Properties

Label 441.4.a.p.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.64575 q^{2} -1.00000 q^{4} +23.8118 q^{8} +O(q^{10})\) \(q-2.64575 q^{2} -1.00000 q^{4} +23.8118 q^{8} -26.4575 q^{11} -55.0000 q^{16} +70.0000 q^{22} +216.952 q^{23} -125.000 q^{25} +264.575 q^{29} -44.9778 q^{32} -450.000 q^{37} +180.000 q^{43} +26.4575 q^{44} -574.000 q^{46} +330.719 q^{50} -497.401 q^{53} -700.000 q^{58} +559.000 q^{64} -740.000 q^{67} -978.928 q^{71} +1190.59 q^{74} -1384.00 q^{79} -476.235 q^{86} -630.000 q^{88} -216.952 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 110q^{16} + 140q^{22} - 250q^{25} - 900q^{37} + 360q^{43} - 1148q^{46} - 1400q^{58} + 1118q^{64} - 1480q^{67} - 2768q^{79} - 1260q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64575 −0.935414 −0.467707 0.883883i \(-0.654920\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.125000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 23.8118 1.05234
\(9\) 0 0
\(10\) 0 0
\(11\) −26.4575 −0.725204 −0.362602 0.931944i \(-0.618111\pi\)
−0.362602 + 0.931944i \(0.618111\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −55.0000 −0.859375
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 70.0000 0.678366
\(23\) 216.952 1.96685 0.983425 0.181317i \(-0.0580360\pi\)
0.983425 + 0.181317i \(0.0580360\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 264.575 1.69415 0.847075 0.531473i \(-0.178361\pi\)
0.847075 + 0.531473i \(0.178361\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −44.9778 −0.248469
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −450.000 −1.99945 −0.999724 0.0235113i \(-0.992515\pi\)
−0.999724 + 0.0235113i \(0.992515\pi\)
\(38\) 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 \(-0.396592\pi\)
0.319183 + 0.947693i \(0.396592\pi\)
\(44\) 26.4575 0.0906505
\(45\) 0 0
\(46\) −574.000 −1.83982
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 330.719 0.935414
\(51\) 0 0
\(52\) 0 0
\(53\) −497.401 −1.28912 −0.644560 0.764554i \(-0.722959\pi\)
−0.644560 + 0.764554i \(0.722959\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −700.000 −1.58473
\(59\) 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
\(64\) 559.000 1.09180
\(65\) 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
\(71\) −978.928 −1.63630 −0.818151 0.575004i \(-0.805000\pi\)
−0.818151 + 0.575004i \(0.805000\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 1190.59 1.87031
\(75\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −476.235 −0.597137
\(87\) 0 0
\(88\) −630.000 −0.763162
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −216.952 −0.245856
\(93\) 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
\(99\) 0 0
\(100\) 125.000 0.125000
\(101\) 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
\(106\) 1316.00 1.20586
\(107\) −1550.41 −1.40078 −0.700392 0.713759i \(-0.746991\pi\)
−0.700392 + 0.713759i \(0.746991\pi\)
\(108\) 0 0
\(109\) 54.0000 0.0474519 0.0237260 0.999718i \(-0.492447\pi\)
0.0237260 + 0.999718i \(0.492447\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2307.10 −1.92065 −0.960324 0.278886i \(-0.910035\pi\)
−0.960324 + 0.278886i \(0.910035\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −264.575 −0.211769
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −631.000 −0.474080
\(122\) 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 \(0.746240\pi\)
\(128\) −1119.15 −0.772813
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1957.86 1.26219
\(135\) 0 0
\(136\) 0 0
\(137\) −783.142 −0.488382 −0.244191 0.969727i \(-0.578522\pi\)
−0.244191 + 0.969727i \(0.578522\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2590.00 1.53062
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 450.000 0.249931
\(149\) 3545.31 1.94928 0.974640 0.223777i \(-0.0718387\pi\)
0.974640 + 0.223777i \(0.0718387\pi\)
\(150\) 0 0
\(151\) 2952.00 1.59093 0.795465 0.606000i \(-0.207227\pi\)
0.795465 + 0.606000i \(0.207227\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 3661.72 1.84374
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 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
\(172\) −180.000 −0.0797958
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1455.16 0.623222
\(177\) 0 0
\(178\) 0 0
\(179\) −4312.57 −1.80077 −0.900383 0.435099i \(-0.856713\pi\)
−0.900383 + 0.435099i \(0.856713\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5166.00 2.06980
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3360.10 1.27292 0.636462 0.771308i \(-0.280397\pi\)
0.636462 + 0.771308i \(0.280397\pi\)
\(192\) 0 0
\(193\) 4590.00 1.71189 0.855947 0.517064i \(-0.172975\pi\)
0.855947 + 0.517064i \(0.172975\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5069.26 1.83335 0.916675 0.399634i \(-0.130863\pi\)
0.916675 + 0.399634i \(0.130863\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −2976.47 −1.05234
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5868.00 −1.91455 −0.957274 0.289181i \(-0.906617\pi\)
−0.957274 + 0.289181i \(0.906617\pi\)
\(212\) 497.401 0.161140
\(213\) 0 0
\(214\) 4102.00 1.31031
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −142.871 −0.0443872
\(219\) 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
\(226\) 6104.00 1.79660
\(227\) 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
\(232\) 6300.00 1.78282
\(233\) −5312.67 −1.49375 −0.746877 0.664963i \(-0.768447\pi\)
−0.746877 + 0.664963i \(0.768447\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 449.778 0.121731 0.0608655 0.998146i \(-0.480614\pi\)
0.0608655 + 0.998146i \(0.480614\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1669.47 0.443461
\(243\) 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
\(253\) −5740.00 −1.42637
\(254\) 5291.50 1.30716
\(255\) 0 0
\(256\) −1511.00 −0.368896
\(257\) 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
\(263\) 4026.83 0.944126 0.472063 0.881565i \(-0.343509\pi\)
0.472063 + 0.881565i \(0.343509\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 740.000 0.168667
\(269\) 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
\(274\) 2072.00 0.456840
\(275\) 3307.19 0.725204
\(276\) 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
\(281\) −8360.57 −1.77491 −0.887456 0.460893i \(-0.847529\pi\)
−0.887456 + 0.460893i \(0.847529\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 978.928 0.204538
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −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
\(296\) −10715.3 −2.10410
\(297\) 0 0
\(298\) −9380.00 −1.82339
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −7810.26 −1.48818
\(303\) 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
\(316\) 1384.00 0.246380
\(317\) −8879.14 −1.57319 −0.786597 0.617467i \(-0.788159\pi\)
−0.786597 + 0.617467i \(0.788159\pi\)
\(318\) 0 0
\(319\) −7000.00 −1.22860
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −4709.44 −0.800097
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10908.0 −1.81135 −0.905677 0.423969i \(-0.860636\pi\)
−0.905677 + 0.423969i \(0.860636\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3330.00 0.538269 0.269135 0.963103i \(-0.413262\pi\)
0.269135 + 0.963103i \(0.413262\pi\)
\(338\) 5812.72 0.935414
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 4286.12 0.671779
\(345\) 0 0
\(346\) 0 0
\(347\) 12260.4 1.89675 0.948377 0.317146i \(-0.102725\pi\)
0.948377 + 0.317146i \(0.102725\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1190.00 0.180191
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 11410.0 1.68446
\(359\) −10927.0 −1.60641 −0.803207 0.595700i \(-0.796875\pi\)
−0.803207 + 0.595700i \(0.796875\pi\)
\(360\) 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 \(0.5\pi\)
\(368\) −11932.3 −1.69026
\(369\) 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 \(-0.799179\pi\)
−0.807498 + 0.589870i \(0.799179\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8890.00 −1.19071
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12144.0 −1.60133
\(387\) 0 0
\(388\) 0 0
\(389\) 11165.1 1.45525 0.727624 0.685976i \(-0.240625\pi\)
0.727624 + 0.685976i \(0.240625\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −13412.0 −1.71494
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6875.00 0.859375
\(401\) 15980.3 1.99007 0.995037 0.0995016i \(-0.0317249\pi\)
0.995037 + 0.0995016i \(0.0317249\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11905.9 1.45001
\(408\) 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
\(414\) 0 0
\(415\) 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 \(0.155247\pi\)
\(422\) 15525.3 1.79090
\(423\) 0 0
\(424\) −11844.0 −1.35659
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1550.41 0.175098
\(429\) 0 0
\(430\) 0 0
\(431\) −15689.3 −1.75343 −0.876714 0.481012i \(-0.840269\pi\)
−0.876714 + 0.481012i \(0.840269\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −54.0000 −0.00593149
\(437\) 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
\(443\) 1592.74 0.170820 0.0854102 0.996346i \(-0.472780\pi\)
0.0854102 + 0.996346i \(0.472780\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18837.7 1.97997 0.989987 0.141158i \(-0.0450827\pi\)
0.989987 + 0.141158i \(0.0450827\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2307.10 0.240081
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8010.00 −0.819895 −0.409947 0.912109i \(-0.634453\pi\)
−0.409947 + 0.912109i \(0.634453\pi\)
\(458\) 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 \(0.639229\pi\)
\(464\) −14551.6 −1.45591
\(465\) 0 0
\(466\) 14056.0 1.39728
\(467\) 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
\(473\) −4762.35 −0.462945
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1190.00 −0.113869
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 631.000 0.0592600
\(485\) 0 0
\(486\) 0 0
\(487\) −21240.0 −1.97634 −0.988169 0.153371i \(-0.950987\pi\)
−0.988169 + 0.153371i \(0.950987\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7646.22 0.702788 0.351394 0.936228i \(-0.385708\pi\)
0.351394 + 0.936228i \(0.385708\pi\)
\(492\) 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 \(-0.394778\pi\)
0.324577 + 0.945859i \(0.394778\pi\)
\(500\) 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
\(506\) 15186.6 1.33424
\(507\) 0 0
\(508\) 2000.00 0.174676
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12951.0 1.11788
\(513\) 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 \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −10654.0 −0.883149
\(527\) 0 0
\(528\) 0 0
\(529\) 34901.0 2.86850
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −17620.7 −1.41996
\(537\) 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 \(0.330644\pi\)
\(548\) 783.142 0.0610478
\(549\) 0 0
\(550\) −8750.00 −0.678366
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −19340.4 −1.48321
\(555\) 0 0
\(556\) 0 0
\(557\) 16498.9 1.25508 0.627541 0.778583i \(-0.284061\pi\)
0.627541 + 0.778583i \(0.284061\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 22120.0 1.66028
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −23310.0 −1.72195
\(569\) −3598.22 −0.265106 −0.132553 0.991176i \(-0.542317\pi\)
−0.132553 + 0.991176i \(0.542317\pi\)
\(570\) 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
\(575\) −27119.0 −1.96685
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 12998.6 0.935414
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 13160.0 0.934874
\(584\) 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
\(592\) 24750.0 1.71827
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3545.31 −0.243660
\(597\) 0 0
\(598\) 0 0
\(599\) 15742.2 1.07381 0.536903 0.843644i \(-0.319594\pi\)
0.536903 + 0.843644i \(0.319594\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2952.00 −0.198866
\(605\) 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
\(617\) 2497.59 0.162965 0.0814823 0.996675i \(-0.474035\pi\)
0.0814823 + 0.996675i \(0.474035\pi\)
\(618\) 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 \(0.809512\pi\)
\(632\) −32955.5 −2.07421
\(633\) 0 0
\(634\) 23492.0 1.47159
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 18520.3 1.14925
\(639\) 0 0
\(640\) 0 0
\(641\) −31219.9 −1.92373 −0.961865 0.273526i \(-0.911810\pi\)
−0.961865 + 0.273526i \(0.911810\pi\)
\(642\) 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 \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1780.00 −0.106917
\(653\) −19546.8 −1.17140 −0.585701 0.810527i \(-0.699181\pi\)
−0.585701 + 0.810527i \(0.699181\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −33786.2 −1.99716 −0.998578 0.0533186i \(-0.983020\pi\)
−0.998578 + 0.0533186i \(0.983020\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 28859.9 1.69437
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 57400.0 3.33214
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 33570.0 1.92278 0.961388 0.275196i \(-0.0887428\pi\)
0.961388 + 0.275196i \(0.0887428\pi\)
\(674\) −8810.35 −0.503505
\(675\) 0 0
\(676\) 2197.00 0.125000
\(677\) 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
\(683\) 10694.1 0.599121 0.299560 0.954077i \(-0.403160\pi\)
0.299560 + 0.954077i \(0.403160\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −9900.00 −0.548596
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −32438.0 −1.77425
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36881.8 −1.98717 −0.993584 0.113093i \(-0.963924\pi\)
−0.993584 + 0.113093i \(0.963924\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −14789.7 −0.791775
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12546.0 −0.664563 −0.332281 0.943180i \(-0.607818\pi\)
−0.332281 + 0.943180i \(0.607818\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4312.57 0.225096
\(717\) 0 0
\(718\) 28910.0 1.50266
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18147.2 0.935414
\(723\) 0 0
\(724\) 0 0
\(725\) −33071.9 −1.69415
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 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
\(736\) −9758.00 −0.488702
\(737\) 19578.6 0.978542
\(738\) 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
\(743\) 31743.7 1.56738 0.783691 0.621151i \(-0.213335\pi\)
0.783691 + 0.621151i \(0.213335\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 36961.1 1.81400
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2448.00 0.118946 0.0594732 0.998230i \(-0.481058\pi\)
0.0594732 + 0.998230i \(0.481058\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34830.0 1.67228 0.836141 0.548514i \(-0.184806\pi\)
0.836141 + 0.548514i \(0.184806\pi\)
\(758\) 31526.8 1.51069
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3360.10 −0.159116
\(765\) 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
\(772\) −4590.00 −0.213987
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −29540.0 −1.36126
\(779\) 0 0
\(780\) 0 0
\(781\) 25900.0 1.18665
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −5069.26 −0.229169
\(789\) 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
\(799\) 0 0
\(800\) 5622.22 0.248469
\(801\) 0 0
\(802\) −42280.0 −1.86154
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 26880.8 1.16821 0.584104 0.811679i \(-0.301446\pi\)
0.584104 + 0.811679i \(0.301446\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −31500.0 −1.35636
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17832.4 0.758044 0.379022 0.925388i \(-0.376261\pi\)
0.379022 + 0.925388i \(0.376261\pi\)
\(822\) 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
\(827\) 41077.9 1.72723 0.863615 0.504151i \(-0.168195\pi\)
0.863615 + 0.504151i \(0.168195\pi\)
\(828\) 0 0
\(829\) 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
\(841\) 45611.0 1.87015
\(842\) −40379.5 −1.65269
\(843\) 0 0
\(844\) 5868.00 0.239319
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 27357.1 1.10784
\(849\) 0 0
\(850\) 0 0
\(851\) −97628.2 −3.93261
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36918.0 −1.47410
\(857\) 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
\(862\) 41510.0 1.64018
\(863\) 46507.0 1.83443 0.917217 0.398387i \(-0.130430\pi\)
0.917217 + 0.398387i \(0.130430\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 36617.2 1.42941
\(870\) 0 0
\(871\) 0 0
\(872\) 1285.84 0.0499356
\(873\) 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 \(-0.694150\pi\)
−0.572820 + 0.819681i \(0.694150\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4214.00 −0.159788
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −49840.0 −1.85210
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −54936.0 −2.02118
\(905\) 0 0
\(906\) 0 0
\(907\) −52740.0 −1.93076 −0.965382 0.260840i \(-0.916000\pi\)
−0.965382 + 0.260840i \(0.916000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38125.3 −1.38655 −0.693275 0.720673i \(-0.743833\pi\)
−0.693275 + 0.720673i \(0.743833\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 21192.5 0.766942
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −21744.0 −0.780488 −0.390244 0.920711i \(-0.627609\pi\)
−0.390244 + 0.920711i \(0.627609\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 56250.0 1.99945
\(926\) 22330.1 0.792456
\(927\) 0 0
\(928\) −11900.0 −0.420945
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5312.67 0.186719
\(933\) 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
\(946\) 12600.0 0.433046
\(947\) −31839.0 −1.09253 −0.546266 0.837612i \(-0.683951\pi\)
−0.546266 + 0.837612i \(0.683951\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −51031.3 −1.73459 −0.867295 0.497794i \(-0.834143\pi\)
−0.867295 + 0.497794i \(0.834143\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −449.778 −0.0152164
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52040.0 1.73060 0.865302 0.501251i \(-0.167127\pi\)
0.865302 + 0.501251i \(0.167127\pi\)
\(968\) −15025.2 −0.498894
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 56195.8 1.84869
\(975\) 0 0
\(976\) 0 0
\(977\) 48216.2 1.57889 0.789443 0.613824i \(-0.210369\pi\)
0.789443 + 0.613824i \(0.210369\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −20230.0 −0.657398
\(983\) 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
\(989\) 39051.3 1.25557
\(990\) 0 0
\(991\) −57528.0 −1.84403 −0.922017 0.387150i \(-0.873460\pi\)
−0.922017 + 0.387150i \(0.873460\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) −19144.7 −0.607228
\(999\) 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 441.4.a.p.1.1 2
3.2 odd 2 inner 441.4.a.p.1.2 yes 2
7.2 even 3 441.4.e.t.361.2 4
7.3 odd 6 441.4.e.t.226.2 4
7.4 even 3 441.4.e.t.226.2 4
7.5 odd 6 441.4.e.t.361.2 4
7.6 odd 2 CM 441.4.a.p.1.1 2
21.2 odd 6 441.4.e.t.361.1 4
21.5 even 6 441.4.e.t.361.1 4
21.11 odd 6 441.4.e.t.226.1 4
21.17 even 6 441.4.e.t.226.1 4
21.20 even 2 inner 441.4.a.p.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.a.p.1.1 2 1.1 even 1 trivial
441.4.a.p.1.1 2 7.6 odd 2 CM
441.4.a.p.1.2 yes 2 3.2 odd 2 inner
441.4.a.p.1.2 yes 2 21.20 even 2 inner
441.4.e.t.226.1 4 21.11 odd 6
441.4.e.t.226.1 4 21.17 even 6
441.4.e.t.226.2 4 7.3 odd 6
441.4.e.t.226.2 4 7.4 even 3
441.4.e.t.361.1 4 21.2 odd 6
441.4.e.t.361.1 4 21.5 even 6
441.4.e.t.361.2 4 7.2 even 3
441.4.e.t.361.2 4 7.5 odd 6