)
−0.718241 + 0.695795i \(0.755052\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −90.0000 −0.456853 −0.228426 0.973561i \(-0.573358\pi\)
−0.228426 + 0.973561i \(0.573358\pi\)
\(198\) 0 0
\(199\) 169.619 0.852356 0.426178 0.904639i \(-0.359860\pi\)
0.426178 + 0.904639i \(0.359860\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −270.000 −1.30435
\(208\) 0 0
\(209\) 387.117 1.85224
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 39.0274 0.181523
\(216\) 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
\(225\) −220.268 −0.978970
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 387.866 1.69374 0.846870 0.531800i \(-0.178484\pi\)
0.846870 + 0.531800i \(0.178484\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 258.924 1.11126 0.555632 0.831428i \(-0.312476\pi\)
0.555632 + 0.831428i \(0.312476\pi\)
\(234\) 0 0
\(235\) 62.7733 0.267120
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 358.622 1.50051 0.750255 0.661148i \(-0.229930\pi\)
0.750255 + 0.661148i \(0.229930\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −103.051 −0.420618
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −420.615 −1.67576 −0.837879 0.545855i \(-0.816205\pi\)
−0.837879 + 0.545855i \(0.816205\pi\)
\(252\) 0 0
\(253\) −611.238 −2.41596
\(254\) 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
\(263\) −88.1686 −0.335242 −0.167621 0.985852i \(-0.553608\pi\)
−0.167621 + 0.985852i \(0.553608\pi\)
\(264\) 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
\(271\) −142.000 −0.523985 −0.261993 0.965070i \(-0.584380\pi\)
−0.261993 + 0.965070i \(0.584380\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −498.653 −1.81328
\(276\) 0 0
\(277\) 392.072 1.41542 0.707712 0.706501i \(-0.249728\pi\)
0.707712 + 0.706501i \(0.249728\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 548.567 1.93840 0.969200 0.246274i \(-0.0792064\pi\)
0.969200 + 0.246274i \(0.0792064\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 69.1337 0.239217
\(290\) 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
\(301\) −744.114 −2.47214
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.71515 0.0121808
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −169.378 −0.544623 −0.272312 0.962209i \(-0.587788\pi\)
−0.272312 + 0.962209i \(0.587788\pi\)
\(312\) 0 0
\(313\) −590.000 −1.88498 −0.942492 0.334229i \(-0.891524\pi\)
−0.942492 + 0.334229i \(0.891524\pi\)
\(314\) 0 0
\(315\) −90.2168 −0.286403
\(316\) 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
\(323\) 359.564 1.11320
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1196.86 −3.63788
\(330\) 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
\(343\) 1287.41 3.75339
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 477.172 1.37514 0.687568 0.726120i \(-0.258678\pi\)
0.687568 + 0.726120i \(0.258678\pi\)
\(348\) 0 0
\(349\) 659.866 1.89073 0.945367 0.326007i \(-0.105703\pi\)
0.945367 + 0.326007i \(0.105703\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −510.000 −1.44476 −0.722380 0.691497i \(-0.756952\pi\)
−0.722380 + 0.691497i \(0.756952\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 463.612 1.29140 0.645699 0.763592i \(-0.276566\pi\)
0.645699 + 0.763592i \(0.276566\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 81.2617 0.222635
\(366\) 0 0
\(367\) 50.0000 0.136240 0.0681199 0.997677i \(-0.478300\pi\)
0.0681199 + 0.997677i \(0.478300\pi\)
\(368\) 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
\(385\) −204.237 −0.530485
\(386\) 0 0
\(387\) −484.423 −1.25174
\(388\) 0 0
\(389\) −737.111 −1.89489 −0.947443 0.319925i \(-0.896342\pi\)
−0.947443 + 0.319925i \(0.896342\pi\)
\(390\) 0 0
\(391\) −567.733 −1.45200
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −610.320 −1.53733 −0.768665 0.639652i \(-0.779079\pi\)
−0.768665 + 0.639652i \(0.779079\pi\)
\(398\) 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
\(405\) −58.7317 −0.145017
\(406\) 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
\(415\) 65.2575 0.157247
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −762.000 −1.81862 −0.909308 0.416124i \(-0.863388\pi\)
−0.909308 + 0.416124i \(0.863388\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −779.165 −1.84200
\(424\) 0 0
\(425\) −463.161 −1.08979
\(426\) 0 0
\(427\) −70.8347 −0.165889
\(428\) 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
\(437\) −570.000 −1.30435
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1279.11 2.90048
\(442\) 0 0
\(443\) −788.808 −1.78061 −0.890303 0.455369i \(-0.849507\pi\)
−0.890303 + 0.455369i \(0.849507\pi\)
\(444\) 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
\(457\) −265.062 −0.580005 −0.290003 0.957026i \(-0.593656\pi\)
−0.290003 + 0.957026i \(0.593656\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 921.860 1.99970 0.999848 0.0174455i \(-0.00555337\pi\)
0.999848 + 0.0174455i \(0.00555337\pi\)
\(462\) 0 0
\(463\) 86.8148 0.187505 0.0937525 0.995596i \(-0.470114\pi\)
0.0937525 + 0.995596i \(0.470114\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 295.179 0.632074 0.316037 0.948747i \(-0.397648\pi\)
0.316037 + 0.948747i \(0.397648\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1096.66 −2.31851
\(474\) 0 0
\(475\) −465.011 −0.978970
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −942.000 −1.96660 −0.983299 0.182000i \(-0.941743\pi\)
−0.983299 + 0.182000i \(0.941743\pi\)
\(480\) 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
\(491\) 918.000 1.86965 0.934827 0.355104i \(-0.115554\pi\)
0.934827 + 0.355104i \(0.115554\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −132.959 −0.268605
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −474.609 −0.951120 −0.475560 0.879683i \(-0.657755\pi\)
−0.475560 + 0.879683i \(0.657755\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 930.000 1.84891 0.924453 0.381295i \(-0.124522\pi\)
0.924453 + 0.381295i \(0.124522\pi\)
\(504\) 0 0
\(505\) −73.9584 −0.146452
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −1549.37 −3.03204
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1763.91 −3.41181
\(518\) 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
\(529\) 371.000 0.701323
\(530\) 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
\(539\) 2895.71 5.37238
\(540\) 0 0
\(541\) 620.856 1.14761 0.573804 0.818992i \(-0.305467\pi\)
0.573804 + 0.818992i \(0.305467\pi\)
\(542\) 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
\(549\) −46.1138 −0.0839960
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 370.079 0.664415 0.332207 0.943206i \(-0.392207\pi\)
0.332207 + 0.943206i \(0.392207\pi\)
\(558\) 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
\(567\) 1119.80 1.97496
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −458.000 −0.802102 −0.401051 0.916056i \(-0.631355\pi\)
−0.401051 + 0.916056i \(0.631355\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 734.228 1.27692
\(576\) 0 0
\(577\) 447.928 0.776305 0.388152 0.921595i \(-0.373113\pi\)
0.388152 + 0.921595i \(0.373113\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1244.23 −2.14153
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −271.831 −0.463086 −0.231543 0.972825i \(-0.574377\pi\)
−0.231543 + 0.972825i \(0.574377\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.0000 −0.0505902 −0.0252951 0.999680i \(-0.508053\pi\)
−0.0252951 + 0.999680i \(0.508053\pi\)
\(594\) 0 0
\(595\) −189.700 −0.318824
\(596\) 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
\(605\) −213.264 −0.352503
\(606\) 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
\(613\) 1178.05 1.92178 0.960891 0.276927i \(-0.0893159\pi\)
0.960891 + 0.276927i \(0.0893159\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1072.31 1.73795 0.868973 0.494859i \(-0.164780\pi\)
0.868973 + 0.494859i \(0.164780\pi\)
\(618\) 0 0
\(619\) 662.000 1.06947 0.534733 0.845021i \(-0.320412\pi\)
0.534733 + 0.845021i \(0.320412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 585.846 0.937353
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −104.361 −0.165390 −0.0826952 0.996575i \(-0.526353\pi\)
−0.0826952 + 0.996575i \(0.526353\pi\)
\(632\) 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
\(643\) 2.58717 0.00402359 0.00201180 0.999998i \(-0.499360\pi\)
0.00201180 + 0.999998i \(0.499360\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1208.41 1.86771 0.933856 0.357650i \(-0.116422\pi\)
0.933856 + 0.357650i \(0.116422\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −895.901 −1.37198 −0.685989 0.727612i \(-0.740630\pi\)
−0.685989 + 0.727612i \(0.740630\pi\)
\(654\) 0 0
\(655\) 173.021 0.264154
\(656\) 0 0
\(657\) −1008.65 −1.53524
\(658\) 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
\(665\) −190.458 −0.286403
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −104.394 −0.155580
\(672\) 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
\(685\) 29.4941 0.0430571
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1267.60 −1.83444 −0.917221 0.398380i \(-0.869573\pi\)
−0.917221 + 0.398380i \(0.869573\pi\)
\(692\) 0 0
\(693\) 2535.06 3.65810
\(694\) 0 0
\(695\) −51.7501 −0.0744606
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1098.00 −1.56633 −0.783167 0.621812i \(-0.786397\pi\)
−0.783167 + 0.621812i \(0.786397\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1410.12 1.99452
\(708\) 0 0
\(709\) 1318.00 1.85896 0.929478 0.368877i \(-0.120258\pi\)
0.929478 + 0.368877i \(0.120258\pi\)
\(710\) 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
\(719\) −1406.35 −1.95599 −0.977994 0.208635i \(-0.933098\pi\)
−0.977994 + 0.208635i \(0.933098\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1214.55 −1.67063 −0.835315 0.549772i \(-0.814715\pi\)
−0.835315 + 0.549772i \(0.814715\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) −1018.60 −1.39344
\(732\) 0 0
\(733\) 1270.00 1.73261 0.866303 0.499519i \(-0.166490\pi\)
0.866303 + 0.499519i \(0.166490\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −915.599 −1.23897 −0.619485 0.785008i \(-0.712659\pi\)
−0.619485 + 0.785008i \(0.712659\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 214.712 0.288204
\(746\) 0 0
\(747\) −810.000 −1.08434
\(748\) 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
\(757\) 728.650 0.962550 0.481275 0.876570i \(-0.340174\pi\)
0.481275 + 0.876570i \(0.340174\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −543.880 −0.714691 −0.357345 0.933972i \(-0.616318\pi\)
−0.357345 + 0.933972i \(0.616318\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −123.496 −0.161433
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 431.104 0.560603 0.280302 0.959912i \(-0.409566\pi\)
0.280302 + 0.959912i \(0.409566\pi\)
\(770\) 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
\(785\) 7.25083 0.00923672
\(786\) 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
\(799\) −1638.36 −2.05051
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2283.43 −2.84362
\(804\) 0 0
\(805\) 300.723 0.373569
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 551.130 0.681249 0.340624 0.940199i \(-0.389362\pi\)
0.340624 + 0.940199i \(0.389362\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 181.271 0.222418
\(816\) 0 0
\(817\) −1022.67 −1.25174
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1583.85 1.92918 0.964588 0.263762i \(-0.0849634\pi\)
0.964588 + 0.263762i \(0.0849634\pi\)
\(822\) 0 0
\(823\) 340.835 0.414137 0.207068 0.978326i \(-0.433608\pi\)
0.207068 + 0.978326i \(0.433608\pi\)
\(824\) 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
\(833\) 2689.61 3.22882
\(834\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) −122.539 −0.145017
\(846\) 0 0
\(847\) 4066.19 4.80069
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1030.00 1.20750 0.603751 0.797173i \(-0.293672\pi\)
0.603751 + 0.797173i \(0.293672\pi\)
\(854\) 0 0
\(855\) −123.989 −0.145017
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1482.61 −1.72597 −0.862985 0.505229i \(-0.831408\pi\)
−0.862985 + 0.505229i \(0.831408\pi\)
\(860\) 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
\(875\) 495.934 0.566782
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1184.84 −1.34488 −0.672442 0.740150i \(-0.734755\pi\)
−0.672442 + 0.740150i \(0.734755\pi\)
\(882\) 0 0
\(883\) 1765.15 1.99903 0.999516 0.0311055i \(-0.00990277\pi\)
0.999516 + 0.0311055i \(0.00990277\pi\)
\(884\) 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
\(891\) 1650.34 1.85224
\(892\) 0 0
\(893\) −1644.90 −1.84200
\(894\) 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
\(909\) 918.000 1.00990
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1833.71 −2.00845
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3298.89 −3.59748
\(918\) 0 0
\(919\) 1762.00 1.91730 0.958651 0.284585i \(-0.0918559\pi\)
0.958651 + 0.284585i \(0.0918559\pi\)
\(920\) 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
\(929\) 642.000 0.691066 0.345533 0.938407i \(-0.387698\pi\)
0.345533 + 0.938407i \(0.387698\pi\)
\(930\) 0 0
\(931\) 2700.35 2.90048
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −279.575 −0.299011
\(936\) 0 0
\(937\) 1429.29 1.52539 0.762695 0.646759i \(-0.223876\pi\)
0.762695 + 0.646759i \(0.223876\pi\)
\(938\) 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
\(947\) 1830.00 1.93242 0.966209 0.257760i \(-0.0829843\pi\)
0.966209 + 0.257760i \(0.0829843\pi\)
\(948\) 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
\(955\) 198.939 0.208314
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −562.348 −0.586390
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1790.00 −1.85109 −0.925543 0.378643i \(-0.876391\pi\)
−0.925543 + 0.378643i \(0.876391\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 986.690 1.01407
\(974\) 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
\(985\) 65.2575 0.0662512
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1614.74 1.63270
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −122.988 −0.123606
\(996\) 0 0
\(997\) 749.680 0.751936 0.375968 0.926633i \(-0.377310\pi\)
0.375968 + 0.926633i \(0.377310\pi\)
\(998\) 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 1216.3.e.f.1025.2 2
4.3 odd 2 1216.3.e.e.1025.2 2
8.3 odd 2 304.3.e.e.113.1 2
8.5 even 2 76.3.c.b.37.1 2
19.18 odd 2 CM 1216.3.e.f.1025.2 2
24.5 odd 2 684.3.h.a.37.2 2
24.11 even 2 2736.3.o.c.721.2 2
40.13 odd 4 1900.3.g.a.949.1 4
40.29 even 2 1900.3.e.a.1101.1 2
40.37 odd 4 1900.3.g.a.949.4 4
76.75 even 2 1216.3.e.e.1025.2 2
152.37 odd 2 76.3.c.b.37.1 2
152.75 even 2 304.3.e.e.113.1 2
456.227 odd 2 2736.3.o.c.721.2 2
456.341 even 2 684.3.h.a.37.2 2
760.37 even 4 1900.3.g.a.949.4 4
760.189 odd 2 1900.3.e.a.1101.1 2
760.493 even 4 1900.3.g.a.949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.c.b.37.1 2 8.5 even 2
76.3.c.b.37.1 2 152.37 odd 2
304.3.e.e.113.1 2 8.3 odd 2
304.3.e.e.113.1 2 152.75 even 2
684.3.h.a.37.2 2 24.5 odd 2
684.3.h.a.37.2 2 456.341 even 2
1216.3.e.e.1025.2 2 4.3 odd 2
1216.3.e.e.1025.2 2 76.75 even 2
1216.3.e.f.1025.2 2 1.1 even 1 trivial
1216.3.e.f.1025.2 2 19.18 odd 2 CM
1900.3.e.a.1101.1 2 40.29 even 2
1900.3.e.a.1101.1 2 760.189 odd 2
1900.3.g.a.949.1 4 40.13 odd 4
1900.3.g.a.949.1 4 760.493 even 4
1900.3.g.a.949.4 4 40.37 odd 4
1900.3.g.a.949.4 4 760.37 even 4
2736.3.o.c.721.2 2 24.11 even 2
2736.3.o.c.721.2 2 456.227 odd 2