Properties

Label 304.3.e.e.113.2
Level $304$
Weight $3$
Character 304.113
Self dual yes
Analytic conductor $8.283$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.28340003655\)
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 113.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 304.113

$q$-expansion

\(f(q)\) \(=\) \(q+8.27492 q^{5} +8.82475 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+8.27492 q^{5} +8.82475 q^{7} +9.00000 q^{9} -17.3746 q^{11} -33.9244 q^{17} +19.0000 q^{19} +30.0000 q^{23} +43.4743 q^{25} +73.0241 q^{35} -31.1752 q^{43} +74.4743 q^{45} -11.5739 q^{47} +28.8762 q^{49} -143.773 q^{55} -108.124 q^{61} +79.4228 q^{63} +137.072 q^{73} -153.326 q^{77} +81.0000 q^{81} -90.0000 q^{83} -280.722 q^{85} +157.223 q^{95} -156.371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{5} - 5 q^{7} + 18 q^{9} + O(q^{10}) \) \( 2 q + 9 q^{5} - 5 q^{7} + 18 q^{9} + 3 q^{11} - 15 q^{17} + 38 q^{19} + 60 q^{23} + 19 q^{25} + 63 q^{35} - 85 q^{43} + 81 q^{45} + 75 q^{47} + 171 q^{49} - 129 q^{55} - 103 q^{61} - 45 q^{63} + 25 q^{73} - 435 q^{77} + 162 q^{81} - 180 q^{83} - 267 q^{85} + 171 q^{95} + 27 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 8.27492 1.65498 0.827492 0.561478i \(-0.189767\pi\)
0.827492 + 0.561478i \(0.189767\pi\)
\(6\) 0 0
\(7\) 8.82475 1.26068 0.630339 0.776320i \(-0.282916\pi\)
0.630339 + 0.776320i \(0.282916\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −17.3746 −1.57951 −0.789754 0.613424i \(-0.789792\pi\)
−0.789754 + 0.613424i \(0.789792\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −33.9244 −1.99555 −0.997777 0.0666402i \(-0.978772\pi\)
−0.997777 + 0.0666402i \(0.978772\pi\)
\(18\) 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 \(-0.273857\pi\)
0.652174 + 0.758069i \(0.273857\pi\)
\(24\) 0 0
\(25\) 43.4743 1.73897
\(26\) 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
\(35\) 73.0241 2.08640
\(36\) 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
\(43\) −31.1752 −0.725006 −0.362503 0.931983i \(-0.618078\pi\)
−0.362503 + 0.931983i \(0.618078\pi\)
\(44\) 0 0
\(45\) 74.4743 1.65498
\(46\) 0 0
\(47\) −11.5739 −0.246254 −0.123127 0.992391i \(-0.539292\pi\)
−0.123127 + 0.992391i \(0.539292\pi\)
\(48\) 0 0
\(49\) 28.8762 0.589311
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −143.773 −2.61406
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −108.124 −1.77252 −0.886260 0.463187i \(-0.846706\pi\)
−0.886260 + 0.463187i \(0.846706\pi\)
\(62\) 0 0
\(63\) 79.4228 1.26068
\(64\) 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
\(73\) 137.072 1.87770 0.938851 0.344323i \(-0.111892\pi\)
0.938851 + 0.344323i \(0.111892\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −153.326 −1.99125
\(78\) 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
\(85\) −280.722 −3.30261
\(86\) 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
\(95\) 157.223 1.65498
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −156.371 −1.57951
\(100\) 0 0
\(101\) −102.000 −1.00990 −0.504950 0.863148i \(-0.668489\pi\)
−0.504950 + 0.863148i \(0.668489\pi\)
\(102\) 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
\(115\) 248.248 2.15867
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −299.375 −2.51575
\(120\) 0 0
\(121\) 180.876 1.49484
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 152.873 1.22298
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 25.6221 0.195589 0.0977943 0.995207i \(-0.468821\pi\)
0.0977943 + 0.995207i \(0.468821\pi\)
\(132\) 0 0
\(133\) 167.670 1.26068
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −214.323 −1.56440 −0.782201 0.623026i \(-0.785903\pi\)
−0.782201 + 0.623026i \(0.785903\pi\)
\(138\) 0 0
\(139\) −268.371 −1.93073 −0.965364 0.260906i \(-0.915979\pi\)
−0.965364 + 0.260906i \(0.915979\pi\)
\(140\) 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
\(149\) −119.120 −0.799466 −0.399733 0.916632i \(-0.630897\pi\)
−0.399733 + 0.916632i \(0.630897\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −305.320 −1.99555
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.0636943 0.0318471 0.999493i \(-0.489861\pi\)
0.0318471 + 0.999493i \(0.489861\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 264.743 1.64436
\(162\) 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
\(175\) 383.650 2.19228
\(176\) 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
\(187\) 589.423 3.15199
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −367.368 −1.92339 −0.961696 0.274117i \(-0.911614\pi\)
−0.961696 + 0.274117i \(0.911614\pi\)
\(192\) 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 \(-0.426642\pi\)
0.228426 + 0.973561i \(0.426642\pi\)
\(198\) 0 0
\(199\) 396.619 1.99306 0.996530 0.0832388i \(-0.0265264\pi\)
0.996530 + 0.0832388i \(0.0265264\pi\)
\(200\) 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
\(209\) −330.117 −1.57951
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −257.973 −1.19987
\(216\) 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
\(225\) 391.268 1.73897
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 404.866 1.76798 0.883988 0.467510i \(-0.154849\pi\)
0.883988 + 0.467510i \(0.154849\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 206.076 0.884445 0.442222 0.896905i \(-0.354190\pi\)
0.442222 + 0.896905i \(0.354190\pi\)
\(234\) 0 0
\(235\) −95.7733 −0.407546
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −94.3779 −0.394887 −0.197443 0.980314i \(-0.563264\pi\)
−0.197443 + 0.980314i \(0.563264\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 238.949 0.975300
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 447.615 1.78333 0.891664 0.452697i \(-0.149538\pi\)
0.891664 + 0.452697i \(0.149538\pi\)
\(252\) 0 0
\(253\) −521.238 −2.06023
\(254\) 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
\(263\) −493.169 −1.87517 −0.937583 0.347762i \(-0.886942\pi\)
−0.937583 + 0.347762i \(0.886942\pi\)
\(264\) 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 \(-0.415620\pi\)
0.261993 + 0.965070i \(0.415620\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −755.347 −2.74672
\(276\) 0 0
\(277\) −142.928 −0.515985 −0.257992 0.966147i \(-0.583061\pi\)
−0.257992 + 0.966147i \(0.583061\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −153.567 −0.542641 −0.271320 0.962489i \(-0.587460\pi\)
−0.271320 + 0.962489i \(0.587460\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 861.866 2.98224
\(290\) 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
\(301\) −275.114 −0.913999
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −894.715 −2.93349
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 433.622 1.39428 0.697142 0.716933i \(-0.254455\pi\)
0.697142 + 0.716933i \(0.254455\pi\)
\(312\) 0 0
\(313\) −590.000 −1.88498 −0.942492 0.334229i \(-0.891524\pi\)
−0.942492 + 0.334229i \(0.891524\pi\)
\(314\) 0 0
\(315\) 657.217 2.08640
\(316\) 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
\(323\) −644.564 −1.99555
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −102.137 −0.310447
\(330\) 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
\(343\) −177.587 −0.517747
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 197.828 0.570110 0.285055 0.958511i \(-0.407988\pi\)
0.285055 + 0.958511i \(0.407988\pi\)
\(348\) 0 0
\(349\) 132.866 0.380706 0.190353 0.981716i \(-0.439037\pi\)
0.190353 + 0.981716i \(0.439037\pi\)
\(350\) 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
\(359\) 706.612 1.96828 0.984140 0.177396i \(-0.0567674\pi\)
0.984140 + 0.177396i \(0.0567674\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1134.26 3.10757
\(366\) 0 0
\(367\) −50.0000 −0.136240 −0.0681199 0.997677i \(-0.521700\pi\)
−0.0681199 + 0.997677i \(0.521700\pi\)
\(368\) 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
\(385\) −1268.76 −3.29549
\(386\) 0 0
\(387\) −280.577 −0.725006
\(388\) 0 0
\(389\) −584.111 −1.50157 −0.750785 0.660547i \(-0.770324\pi\)
−0.750785 + 0.660547i \(0.770324\pi\)
\(390\) 0 0
\(391\) −1017.73 −2.60290
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 134.680 0.339245 0.169622 0.985509i \(-0.445745\pi\)
0.169622 + 0.985509i \(0.445745\pi\)
\(398\) 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
\(405\) 670.268 1.65498
\(406\) 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
\(415\) −744.743 −1.79456
\(416\) 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
\(423\) −104.165 −0.246254
\(424\) 0 0
\(425\) −1474.84 −3.47021
\(426\) 0 0
\(427\) −954.165 −2.23458
\(428\) 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
\(441\) 259.886 0.589311
\(442\) 0 0
\(443\) 743.808 1.67903 0.839513 0.543340i \(-0.182841\pi\)
0.839513 + 0.543340i \(0.182841\pi\)
\(444\) 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
\(457\) 890.062 1.94762 0.973810 0.227363i \(-0.0730105\pi\)
0.973810 + 0.227363i \(0.0730105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 474.860 1.03006 0.515032 0.857171i \(-0.327780\pi\)
0.515032 + 0.857171i \(0.327780\pi\)
\(462\) 0 0
\(463\) 841.815 1.81817 0.909087 0.416606i \(-0.136780\pi\)
0.909087 + 0.416606i \(0.136780\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 619.821 1.32724 0.663620 0.748070i \(-0.269019\pi\)
0.663620 + 0.748070i \(0.269019\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 541.657 1.14515
\(474\) 0 0
\(475\) 826.011 1.73897
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 942.000 1.96660 0.983299 0.182000i \(-0.0582571\pi\)
0.983299 + 0.182000i \(0.0582571\pi\)
\(480\) 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
\(495\) −1293.96 −2.61406
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 997.609 1.99922 0.999608 0.0279946i \(-0.00891213\pi\)
0.999608 + 0.0279946i \(0.00891213\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −930.000 −1.84891 −0.924453 0.381295i \(-0.875478\pi\)
−0.924453 + 0.381295i \(0.875478\pi\)
\(504\) 0 0
\(505\) −844.042 −1.67137
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1209.63 2.36718
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 201.092 0.388960
\(518\) 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
\(539\) −501.713 −0.930821
\(540\) 0 0
\(541\) 1077.86 1.99234 0.996170 0.0874330i \(-0.0278664\pi\)
0.996170 + 0.0874330i \(0.0278664\pi\)
\(542\) 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
\(549\) −973.114 −1.77252
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −724.921 −1.30147 −0.650737 0.759303i \(-0.725540\pi\)
−0.650737 + 0.759303i \(0.725540\pi\)
\(558\) 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
\(567\) 714.805 1.26068
\(568\) 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
\(575\) 1304.23 2.26822
\(576\) 0 0
\(577\) 697.072 1.20810 0.604049 0.796947i \(-0.293553\pi\)
0.604049 + 0.796947i \(0.293553\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −794.228 −1.36700
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −853.169 −1.45344 −0.726719 0.686934i \(-0.758956\pi\)
−0.726719 + 0.686934i \(0.758956\pi\)
\(588\) 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
\(595\) −2477.30 −4.16353
\(596\) 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
\(605\) 1496.74 2.47394
\(606\) 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
\(613\) 883.052 1.44054 0.720271 0.693693i \(-0.244017\pi\)
0.720271 + 0.693693i \(0.244017\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.31316 −0.0118528 −0.00592639 0.999982i \(-0.501886\pi\)
−0.00592639 + 0.999982i \(0.501886\pi\)
\(618\) 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
\(625\) 178.154 0.285047
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1141.36 −1.80881 −0.904407 0.426671i \(-0.859686\pi\)
−0.904407 + 0.426671i \(0.859686\pi\)
\(632\) 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
\(643\) 1112.41 1.73004 0.865018 0.501741i \(-0.167307\pi\)
0.865018 + 0.501741i \(0.167307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 203.410 0.314389 0.157194 0.987568i \(-0.449755\pi\)
0.157194 + 0.987568i \(0.449755\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1270.90 −1.94625 −0.973125 0.230278i \(-0.926037\pi\)
−0.973125 + 0.230278i \(0.926037\pi\)
\(654\) 0 0
\(655\) 212.021 0.323696
\(656\) 0 0
\(657\) 1233.65 1.87770
\(658\) 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
\(665\) 1387.46 2.08640
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1878.61 2.79971
\(672\) 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
\(685\) −1773.51 −2.58906
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1110.60 1.60723 0.803617 0.595147i \(-0.202906\pi\)
0.803617 + 0.595147i \(0.202906\pi\)
\(692\) 0 0
\(693\) −1379.94 −1.99125
\(694\) 0 0
\(695\) −2220.75 −3.19532
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1098.00 1.56633 0.783167 0.621812i \(-0.213603\pi\)
0.783167 + 0.621812i \(0.213603\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −900.125 −1.27316
\(708\) 0 0
\(709\) −1318.00 −1.85896 −0.929478 0.368877i \(-0.879742\pi\)
−0.929478 + 0.368877i \(0.879742\pi\)
\(710\) 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
\(719\) −443.355 −0.616627 −0.308313 0.951285i \(-0.599765\pi\)
−0.308313 + 0.951285i \(0.599765\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1299.55 −1.78755 −0.893774 0.448518i \(-0.851952\pi\)
−0.893774 + 0.448518i \(0.851952\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 1057.60 1.44679
\(732\) 0 0
\(733\) −1270.00 −1.73261 −0.866303 0.499519i \(-0.833510\pi\)
−0.866303 + 0.499519i \(0.833510\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1462.60 1.97916 0.989580 0.143986i \(-0.0459919\pi\)
0.989580 + 0.143986i \(0.0459919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −985.712 −1.32310
\(746\) 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
\(757\) 1513.65 1.99954 0.999769 0.0214884i \(-0.00684050\pi\)
0.999769 + 0.0214884i \(0.00684050\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −959.120 −1.26034 −0.630171 0.776456i \(-0.717015\pi\)
−0.630171 + 0.776456i \(0.717015\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2526.50 −3.30261
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1494.10 −1.94292 −0.971459 0.237208i \(-0.923768\pi\)
−0.971459 + 0.237208i \(0.923768\pi\)
\(770\) 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
\(785\) 82.7492 0.105413
\(786\) 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
\(799\) 392.639 0.491413
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2381.57 −2.96585
\(804\) 0 0
\(805\) 2190.72 2.72139
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1041.87 1.28785 0.643924 0.765089i \(-0.277305\pi\)
0.643924 + 0.765089i \(0.277305\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2068.73 −2.53832
\(816\) 0 0
\(817\) −592.330 −0.725006
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 416.853 0.507738 0.253869 0.967239i \(-0.418297\pi\)
0.253869 + 0.967239i \(0.418297\pi\)
\(822\) 0 0
\(823\) −1224.17 −1.48744 −0.743721 0.668490i \(-0.766941\pi\)
−0.743721 + 0.668490i \(0.766941\pi\)
\(824\) 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
\(833\) −979.610 −1.17600
\(834\) 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
\(845\) 1398.46 1.65498
\(846\) 0 0
\(847\) 1596.19 1.88452
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1030.00 −1.20750 −0.603751 0.797173i \(-0.706328\pi\)
−0.603751 + 0.797173i \(0.706328\pi\)
\(854\) 0 0
\(855\) 1415.01 1.65498
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −10.3911 −0.0120968 −0.00604839 0.999982i \(-0.501925\pi\)
−0.00604839 + 0.999982i \(0.501925\pi\)
\(860\) 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
\(875\) 1349.07 1.54179
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1721.84 1.95442 0.977209 0.212277i \(-0.0680880\pi\)
0.977209 + 0.212277i \(0.0680880\pi\)
\(882\) 0 0
\(883\) −930.145 −1.05339 −0.526696 0.850054i \(-0.676569\pi\)
−0.526696 + 0.850054i \(0.676569\pi\)
\(884\) 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
\(891\) −1407.34 −1.57951
\(892\) 0 0
\(893\) −219.905 −0.246254
\(894\) 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
\(913\) 1563.71 1.71272
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 226.109 0.246574
\(918\) 0 0
\(919\) −1762.00 −1.91730 −0.958651 0.284585i \(-0.908144\pi\)
−0.958651 + 0.284585i \(0.908144\pi\)
\(920\) 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
\(931\) 548.649 0.589311
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4877.42 5.21650
\(936\) 0 0
\(937\) −1764.29 −1.88291 −0.941457 0.337134i \(-0.890543\pi\)
−0.941457 + 0.337134i \(0.890543\pi\)
\(938\) 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
\(955\) −3039.94 −3.18318
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1891.35 −1.97221
\(960\) 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 \(-0.123609\pi\)
0.925543 + 0.378643i \(0.123609\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −2368.31 −2.43403
\(974\) 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
\(985\) 744.743 0.756084
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −935.257 −0.945660
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3281.99 3.29848
\(996\) 0 0
\(997\) −1225.32 −1.22901 −0.614503 0.788914i \(-0.710644\pi\)
−0.614503 + 0.788914i \(0.710644\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 304.3.e.e.113.2 2
3.2 odd 2 2736.3.o.c.721.1 2
4.3 odd 2 76.3.c.b.37.2 2
8.3 odd 2 1216.3.e.f.1025.1 2
8.5 even 2 1216.3.e.e.1025.1 2
12.11 even 2 684.3.h.a.37.1 2
19.18 odd 2 CM 304.3.e.e.113.2 2
20.3 even 4 1900.3.g.a.949.3 4
20.7 even 4 1900.3.g.a.949.2 4
20.19 odd 2 1900.3.e.a.1101.2 2
57.56 even 2 2736.3.o.c.721.1 2
76.75 even 2 76.3.c.b.37.2 2
152.37 odd 2 1216.3.e.e.1025.1 2
152.75 even 2 1216.3.e.f.1025.1 2
228.227 odd 2 684.3.h.a.37.1 2
380.227 odd 4 1900.3.g.a.949.2 4
380.303 odd 4 1900.3.g.a.949.3 4
380.379 even 2 1900.3.e.a.1101.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.c.b.37.2 2 4.3 odd 2
76.3.c.b.37.2 2 76.75 even 2
304.3.e.e.113.2 2 1.1 even 1 trivial
304.3.e.e.113.2 2 19.18 odd 2 CM
684.3.h.a.37.1 2 12.11 even 2
684.3.h.a.37.1 2 228.227 odd 2
1216.3.e.e.1025.1 2 8.5 even 2
1216.3.e.e.1025.1 2 152.37 odd 2
1216.3.e.f.1025.1 2 8.3 odd 2
1216.3.e.f.1025.1 2 152.75 even 2
1900.3.e.a.1101.2 2 20.19 odd 2
1900.3.e.a.1101.2 2 380.379 even 2
1900.3.g.a.949.2 4 20.7 even 4
1900.3.g.a.949.2 4 380.227 odd 4
1900.3.g.a.949.3 4 20.3 even 4
1900.3.g.a.949.3 4 380.303 odd 4
2736.3.o.c.721.1 2 3.2 odd 2
2736.3.o.c.721.1 2 57.56 even 2