Properties

Label 1900.3.g.a.949.3
Level $1900$
Weight $3$
Character 1900.949
Analytic conductor $51.771$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
Defining polynomial: \(x^{4} + 29 x^{2} + 196\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 949.3
Root \(3.27492i\) of defining polynomial
Character \(\chi\) \(=\) 1900.949
Dual form 1900.3.g.a.949.2

$q$-expansion

\(f(q)\) \(=\) \(q+8.82475i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+8.82475i q^{7} -9.00000 q^{9} +17.3746 q^{11} +33.9244i q^{17} +19.0000 q^{19} -30.0000i q^{23} +31.1752i q^{43} -11.5739i q^{47} -28.8762 q^{49} -108.124 q^{61} -79.4228i q^{63} +137.072i q^{73} +153.326i q^{77} +81.0000 q^{81} +90.0000i q^{83} -156.371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 36q^{9} + O(q^{10}) \) \( 4q - 36q^{9} - 6q^{11} + 76q^{19} - 342q^{49} - 206q^{61} + 324q^{81} + 54q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.82475i 1.26068i 0.776320 + 0.630339i \(0.217084\pi\)
−0.776320 + 0.630339i \(0.782916\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 17.3746 1.57951 0.789754 0.613424i \(-0.210208\pi\)
0.789754 + 0.613424i \(0.210208\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 33.9244i 1.99555i 0.0666402 + 0.997777i \(0.478772\pi\)
−0.0666402 + 0.997777i \(0.521228\pi\)
\(18\) 0 0
\(19\) 19.0000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 30.0000i − 1.30435i −0.758069 0.652174i \(-0.773857\pi\)
0.758069 0.652174i \(-0.226143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 31.1752i 0.725006i 0.931983 + 0.362503i \(0.118078\pi\)
−0.931983 + 0.362503i \(0.881922\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.5739i − 0.246254i −0.992391 0.123127i \(-0.960708\pi\)
0.992391 0.123127i \(-0.0392922\pi\)
\(48\) 0 0
\(49\) −28.8762 −0.589311
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(63\) − 79.4228i − 1.26068i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 137.072i 1.87770i 0.344323 + 0.938851i \(0.388108\pi\)
−0.344323 + 0.938851i \(0.611892\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 153.326i 1.99125i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 90.0000i 1.08434i 0.840270 + 0.542169i \(0.182397\pi\)
−0.840270 + 0.542169i \(0.817603\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −25.6221 −0.195589 −0.0977943 0.995207i \(-0.531179\pi\)
−0.0977943 + 0.995207i \(0.531179\pi\)
\(132\) 0 0
\(133\) 167.670i 1.26068i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 214.323i 1.56440i 0.623026 + 0.782201i \(0.285903\pi\)
−0.623026 + 0.782201i \(0.714097\pi\)
\(138\) 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 \(-0.369103\pi\)
0.399733 + 0.916632i \(0.369103\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) − 305.320i − 1.99555i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.0636943i −0.999493 0.0318471i \(-0.989861\pi\)
0.999493 0.0318471i \(-0.0101390\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 264.743 1.64436
\(162\) 0 0
\(163\) 250.000i 1.53374i 0.641801 + 0.766871i \(0.278187\pi\)
−0.641801 + 0.766871i \(0.721813\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −169.000 −1.00000
\(170\) 0 0
\(171\) −171.000 −1.00000
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 589.423i 3.15199i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 367.368 1.92339 0.961696 0.274117i \(-0.0883857\pi\)
0.961696 + 0.274117i \(0.0883857\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 90.0000i − 0.456853i −0.973561 0.228426i \(-0.926642\pi\)
0.973561 0.228426i \(-0.0733580\pi\)
\(198\) 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
\(207\) 270.000i 1.30435i
\(208\) 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
\(215\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −404.866 −1.76798 −0.883988 0.467510i \(-0.845151\pi\)
−0.883988 + 0.467510i \(0.845151\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 206.076i 0.884445i 0.896905 + 0.442222i \(0.145810\pi\)
−0.896905 + 0.442222i \(0.854190\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −447.615 −1.78333 −0.891664 0.452697i \(-0.850462\pi\)
−0.891664 + 0.452697i \(0.850462\pi\)
\(252\) 0 0
\(253\) − 521.238i − 2.06023i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 493.169i 1.87517i 0.347762 + 0.937583i \(0.386942\pi\)
−0.347762 + 0.937583i \(0.613058\pi\)
\(264\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 142.928i 0.515985i 0.966147 + 0.257992i \(0.0830609\pi\)
−0.966147 + 0.257992i \(0.916939\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 153.567i 0.542641i 0.962489 + 0.271320i \(0.0874603\pi\)
−0.962489 + 0.271320i \(0.912540\pi\)
\(284\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −433.622 −1.39428 −0.697142 0.716933i \(-0.745545\pi\)
−0.697142 + 0.716933i \(0.745545\pi\)
\(312\) 0 0
\(313\) − 590.000i − 1.88498i −0.334229 0.942492i \(-0.608476\pi\)
0.334229 0.942492i \(-0.391524\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 644.564i 1.99555i
\(324\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 177.587i 0.517747i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 197.828i 0.570110i 0.958511 + 0.285055i \(0.0920118\pi\)
−0.958511 + 0.285055i \(0.907988\pi\)
\(348\) 0 0
\(349\) −132.866 −0.380706 −0.190353 0.981716i \(-0.560963\pi\)
−0.190353 + 0.981716i \(0.560963\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 510.000i − 1.44476i −0.691497 0.722380i \(-0.743048\pi\)
0.691497 0.722380i \(-0.256952\pi\)
\(354\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) − 50.0000i − 0.136240i −0.997677 0.0681199i \(-0.978300\pi\)
0.997677 0.0681199i \(-0.0217000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 280.577i − 0.725006i
\(388\) 0 0
\(389\) 584.111 1.50157 0.750785 0.660547i \(-0.229676\pi\)
0.750785 + 0.660547i \(0.229676\pi\)
\(390\) 0 0
\(391\) 1017.73 2.60290
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 134.680i − 0.339245i −0.985509 0.169622i \(-0.945745\pi\)
0.985509 0.169622i \(-0.0542548\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 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
\(415\) 0 0
\(416\) 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
\(423\) 104.165i 0.246254i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 954.165i − 2.23458i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 570.000i − 1.30435i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 259.886 0.589311
\(442\) 0 0
\(443\) − 743.808i − 1.67903i −0.543340 0.839513i \(-0.682841\pi\)
0.543340 0.839513i \(-0.317159\pi\)
\(444\) 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
\(457\) − 890.062i − 1.94762i −0.227363 0.973810i \(-0.573011\pi\)
0.227363 0.973810i \(-0.426989\pi\)
\(458\) 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
\(463\) − 841.815i − 1.81817i −0.416606 0.909087i \(-0.636780\pi\)
0.416606 0.909087i \(-0.363220\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 619.821i 1.32724i 0.748070 + 0.663620i \(0.230981\pi\)
−0.748070 + 0.663620i \(0.769019\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 541.657i 1.14515i
\(474\) 0 0
\(475\) 0 0
\(476\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −918.000 −1.86965 −0.934827 0.355104i \(-0.884446\pi\)
−0.934827 + 0.355104i \(0.884446\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 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
\(503\) 930.000i 1.84891i 0.381295 + 0.924453i \(0.375478\pi\)
−0.381295 + 0.924453i \(0.624522\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(517\) − 201.092i − 0.388960i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(557\) 724.921i 1.30147i 0.759303 + 0.650737i \(0.225540\pi\)
−0.759303 + 0.650737i \(0.774460\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 714.805i 1.26068i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 458.000 0.802102 0.401051 0.916056i \(-0.368645\pi\)
0.401051 + 0.916056i \(0.368645\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 697.072i − 1.20810i −0.796947 0.604049i \(-0.793553\pi\)
0.796947 0.604049i \(-0.206447\pi\)
\(578\) 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
\(587\) − 853.169i − 1.45344i −0.686934 0.726719i \(-0.741044\pi\)
0.686934 0.726719i \(-0.258956\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 30.0000i − 0.0505902i −0.999680 0.0252951i \(-0.991947\pi\)
0.999680 0.0252951i \(-0.00805254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 883.052i 1.44054i 0.693693 + 0.720271i \(0.255983\pi\)
−0.693693 + 0.720271i \(0.744017\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.31316i 0.0118528i 0.999982 + 0.00592639i \(0.00188644\pi\)
−0.999982 + 0.00592639i \(0.998114\pi\)
\(618\) 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
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1141.36 1.80881 0.904407 0.426671i \(-0.140314\pi\)
0.904407 + 0.426671i \(0.140314\pi\)
\(632\) 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
\(643\) − 1112.41i − 1.73004i −0.501741 0.865018i \(-0.667307\pi\)
0.501741 0.865018i \(-0.332693\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 203.410i 0.314389i 0.987568 + 0.157194i \(0.0502449\pi\)
−0.987568 + 0.157194i \(0.949755\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1270.90i − 1.94625i −0.230278 0.973125i \(-0.573963\pi\)
0.230278 0.973125i \(-0.426037\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1233.65i − 1.87770i
\(658\) 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
\(665\) 0 0
\(666\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1110.60 −1.60723 −0.803617 0.595147i \(-0.797094\pi\)
−0.803617 + 0.595147i \(0.797094\pi\)
\(692\) 0 0
\(693\) − 1379.94i − 1.99125i
\(694\) 0 0
\(695\) 0 0
\(696\) 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
\(707\) − 900.125i − 1.27316i
\(708\) 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
\(719\) −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
\(727\) − 1299.55i − 1.78755i −0.448518 0.893774i \(-0.648048\pi\)
0.448518 0.893774i \(-0.351952\pi\)
\(728\) 0 0
\(729\) −729.000 −1.00000
\(730\) 0 0
\(731\) −1057.60 −1.44679
\(732\) 0 0
\(733\) − 1270.00i − 1.73261i −0.499519 0.866303i \(-0.666490\pi\)
0.499519 0.866303i \(-0.333510\pi\)
\(734\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 810.000i − 1.08434i
\(748\) 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
\(757\) − 1513.65i − 1.99954i −0.0214884 0.999769i \(-0.506840\pi\)
0.0214884 0.999769i \(-0.493160\pi\)
\(758\) 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
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1494.10 1.94292 0.971459 0.237208i \(-0.0762322\pi\)
0.971459 + 0.237208i \(0.0762322\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 392.639 0.491413
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2381.57i 2.96585i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1041.87 −1.28785 −0.643924 0.765089i \(-0.722695\pi\)
−0.643924 + 0.765089i \(0.722695\pi\)
\(810\) 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
\(817\) 592.330i 0.725006i
\(818\) 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
\(823\) 1224.17i 1.48744i 0.668490 + 0.743721i \(0.266941\pi\)
−0.668490 + 0.743721i \(0.733059\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 979.610i − 1.17600i
\(834\) 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
\(845\) 0 0
\(846\) 0 0
\(847\) 1596.19i 1.88452i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 1030.00i − 1.20750i −0.797173 0.603751i \(-0.793672\pi\)
0.797173 0.603751i \(-0.206328\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 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
\(883\) 930.145i 1.05339i 0.850054 + 0.526696i \(0.176569\pi\)
−0.850054 + 0.526696i \(0.823431\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1407.34 1.57951
\(892\) 0 0
\(893\) − 219.905i − 0.246254i
\(894\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 918.000 1.00990
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1563.71i 1.71272i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 226.109i − 0.246574i
\(918\) 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 \(-0.612302\pi\)
−0.345533 + 0.938407i \(0.612302\pi\)
\(930\) 0 0
\(931\) −548.649 −0.589311
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1764.29i 1.88291i 0.337134 + 0.941457i \(0.390543\pi\)
−0.337134 + 0.941457i \(0.609457\pi\)
\(938\) 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
\(947\) 1830.00i 1.93242i 0.257760 + 0.966209i \(0.417016\pi\)
−0.257760 + 0.966209i \(0.582984\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(967\) 1790.00i 1.85109i 0.378643 + 0.925543i \(0.376391\pi\)
−0.378643 + 0.925543i \(0.623609\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) − 2368.31i − 2.43403i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) 1225.32i 1.22901i 0.788914 + 0.614503i \(0.210644\pi\)
−0.788914 + 0.614503i \(0.789356\pi\)
\(998\) 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 1900.3.g.a.949.3 4
5.2 odd 4 76.3.c.b.37.2 2
5.3 odd 4 1900.3.e.a.1101.2 2
5.4 even 2 inner 1900.3.g.a.949.2 4
15.2 even 4 684.3.h.a.37.1 2
19.18 odd 2 CM 1900.3.g.a.949.3 4
20.7 even 4 304.3.e.e.113.2 2
40.27 even 4 1216.3.e.e.1025.1 2
40.37 odd 4 1216.3.e.f.1025.1 2
60.47 odd 4 2736.3.o.c.721.1 2
95.18 even 4 1900.3.e.a.1101.2 2
95.37 even 4 76.3.c.b.37.2 2
95.94 odd 2 inner 1900.3.g.a.949.2 4
285.227 odd 4 684.3.h.a.37.1 2
380.227 odd 4 304.3.e.e.113.2 2
760.37 even 4 1216.3.e.f.1025.1 2
760.227 odd 4 1216.3.e.e.1025.1 2
1140.227 even 4 2736.3.o.c.721.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.c.b.37.2 2 5.2 odd 4
76.3.c.b.37.2 2 95.37 even 4
304.3.e.e.113.2 2 20.7 even 4
304.3.e.e.113.2 2 380.227 odd 4
684.3.h.a.37.1 2 15.2 even 4
684.3.h.a.37.1 2 285.227 odd 4
1216.3.e.e.1025.1 2 40.27 even 4
1216.3.e.e.1025.1 2 760.227 odd 4
1216.3.e.f.1025.1 2 40.37 odd 4
1216.3.e.f.1025.1 2 760.37 even 4
1900.3.e.a.1101.2 2 5.3 odd 4
1900.3.e.a.1101.2 2 95.18 even 4
1900.3.g.a.949.2 4 5.4 even 2 inner
1900.3.g.a.949.2 4 95.94 odd 2 inner
1900.3.g.a.949.3 4 1.1 even 1 trivial
1900.3.g.a.949.3 4 19.18 odd 2 CM
2736.3.o.c.721.1 2 60.47 odd 4
2736.3.o.c.721.1 2 1140.227 even 4