Properties

Label 441.6.a.a.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-11.0000 q^{2} +89.0000 q^{4} -627.000 q^{8} +O(q^{10})\) \(q-11.0000 q^{2} +89.0000 q^{4} -627.000 q^{8} +76.0000 q^{11} +4049.00 q^{16} -836.000 q^{22} +4952.00 q^{23} -3125.00 q^{25} -7282.00 q^{29} -24475.0 q^{32} -8886.00 q^{37} +11748.0 q^{43} +6764.00 q^{44} -54472.0 q^{46} +34375.0 q^{50} -24550.0 q^{53} +80102.0 q^{58} +139657. q^{64} +69364.0 q^{67} +2224.00 q^{71} +97746.0 q^{74} +80168.0 q^{79} -129228. q^{86} -47652.0 q^{88} +440728. q^{92} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.0000 −1.94454 −0.972272 0.233854i \(-0.924866\pi\)
−0.972272 + 0.233854i \(0.924866\pi\)
\(3\) 0 0
\(4\) 89.0000 2.78125
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −627.000 −3.46372
\(9\) 0 0
\(10\) 0 0
\(11\) 76.0000 0.189379 0.0946895 0.995507i \(-0.469814\pi\)
0.0946895 + 0.995507i \(0.469814\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4049.00 3.95410
\(17\) 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
\(22\) −836.000 −0.368256
\(23\) 4952.00 1.95192 0.975958 0.217959i \(-0.0699401\pi\)
0.975958 + 0.217959i \(0.0699401\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7282.00 −1.60789 −0.803944 0.594705i \(-0.797269\pi\)
−0.803944 + 0.594705i \(0.797269\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −24475.0 −4.22520
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8886.00 −1.06709 −0.533546 0.845771i \(-0.679141\pi\)
−0.533546 + 0.845771i \(0.679141\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 11748.0 0.968931 0.484465 0.874810i \(-0.339014\pi\)
0.484465 + 0.874810i \(0.339014\pi\)
\(44\) 6764.00 0.526710
\(45\) 0 0
\(46\) −54472.0 −3.79559
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 34375.0 1.94454
\(51\) 0 0
\(52\) 0 0
\(53\) −24550.0 −1.20050 −0.600250 0.799813i \(-0.704932\pi\)
−0.600250 + 0.799813i \(0.704932\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 80102.0 3.12661
\(59\) 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
\(64\) 139657. 4.26199
\(65\) 0 0
\(66\) 0 0
\(67\) 69364.0 1.88776 0.943881 0.330286i \(-0.107145\pi\)
0.943881 + 0.330286i \(0.107145\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2224.00 0.0523587 0.0261794 0.999657i \(-0.491666\pi\)
0.0261794 + 0.999657i \(0.491666\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 97746.0 2.07501
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 80168.0 1.44522 0.722609 0.691257i \(-0.242943\pi\)
0.722609 + 0.691257i \(0.242943\pi\)
\(80\) 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
\(86\) −129228. −1.88413
\(87\) 0 0
\(88\) −47652.0 −0.655956
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 440728. 5.42877
\(93\) 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
\(100\) −278125. −2.78125
\(101\) 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
\(106\) 270050. 2.33442
\(107\) 64900.0 0.548006 0.274003 0.961729i \(-0.411652\pi\)
0.274003 + 0.961729i \(0.411652\pi\)
\(108\) 0 0
\(109\) −219582. −1.77023 −0.885117 0.465369i \(-0.845922\pi\)
−0.885117 + 0.465369i \(0.845922\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −123202. −0.907657 −0.453828 0.891089i \(-0.649942\pi\)
−0.453828 + 0.891089i \(0.649942\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −648098. −4.47194
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −155275. −0.964136
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −262064. −1.44178 −0.720888 0.693051i \(-0.756266\pi\)
−0.720888 + 0.693051i \(0.756266\pi\)
\(128\) −753027. −4.06243
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −763004. −3.67083
\(135\) 0 0
\(136\) 0 0
\(137\) 353450. 1.60889 0.804445 0.594027i \(-0.202463\pi\)
0.804445 + 0.594027i \(0.202463\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −24464.0 −0.101814
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −790854. −2.96785
\(149\) 337018. 1.24362 0.621810 0.783168i \(-0.286398\pi\)
0.621810 + 0.783168i \(0.286398\pi\)
\(150\) 0 0
\(151\) −261624. −0.933760 −0.466880 0.884321i \(-0.654622\pi\)
−0.466880 + 0.884321i \(0.654622\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −881848. −2.81029
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 663100. 1.95483 0.977417 0.211318i \(-0.0677757\pi\)
0.977417 + 0.211318i \(0.0677757\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −371293. −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 1.04557e6 2.69484
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 307724. 0.748824
\(177\) 0 0
\(178\) 0 0
\(179\) −584564. −1.36364 −0.681820 0.731520i \(-0.738811\pi\)
−0.681820 + 0.731520i \(0.738811\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.10490e6 −6.76089
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.00305e6 −1.98947 −0.994737 0.102461i \(-0.967328\pi\)
−0.994737 + 0.102461i \(0.967328\pi\)
\(192\) 0 0
\(193\) −385902. −0.745734 −0.372867 0.927885i \(-0.621625\pi\)
−0.372867 + 0.927885i \(0.621625\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −385814. −0.708292 −0.354146 0.935190i \(-0.615228\pi\)
−0.354146 + 0.935190i \(0.615228\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.95938e6 3.46372
\(201\) 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
\(211\) −1.09705e6 −1.69637 −0.848186 0.529699i \(-0.822305\pi\)
−0.848186 + 0.529699i \(0.822305\pi\)
\(212\) −2.18495e6 −3.33889
\(213\) 0 0
\(214\) −713900. −1.06562
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.41540e6 3.44230
\(219\) 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
\(226\) 1.35522e6 1.76498
\(227\) 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
\(232\) 4.56581e6 5.56927
\(233\) 1.27950e6 1.54401 0.772004 0.635617i \(-0.219254\pi\)
0.772004 + 0.635617i \(0.219254\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −723536. −0.819342 −0.409671 0.912233i \(-0.634357\pi\)
−0.409671 + 0.912233i \(0.634357\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1.70802e6 1.87480
\(243\) 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
\(253\) 376352. 0.369652
\(254\) 2.88270e6 2.80360
\(255\) 0 0
\(256\) 3.81427e6 3.63757
\(257\) 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
\(263\) −1.53155e6 −1.36534 −0.682672 0.730725i \(-0.739182\pi\)
−0.682672 + 0.730725i \(0.739182\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 6.17340e6 5.25034
\(269\) 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
\(274\) −3.88795e6 −3.12856
\(275\) −237500. −0.189379
\(276\) 0 0
\(277\) −2.55145e6 −1.99796 −0.998982 0.0451116i \(-0.985636\pi\)
−0.998982 + 0.0451116i \(0.985636\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.54797e6 −1.92499 −0.962497 0.271294i \(-0.912548\pi\)
−0.962497 + 0.271294i \(0.912548\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 197936. 0.145623
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −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
\(296\) 5.57152e6 3.69611
\(297\) 0 0
\(298\) −3.70720e6 −2.41827
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 2.87786e6 1.81574
\(303\) 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
\(316\) 7.13495e6 4.01951
\(317\) 221714. 0.123921 0.0619605 0.998079i \(-0.480265\pi\)
0.0619605 + 0.998079i \(0.480265\pi\)
\(318\) 0 0
\(319\) −553432. −0.304500
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −7.29410e6 −3.80126
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.97148e6 1.49074 0.745371 0.666650i \(-0.232273\pi\)
0.745371 + 0.666650i \(0.232273\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.15965e6 −1.99518 −0.997590 0.0693859i \(-0.977896\pi\)
−0.997590 + 0.0693859i \(0.977896\pi\)
\(338\) 4.08422e6 1.94454
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −7.36600e6 −3.35610
\(345\) 0 0
\(346\) 0 0
\(347\) −2.29816e6 −1.02461 −0.512304 0.858804i \(-0.671208\pi\)
−0.512304 + 0.858804i \(0.671208\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.86010e6 −0.800165
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 6.43020e6 2.65166
\(359\) −4.26897e6 −1.74818 −0.874091 0.485762i \(-0.838542\pi\)
−0.874091 + 0.485762i \(0.838542\pi\)
\(360\) 0 0
\(361\) −2.47610e6 −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 \(0.5\pi\)
\(368\) 2.00506e7 7.71807
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 599302. 0.223035 0.111518 0.993762i \(-0.464429\pi\)
0.111518 + 0.993762i \(0.464429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.59273e6 −1.99998 −0.999991 0.00429827i \(-0.998632\pi\)
−0.999991 + 0.00429827i \(0.998632\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.10335e7 3.86862
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.24492e6 1.45011
\(387\) 0 0
\(388\) 0 0
\(389\) 1.26012e6 0.422218 0.211109 0.977462i \(-0.432292\pi\)
0.211109 + 0.977462i \(0.432292\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 4.24395e6 1.37730
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.26531e7 −3.95410
\(401\) −6.03293e6 −1.87356 −0.936779 0.349922i \(-0.886208\pi\)
−0.936779 + 0.349922i \(0.886208\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −675336. −0.202085
\(408\) 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
\(421\) 2.07477e6 0.570513 0.285257 0.958451i \(-0.407921\pi\)
0.285257 + 0.958451i \(0.407921\pi\)
\(422\) 1.20676e7 3.29867
\(423\) 0 0
\(424\) 1.53929e7 4.15819
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 5.77610e6 1.52414
\(429\) 0 0
\(430\) 0 0
\(431\) 7.33885e6 1.90298 0.951491 0.307676i \(-0.0995514\pi\)
0.951491 + 0.307676i \(0.0995514\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.95428e7 −4.92346
\(437\) 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
\(443\) 5.10490e6 1.23588 0.617942 0.786223i \(-0.287967\pi\)
0.617942 + 0.786223i \(0.287967\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.99808e6 −0.467732 −0.233866 0.972269i \(-0.575138\pi\)
−0.233866 + 0.972269i \(0.575138\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.09650e7 −2.52442
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.77969e6 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 2.88620e6 0.625711 0.312856 0.949801i \(-0.398714\pi\)
0.312856 + 0.949801i \(0.398714\pi\)
\(464\) −2.94848e7 −6.35775
\(465\) 0 0
\(466\) −1.40745e7 −3.00239
\(467\) 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
\(473\) 892848. 0.183495
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 7.95890e6 1.59325
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.38195e7 −2.68150
\(485\) 0 0
\(486\) 0 0
\(487\) −2.76146e6 −0.527615 −0.263807 0.964575i \(-0.584978\pi\)
−0.263807 + 0.964575i \(0.584978\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.82732e6 −1.65244 −0.826219 0.563349i \(-0.809513\pi\)
−0.826219 + 0.563349i \(0.809513\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.11204e7 −1.99925 −0.999626 0.0273386i \(-0.991297\pi\)
−0.999626 + 0.0273386i \(0.991297\pi\)
\(500\) 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
\(506\) −4.13987e6 −0.718804
\(507\) 0 0
\(508\) −2.33237e7 −4.00994
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.78601e7 −3.01099
\(513\) 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
\(526\) 1.68471e7 2.65497
\(527\) 0 0
\(528\) 0 0
\(529\) 1.80860e7 2.80997
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −4.34912e7 −6.53867
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.11261e7 1.63437 0.817186 0.576374i \(-0.195533\pi\)
0.817186 + 0.576374i \(0.195533\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.23604e6 −0.319529 −0.159765 0.987155i \(-0.551074\pi\)
−0.159765 + 0.987155i \(0.551074\pi\)
\(548\) 3.14570e7 4.47473
\(549\) 0 0
\(550\) 2.61250e6 0.368256
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 2.80660e7 3.88513
\(555\) 0 0
\(556\) 0 0
\(557\) −6.23949e6 −0.852140 −0.426070 0.904690i \(-0.640102\pi\)
−0.426070 + 0.904690i \(0.640102\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.80277e7 3.74323
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −1.39445e6 −0.181356
\(569\) −1.04738e7 −1.35620 −0.678099 0.734971i \(-0.737196\pi\)
−0.678099 + 0.734971i \(0.737196\pi\)
\(570\) 0 0
\(571\) 6.33912e6 0.813653 0.406826 0.913506i \(-0.366635\pi\)
0.406826 + 0.913506i \(0.366635\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.54750e7 −1.95192
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 1.56184e7 1.94454
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.86580e6 −0.227349
\(584\) 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
\(592\) −3.59794e7 −4.21939
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.99946e7 3.45882
\(597\) 0 0
\(598\) 0 0
\(599\) 7.17757e6 0.817354 0.408677 0.912679i \(-0.365990\pi\)
0.408677 + 0.912679i \(0.365990\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.32845e7 −2.59702
\(605\) 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
\(613\) 1.75514e7 1.88652 0.943258 0.332060i \(-0.107744\pi\)
0.943258 + 0.332060i \(0.107744\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.14899e6 0.756017 0.378008 0.925802i \(-0.376609\pi\)
0.378008 + 0.925802i \(0.376609\pi\)
\(618\) 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
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.99786e7 1.99752 0.998760 0.0497844i \(-0.0158534\pi\)
0.998760 + 0.0497844i \(0.0158534\pi\)
\(632\) −5.02653e7 −5.00583
\(633\) 0 0
\(634\) −2.43885e6 −0.240970
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 6.08775e6 0.592114
\(639\) 0 0
\(640\) 0 0
\(641\) 1.14963e7 1.10513 0.552563 0.833471i \(-0.313650\pi\)
0.552563 + 0.833471i \(0.313650\pi\)
\(642\) 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
\(652\) 5.90159e7 5.43688
\(653\) −1.09772e7 −1.00742 −0.503710 0.863873i \(-0.668032\pi\)
−0.503710 + 0.863873i \(0.668032\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.02232e7 1.81400 0.907000 0.421131i \(-0.138367\pi\)
0.907000 + 0.421131i \(0.138367\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −3.26862e7 −2.89881
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.60605e7 −3.13846
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.38435e6 0.117817 0.0589085 0.998263i \(-0.481238\pi\)
0.0589085 + 0.998263i \(0.481238\pi\)
\(674\) 4.57562e7 3.87971
\(675\) 0 0
\(676\) −3.30451e7 −2.78125
\(677\) 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
\(683\) 2.13149e7 1.74836 0.874179 0.485603i \(-0.161400\pi\)
0.874179 + 0.485603i \(0.161400\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.75677e7 3.83125
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.52798e7 1.99239
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.88600e6 −0.375542 −0.187771 0.982213i \(-0.560126\pi\)
−0.187771 + 0.982213i \(0.560126\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.06139e7 0.807132
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.43225e7 −1.07005 −0.535023 0.844837i \(-0.679697\pi\)
−0.535023 + 0.844837i \(0.679697\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −5.20262e7 −3.79262
\(717\) 0 0
\(718\) 4.69586e7 3.39942
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.72371e7 1.94454
\(723\) 0 0
\(724\) 0 0
\(725\) 2.27562e7 1.60789
\(726\) 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
\(736\) −1.21200e8 −8.24724
\(737\) 5.27166e6 0.357502
\(738\) 0 0
\(739\) −2.64893e6 −0.178427 −0.0892133 0.996013i \(-0.528435\pi\)
−0.0892133 + 0.996013i \(0.528435\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.49502e7 −1.65807 −0.829033 0.559199i \(-0.811109\pi\)
−0.829033 + 0.559199i \(0.811109\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6.59232e6 −0.433702
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.51088e7 1.62452 0.812260 0.583295i \(-0.198237\pi\)
0.812260 + 0.583295i \(0.198237\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.78870e7 −1.13448 −0.567242 0.823551i \(-0.691989\pi\)
−0.567242 + 0.823551i \(0.691989\pi\)
\(758\) 6.15201e7 3.88905
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.92713e7 −5.53322
\(765\) 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
\(772\) −3.43453e7 −2.07407
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.38613e7 −0.821022
\(779\) 0 0
\(780\) 0 0
\(781\) 169024. 0.00991564
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −3.43374e7 −1.96994
\(789\) 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
\(800\) 7.64844e7 4.22520
\(801\) 0 0
\(802\) 6.63622e7 3.64321
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.83707e7 −0.986856 −0.493428 0.869787i \(-0.664256\pi\)
−0.493428 + 0.869787i \(0.664256\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 7.42870e6 0.392963
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.55881e7 −1.84267 −0.921334 0.388773i \(-0.872899\pi\)
−0.921334 + 0.388773i \(0.872899\pi\)
\(822\) 0 0
\(823\) 7.08675e6 0.364710 0.182355 0.983233i \(-0.441628\pi\)
0.182355 + 0.983233i \(0.441628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.68552e7 1.87385 0.936926 0.349527i \(-0.113658\pi\)
0.936926 + 0.349527i \(0.113658\pi\)
\(828\) 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
\(841\) 3.25164e7 1.58530
\(842\) −2.28225e7 −1.10939
\(843\) 0 0
\(844\) −9.76376e7 −4.71803
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −9.94030e7 −4.74690
\(849\) 0 0
\(850\) 0 0
\(851\) −4.40035e7 −2.08287
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.06923e7 −1.89814
\(857\) 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
\(862\) −8.07273e7 −3.70043
\(863\) −2.76142e7 −1.26214 −0.631068 0.775727i \(-0.717383\pi\)
−0.631068 + 0.775727i \(0.717383\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.09277e6 0.273694
\(870\) 0 0
\(871\) 0 0
\(872\) 1.37678e8 6.13159
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.33428e7 −1.90291 −0.951453 0.307793i \(-0.900410\pi\)
−0.951453 + 0.307793i \(0.900410\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −3.94011e7 −1.70062 −0.850308 0.526286i \(-0.823584\pi\)
−0.850308 + 0.526286i \(0.823584\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −5.61539e7 −2.40323
\(887\) 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
\(898\) 2.19789e7 0.909526
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 7.72477e7 3.14387
\(905\) 0 0
\(906\) 0 0
\(907\) 4.48347e7 1.80966 0.904828 0.425777i \(-0.139999\pi\)
0.904828 + 0.425777i \(0.139999\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.87850e7 −1.94756 −0.973778 0.227498i \(-0.926945\pi\)
−0.973778 + 0.227498i \(0.926945\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.35765e7 2.51728
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.06716e7 1.97914 0.989569 0.144059i \(-0.0460156\pi\)
0.989569 + 0.144059i \(0.0460156\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.77688e7 1.06709
\(926\) −3.17482e7 −1.21672
\(927\) 0 0
\(928\) 1.78227e8 6.79365
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.13875e8 4.29427
\(933\) 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
\(946\) −9.82133e6 −0.356814
\(947\) 5.50009e7 1.99294 0.996471 0.0839326i \(-0.0267480\pi\)
0.996471 + 0.0839326i \(0.0267480\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.97130e6 0.355647 0.177824 0.984062i \(-0.443094\pi\)
0.177824 + 0.984062i \(0.443094\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.43947e7 −2.27880
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.86292e7 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00222e7 0.688566 0.344283 0.938866i \(-0.388122\pi\)
0.344283 + 0.938866i \(0.388122\pi\)
\(968\) 9.73574e7 3.33949
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 3.03761e7 1.02597
\(975\) 0 0
\(976\) 0 0
\(977\) 4.99817e7 1.67523 0.837616 0.546260i \(-0.183949\pi\)
0.837616 + 0.546260i \(0.183949\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 9.71006e7 3.21324
\(983\) 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
\(989\) 5.81761e7 1.89127
\(990\) 0 0
\(991\) −5.73144e7 −1.85387 −0.926936 0.375219i \(-0.877567\pi\)
−0.926936 + 0.375219i \(0.877567\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 1.22324e8 3.88763
\(999\) 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 441.6.a.a.1.1 1
3.2 odd 2 49.6.a.b.1.1 1
7.6 odd 2 CM 441.6.a.a.1.1 1
12.11 even 2 784.6.a.g.1.1 1
21.2 odd 6 49.6.c.a.18.1 2
21.5 even 6 49.6.c.a.18.1 2
21.11 odd 6 49.6.c.a.30.1 2
21.17 even 6 49.6.c.a.30.1 2
21.20 even 2 49.6.a.b.1.1 1
84.83 odd 2 784.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.b.1.1 1 3.2 odd 2
49.6.a.b.1.1 1 21.20 even 2
49.6.c.a.18.1 2 21.2 odd 6
49.6.c.a.18.1 2 21.5 even 6
49.6.c.a.30.1 2 21.11 odd 6
49.6.c.a.30.1 2 21.17 even 6
441.6.a.a.1.1 1 1.1 even 1 trivial
441.6.a.a.1.1 1 7.6 odd 2 CM
784.6.a.g.1.1 1 12.11 even 2
784.6.a.g.1.1 1 84.83 odd 2