Properties

Label 3.127.b.a.2.1
Level $3$
Weight $127$
Character 3.2
Self dual yes
Analytic conductor $301.614$
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] = [3,127,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 127, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 127);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 127 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(301.614165769\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2.1
Character \(\chi\) \(=\) 3.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.14456e30 q^{3} +8.50706e37 q^{4} -1.33747e53 q^{7} +1.31002e60 q^{9} +O(q^{10})\) \(q-1.14456e30 q^{3} +8.50706e37 q^{4} -1.33747e53 q^{7} +1.31002e60 q^{9} -9.73685e67 q^{12} +1.98412e70 q^{13} +7.23701e75 q^{16} +2.48414e80 q^{19} +1.53081e83 q^{21} +1.17549e88 q^{25} -1.49940e90 q^{27} -1.13779e91 q^{28} +1.22080e94 q^{31} +1.11444e98 q^{36} +7.34533e98 q^{37} -2.27094e100 q^{39} -9.00662e102 q^{43} -8.28320e105 q^{48} -1.24754e106 q^{49} +1.68790e108 q^{52} -2.84325e110 q^{57} +1.29285e112 q^{61} -1.75211e113 q^{63} +6.15656e113 q^{64} -3.55088e114 q^{67} +4.77617e117 q^{73} -1.34543e118 q^{75} +2.11327e118 q^{76} +6.73769e119 q^{79} +1.71615e120 q^{81} +1.30227e121 q^{84} -2.65369e123 q^{91} -1.39728e124 q^{93} -1.77521e125 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\pi\)
\(3\) −1.14456e30 −1.00000
\(4\) 8.50706e37 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.33747e53 −0.767550 −0.383775 0.923427i \(-0.625376\pi\)
−0.383775 + 0.923427i \(0.625376\pi\)
\(8\) 0 0
\(9\) 1.31002e60 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −9.73685e67 −1.00000
\(13\) 1.98412e70 1.31564 0.657818 0.753177i \(-0.271480\pi\)
0.657818 + 0.753177i \(0.271480\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.23701e75 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 2.48414e80 0.681866 0.340933 0.940088i \(-0.389257\pi\)
0.340933 + 0.940088i \(0.389257\pi\)
\(20\) 0 0
\(21\) 1.53081e83 0.767550
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.17549e88 1.00000
\(26\) 0 0
\(27\) −1.49940e90 −1.00000
\(28\) −1.13779e91 −0.767550
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.22080e94 1.35163 0.675813 0.737074i \(-0.263793\pi\)
0.675813 + 0.737074i \(0.263793\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.11444e98 1.00000
\(37\) 7.34533e98 1.17301 0.586507 0.809944i \(-0.300503\pi\)
0.586507 + 0.809944i \(0.300503\pi\)
\(38\) 0 0
\(39\) −2.27094e100 −1.31564
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −9.00662e102 −1.11186 −0.555929 0.831230i \(-0.687637\pi\)
−0.555929 + 0.831230i \(0.687637\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −8.28320e105 −1.00000
\(49\) −1.24754e106 −0.410866
\(50\) 0 0
\(51\) 0 0
\(52\) 1.68790e108 1.31564
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.84325e110 −0.681866
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.29285e112 0.432285 0.216142 0.976362i \(-0.430652\pi\)
0.216142 + 0.976362i \(0.430652\pi\)
\(62\) 0 0
\(63\) −1.75211e113 −0.767550
\(64\) 6.15656e113 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.55088e114 −0.321828 −0.160914 0.986968i \(-0.551444\pi\)
−0.160914 + 0.986968i \(0.551444\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 4.77617e117 1.94868 0.974338 0.225091i \(-0.0722680\pi\)
0.974338 + 0.225091i \(0.0722680\pi\)
\(74\) 0 0
\(75\) −1.34543e118 −1.00000
\(76\) 2.11327e118 0.681866
\(77\) 0 0
\(78\) 0 0
\(79\) 6.73769e119 1.89673 0.948363 0.317187i \(-0.102738\pi\)
0.948363 + 0.317187i \(0.102738\pi\)
\(80\) 0 0
\(81\) 1.71615e120 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.30227e121 0.767550
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.65369e123 −1.00982
\(92\) 0 0
\(93\) −1.39728e124 −1.35163
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.77521e125 −1.20957 −0.604785 0.796389i \(-0.706741\pi\)
−0.604785 + 0.796389i \(0.706741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e126 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.19492e127 −1.85606 −0.928032 0.372499i \(-0.878501\pi\)
−0.928032 + 0.372499i \(0.878501\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.27555e128 −1.00000
\(109\) −4.04571e128 −1.77470 −0.887351 0.461095i \(-0.847457\pi\)
−0.887351 + 0.461095i \(0.847457\pi\)
\(110\) 0 0
\(111\) −8.40719e128 −1.17301
\(112\) −9.67926e128 −0.767550
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.59923e130 1.31564
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.64240e131 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.03854e132 1.35163
\(125\) 0 0
\(126\) 0 0
\(127\) −5.33946e132 −1.54121 −0.770607 0.637311i \(-0.780047\pi\)
−0.770607 + 0.637311i \(0.780047\pi\)
\(128\) 0 0
\(129\) 1.03086e133 1.11186
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −3.32246e133 −0.523366
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.96746e135 −1.92297 −0.961485 0.274857i \(-0.911369\pi\)
−0.961485 + 0.274857i \(0.911369\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 9.48063e135 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 1.42788e136 0.410866
\(148\) 6.24872e136 1.17301
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 6.70685e136 0.355608 0.177804 0.984066i \(-0.443101\pi\)
0.177804 + 0.984066i \(0.443101\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.93190e138 −1.31564
\(157\) −3.33345e138 −1.51781 −0.758903 0.651203i \(-0.774264\pi\)
−0.758903 + 0.651203i \(0.774264\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.60308e139 1.54474 0.772369 0.635174i \(-0.219072\pi\)
0.772369 + 0.635174i \(0.219072\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.66234e140 0.730898
\(170\) 0 0
\(171\) 3.25427e140 0.681866
\(172\) −7.66199e140 −1.11186
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.57219e141 −0.767550
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 5.93223e141 0.346312 0.173156 0.984894i \(-0.444603\pi\)
0.173156 + 0.984894i \(0.444603\pi\)
\(182\) 0 0
\(183\) −1.47975e142 −0.432285
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00540e143 0.767550
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −7.04656e143 −1.00000
\(193\) 1.83828e144 1.88062 0.940312 0.340312i \(-0.110533\pi\)
0.940312 + 0.340312i \(0.110533\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.06129e144 −0.410866
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.31879e145 −1.96080 −0.980400 0.197019i \(-0.936874\pi\)
−0.980400 + 0.197019i \(0.936874\pi\)
\(200\) 0 0
\(201\) 4.06420e144 0.321828
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.43591e146 1.31564
\(209\) 0 0
\(210\) 0 0
\(211\) 1.78139e146 0.662161 0.331081 0.943602i \(-0.392587\pi\)
0.331081 + 0.943602i \(0.392587\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.63278e147 −1.03744
\(218\) 0 0
\(219\) −5.46662e147 −1.94868
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.22589e148 1.39715 0.698576 0.715536i \(-0.253818\pi\)
0.698576 + 0.715536i \(0.253818\pi\)
\(224\) 0 0
\(225\) 1.53992e148 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −2.41877e148 −0.681866
\(229\) 3.95369e148 0.845994 0.422997 0.906131i \(-0.360978\pi\)
0.422997 + 0.906131i \(0.360978\pi\)
\(230\) 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
\(237\) −7.71170e149 −1.89673
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.70865e150 1.46413 0.732063 0.681237i \(-0.238558\pi\)
0.732063 + 0.681237i \(0.238558\pi\)
\(242\) 0 0
\(243\) −1.96424e150 −1.00000
\(244\) 1.09984e150 0.432285
\(245\) 0 0
\(246\) 0 0
\(247\) 4.92882e150 0.897087
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.49053e151 −0.767550
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 5.23742e151 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −9.82415e151 −0.900347
\(260\) 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
\(268\) −3.02076e152 −0.321828
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 2.92904e153 1.54761 0.773803 0.633427i \(-0.218352\pi\)
0.773803 + 0.633427i \(0.218352\pi\)
\(272\) 0 0
\(273\) 3.03731e153 1.00982
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.48739e154 −1.97788 −0.988940 0.148317i \(-0.952614\pi\)
−0.988940 + 0.148317i \(0.952614\pi\)
\(278\) 0 0
\(279\) 1.59927e154 1.35163
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.69188e154 −0.583210 −0.291605 0.956539i \(-0.594189\pi\)
−0.291605 + 0.956539i \(0.594189\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.08784e155 1.00000
\(290\) 0 0
\(291\) 2.03184e155 1.20957
\(292\) 4.06312e155 1.94868
\(293\) 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
\(300\) −1.14456e156 −1.00000
\(301\) 1.20461e156 0.853407
\(302\) 0 0
\(303\) 0 0
\(304\) 1.79777e156 0.681866
\(305\) 0 0
\(306\) 0 0
\(307\) 9.77552e156 1.99720 0.998602 0.0528604i \(-0.0168338\pi\)
0.998602 + 0.0528604i \(0.0168338\pi\)
\(308\) 0 0
\(309\) 1.36766e157 1.85606
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.18559e157 1.92263 0.961317 0.275443i \(-0.0888246\pi\)
0.961317 + 0.275443i \(0.0888246\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.73180e157 1.89673
\(317\) 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
\(324\) 1.45994e158 1.00000
\(325\) 2.33232e158 1.31564
\(326\) 0 0
\(327\) 4.63056e158 1.77470
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.06496e159 1.89744 0.948719 0.316120i \(-0.102380\pi\)
0.948719 + 0.316120i \(0.102380\pi\)
\(332\) 0 0
\(333\) 9.62254e158 1.17301
\(334\) 0 0
\(335\) 0 0
\(336\) 1.10785e159 0.767550
\(337\) 2.30035e158 0.132163 0.0660815 0.997814i \(-0.478950\pi\)
0.0660815 + 0.997814i \(0.478950\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 5.72957e159 1.08291
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 9.36275e159 0.593474 0.296737 0.954959i \(-0.404102\pi\)
0.296737 + 0.954959i \(0.404102\pi\)
\(350\) 0 0
\(351\) −2.97498e160 −1.31564
\(352\) 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
\(361\) −7.10159e160 −0.535059
\(362\) 0 0
\(363\) −1.87982e161 −1.00000
\(364\) −2.25751e161 −1.00982
\(365\) 0 0
\(366\) 0 0
\(367\) −6.13805e161 −1.63708 −0.818538 0.574453i \(-0.805215\pi\)
−0.818538 + 0.574453i \(0.805215\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.18867e162 −1.35163
\(373\) −1.54978e162 −1.48804 −0.744019 0.668159i \(-0.767083\pi\)
−0.744019 + 0.668159i \(0.767083\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.34951e162 −1.17682 −0.588411 0.808562i \(-0.700246\pi\)
−0.588411 + 0.808562i \(0.700246\pi\)
\(380\) 0 0
\(381\) 6.11134e162 1.54121
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.17989e163 −1.11186
\(388\) −1.51018e163 −1.20957
\(389\) 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
\(397\) 1.05862e164 1.99957 0.999787 0.0206172i \(-0.00656312\pi\)
0.999787 + 0.0206172i \(0.00656312\pi\)
\(398\) 0 0
\(399\) 3.80275e163 0.523366
\(400\) 8.50706e163 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 2.42220e164 1.77825
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.89261e164 1.99441 0.997207 0.0746861i \(-0.0237955\pi\)
0.997207 + 0.0746861i \(0.0237955\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.01652e165 −1.85606
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.25187e165 1.92297
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 2.43891e165 1.14136 0.570682 0.821171i \(-0.306679\pi\)
0.570682 + 0.821171i \(0.306679\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.72915e165 −0.331800
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.08512e166 −1.00000
\(433\) −2.44688e166 −1.94928 −0.974642 0.223768i \(-0.928164\pi\)
−0.974642 + 0.223768i \(0.928164\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.44171e166 −1.77470
\(437\) 0 0
\(438\) 0 0
\(439\) 2.73042e166 0.914039 0.457020 0.889457i \(-0.348917\pi\)
0.457020 + 0.889457i \(0.348917\pi\)
\(440\) 0 0
\(441\) −1.63430e166 −0.410866
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −7.15204e166 −1.17301
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −8.23421e166 −0.767550
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −7.67640e166 −0.355608
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.45776e166 0.198558 0.0992791 0.995060i \(-0.468346\pi\)
0.0992791 + 0.995060i \(0.468346\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 6.02173e167 0.704886 0.352443 0.935833i \(-0.385351\pi\)
0.352443 + 0.935833i \(0.385351\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 2.21118e168 1.31564
\(469\) 4.74919e167 0.247019
\(470\) 0 0
\(471\) 3.81534e168 1.51781
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.92009e168 0.681866
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.45740e169 1.54326
\(482\) 0 0
\(483\) 0 0
\(484\) 1.39720e169 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −3.66826e169 −1.77883 −0.889416 0.457098i \(-0.848889\pi\)
−0.889416 + 0.457098i \(0.848889\pi\)
\(488\) 0 0
\(489\) −4.12395e169 −1.54474
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.83490e169 1.35163
\(497\) 0 0
\(498\) 0 0
\(499\) −1.90668e170 −1.99501 −0.997504 0.0706131i \(-0.977504\pi\)
−0.997504 + 0.0706131i \(0.977504\pi\)
\(500\) 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
\(507\) −1.90265e170 −0.730898
\(508\) −4.54231e170 −1.54121
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.38797e170 −1.49571
\(512\) 0 0
\(513\) −3.72471e170 −0.681866
\(514\) 0 0
\(515\) 0 0
\(516\) 8.76962e170 1.11186
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 3.63890e171 1.97408 0.987039 0.160481i \(-0.0513044\pi\)
0.987039 + 0.160481i \(0.0513044\pi\)
\(524\) 0 0
\(525\) 1.79946e171 0.767550
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.78188e171 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.82643e171 −0.523366
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3.10753e172 1.99981 0.999904 0.0138765i \(-0.00441716\pi\)
0.999904 + 0.0138765i \(0.00441716\pi\)
\(542\) 0 0
\(543\) −6.78980e171 −0.346312
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.61268e172 1.80289 0.901444 0.432895i \(-0.142508\pi\)
0.901444 + 0.432895i \(0.142508\pi\)
\(548\) 0 0
\(549\) 1.69367e172 0.432285
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −9.01145e172 −1.45583
\(554\) 0 0
\(555\) 0 0
\(556\) −1.67373e173 −1.92297
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.78702e173 −1.46280
\(560\) 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
\(567\) −2.29530e173 −0.767550
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −5.27882e173 −1.13361 −0.566803 0.823853i \(-0.691820\pi\)
−0.566803 + 0.823853i \(0.691820\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 8.06522e173 1.00000
\(577\) 1.20918e174 1.34405 0.672025 0.740528i \(-0.265425\pi\)
0.672025 + 0.740528i \(0.265425\pi\)
\(578\) 0 0
\(579\) −2.10403e174 −1.88062
\(580\) 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 \(\pi\)
\(588\) 1.21471e174 0.410866
\(589\) 3.03262e174 0.921627
\(590\) 0 0
\(591\) 0 0
\(592\) 5.31582e174 1.17301
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.50944e175 1.96080
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −6.03726e174 −0.514930 −0.257465 0.966288i \(-0.582887\pi\)
−0.257465 + 0.966288i \(0.582887\pi\)
\(602\) 0 0
\(603\) −4.65173e174 −0.321828
\(604\) 5.70556e174 0.355608
\(605\) 0 0
\(606\) 0 0
\(607\) 3.22174e175 1.46961 0.734807 0.678276i \(-0.237273\pi\)
0.734807 + 0.678276i \(0.237273\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7.19316e175 −1.76567 −0.882836 0.469681i \(-0.844369\pi\)
−0.882836 + 0.469681i \(0.844369\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.50310e176 −1.99746 −0.998728 0.0504314i \(-0.983940\pi\)
−0.998728 + 0.0504314i \(0.983940\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.64348e176 −1.31564
\(625\) 1.38179e176 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −2.83579e176 −1.51781
\(629\) 0 0
\(630\) 0 0
\(631\) 4.06643e176 1.61200 0.805998 0.591918i \(-0.201629\pi\)
0.805998 + 0.591918i \(0.201629\pi\)
\(632\) 0 0
\(633\) −2.03891e176 −0.662161
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.47526e176 −0.540550
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −2.76891e176 −0.334980 −0.167490 0.985874i \(-0.553566\pi\)
−0.167490 + 0.985874i \(0.553566\pi\)
\(644\) 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
\(651\) 1.86881e177 1.03744
\(652\) 3.06516e177 1.54474
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.25688e177 1.94868
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 4.18711e177 0.889657 0.444829 0.895616i \(-0.353265\pi\)
0.444829 + 0.895616i \(0.353265\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.40310e178 −1.39715
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.57412e178 −1.76068 −0.880339 0.474346i \(-0.842685\pi\)
−0.880339 + 0.474346i \(0.842685\pi\)
\(674\) 0 0
\(675\) −1.76253e178 −1.00000
\(676\) 1.41416e178 0.730898
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 2.37429e178 0.928406
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 2.76843e178 0.681866
\(685\) 0 0
\(686\) 0 0
\(687\) −4.52524e178 −0.845994
\(688\) −6.51810e178 −1.11186
\(689\) 0 0
\(690\) 0 0
\(691\) −1.42151e179 −1.84346 −0.921729 0.387834i \(-0.873223\pi\)
−0.921729 + 0.387834i \(0.873223\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.33747e179 −0.767550
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.82468e179 0.799838
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.54635e179 1.93653 0.968264 0.249928i \(-0.0804070\pi\)
0.968264 + 0.249928i \(0.0804070\pi\)
\(710\) 0 0
\(711\) 8.82652e179 1.89673
\(712\) 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
\(721\) 1.59817e180 1.42462
\(722\) 0 0
\(723\) −1.95565e180 −1.46413
\(724\) 5.04659e179 0.346312
\(725\) 0 0
\(726\) 0 0
\(727\) −2.44766e180 −1.29444 −0.647221 0.762302i \(-0.724069\pi\)
−0.647221 + 0.762302i \(0.724069\pi\)
\(728\) 0 0
\(729\) 2.24820e180 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1.25883e180 −0.432285
\(733\) 5.14190e180 1.62022 0.810109 0.586279i \(-0.199408\pi\)
0.810109 + 0.586279i \(0.199408\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.05875e181 −1.99615 −0.998077 0.0619839i \(-0.980257\pi\)
−0.998077 + 0.0619839i \(0.980257\pi\)
\(740\) 0 0
\(741\) −5.64133e180 −0.897087
\(742\) 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
\(751\) −2.77037e181 −1.89331 −0.946655 0.322250i \(-0.895561\pi\)
−0.946655 + 0.322250i \(0.895561\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 1.70600e181 0.767550
\(757\) −3.27812e181 −1.35701 −0.678507 0.734593i \(-0.737373\pi\)
−0.678507 + 0.734593i \(0.737373\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 5.41101e181 1.36217
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −5.99455e181 −1.00000
\(769\) 2.17248e181 0.333884 0.166942 0.985967i \(-0.446611\pi\)
0.166942 + 0.985967i \(0.446611\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.56384e182 1.88062
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1.43504e182 1.35163
\(776\) 0 0
\(777\) 1.12443e182 0.900347
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −9.02843e181 −0.410866
\(785\) 0 0
\(786\) 0 0
\(787\) 3.89174e181 0.139231 0.0696155 0.997574i \(-0.477823\pi\)
0.0696155 + 0.997574i \(0.477823\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.56517e182 0.568729
\(794\) 0 0
\(795\) 0 0
\(796\) −1.12191e183 −1.96080
\(797\) 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
\(804\) 3.45744e182 0.321828
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1.58774e183 −0.855983 −0.427991 0.903783i \(-0.640779\pi\)
−0.427991 + 0.903783i \(0.640779\pi\)
\(812\) 0 0
\(813\) −3.35247e183 −1.54761
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.23737e183 −0.758138
\(818\) 0 0
\(819\) −3.47639e183 −1.00982
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.32824e181 0.00283848 0.00141924 0.999999i \(-0.499548\pi\)
0.00141924 + 0.999999i \(0.499548\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.38758e184 1.87639 0.938193 0.346113i \(-0.112499\pi\)
0.938193 + 0.346113i \(0.112499\pi\)
\(830\) 0 0
\(831\) 1.70241e184 1.97788
\(832\) 1.22153e184 1.31564
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.83046e184 −1.35163
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.82872e184 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.51544e184 0.662161
\(845\) 0 0
\(846\) 0 0
\(847\) −2.19665e184 −0.767550
\(848\) 0 0
\(849\) 1.93646e184 0.583210
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −6.66570e184 −1.49299 −0.746495 0.665392i \(-0.768265\pi\)
−0.746495 + 0.665392i \(0.768265\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1.38226e185 1.99077 0.995384 0.0959721i \(-0.0305960\pi\)
0.995384 + 0.0959721i \(0.0305960\pi\)
\(860\) 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
\(867\) −1.24510e185 −1.00000
\(868\) −1.38901e185 −1.03744
\(869\) 0 0
\(870\) 0 0
\(871\) −7.04536e184 −0.423409
\(872\) 0 0
\(873\) −2.32556e185 −1.20957
\(874\) 0 0
\(875\) 0 0
\(876\) −4.65048e185 −1.94868
\(877\) 3.72709e184 0.145343 0.0726717 0.997356i \(-0.476847\pi\)
0.0726717 + 0.997356i \(0.476847\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.07284e185 0.526066 0.263033 0.964787i \(-0.415277\pi\)
0.263033 + 0.964787i \(0.415277\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 7.14136e185 1.18296
\(890\) 0 0
\(891\) 0 0
\(892\) 1.04287e186 1.39715
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.31002e186 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −1.37875e186 −0.853407
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.92426e186 1.83865 0.919323 0.393503i \(-0.128737\pi\)
0.919323 + 0.393503i \(0.128737\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −2.05766e186 −0.681866
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.36343e186 0.845994
\(917\) 0 0
\(918\) 0 0
\(919\) 4.74809e186 0.971946 0.485973 0.873974i \(-0.338465\pi\)
0.485973 + 0.873974i \(0.338465\pi\)
\(920\) 0 0
\(921\) −1.11887e187 −1.99720
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.63440e186 1.17301
\(926\) 0 0
\(927\) −1.56537e187 −1.85606
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −3.09906e186 −0.280156
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −9.43108e186 −0.568810 −0.284405 0.958704i \(-0.591796\pi\)
−0.284405 + 0.958704i \(0.591796\pi\)
\(938\) 0 0
\(939\) −3.64610e187 −1.92263
\(940\) 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 \(\pi\)
\(948\) −6.56039e187 −1.89673
\(949\) 9.47647e187 2.56375
\(950\) 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
\(961\) 6.74561e187 0.826890
\(962\) 0 0
\(963\) 0 0
\(964\) 1.45356e188 1.46413
\(965\) 0 0
\(966\) 0 0
\(967\) 2.21362e188 1.83330 0.916651 0.399687i \(-0.130881\pi\)
0.916651 + 0.399687i \(0.130881\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −1.67099e188 −1.00000
\(973\) 2.63141e188 1.47598
\(974\) 0 0
\(975\) −2.66948e188 −1.31564
\(976\) 9.35639e187 0.432285
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.29996e188 −1.77470
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.19297e188 0.897087
\(989\) 0 0
\(990\) 0 0
\(991\) 1.12489e189 1.98824 0.994119 0.108293i \(-0.0345384\pi\)
0.994119 + 0.108293i \(0.0345384\pi\)
\(992\) 0 0
\(993\) −1.21891e189 −1.89744
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.19613e188 −0.748730 −0.374365 0.927281i \(-0.622139\pi\)
−0.374365 + 0.927281i \(0.622139\pi\)
\(998\) 0 0
\(999\) −1.10136e189 −1.17301
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 3.127.b.a.2.1 1
3.2 odd 2 CM 3.127.b.a.2.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.127.b.a.2.1 1 1.1 even 1 trivial
3.127.b.a.2.1 1 3.2 odd 2 CM