Properties

Label 147.6.c.a.146.1
Level $147$
Weight $6$
Character 147.146
Analytic conductor $23.576$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 146.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 147.146
Dual form 147.6.c.a.146.2

$q$-expansion

\(f(q)\) \(=\) \(q-15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} +O(q^{10})\) \(q-15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} -498.831i q^{12} -1141.42i q^{13} +1024.00 q^{16} -161.081i q^{19} -3125.00 q^{25} +3788.00i q^{27} -10349.0i q^{31} -7776.00 q^{36} +6661.00 q^{37} -17793.0 q^{39} -22475.0 q^{43} -15962.6i q^{48} -36525.5i q^{52} -2511.00 q^{57} -43432.9i q^{61} +32768.0 q^{64} -37939.0 q^{67} +46781.0i q^{73} +48713.9i q^{75} -5154.58i q^{76} +90857.0 q^{79} +59049.0 q^{81} -161325. q^{93} +127631. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{4} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{4} - 486 q^{9} + 2048 q^{16} - 6250 q^{25} - 15552 q^{36} + 13322 q^{37} - 35586 q^{39} - 44950 q^{43} - 5022 q^{57} + 65536 q^{64} - 75878 q^{67} + 181714 q^{79} + 118098 q^{81} - 322650 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 15.5885i − 1.00000i
\(4\) 32.0000 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 498.831i − 1.00000i
\(13\) − 1141.42i − 1.87322i −0.350380 0.936608i \(-0.613948\pi\)
0.350380 0.936608i \(-0.386052\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1024.00 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) − 161.081i − 0.102367i −0.998689 0.0511835i \(-0.983701\pi\)
0.998689 0.0511835i \(-0.0162993\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 3788.00i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) − 10349.0i − 1.93417i −0.254456 0.967084i \(-0.581897\pi\)
0.254456 0.967084i \(-0.418103\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −7776.00 −1.00000
\(37\) 6661.00 0.799899 0.399949 0.916537i \(-0.369028\pi\)
0.399949 + 0.916537i \(0.369028\pi\)
\(38\) 0 0
\(39\) −17793.0 −1.87322
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −22475.0 −1.85365 −0.926827 0.375489i \(-0.877475\pi\)
−0.926827 + 0.375489i \(0.877475\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 15962.6i − 1.00000i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) − 36525.5i − 1.87322i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2511.00 −0.102367
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) − 43432.9i − 1.49449i −0.664546 0.747247i \(-0.731375\pi\)
0.664546 0.747247i \(-0.268625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 32768.0 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −37939.0 −1.03252 −0.516260 0.856432i \(-0.672676\pi\)
−0.516260 + 0.856432i \(0.672676\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 46781.0i 1.02745i 0.857954 + 0.513727i \(0.171735\pi\)
−0.857954 + 0.513727i \(0.828265\pi\)
\(74\) 0 0
\(75\) 48713.9i 1.00000i
\(76\) − 5154.58i − 0.102367i
\(77\) 0 0
\(78\) 0 0
\(79\) 90857.0 1.63791 0.818956 0.573856i \(-0.194553\pi\)
0.818956 + 0.573856i \(0.194553\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −161325. −1.93417
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 127631.i 1.37730i 0.725095 + 0.688649i \(0.241796\pi\)
−0.725095 + 0.688649i \(0.758204\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −100000. −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 40601.0i 0.377089i 0.982065 + 0.188544i \(0.0603769\pi\)
−0.982065 + 0.188544i \(0.939623\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 121216.i 1.00000i
\(109\) 247843. 1.99807 0.999034 0.0439362i \(-0.0139898\pi\)
0.999034 + 0.0439362i \(0.0139898\pi\)
\(110\) 0 0
\(111\) − 103835.i − 0.799899i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 277365.i 1.87322i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 161051. 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) − 331168.i − 1.93417i
\(125\) 0 0
\(126\) 0 0
\(127\) 347111. 1.90967 0.954837 0.297131i \(-0.0960299\pi\)
0.954837 + 0.297131i \(0.0960299\pi\)
\(128\) 0 0
\(129\) 350351.i 1.85365i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 29024.0i 0.127415i 0.997969 + 0.0637074i \(0.0202924\pi\)
−0.997969 + 0.0637074i \(0.979708\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −248832. −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 213152. 0.799899
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 408724. 1.45877 0.729387 0.684102i \(-0.239806\pi\)
0.729387 + 0.684102i \(0.239806\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −569376. −1.87322
\(157\) − 607971.i − 1.96849i −0.176807 0.984246i \(-0.556577\pi\)
0.176807 0.984246i \(-0.443423\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 678248. 1.99949 0.999746 0.0225538i \(-0.00717969\pi\)
0.999746 + 0.0225538i \(0.00717969\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −931550. −2.50893
\(170\) 0 0
\(171\) 39142.6i 0.102367i
\(172\) −719200. −1.85365
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) − 222265.i − 0.504284i −0.967690 0.252142i \(-0.918865\pi\)
0.967690 0.252142i \(-0.0811351\pi\)
\(182\) 0 0
\(183\) −677052. −1.49449
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) − 510803.i − 1.00000i
\(193\) −656375. −1.26841 −0.634204 0.773166i \(-0.718672\pi\)
−0.634204 + 0.773166i \(0.718672\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) − 645047.i − 1.15467i −0.816507 0.577336i \(-0.804092\pi\)
0.816507 0.577336i \(-0.195908\pi\)
\(200\) 0 0
\(201\) 591410.i 1.03252i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) − 1.16882e6i − 1.87322i
\(209\) 0 0
\(210\) 0 0
\(211\) 288976. 0.446844 0.223422 0.974722i \(-0.428277\pi\)
0.223422 + 0.974722i \(0.428277\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 729243. 1.02745
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.45377e6i 1.95764i 0.204718 + 0.978821i \(0.434372\pi\)
−0.204718 + 0.978821i \(0.565628\pi\)
\(224\) 0 0
\(225\) 759375. 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −80352.0 −0.102367
\(229\) 623613.i 0.785826i 0.919576 + 0.392913i \(0.128533\pi\)
−0.919576 + 0.392913i \(0.871467\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.41632e6i − 1.63791i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.25292e6i 1.38958i 0.719215 + 0.694788i \(0.244502\pi\)
−0.719215 + 0.694788i \(0.755498\pi\)
\(242\) 0 0
\(243\) − 920483.i − 1.00000i
\(244\) − 1.38985e6i − 1.49449i
\(245\) 0 0
\(246\) 0 0
\(247\) −183861. −0.191755
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.04858e6 1.00000
\(257\) 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
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.21405e6 −1.03252
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 878236.i 0.726421i 0.931707 + 0.363210i \(0.118319\pi\)
−0.931707 + 0.363210i \(0.881681\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 738389. 0.578210 0.289105 0.957297i \(-0.406642\pi\)
0.289105 + 0.957297i \(0.406642\pi\)
\(278\) 0 0
\(279\) 2.51481e6i 1.93417i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.34561e6i 0.998745i 0.866387 + 0.499373i \(0.166436\pi\)
−0.866387 + 0.499373i \(0.833564\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −1.00000
\(290\) 0 0
\(291\) 1.98958e6 1.37730
\(292\) 1.49699e6i 1.02745i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1.55885e6i 1.00000i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) − 164947.i − 0.102367i
\(305\) 0 0
\(306\) 0 0
\(307\) − 808260.i − 0.489446i −0.969593 0.244723i \(-0.921303\pi\)
0.969593 0.244723i \(-0.0786971\pi\)
\(308\) 0 0
\(309\) 632907. 0.377089
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 2.32953e6i 1.34403i 0.740539 + 0.672014i \(0.234570\pi\)
−0.740539 + 0.672014i \(0.765430\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.90742e6 1.63791
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.88957e6 1.00000
\(325\) 3.56694e6i 1.87322i
\(326\) 0 0
\(327\) − 3.86349e6i − 1.99807i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 546151. 0.273995 0.136998 0.990571i \(-0.456255\pi\)
0.136998 + 0.990571i \(0.456255\pi\)
\(332\) 0 0
\(333\) −1.61862e6 −0.799899
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.63172e6 −1.26231 −0.631155 0.775657i \(-0.717419\pi\)
−0.631155 + 0.775657i \(0.717419\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) − 1.55867e6i − 0.685001i −0.939518 0.342501i \(-0.888726\pi\)
0.939518 0.342501i \(-0.111274\pi\)
\(350\) 0 0
\(351\) 4.32370e6 1.87322
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 2.45015e6 0.989521
\(362\) 0 0
\(363\) − 2.51054e6i − 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.46748e6i − 1.73140i −0.500563 0.865700i \(-0.666874\pi\)
0.500563 0.865700i \(-0.333126\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.16240e6 −1.93417
\(373\) −5.31727e6 −1.97887 −0.989434 0.144983i \(-0.953687\pi\)
−0.989434 + 0.144983i \(0.953687\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −69893.0 −0.0249940 −0.0124970 0.999922i \(-0.503978\pi\)
−0.0124970 + 0.999922i \(0.503978\pi\)
\(380\) 0 0
\(381\) − 5.41092e6i − 1.90967i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.46142e6 1.85365
\(388\) 4.08420e6i 1.37730i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 3.79639e6i − 1.20891i −0.796638 0.604456i \(-0.793390\pi\)
0.796638 0.604456i \(-0.206610\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.20000e6 −1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.18126e7 −3.62311
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 4.36452e6i − 1.29011i −0.764134 0.645057i \(-0.776834\pi\)
0.764134 0.645057i \(-0.223166\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.29923e6i 0.377089i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 452439. 0.127415
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −4.87580e6 −1.34073 −0.670364 0.742033i \(-0.733862\pi\)
−0.670364 + 0.742033i \(0.733862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 3.87891e6i 1.00000i
\(433\) 2.18049e6i 0.558901i 0.960160 + 0.279450i \(0.0901522\pi\)
−0.960160 + 0.279450i \(0.909848\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.93098e6 1.99807
\(437\) 0 0
\(438\) 0 0
\(439\) 7.01951e6i 1.73838i 0.494475 + 0.869192i \(0.335360\pi\)
−0.494475 + 0.869192i \(0.664640\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) − 3.32271e6i − 0.799899i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 6.37138e6i − 1.45877i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.18576e6 −0.265587 −0.132793 0.991144i \(-0.542395\pi\)
−0.132793 + 0.991144i \(0.542395\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 2.92217e6 0.633510 0.316755 0.948507i \(-0.397407\pi\)
0.316755 + 0.948507i \(0.397407\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 8.87569e6i 1.87322i
\(469\) 0 0
\(470\) 0 0
\(471\) −9.47732e6 −1.96849
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 503377.i 0.102367i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) − 7.60301e6i − 1.49838i
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.36487e6 0.260778 0.130389 0.991463i \(-0.458377\pi\)
0.130389 + 0.991463i \(0.458377\pi\)
\(488\) 0 0
\(489\) − 1.05728e7i − 1.99949i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) − 1.05974e7i − 1.93417i
\(497\) 0 0
\(498\) 0 0
\(499\) 2.17369e6 0.390793 0.195397 0.980724i \(-0.437401\pi\)
0.195397 + 0.980724i \(0.437401\pi\)
\(500\) 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
\(506\) 0 0
\(507\) 1.45214e7i 2.50893i
\(508\) 1.11076e7 1.90967
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 610173. 0.102367
\(514\) 0 0
\(515\) 0 0
\(516\) 1.12112e7i 1.85365i
\(517\) 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
\(523\) − 4.35404e6i − 0.696047i −0.937486 0.348023i \(-0.886853\pi\)
0.937486 0.348023i \(-0.113147\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.43634e6 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.02235e6 −0.443968 −0.221984 0.975050i \(-0.571253\pi\)
−0.221984 + 0.975050i \(0.571253\pi\)
\(542\) 0 0
\(543\) −3.46478e6 −0.504284
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27982e7 1.82886 0.914430 0.404744i \(-0.132639\pi\)
0.914430 + 0.404744i \(0.132639\pi\)
\(548\) 0 0
\(549\) 1.05542e7i 1.49449i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 928767.i 0.127415i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 2.56534e7i 3.47229i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 5.88780e6 0.755723 0.377862 0.925862i \(-0.376660\pi\)
0.377862 + 0.925862i \(0.376660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −7.96262e6 −1.00000
\(577\) − 9.44227e6i − 1.18069i −0.807150 0.590347i \(-0.798991\pi\)
0.807150 0.590347i \(-0.201009\pi\)
\(578\) 0 0
\(579\) 1.02319e7i 1.26841i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −1.66702e6 −0.197995
\(590\) 0 0
\(591\) 0 0
\(592\) 6.82086e6 0.799899
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.00553e7 −1.15467
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) − 3.11249e6i − 0.351498i −0.984435 0.175749i \(-0.943765\pi\)
0.984435 0.175749i \(-0.0562346\pi\)
\(602\) 0 0
\(603\) 9.21918e6 1.03252
\(604\) 1.30792e7 1.45877
\(605\) 0 0
\(606\) 0 0
\(607\) 9.15058e6i 1.00804i 0.863693 + 0.504019i \(0.168146\pi\)
−0.863693 + 0.504019i \(0.831854\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.58234e7 −1.70079 −0.850394 0.526147i \(-0.823636\pi\)
−0.850394 + 0.526147i \(0.823636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.46298e7i 1.53466i 0.641253 + 0.767330i \(0.278415\pi\)
−0.641253 + 0.767330i \(0.721585\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.82200e7 −1.87322
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) − 1.94551e7i − 1.96849i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.62439e7 −1.62411 −0.812057 0.583579i \(-0.801652\pi\)
−0.812057 + 0.583579i \(0.801652\pi\)
\(632\) 0 0
\(633\) − 4.50469e6i − 0.446844i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) − 1.61402e7i − 1.53951i −0.638342 0.769753i \(-0.720380\pi\)
0.638342 0.769753i \(-0.279620\pi\)
\(644\) 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
\(652\) 2.17039e7 1.99949
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.13678e7i − 1.02745i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) − 2.24558e7i − 1.99905i −0.0307548 0.999527i \(-0.509791\pi\)
0.0307548 0.999527i \(-0.490209\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 2.26620e7 1.95764
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.73115e7 −1.47332 −0.736661 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(674\) 0 0
\(675\) − 1.18375e7i − 1.00000i
\(676\) −2.98096e7 −2.50893
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 1.25256e6i 0.102367i
\(685\) 0 0
\(686\) 0 0
\(687\) 9.72116e6 0.785826
\(688\) −2.30144e7 −1.85365
\(689\) 0 0
\(690\) 0 0
\(691\) − 2.41017e7i − 1.92022i −0.279616 0.960112i \(-0.590207\pi\)
0.279616 0.960112i \(-0.409793\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) − 1.07296e6i − 0.0818832i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.27014e7 1.69604 0.848021 0.529962i \(-0.177794\pi\)
0.848021 + 0.529962i \(0.177794\pi\)
\(710\) 0 0
\(711\) −2.20783e7 −1.63791
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.95312e7 1.38958
\(724\) − 7.11249e6i − 0.504284i
\(725\) 0 0
\(726\) 0 0
\(727\) 2.37945e7i 1.66971i 0.550474 + 0.834853i \(0.314447\pi\)
−0.550474 + 0.834853i \(0.685553\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −2.16657e7 −1.49449
\(733\) 2.31923e7i 1.59435i 0.603746 + 0.797177i \(0.293674\pi\)
−0.603746 + 0.797177i \(0.706326\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.28170e7 0.863330 0.431665 0.902034i \(-0.357926\pi\)
0.431665 + 0.902034i \(0.357926\pi\)
\(740\) 0 0
\(741\) 2.86611e6i 0.191755i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.73219e7 1.76771 0.883857 0.467758i \(-0.154938\pi\)
0.883857 + 0.467758i \(0.154938\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.98526e7 −1.25915 −0.629575 0.776940i \(-0.716771\pi\)
−0.629575 + 0.776940i \(0.716771\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) − 1.63457e7i − 1.00000i
\(769\) − 2.12277e7i − 1.29446i −0.762296 0.647228i \(-0.775928\pi\)
0.762296 0.647228i \(-0.224072\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.10040e7 −1.26841
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 3.23406e7i 1.93417i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.05647e7i − 0.608024i −0.952668 0.304012i \(-0.901674\pi\)
0.952668 0.304012i \(-0.0983263\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.95753e7 −2.79951
\(794\) 0 0
\(795\) 0 0
\(796\) − 2.06415e7i − 1.15467i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.89251e7i 1.03252i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 2.97462e7i 1.58811i 0.607849 + 0.794053i \(0.292033\pi\)
−0.607849 + 0.794053i \(0.707967\pi\)
\(812\) 0 0
\(813\) 1.36904e7 0.726421
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.62029e6i 0.189753i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 2.22923e7 1.14725 0.573623 0.819120i \(-0.305538\pi\)
0.573623 + 0.819120i \(0.305538\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 3.24529e7i 1.64009i 0.572299 + 0.820045i \(0.306052\pi\)
−0.572299 + 0.820045i \(0.693948\pi\)
\(830\) 0 0
\(831\) − 1.15103e7i − 0.578210i
\(832\) − 3.74021e7i − 1.87322i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.92020e7 1.93417
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 2.05111e7 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 9.24723e6 0.446844
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.09761e7 0.998745
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 4.23947e7i − 1.99498i −0.0707842 0.997492i \(-0.522550\pi\)
0.0707842 0.997492i \(-0.477450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 4.31437e7i 1.99496i 0.0709259 + 0.997482i \(0.477405\pi\)
−0.0709259 + 0.997482i \(0.522595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.21334e7i 1.00000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.33044e7i 1.93413i
\(872\) 0 0
\(873\) − 3.10144e7i − 1.37730i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.33358e7 1.02745
\(877\) −3.83931e6 −0.168560 −0.0842800 0.996442i \(-0.526859\pi\)
−0.0842800 + 0.996442i \(0.526859\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 4.58066e7 1.97709 0.988545 0.150925i \(-0.0482253\pi\)
0.988545 + 0.150925i \(0.0482253\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 4.65206e7i 1.95764i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.43000e7 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −2.62120e7 −1.05799 −0.528995 0.848625i \(-0.677431\pi\)
−0.528995 + 0.848625i \(0.677431\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −2.57126e6 −0.102367
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.99556e7i 0.785826i
\(917\) 0 0
\(918\) 0 0
\(919\) 4.21821e6 0.164755 0.0823777 0.996601i \(-0.473749\pi\)
0.0823777 + 0.996601i \(0.473749\pi\)
\(920\) 0 0
\(921\) −1.25995e7 −0.489446
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.08156e7 −0.799899
\(926\) 0 0
\(927\) − 9.86604e6i − 0.377089i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.34884e7i 1.99026i 0.0985501 + 0.995132i \(0.468580\pi\)
−0.0985501 + 0.995132i \(0.531420\pi\)
\(938\) 0 0
\(939\) 3.63138e7 1.34403
\(940\) 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
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) − 4.53223e7i − 1.63791i
\(949\) 5.33968e7 1.92464
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −7.84727e7 −2.74101
\(962\) 0 0
\(963\) 0 0
\(964\) 4.00936e7i 1.38958i
\(965\) 0 0
\(966\) 0 0
\(967\) −3.23453e7 −1.11236 −0.556180 0.831062i \(-0.687733\pi\)
−0.556180 + 0.831062i \(0.687733\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) − 2.94555e7i − 1.00000i
\(973\) 0 0
\(974\) 0 0
\(975\) 5.56031e7 1.87322
\(976\) − 4.44753e7i − 1.49449i
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.02258e7 −1.99807
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −5.88355e6 −0.191755
\(989\) 0 0
\(990\) 0 0
\(991\) −6.05528e7 −1.95862 −0.979310 0.202365i \(-0.935137\pi\)
−0.979310 + 0.202365i \(0.935137\pi\)
\(992\) 0 0
\(993\) − 8.51365e6i − 0.273995i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.68344e7i 0.536365i 0.963368 + 0.268183i \(0.0864230\pi\)
−0.963368 + 0.268183i \(0.913577\pi\)
\(998\) 0 0
\(999\) 2.52318e7i 0.799899i
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 147.6.c.a.146.1 2
3.2 odd 2 CM 147.6.c.a.146.1 2
7.2 even 3 21.6.g.a.17.1 yes 2
7.3 odd 6 21.6.g.a.5.1 2
7.6 odd 2 inner 147.6.c.a.146.2 2
21.2 odd 6 21.6.g.a.17.1 yes 2
21.17 even 6 21.6.g.a.5.1 2
21.20 even 2 inner 147.6.c.a.146.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.g.a.5.1 2 7.3 odd 6
21.6.g.a.5.1 2 21.17 even 6
21.6.g.a.17.1 yes 2 7.2 even 3
21.6.g.a.17.1 yes 2 21.2 odd 6
147.6.c.a.146.1 2 1.1 even 1 trivial
147.6.c.a.146.1 2 3.2 odd 2 CM
147.6.c.a.146.2 2 7.6 odd 2 inner
147.6.c.a.146.2 2 21.20 even 2 inner