Properties

Label 49.4.a.a.1.1
Level $49$
Weight $4$
Character 49.1
Self dual yes
Analytic conductor $2.891$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89109359028\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{2} +17.0000 q^{4} -45.0000 q^{8} -27.0000 q^{9} +O(q^{10})\) \(q-5.00000 q^{2} +17.0000 q^{4} -45.0000 q^{8} -27.0000 q^{9} -68.0000 q^{11} +89.0000 q^{16} +135.000 q^{18} +340.000 q^{22} -40.0000 q^{23} -125.000 q^{25} -166.000 q^{29} -85.0000 q^{32} -459.000 q^{36} +450.000 q^{37} -180.000 q^{43} -1156.00 q^{44} +200.000 q^{46} +625.000 q^{50} +590.000 q^{53} +830.000 q^{58} -287.000 q^{64} -740.000 q^{67} +688.000 q^{71} +1215.00 q^{72} -2250.00 q^{74} -1384.00 q^{79} +729.000 q^{81} +900.000 q^{86} +3060.00 q^{88} -680.000 q^{92} +1836.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 17.0000 2.12500
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −45.0000 −1.98874
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) −68.0000 −1.86389 −0.931944 0.362602i \(-0.881889\pi\)
−0.931944 + 0.362602i \(0.881889\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 135.000 1.76777
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 340.000 3.29492
\(23\) −40.0000 −0.362634 −0.181317 0.983425i \(-0.558036\pi\)
−0.181317 + 0.983425i \(0.558036\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −85.0000 −0.469563
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −459.000 −2.12500
\(37\) 450.000 1.99945 0.999724 0.0235113i \(-0.00748457\pi\)
0.999724 + 0.0235113i \(0.00748457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −180.000 −0.638366 −0.319183 0.947693i \(-0.603408\pi\)
−0.319183 + 0.947693i \(0.603408\pi\)
\(44\) −1156.00 −3.96076
\(45\) 0 0
\(46\) 200.000 0.641052
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 625.000 1.76777
\(51\) 0 0
\(52\) 0 0
\(53\) 590.000 1.52911 0.764554 0.644560i \(-0.222959\pi\)
0.764554 + 0.644560i \(0.222959\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 830.000 1.87904
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) −740.000 −1.34933 −0.674667 0.738122i \(-0.735713\pi\)
−0.674667 + 0.738122i \(0.735713\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 688.000 1.15001 0.575004 0.818151i \(-0.305000\pi\)
0.575004 + 0.818151i \(0.305000\pi\)
\(72\) 1215.00 1.98874
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −2250.00 −3.53456
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1384.00 −1.97104 −0.985520 0.169559i \(-0.945766\pi\)
−0.985520 + 0.169559i \(0.945766\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 900.000 1.12848
\(87\) 0 0
\(88\) 3060.00 3.70679
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −680.000 −0.770597
\(93\) 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
\(99\) 1836.00 1.86389
\(100\) −2125.00 −2.12500
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2950.00 −2.70311
\(107\) −1580.00 −1.42752 −0.713759 0.700392i \(-0.753009\pi\)
−0.713759 + 0.700392i \(0.753009\pi\)
\(108\) 0 0
\(109\) −54.0000 −0.0474519 −0.0237260 0.999718i \(-0.507553\pi\)
−0.0237260 + 0.999718i \(0.507553\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −670.000 −0.557773 −0.278886 0.960324i \(-0.589965\pi\)
−0.278886 + 0.960324i \(0.589965\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2822.00 −2.25876
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3293.00 2.47408
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2000.00 −1.39741 −0.698706 0.715409i \(-0.746240\pi\)
−0.698706 + 0.715409i \(0.746240\pi\)
\(128\) 2115.00 1.46048
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3700.00 2.38531
\(135\) 0 0
\(136\) 0 0
\(137\) 3110.00 1.93945 0.969727 0.244191i \(-0.0785224\pi\)
0.969727 + 0.244191i \(0.0785224\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3440.00 −2.03295
\(143\) 0 0
\(144\) −2403.00 −1.39062
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 7650.00 4.24883
\(149\) 814.000 0.447554 0.223777 0.974640i \(-0.428161\pi\)
0.223777 + 0.974640i \(0.428161\pi\)
\(150\) 0 0
\(151\) −2952.00 −1.59093 −0.795465 0.606000i \(-0.792773\pi\)
−0.795465 + 0.606000i \(0.792773\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 6920.00 3.48434
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −3645.00 −1.76777
\(163\) 1780.00 0.855340 0.427670 0.903935i \(-0.359335\pi\)
0.427670 + 0.903935i \(0.359335\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2197.00 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −3060.00 −1.35653
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6052.00 −2.59197
\(177\) 0 0
\(178\) 0 0
\(179\) −2084.00 −0.870198 −0.435099 0.900383i \(-0.643287\pi\)
−0.435099 + 0.900383i \(0.643287\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1800.00 0.721183
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4072.00 −1.54262 −0.771308 0.636462i \(-0.780397\pi\)
−0.771308 + 0.636462i \(0.780397\pi\)
\(192\) 0 0
\(193\) −4590.00 −1.71189 −0.855947 0.517064i \(-0.827025\pi\)
−0.855947 + 0.517064i \(0.827025\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2210.00 −0.799269 −0.399634 0.916675i \(-0.630863\pi\)
−0.399634 + 0.916675i \(0.630863\pi\)
\(198\) −9180.00 −3.29492
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 5625.00 1.98874
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1080.00 0.362634
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5868.00 1.91455 0.957274 0.289181i \(-0.0933830\pi\)
0.957274 + 0.289181i \(0.0933830\pi\)
\(212\) 10030.0 3.24935
\(213\) 0 0
\(214\) 7900.00 2.52352
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 270.000 0.0838840
\(219\) 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
\(225\) 3375.00 1.00000
\(226\) 3350.00 0.986012
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7470.00 2.11392
\(233\) −4730.00 −1.32993 −0.664963 0.746877i \(-0.731553\pi\)
−0.664963 + 0.746877i \(0.731553\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7376.00 −1.99629 −0.998146 0.0608655i \(-0.980614\pi\)
−0.998146 + 0.0608655i \(0.980614\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −16465.0 −4.37360
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 2720.00 0.675909
\(254\) 10000.0 2.47030
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 4482.00 1.06295
\(262\) 0 0
\(263\) 7520.00 1.76313 0.881565 0.472063i \(-0.156491\pi\)
0.881565 + 0.472063i \(0.156491\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −12580.0 −2.86734
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −15550.0 −3.42850
\(275\) 8500.00 1.86389
\(276\) 0 0
\(277\) 7310.00 1.58561 0.792807 0.609472i \(-0.208619\pi\)
0.792807 + 0.609472i \(0.208619\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4342.00 0.921786 0.460893 0.887456i \(-0.347529\pi\)
0.460893 + 0.887456i \(0.347529\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 11696.0 2.44377
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 2295.00 0.469563
\(289\) −4913.00 −1.00000
\(290\) 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
\(296\) −20250.0 −3.97638
\(297\) 0 0
\(298\) −4070.00 −0.791170
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 14760.0 2.81239
\(303\) 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
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −23528.0 −4.18846
\(317\) −6970.00 −1.23493 −0.617467 0.786597i \(-0.711841\pi\)
−0.617467 + 0.786597i \(0.711841\pi\)
\(318\) 0 0
\(319\) 11288.0 1.98121
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 12393.0 2.12500
\(325\) 0 0
\(326\) −8900.00 −1.51204
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10908.0 1.81135 0.905677 0.423969i \(-0.139364\pi\)
0.905677 + 0.423969i \(0.139364\pi\)
\(332\) 0 0
\(333\) −12150.0 −1.99945
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3330.00 −0.538269 −0.269135 0.963103i \(-0.586738\pi\)
−0.269135 + 0.963103i \(0.586738\pi\)
\(338\) 10985.0 1.76777
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 8100.00 1.26954
\(345\) 0 0
\(346\) 0 0
\(347\) −4100.00 −0.634293 −0.317146 0.948377i \(-0.602725\pi\)
−0.317146 + 0.948377i \(0.602725\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5780.00 0.875213
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 10420.0 1.53831
\(359\) −8104.00 −1.19140 −0.595700 0.803207i \(-0.703125\pi\)
−0.595700 + 0.803207i \(0.703125\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −3560.00 −0.504288
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13970.0 −1.93925 −0.969624 0.244602i \(-0.921343\pi\)
−0.969624 + 0.244602i \(0.921343\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11916.0 1.61500 0.807498 0.589870i \(-0.200821\pi\)
0.807498 + 0.589870i \(0.200821\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 20360.0 2.72698
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 22950.0 3.02623
\(387\) 4860.00 0.638366
\(388\) 0 0
\(389\) −10526.0 −1.37195 −0.685976 0.727624i \(-0.740625\pi\)
−0.685976 + 0.727624i \(0.740625\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 11050.0 1.41292
\(395\) 0 0
\(396\) 31212.0 3.96076
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −11125.0 −1.39062
\(401\) 1598.00 0.199003 0.0995016 0.995037i \(-0.468275\pi\)
0.0995016 + 0.995037i \(0.468275\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30600.0 −3.72675
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −5400.00 −0.641052
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 15262.0 1.76680 0.883402 0.468616i \(-0.155247\pi\)
0.883402 + 0.468616i \(0.155247\pi\)
\(422\) −29340.0 −3.38448
\(423\) 0 0
\(424\) −26550.0 −3.04100
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −26860.0 −3.03347
\(429\) 0 0
\(430\) 0 0
\(431\) −8608.00 −0.962025 −0.481012 0.876714i \(-0.659731\pi\)
−0.481012 + 0.876714i \(0.659731\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −918.000 −0.100835
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18580.0 1.99269 0.996346 0.0854102i \(-0.0272201\pi\)
0.996346 + 0.0854102i \(0.0272201\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2686.00 −0.282317 −0.141158 0.989987i \(-0.545083\pi\)
−0.141158 + 0.989987i \(0.545083\pi\)
\(450\) −16875.0 −1.76777
\(451\) 0 0
\(452\) −11390.0 −1.18527
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8010.00 0.819895 0.409947 0.912109i \(-0.365547\pi\)
0.409947 + 0.912109i \(0.365547\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −8440.00 −0.847171 −0.423585 0.905856i \(-0.639229\pi\)
−0.423585 + 0.905856i \(0.639229\pi\)
\(464\) −14774.0 −1.47816
\(465\) 0 0
\(466\) 23650.0 2.35100
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12240.0 1.18984
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15930.0 −1.52911
\(478\) 36880.0 3.52898
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 55981.0 5.25742
\(485\) 0 0
\(486\) 0 0
\(487\) 21240.0 1.97634 0.988169 0.153371i \(-0.0490130\pi\)
0.988169 + 0.153371i \(0.0490130\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20372.0 1.87246 0.936228 0.351394i \(-0.114292\pi\)
0.936228 + 0.351394i \(0.114292\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −7236.00 −0.649154 −0.324577 0.945859i \(-0.605222\pi\)
−0.324577 + 0.945859i \(0.605222\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −13600.0 −1.19485
\(507\) 0 0
\(508\) −34000.0 −2.96950
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 24475.0 2.11260
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −22410.0 −1.87904
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −37600.0 −3.11680
\(527\) 0 0
\(528\) 0 0
\(529\) −10567.0 −0.868497
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 33300.0 2.68347
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15878.0 1.26183 0.630914 0.775853i \(-0.282680\pi\)
0.630914 + 0.775853i \(0.282680\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12980.0 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(548\) 52870.0 4.12134
\(549\) 0 0
\(550\) −42500.0 −3.29492
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −36550.0 −2.80300
\(555\) 0 0
\(556\) 0 0
\(557\) 20470.0 1.55717 0.778583 0.627541i \(-0.215939\pi\)
0.778583 + 0.627541i \(0.215939\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −21710.0 −1.62950
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −30960.0 −2.28706
\(569\) −26906.0 −1.98235 −0.991176 0.132553i \(-0.957683\pi\)
−0.991176 + 0.132553i \(0.957683\pi\)
\(570\) 0 0
\(571\) −6788.00 −0.497494 −0.248747 0.968569i \(-0.580019\pi\)
−0.248747 + 0.968569i \(0.580019\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5000.00 0.362634
\(576\) 7749.00 0.560547
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 24565.0 1.76777
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −40120.0 −2.85009
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 40050.0 2.78048
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13838.0 0.951051
\(597\) 0 0
\(598\) 0 0
\(599\) −24736.0 −1.68729 −0.843644 0.536903i \(-0.819594\pi\)
−0.843644 + 0.536903i \(0.819594\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 19980.0 1.34933
\(604\) −50184.0 −3.38073
\(605\) 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
\(613\) 15010.0 0.988986 0.494493 0.869182i \(-0.335354\pi\)
0.494493 + 0.869182i \(0.335354\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30550.0 1.99335 0.996675 0.0814823i \(-0.0259654\pi\)
0.996675 + 0.0814823i \(0.0259654\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −26192.0 −1.65244 −0.826218 0.563351i \(-0.809512\pi\)
−0.826218 + 0.563351i \(0.809512\pi\)
\(632\) 62280.0 3.91988
\(633\) 0 0
\(634\) 34850.0 2.18308
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −56440.0 −3.50232
\(639\) −18576.0 −1.15001
\(640\) 0 0
\(641\) 8878.00 0.547051 0.273526 0.961865i \(-0.411810\pi\)
0.273526 + 0.961865i \(0.411810\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −32805.0 −1.98874
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 30260.0 1.81760
\(653\) 27050.0 1.62105 0.810527 0.585701i \(-0.199181\pi\)
0.810527 + 0.585701i \(0.199181\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1804.00 −0.106637 −0.0533186 0.998578i \(-0.516980\pi\)
−0.0533186 + 0.998578i \(0.516980\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −54540.0 −3.20205
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 60750.0 3.53456
\(667\) 6640.00 0.385460
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33570.0 −1.92278 −0.961388 0.275196i \(-0.911257\pi\)
−0.961388 + 0.275196i \(0.911257\pi\)
\(674\) 16650.0 0.951534
\(675\) 0 0
\(676\) −37349.0 −2.12500
\(677\) 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
\(683\) −34060.0 −1.90815 −0.954077 0.299560i \(-0.903160\pi\)
−0.954077 + 0.299560i \(0.903160\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −16020.0 −0.887728
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 20500.0 1.12128
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4198.00 −0.226186 −0.113093 0.993584i \(-0.536076\pi\)
−0.113093 + 0.993584i \(0.536076\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 19516.0 1.04480
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12546.0 0.664563 0.332281 0.943180i \(-0.392182\pi\)
0.332281 + 0.943180i \(0.392182\pi\)
\(710\) 0 0
\(711\) 37368.0 1.97104
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −35428.0 −1.84917
\(717\) 0 0
\(718\) 40520.0 2.10612
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 34295.0 1.76777
\(723\) 0 0
\(724\) 0 0
\(725\) 20750.0 1.06295
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 3400.00 0.170279
\(737\) 50320.0 2.51501
\(738\) 0 0
\(739\) −25324.0 −1.26057 −0.630283 0.776365i \(-0.717061\pi\)
−0.630283 + 0.776365i \(0.717061\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25160.0 1.24230 0.621151 0.783691i \(-0.286665\pi\)
0.621151 + 0.783691i \(0.286665\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 69850.0 3.42814
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2448.00 −0.118946 −0.0594732 0.998230i \(-0.518942\pi\)
−0.0594732 + 0.998230i \(0.518942\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −34830.0 −1.67228 −0.836141 0.548514i \(-0.815194\pi\)
−0.836141 + 0.548514i \(0.815194\pi\)
\(758\) −59580.0 −2.85494
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −69224.0 −3.27806
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −78030.0 −3.63777
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −24300.0 −1.12848
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 52630.0 2.42529
\(779\) 0 0
\(780\) 0 0
\(781\) −46784.0 −2.14349
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −37570.0 −1.69845
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −82620.0 −3.70679
\(793\) 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
\(799\) 0 0
\(800\) 10625.0 0.469563
\(801\) 0 0
\(802\) −7990.00 −0.351791
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 37354.0 1.62336 0.811679 0.584104i \(-0.198554\pi\)
0.811679 + 0.584104i \(0.198554\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 153000. 6.58802
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43538.0 −1.85078 −0.925388 0.379022i \(-0.876261\pi\)
−0.925388 + 0.379022i \(0.876261\pi\)
\(822\) 0 0
\(823\) −46240.0 −1.95848 −0.979238 0.202716i \(-0.935023\pi\)
−0.979238 + 0.202716i \(0.935023\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23980.0 −1.00830 −0.504151 0.863615i \(-0.668195\pi\)
−0.504151 + 0.863615i \(0.668195\pi\)
\(828\) 18360.0 0.770597
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) −76310.0 −3.12330
\(843\) 0 0
\(844\) 99756.0 4.06842
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 52510.0 2.12642
\(849\) 0 0
\(850\) 0 0
\(851\) −18000.0 −0.725067
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 71100.0 2.83896
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 43040.0 1.70064
\(863\) −20200.0 −0.796774 −0.398387 0.917217i \(-0.630430\pi\)
−0.398387 + 0.917217i \(0.630430\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 94112.0 3.67380
\(870\) 0 0
\(871\) 0 0
\(872\) 2430.00 0.0943695
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6550.00 −0.252198 −0.126099 0.992018i \(-0.540246\pi\)
−0.126099 + 0.992018i \(0.540246\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 30060.0 1.14564 0.572820 0.819681i \(-0.305850\pi\)
0.572820 + 0.819681i \(0.305850\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −92900.0 −3.52261
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −49572.0 −1.86389
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 13430.0 0.499070
\(899\) 0 0
\(900\) 57375.0 2.12500
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 30150.0 1.10926
\(905\) 0 0
\(906\) 0 0
\(907\) 52740.0 1.93076 0.965382 0.260840i \(-0.0839996\pi\)
0.965382 + 0.260840i \(0.0839996\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −39632.0 −1.44135 −0.720673 0.693275i \(-0.756167\pi\)
−0.720673 + 0.693275i \(0.756167\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −40050.0 −1.44938
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 21744.0 0.780488 0.390244 0.920711i \(-0.372391\pi\)
0.390244 + 0.920711i \(0.372391\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −56250.0 −1.99945
\(926\) 42200.0 1.49760
\(927\) 0 0
\(928\) 14110.0 0.499120
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −80410.0 −2.82609
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −61200.0 −2.10337
\(947\) 48820.0 1.67522 0.837612 0.546266i \(-0.183951\pi\)
0.837612 + 0.546266i \(0.183951\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29290.0 0.995589 0.497794 0.867295i \(-0.334143\pi\)
0.497794 + 0.867295i \(0.334143\pi\)
\(954\) 79650.0 2.70311
\(955\) 0 0
\(956\) −125392. −4.24212
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 42660.0 1.42752
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52040.0 1.73060 0.865302 0.501251i \(-0.167127\pi\)
0.865302 + 0.501251i \(0.167127\pi\)
\(968\) −148185. −4.92030
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −106200. −3.49370
\(975\) 0 0
\(976\) 0 0
\(977\) −37490.0 −1.22765 −0.613824 0.789443i \(-0.710369\pi\)
−0.613824 + 0.789443i \(0.710369\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1458.00 0.0474519
\(982\) −101860. −3.31006
\(983\) 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
\(989\) 7200.00 0.231493
\(990\) 0 0
\(991\) 57528.0 1.84403 0.922017 0.387150i \(-0.126540\pi\)
0.922017 + 0.387150i \(0.126540\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 36180.0 1.14755
\(999\) 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 49.4.a.a.1.1 1
3.2 odd 2 441.4.a.m.1.1 1
4.3 odd 2 784.4.a.k.1.1 1
5.4 even 2 1225.4.a.l.1.1 1
7.2 even 3 49.4.c.d.18.1 2
7.3 odd 6 49.4.c.d.30.1 2
7.4 even 3 49.4.c.d.30.1 2
7.5 odd 6 49.4.c.d.18.1 2
7.6 odd 2 CM 49.4.a.a.1.1 1
21.2 odd 6 441.4.e.a.361.1 2
21.5 even 6 441.4.e.a.361.1 2
21.11 odd 6 441.4.e.a.226.1 2
21.17 even 6 441.4.e.a.226.1 2
21.20 even 2 441.4.a.m.1.1 1
28.27 even 2 784.4.a.k.1.1 1
35.34 odd 2 1225.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.a.1.1 1 1.1 even 1 trivial
49.4.a.a.1.1 1 7.6 odd 2 CM
49.4.c.d.18.1 2 7.2 even 3
49.4.c.d.18.1 2 7.5 odd 6
49.4.c.d.30.1 2 7.3 odd 6
49.4.c.d.30.1 2 7.4 even 3
441.4.a.m.1.1 1 3.2 odd 2
441.4.a.m.1.1 1 21.20 even 2
441.4.e.a.226.1 2 21.11 odd 6
441.4.e.a.226.1 2 21.17 even 6
441.4.e.a.361.1 2 21.2 odd 6
441.4.e.a.361.1 2 21.5 even 6
784.4.a.k.1.1 1 4.3 odd 2
784.4.a.k.1.1 1 28.27 even 2
1225.4.a.l.1.1 1 5.4 even 2
1225.4.a.l.1.1 1 35.34 odd 2