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Properties

Label 147.5.b.b.50.1
Level $147$
Weight $5$
Character 147.50
Self dual yes
Analytic conductor $15.195$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,5,Mod(50,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.50");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 147.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.1953845733\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 50.1
Character \(\chi\) \(=\) 147.50

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} +16.0000 q^{4} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} +16.0000 q^{4} +81.0000 q^{9} +144.000 q^{12} +191.000 q^{13} +256.000 q^{16} -601.000 q^{19} +625.000 q^{25} +729.000 q^{27} -1753.00 q^{31} +1296.00 q^{36} +2591.00 q^{37} +1719.00 q^{39} +23.0000 q^{43} +2304.00 q^{48} +3056.00 q^{52} -5409.00 q^{57} -1966.00 q^{61} +4096.00 q^{64} -8809.00 q^{67} -1249.00 q^{73} +5625.00 q^{75} -9616.00 q^{76} -12361.0 q^{79} +6561.00 q^{81} -15777.0 q^{93} -18814.0 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 \(\pi\)
\(3\) 9.00000 1.00000
\(4\) 16.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 144.000 1.00000
\(13\) 191.000 1.13018 0.565089 − 0.825030i \(-0.308842\pi\)
0.565089 + 0.825030i \(0.308842\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −601.000 −1.66482 −0.832410 − 0.554160i \(-0.813039\pi\)
−0.832410 + 0.554160i \(0.813039\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 729.000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1753.00 −1.82414 −0.912071 − 0.410033i \(-0.865517\pi\)
−0.912071 + 0.410033i \(0.865517\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1296.00 1.00000
\(37\) 2591.00 1.89262 0.946311 − 0.323257i \(-0.104778\pi\)
0.946311 + 0.323257i \(0.104778\pi\)
\(38\) 0 0
\(39\) 1719.00 1.13018
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 23.0000 0.0124392 0.00621958 − 0.999981i \(-0.498020\pi\)
0.00621958 + 0.999981i \(0.498020\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 2304.00 1.00000
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 3056.00 1.13018
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5409.00 −1.66482
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1966.00 −0.528353 −0.264176 − 0.964474i \(-0.585100\pi\)
−0.264176 + 0.964474i \(0.585100\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8809.00 −1.96235 −0.981176 − 0.193115i \(-0.938141\pi\)
−0.981176 + 0.193115i \(0.938141\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1249.00 −0.234378 −0.117189 − 0.993110i \(-0.537388\pi\)
−0.117189 + 0.993110i \(0.537388\pi\)
\(74\) 0 0
\(75\) 5625.00 1.00000
\(76\) −9616.00 −1.66482
\(77\) 0 0
\(78\) 0 0
\(79\) −12361.0 −1.98061 −0.990306 − 0.138903i \(-0.955642\pi\)
−0.990306 + 0.138903i \(0.955642\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −15777.0 −1.82414
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18814.0 −1.99957 −0.999787 − 0.0206175i \(-0.993437\pi\)
−0.999787 + 0.0206175i \(0.993437\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10000.0 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 3431.00 0.323405 0.161702 − 0.986840i \(-0.448302\pi\)
0.161702 + 0.986840i \(0.448302\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 11664.0 1.00000
\(109\) −18721.0 −1.57571 −0.787855 − 0.615861i \(-0.788808\pi\)
−0.787855 + 0.615861i \(0.788808\pi\)
\(110\) 0 0
\(111\) 23319.0 1.89262
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 15471.0 1.13018
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −28048.0 −1.82414
\(125\) 0 0
\(126\) 0 0
\(127\) −20809.0 −1.29016 −0.645080 − 0.764115i \(-0.723176\pi\)
−0.645080 + 0.764115i \(0.723176\pi\)
\(128\) 0 0
\(129\) 207.000 0.0124392
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 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
\(139\) 13799.0 0.714197 0.357098 − 0.934067i \(-0.383766\pi\)
0.357098 + 0.934067i \(0.383766\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 20736.0 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 41456.0 1.89262
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 36194.0 1.58739 0.793693 − 0.608318i \(-0.208156\pi\)
0.793693 + 0.608318i \(0.208156\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 27504.0 1.13018
\(157\) −35374.0 −1.43511 −0.717554 − 0.696502i \(-0.754739\pi\)
−0.717554 + 0.696502i \(0.754739\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15506.0 0.583612 0.291806 − 0.956477i \(-0.405744\pi\)
0.291806 + 0.956477i \(0.405744\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7920.00 0.277301
\(170\) 0 0
\(171\) −48681.0 −1.66482
\(172\) 368.000 0.0124392
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 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
\(181\) 32447.0 0.990415 0.495208 − 0.868775i \(-0.335092\pi\)
0.495208 + 0.868775i \(0.335092\pi\)
\(182\) 0 0
\(183\) −17694.0 −0.528353
\(184\) 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 \(\pi\)
\(192\) 36864.0 1.00000
\(193\) −17377.0 −0.466509 −0.233255 − 0.972416i \(-0.574938\pi\)
−0.233255 + 0.972416i \(0.574938\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 69794.0 1.76243 0.881215 − 0.472715i \(-0.156726\pi\)
0.881215 + 0.472715i \(0.156726\pi\)
\(200\) 0 0
\(201\) −79281.0 −1.96235
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 48896.0 1.13018
\(209\) 0 0
\(210\) 0 0
\(211\) −61486.0 −1.38106 −0.690528 − 0.723306i \(-0.742622\pi\)
−0.690528 + 0.723306i \(0.742622\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11241.0 −0.234378
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14786.0 0.297332 0.148666 − 0.988888i \(-0.452502\pi\)
0.148666 + 0.988888i \(0.452502\pi\)
\(224\) 0 0
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −86544.0 −1.66482
\(229\) 62399.0 1.18989 0.594945 − 0.803767i \(-0.297174\pi\)
0.594945 + 0.803767i \(0.297174\pi\)
\(230\) 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
\(237\) −111249. −1.98061
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −34366.0 −0.591691 −0.295845 − 0.955236i \(-0.595601\pi\)
−0.295845 + 0.955236i \(0.595601\pi\)
\(242\) 0 0
\(243\) 59049.0 1.00000
\(244\) −31456.0 −0.528353
\(245\) 0 0
\(246\) 0 0
\(247\) −114791. −1.88154
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 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
\(268\) −140944. −1.96235
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −88318.0 −1.20257 −0.601285 − 0.799034i \(-0.705345\pi\)
−0.601285 + 0.799034i \(0.705345\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12191.0 0.158884 0.0794419 − 0.996839i \(-0.474686\pi\)
0.0794419 + 0.996839i \(0.474686\pi\)
\(278\) 0 0
\(279\) −141993. −1.82414
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 107111. 1.33740 0.668700 − 0.743532i \(-0.266851\pi\)
0.668700 + 0.743532i \(0.266851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) −169326. −1.99957
\(292\) −19984.0 −0.234378
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 90000.0 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −153856. −1.66482
\(305\) 0 0
\(306\) 0 0
\(307\) 184823. 1.96101 0.980504 − 0.196500i \(-0.0629577\pi\)
0.980504 + 0.196500i \(0.0629577\pi\)
\(308\) 0 0
\(309\) 30879.0 0.323405
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 162863. 1.66239 0.831197 − 0.555979i \(-0.187656\pi\)
0.831197 + 0.555979i \(0.187656\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −197776. −1.98061
\(317\) 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
\(324\) 104976. 1.00000
\(325\) 119375. 1.13018
\(326\) 0 0
\(327\) −168489. −1.57571
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 217799. 1.98792 0.993962 − 0.109722i \(-0.0349962\pi\)
0.993962 + 0.109722i \(0.0349962\pi\)
\(332\) 0 0
\(333\) 209871. 1.89262
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 194063. 1.70877 0.854384 − 0.519643i \(-0.173935\pi\)
0.854384 + 0.519643i \(0.173935\pi\)
\(338\) 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
\(349\) 8402.00 0.0689814 0.0344907 − 0.999405i \(-0.489019\pi\)
0.0344907 + 0.999405i \(0.489019\pi\)
\(350\) 0 0
\(351\) 139239. 1.13018
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(361\) 230880. 1.77163
\(362\) 0 0
\(363\) 131769. 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −28297.0 −0.210091 −0.105046 − 0.994467i \(-0.533499\pi\)
−0.105046 + 0.994467i \(0.533499\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −252432. −1.82414
\(373\) 54671.0 0.392952 0.196476 − 0.980509i \(-0.437050\pi\)
0.196476 + 0.980509i \(0.437050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 86039.0 0.598986 0.299493 − 0.954098i \(-0.403182\pi\)
0.299493 + 0.954098i \(0.403182\pi\)
\(380\) 0 0
\(381\) −187281. −1.29016
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1863.00 0.0124392
\(388\) −301024. −1.99957
\(389\) 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
\(397\) −129457. −0.821381 −0.410690 − 0.911775i \(-0.634712\pi\)
−0.410690 + 0.911775i \(0.634712\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 160000. 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −334823. −2.06160
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 256799. 1.53514 0.767568 − 0.640968i \(-0.221467\pi\)
0.767568 + 0.640968i \(0.221467\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 54896.0 0.323405
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 124191. 0.714197
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −123121. −0.694653 −0.347327 − 0.937744i \(-0.612910\pi\)
−0.347327 + 0.937744i \(0.612910\pi\)
\(422\) 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 \(\pi\)
\(432\) 186624. 1.00000
\(433\) −246097. −1.31259 −0.656297 − 0.754503i \(-0.727878\pi\)
−0.656297 + 0.754503i \(0.727878\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −299536. −1.57571
\(437\) 0 0
\(438\) 0 0
\(439\) −376606. −1.95415 −0.977076 − 0.212892i \(-0.931712\pi\)
−0.977076 + 0.212892i \(0.931712\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 373104. 1.89262
\(445\) 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
\(453\) 325746. 1.58739
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 170591. 0.816815 0.408408 − 0.912800i \(-0.366084\pi\)
0.408408 + 0.912800i \(0.366084\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −271129. −1.26478 −0.632389 − 0.774651i \(-0.717925\pi\)
−0.632389 + 0.774651i \(0.717925\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 247536. 1.13018
\(469\) 0 0
\(470\) 0 0
\(471\) −318366. −1.43511
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −375625. −1.66482
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 494881. 2.13900
\(482\) 0 0
\(483\) 0 0
\(484\) 234256. 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −352537. −1.48644 −0.743219 − 0.669048i \(-0.766702\pi\)
−0.743219 + 0.669048i \(0.766702\pi\)
\(488\) 0 0
\(489\) 139554. 0.583612
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −448768. −1.82414
\(497\) 0 0
\(498\) 0 0
\(499\) 226199. 0.908426 0.454213 − 0.890893i \(-0.349920\pi\)
0.454213 + 0.890893i \(0.349920\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 71280.0 0.277301
\(508\) −332944. −1.29016
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −438129. −1.66482
\(514\) 0 0
\(515\) 0 0
\(516\) 3312.00 0.0124392
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −515017. −1.88286 −0.941430 − 0.337208i \(-0.890518\pi\)
−0.941430 + 0.337208i \(0.890518\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 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
\(541\) −527281. −1.80156 −0.900778 − 0.434281i \(-0.857003\pi\)
−0.900778 + 0.434281i \(0.857003\pi\)
\(542\) 0 0
\(543\) 292023. 0.990415
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −342382. −1.14429 −0.572145 − 0.820152i \(-0.693889\pi\)
−0.572145 + 0.820152i \(0.693889\pi\)
\(548\) 0 0
\(549\) −159246. −0.528353
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 220784. 0.714197
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 4393.00 0.0140585
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 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
\(571\) −619321. −1.89952 −0.949759 − 0.312981i \(-0.898672\pi\)
−0.949759 + 0.312981i \(0.898672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 331776. 1.00000
\(577\) −660817. −1.98486 −0.992429 − 0.122817i \(-0.960807\pi\)
−0.992429 + 0.122817i \(0.960807\pi\)
\(578\) 0 0
\(579\) −156393. −0.466509
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 1.05355e6 3.03687
\(590\) 0 0
\(591\) 0 0
\(592\) 663296. 1.89262
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 628146. 1.76243
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 715199. 1.98006 0.990029 − 0.140863i \(-0.0449877\pi\)
0.990029 + 0.140863i \(0.0449877\pi\)
\(602\) 0 0
\(603\) −713529. −1.96235
\(604\) 579104. 1.58739
\(605\) 0 0
\(606\) 0 0
\(607\) 672071. 1.82405 0.912027 − 0.410130i \(-0.134517\pi\)
0.912027 + 0.410130i \(0.134517\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 516338. 1.37408 0.687042 − 0.726618i \(-0.258909\pi\)
0.687042 + 0.726618i \(0.258909\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 586247. 1.53003 0.765014 − 0.644014i \(-0.222732\pi\)
0.765014 + 0.644014i \(0.222732\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 440064. 1.13018
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −565984. −1.43511
\(629\) 0 0
\(630\) 0 0
\(631\) −342046. −0.859065 −0.429532 − 0.903052i \(-0.641322\pi\)
−0.429532 + 0.903052i \(0.641322\pi\)
\(632\) 0 0
\(633\) −553374. −1.38106
\(634\) 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
\(643\) 703271. 1.70099 0.850493 − 0.525986i \(-0.176304\pi\)
0.850493 + 0.525986i \(0.176304\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 248096. 0.583612
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −101169. −0.234378
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 290399. 0.664649 0.332324 − 0.943165i \(-0.392167\pi\)
0.332324 + 0.943165i \(0.392167\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 133074. 0.297332
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −905329. −1.99883 −0.999416 − 0.0341703i \(-0.989121\pi\)
−0.999416 + 0.0341703i \(0.989121\pi\)
\(674\) 0 0
\(675\) 455625. 1.00000
\(676\) 126720. 0.277301
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −778896. −1.66482
\(685\) 0 0
\(686\) 0 0
\(687\) 561591. 1.18989
\(688\) 5888.00 0.0124392
\(689\) 0 0
\(690\) 0 0
\(691\) 83399.0 0.174665 0.0873323 − 0.996179i \(-0.472166\pi\)
0.0873323 + 0.996179i \(0.472166\pi\)
\(692\) 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
\(703\) −1.55719e6 −3.15088
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −133006. −0.264593 −0.132297 − 0.991210i \(-0.542235\pi\)
−0.132297 + 0.991210i \(0.542235\pi\)
\(710\) 0 0
\(711\) −1.00124e6 −1.98061
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −309294. −0.591691
\(724\) 519152. 0.990415
\(725\) 0 0
\(726\) 0 0
\(727\) −160249. −0.303198 −0.151599 − 0.988442i \(-0.548442\pi\)
−0.151599 + 0.988442i \(0.548442\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −283104. −0.528353
\(733\) 933311. 1.73707 0.868537 − 0.495624i \(-0.165061\pi\)
0.868537 + 0.495624i \(0.165061\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 679319. 1.24390 0.621949 − 0.783058i \(-0.286341\pi\)
0.621949 + 0.783058i \(0.286341\pi\)
\(740\) 0 0
\(741\) −1.03312e6 −1.88154
\(742\) 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
\(751\) −1.09596e6 −1.94319 −0.971595 − 0.236650i \(-0.923950\pi\)
−0.971595 + 0.236650i \(0.923950\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −443854. −0.774548 −0.387274 − 0.921965i \(-0.626583\pi\)
−0.387274 + 0.921965i \(0.626583\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 589824. 1.00000
\(769\) −437953. −0.740585 −0.370292 − 0.928915i \(-0.620743\pi\)
−0.370292 + 0.928915i \(0.620743\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −278032. −0.466509
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.09562e6 −1.82414
\(776\) 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
\(787\) 1.20111e6 1.93924 0.969621 − 0.244613i \(-0.0786610\pi\)
0.969621 + 0.244613i \(0.0786610\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −375506. −0.597132
\(794\) 0 0
\(795\) 0 0
\(796\) 1.11670e6 1.76243
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −1.26850e6 −1.96235
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 976754. 1.48506 0.742529 − 0.669814i \(-0.233626\pi\)
0.742529 + 0.669814i \(0.233626\pi\)
\(812\) 0 0
\(813\) −794862. −1.20257
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −13823.0 −0.0207090
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −235294. −0.347385 −0.173693 − 0.984800i \(-0.555570\pi\)
−0.173693 + 0.984800i \(0.555570\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.21440e6 1.76706 0.883532 − 0.468371i \(-0.155159\pi\)
0.883532 + 0.468371i \(0.155159\pi\)
\(830\) 0 0
\(831\) 109719. 0.158884
\(832\) 782336. 1.13018
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.27794e6 −1.82414
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −983776. −1.38106
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 963999. 1.33740
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.22386e6 −1.68203 −0.841013 − 0.541015i \(-0.818040\pi\)
−0.841013 + 0.541015i \(0.818040\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 534962. 0.724998 0.362499 − 0.931984i \(-0.381924\pi\)
0.362499 + 0.931984i \(0.381924\pi\)
\(860\) 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
\(867\) 751689. 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.68252e6 −2.21781
\(872\) 0 0
\(873\) −1.52393e6 −1.99957
\(874\) 0 0
\(875\) 0 0
\(876\) −179856. −0.234378
\(877\) −1.18065e6 −1.53505 −0.767527 − 0.641017i \(-0.778513\pi\)
−0.767527 + 0.641017i \(0.778513\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.36313e6 −1.74830 −0.874149 − 0.485658i \(-0.838580\pi\)
−0.874149 + 0.485658i \(0.838580\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 236576. 0.297332
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 810000. 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.13359e6 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.38470e6 −1.66482
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 998384. 1.18989
\(917\) 0 0
\(918\) 0 0
\(919\) 1.68545e6 1.99565 0.997824 − 0.0659290i \(-0.0210011\pi\)
0.997824 + 0.0659290i \(0.0210011\pi\)
\(920\) 0 0
\(921\) 1.66341e6 1.96101
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.61938e6 1.89262
\(926\) 0 0
\(927\) 277911. 0.323405
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.33474e6 −1.52026 −0.760128 − 0.649774i \(-0.774864\pi\)
−0.760128 + 0.649774i \(0.774864\pi\)
\(938\) 0 0
\(939\) 1.46577e6 1.66239
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.77998e6 −1.98061
\(949\) −238559. −0.264889
\(950\) 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
\(961\) 2.14949e6 2.32749
\(962\) 0 0
\(963\) 0 0
\(964\) −549856. −0.591691
\(965\) 0 0
\(966\) 0 0
\(967\) 1.32319e6 1.41504 0.707521 − 0.706692i \(-0.249813\pi\)
0.707521 + 0.706692i \(0.249813\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 944784. 1.00000
\(973\) 0 0
\(974\) 0 0
\(975\) 1.07438e6 1.13018
\(976\) −503296. −0.528353
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.51640e6 −1.57571
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −1.83666e6 −1.88154
\(989\) 0 0
\(990\) 0 0
\(991\) 902087. 0.918546 0.459273 − 0.888295i \(-0.348110\pi\)
0.459273 + 0.888295i \(0.348110\pi\)
\(992\) 0 0
\(993\) 1.96019e6 1.98792
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.82237e6 −1.83335 −0.916676 − 0.399631i \(-0.869138\pi\)
−0.916676 + 0.399631i \(0.869138\pi\)
\(998\) 0 0
\(999\) 1.88884e6 1.89262
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 147.5.b.b.50.1 1
3.2 odd 2 CM 147.5.b.b.50.1 1
7.2 even 3 21.5.h.a.11.1 yes 2
7.3 odd 6 147.5.h.a.128.1 2
7.4 even 3 21.5.h.a.2.1 ✓ 2
7.5 odd 6 147.5.h.a.116.1 2
7.6 odd 2 147.5.b.a.50.1 1
21.2 odd 6 21.5.h.a.11.1 yes 2
21.5 even 6 147.5.h.a.116.1 2
21.11 odd 6 21.5.h.a.2.1 ✓ 2
21.17 even 6 147.5.h.a.128.1 2
21.20 even 2 147.5.b.a.50.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.5.h.a.2.1 ✓ 2 7.4 even 3
21.5.h.a.2.1 ✓ 2 21.11 odd 6
21.5.h.a.11.1 yes 2 7.2 even 3
21.5.h.a.11.1 yes 2 21.2 odd 6
147.5.b.a.50.1 1 7.6 odd 2
147.5.b.a.50.1 1 21.20 even 2
147.5.b.b.50.1 1 1.1 even 1 trivial
147.5.b.b.50.1 1 3.2 odd 2 CM
147.5.h.a.116.1 2 7.5 odd 6
147.5.h.a.116.1 2 21.5 even 6
147.5.h.a.128.1 2 7.3 odd 6
147.5.h.a.128.1 2 21.17 even 6