Properties

Label 441.6.a.o.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
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

\(f(q)\) \(=\) \(q-2.64575 q^{2} -25.0000 q^{4} +150.808 q^{8} +O(q^{10})\) \(q-2.64575 q^{2} -25.0000 q^{4} +150.808 q^{8} +799.017 q^{11} +401.000 q^{16} -2114.00 q^{22} +1105.92 q^{23} -3125.00 q^{25} -5386.75 q^{29} -5886.80 q^{32} +8886.00 q^{37} -11748.0 q^{43} -19975.4 q^{44} -2926.00 q^{46} +8267.97 q^{50} +32712.1 q^{53} +14252.0 q^{58} +2743.00 q^{64} +69364.0 q^{67} -84923.3 q^{71} -23510.1 q^{74} +80168.0 q^{79} +31082.3 q^{86} +120498. q^{88} -27648.1 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 50 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88}+O(q^{100}) \) Copy content Toggle raw display

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 −0.467707 −0.233854 0.972272i \(-0.575134\pi\)
−0.233854 + 0.972272i \(0.575134\pi\)
\(3\) 0 0
\(4\) −25.0000 −0.781250
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 150.808 0.833103
\(9\) 0 0
\(10\) 0 0
\(11\) 799.017 1.99101 0.995507 0.0946895i \(-0.0301858\pi\)
0.995507 + 0.0946895i \(0.0301858\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 401.000 0.391602
\(17\) 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
\(22\) −2114.00 −0.931211
\(23\) 1105.92 0.435919 0.217959 0.975958i \(-0.430060\pi\)
0.217959 + 0.975958i \(0.430060\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5386.75 −1.18941 −0.594705 0.803944i \(-0.702731\pi\)
−0.594705 + 0.803944i \(0.702731\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5886.80 −1.01626
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8886.00 1.06709 0.533546 0.845771i \(-0.320859\pi\)
0.533546 + 0.845771i \(0.320859\pi\)
\(38\) 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 \(-0.660986\pi\)
−0.484465 + 0.874810i \(0.660986\pi\)
\(44\) −19975.4 −1.55548
\(45\) 0 0
\(46\) −2926.00 −0.203882
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 8267.97 0.467707
\(51\) 0 0
\(52\) 0 0
\(53\) 32712.1 1.59963 0.799813 0.600250i \(-0.204932\pi\)
0.799813 + 0.600250i \(0.204932\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 14252.0 0.556296
\(59\) 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
\(64\) 2743.00 0.0837097
\(65\) 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
\(71\) −84923.3 −1.99931 −0.999657 0.0261794i \(-0.991666\pi\)
−0.999657 + 0.0261794i \(0.991666\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −23510.1 −0.499087
\(75\) 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
\(86\) 31082.3 0.453176
\(87\) 0 0
\(88\) 120498. 1.65872
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −27648.1 −0.340562
\(93\) 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
\(100\) 78125.0 0.781250
\(101\) 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
\(106\) −86548.0 −0.748156
\(107\) −227794. −1.92346 −0.961729 0.274003i \(-0.911652\pi\)
−0.961729 + 0.274003i \(0.911652\pi\)
\(108\) 0 0
\(109\) 219582. 1.77023 0.885117 0.465369i \(-0.154078\pi\)
0.885117 + 0.465369i \(0.154078\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 241906. 1.78218 0.891089 0.453828i \(-0.149942\pi\)
0.891089 + 0.453828i \(0.149942\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 134669. 0.929227
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 477377. 2.96414
\(122\) 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 \(0.756266\pi\)
\(128\) 181120. 0.977107
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −183520. −0.882920
\(135\) 0 0
\(136\) 0 0
\(137\) −260998. −1.18805 −0.594027 0.804445i \(-0.702463\pi\)
−0.594027 + 0.804445i \(0.702463\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 224686. 0.935094
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −222150. −0.833666
\(149\) 424474. 1.56634 0.783168 0.621810i \(-0.213602\pi\)
0.783168 + 0.621810i \(0.213602\pi\)
\(150\) 0 0
\(151\) 261624. 0.933760 0.466880 0.884321i \(-0.345378\pi\)
0.466880 + 0.884321i \(0.345378\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −212105. −0.675939
\(159\) 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
\(172\) 293700. 0.756977
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 320406. 0.779684
\(177\) 0 0
\(178\) 0 0
\(179\) 627175. 1.46304 0.731520 0.681820i \(-0.238811\pi\)
0.731520 + 0.681820i \(0.238811\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 166782. 0.363166
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −103317. −0.204921 −0.102461 0.994737i \(-0.532672\pi\)
−0.102461 + 0.994737i \(0.532672\pi\)
\(192\) 0 0
\(193\) 385902. 0.745734 0.372867 0.927885i \(-0.378375\pi\)
0.372867 + 0.927885i \(0.378375\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.01882e6 1.87038 0.935190 0.354146i \(-0.115228\pi\)
0.935190 + 0.354146i \(0.115228\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −471274. −0.833103
\(201\) 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 \(-0.177695\pi\)
0.848186 + 0.529699i \(0.177695\pi\)
\(212\) −817802. −1.24971
\(213\) 0 0
\(214\) 602686. 0.899615
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −580959. −0.827951
\(219\) 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
\(226\) −640024. −0.833538
\(227\) 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
\(232\) −812364. −0.990902
\(233\) 1.05345e6 1.27123 0.635617 0.772004i \(-0.280746\pi\)
0.635617 + 0.772004i \(0.280746\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.61113e6 −1.82447 −0.912233 0.409671i \(-0.865643\pi\)
−0.912233 + 0.409671i \(0.865643\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −1.26302e6 −1.38635
\(243\) 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
\(253\) 883652. 0.867921
\(254\) 693356. 0.674329
\(255\) 0 0
\(256\) −566975. −0.540709
\(257\) 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
\(263\) 1.63936e6 1.46145 0.730725 0.682672i \(-0.239182\pi\)
0.730725 + 0.682672i \(0.239182\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.73410e6 −1.47481
\(269\) 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
\(274\) 690536. 0.555661
\(275\) −2.49693e6 −1.99101
\(276\) 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
\(281\) 718184. 0.542588 0.271294 0.962497i \(-0.412548\pi\)
0.271294 + 0.962497i \(0.412548\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 2.12308e6 1.56196
\(285\) 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
\(296\) 1.34008e6 0.888998
\(297\) 0 0
\(298\) −1.12305e6 −0.732587
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −692192. −0.436726
\(303\) 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
\(316\) −2.00420e6 −1.12908
\(317\) −3.57144e6 −1.99616 −0.998079 0.0619605i \(-0.980265\pi\)
−0.998079 + 0.0619605i \(0.980265\pi\)
\(318\) 0 0
\(319\) −4.30410e6 −2.36813
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −1.75440e6 −0.914290
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.97148e6 −1.49074 −0.745371 0.666650i \(-0.767727\pi\)
−0.745371 + 0.666650i \(0.767727\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.15965e6 1.99518 0.997590 0.0693859i \(-0.0221040\pi\)
0.997590 + 0.0693859i \(0.0221040\pi\)
\(338\) 982349. 0.467707
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.77169e6 −0.807220
\(345\) 0 0
\(346\) 0 0
\(347\) −3.85255e6 −1.71761 −0.858804 0.512304i \(-0.828792\pi\)
−0.858804 + 0.512304i \(0.828792\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.70365e6 −2.02338
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.65935e6 −0.684275
\(359\) 2.37241e6 0.971523 0.485762 0.874091i \(-0.338542\pi\)
0.485762 + 0.874091i \(0.338542\pi\)
\(360\) 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 \(0.5\pi\)
\(368\) 443476. 0.170707
\(369\) 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 \(-0.00136819\pi\)
0.999991 + 0.00429827i \(0.00136819\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 273350. 0.0958431
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.02100e6 −0.348785
\(387\) 0 0
\(388\) 0 0
\(389\) 5.83451e6 1.95492 0.977462 0.211109i \(-0.0677076\pi\)
0.977462 + 0.211109i \(0.0677076\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −2.69553e6 −0.874790
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.25312e6 −0.391602
\(401\) 2.25352e6 0.699844 0.349922 0.936779i \(-0.386208\pi\)
0.349922 + 0.936779i \(0.386208\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.10006e6 2.12460
\(408\) 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 \(0.407921\pi\)
\(422\) −2.90253e6 −0.793405
\(423\) 0 0
\(424\) 4.93324e6 1.33265
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 5.69485e6 1.50270
\(429\) 0 0
\(430\) 0 0
\(431\) 2.37311e6 0.615353 0.307676 0.951491i \(-0.400449\pi\)
0.307676 + 0.951491i \(0.400449\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.48955e6 −1.38299
\(437\) 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
\(443\) 6.49509e6 1.57245 0.786223 0.617942i \(-0.212033\pi\)
0.786223 + 0.617942i \(0.212033\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.30677e6 −1.94454 −0.972269 0.233866i \(-0.924862\pi\)
−0.972269 + 0.233866i \(0.924862\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.04766e6 −1.39233
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.77969e6 1.29453 0.647267 0.762263i \(-0.275912\pi\)
0.647267 + 0.762263i \(0.275912\pi\)
\(458\) 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 \(0.398714\pi\)
\(464\) −2.16009e6 −0.465775
\(465\) 0 0
\(466\) −2.78718e6 −0.594565
\(467\) 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
\(473\) −9.38685e6 −1.92915
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 4.26265e6 0.853316
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.19344e7 −2.31573
\(485\) 0 0
\(486\) 0 0
\(487\) 2.76146e6 0.527615 0.263807 0.964575i \(-0.415022\pi\)
0.263807 + 0.964575i \(0.415022\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.01881e6 1.12670 0.563349 0.826219i \(-0.309513\pi\)
0.563349 + 0.826219i \(0.309513\pi\)
\(492\) 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 \(-0.00870324\pi\)
0.999626 + 0.0273386i \(0.00870324\pi\)
\(500\) 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
\(506\) −2.33792e6 −0.405933
\(507\) 0 0
\(508\) 6.55160e6 1.12639
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4.29577e6 −0.724213
\(513\) 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
\(526\) −4.33733e6 −0.683530
\(527\) 0 0
\(528\) 0 0
\(529\) −5.21328e6 −0.809975
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.04606e7 1.57270
\(537\) 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 \(0.551074\pi\)
\(548\) 6.52495e6 0.928167
\(549\) 0 0
\(550\) 6.60625e6 0.931211
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 6.75050e6 0.934462
\(555\) 0 0
\(556\) 0 0
\(557\) 1.32485e7 1.80938 0.904690 0.426070i \(-0.140102\pi\)
0.904690 + 0.426070i \(0.140102\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.90014e6 −0.253772
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −1.28071e7 −1.66564
\(569\) −1.13522e7 −1.46994 −0.734971 0.678099i \(-0.762804\pi\)
−0.734971 + 0.678099i \(0.762804\pi\)
\(570\) 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
\(575\) −3.45601e6 −0.435919
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 3.75659e6 0.467707
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.61375e7 3.18488
\(584\) 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
\(592\) 3.56329e6 0.417875
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.06118e7 −1.22370
\(597\) 0 0
\(598\) 0 0
\(599\) −1.60293e7 −1.82536 −0.912679 0.408677i \(-0.865990\pi\)
−0.912679 + 0.408677i \(0.865990\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −6.54060e6 −0.729500
\(605\) 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
\(617\) −1.75090e7 −1.85160 −0.925802 0.378008i \(-0.876609\pi\)
−0.925802 + 0.378008i \(0.876609\pi\)
\(618\) 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 \(0.0158534\pi\)
\(632\) 1.20900e7 1.20402
\(633\) 0 0
\(634\) 9.44913e6 0.933617
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.13876e7 1.10759
\(639\) 0 0
\(640\) 0 0
\(641\) −1.73407e7 −1.66694 −0.833471 0.552563i \(-0.813650\pi\)
−0.833471 + 0.552563i \(0.813650\pi\)
\(642\) 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
\(652\) −1.65775e7 −1.52721
\(653\) −1.88262e7 −1.72775 −0.863873 0.503710i \(-0.831968\pi\)
−0.863873 + 0.503710i \(0.831968\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.38990e6 −0.842263 −0.421131 0.907000i \(-0.638367\pi\)
−0.421131 + 0.907000i \(0.638367\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 7.86179e6 0.697230
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.95734e6 −0.518487
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.38435e6 −0.117817 −0.0589085 0.998263i \(-0.518762\pi\)
−0.0589085 + 0.998263i \(0.518762\pi\)
\(674\) −1.10054e7 −0.933160
\(675\) 0 0
\(676\) 9.28232e6 0.781250
\(677\) 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
\(683\) 1.18403e7 0.971206 0.485603 0.874179i \(-0.338600\pi\)
0.485603 + 0.874179i \(0.338600\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −4.71095e6 −0.379435
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.01929e7 0.803338
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.55582e7 1.96443 0.982213 0.187771i \(-0.0601263\pi\)
0.982213 + 0.187771i \(0.0601263\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.19170e6 0.166667
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.43225e7 1.07005 0.535023 0.844837i \(-0.320303\pi\)
0.535023 + 0.844837i \(0.320303\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.56794e7 −1.14300
\(717\) 0 0
\(718\) −6.27680e6 −0.454388
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.55114e6 0.467707
\(723\) 0 0
\(724\) 0 0
\(725\) 1.68336e7 1.18941
\(726\) 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
\(736\) −6.51035e6 −0.443006
\(737\) 5.54230e7 3.75856
\(738\) 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
\(743\) −1.68294e7 −1.11840 −0.559199 0.829033i \(-0.688891\pi\)
−0.559199 + 0.829033i \(0.688891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.58560e6 −0.104315
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.51088e7 −1.62452 −0.812260 0.583295i \(-0.801763\pi\)
−0.812260 + 0.583295i \(0.801763\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.78870e7 1.13448 0.567242 0.823551i \(-0.308011\pi\)
0.567242 + 0.823551i \(0.308011\pi\)
\(758\) −1.47970e7 −0.935406
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.58291e6 0.160095
\(765\) 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
\(772\) −9.64755e6 −0.582604
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.54367e7 −0.914332
\(779\) 0 0
\(780\) 0 0
\(781\) −6.78552e7 −3.98066
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −2.54704e7 −1.46123
\(789\) 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
\(800\) 1.83962e7 1.01626
\(801\) 0 0
\(802\) −5.96226e6 −0.327322
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.23828e7 −1.73957 −0.869787 0.493428i \(-0.835744\pi\)
−0.869787 + 0.493428i \(0.835744\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.87850e7 −0.993689
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.50170e7 0.777546 0.388773 0.921334i \(-0.372899\pi\)
0.388773 + 0.921334i \(0.372899\pi\)
\(822\) 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
\(827\) −1.37491e7 −0.699054 −0.349527 0.936926i \(-0.613658\pi\)
−0.349527 + 0.936926i \(0.613658\pi\)
\(828\) 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
\(841\) 8.50592e6 0.414698
\(842\) −5.48934e6 −0.266833
\(843\) 0 0
\(844\) −2.74263e7 −1.32529
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 1.31175e7 0.626416
\(849\) 0 0
\(850\) 0 0
\(851\) 9.82724e6 0.465166
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.43531e7 −1.60244
\(857\) 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
\(862\) −6.27865e6 −0.287805
\(863\) −3.39442e7 −1.55145 −0.775727 0.631068i \(-0.782617\pi\)
−0.775727 + 0.631068i \(0.782617\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.40556e7 2.87745
\(870\) 0 0
\(871\) 0 0
\(872\) 3.31147e7 1.47479
\(873\) 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 \(-0.176416\pi\)
0.850308 + 0.526286i \(0.176416\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.71844e7 −0.735445
\(887\) 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
\(898\) 2.19776e7 0.909474
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 3.64814e7 1.48474
\(905\) 0 0
\(906\) 0 0
\(907\) −4.48347e7 −1.80966 −0.904828 0.425777i \(-0.860001\pi\)
−0.904828 + 0.425777i \(0.860001\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.13974e7 0.454997 0.227498 0.973778i \(-0.426945\pi\)
0.227498 + 0.973778i \(0.426945\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.52916e7 −0.605463
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5.06716e7 −1.97914 −0.989569 0.144059i \(-0.953984\pi\)
−0.989569 + 0.144059i \(0.953984\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.77688e7 −1.06709
\(926\) −7.63617e6 −0.292650
\(927\) 0 0
\(928\) 3.17107e7 1.20875
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.63363e7 −0.993152
\(933\) 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
\(946\) 2.48353e7 0.902279
\(947\) −4.63272e6 −0.167865 −0.0839326 0.996471i \(-0.526748\pi\)
−0.0839326 + 0.996471i \(0.526748\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.51804e7 −1.96812 −0.984062 0.177824i \(-0.943094\pi\)
−0.984062 + 0.177824i \(0.943094\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.02783e7 1.42536
\(957\) 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 \(0.388122\pi\)
\(968\) 7.19922e7 2.46943
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7.30615e6 −0.246769
\(975\) 0 0
\(976\) 0 0
\(977\) −3.25961e7 −1.09252 −0.546260 0.837616i \(-0.683949\pi\)
−0.546260 + 0.837616i \(0.683949\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.59243e7 −0.526964
\(983\) 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
\(989\) −1.29924e7 −0.422375
\(990\) 0 0
\(991\) 5.73144e7 1.85387 0.926936 0.375219i \(-0.122433\pi\)
0.926936 + 0.375219i \(0.122433\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) −2.94217e7 −0.935065
\(999\) 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.o.1.1 2
3.2 odd 2 inner 441.6.a.o.1.2 yes 2
7.6 odd 2 CM 441.6.a.o.1.1 2
21.20 even 2 inner 441.6.a.o.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.o.1.1 2 1.1 even 1 trivial
441.6.a.o.1.1 2 7.6 odd 2 CM
441.6.a.o.1.2 yes 2 3.2 odd 2 inner
441.6.a.o.1.2 yes 2 21.20 even 2 inner