Properties

Label 3.115.b.a.2.1
Level $3$
Weight $115$
Character 3.2
Self dual yes
Analytic conductor $246.901$
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] = [3,115,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, 115, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 115);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 115 \)
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: \(246.900726555\)
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)
 
\(f(q)\) \(=\) \(q-1.57004e27 q^{3} +2.07692e34 q^{4} +2.81479e48 q^{7} +2.46503e54 q^{9} +O(q^{10})\) \(q-1.57004e27 q^{3} +2.07692e34 q^{4} +2.81479e48 q^{7} +2.46503e54 q^{9} -3.26085e61 q^{12} -5.07835e63 q^{13} +4.31359e68 q^{16} -1.32905e73 q^{19} -4.41934e75 q^{21} +4.81482e79 q^{25} -3.87021e81 q^{27} +5.84609e82 q^{28} +1.02087e85 q^{31} +5.11968e88 q^{36} +4.87661e89 q^{37} +7.97323e90 q^{39} -2.10341e93 q^{43} -6.77252e95 q^{48} +5.72935e96 q^{49} -1.05473e98 q^{52} +2.08666e100 q^{57} -6.07714e101 q^{61} +6.93856e102 q^{63} +8.95898e102 q^{64} +1.65321e104 q^{67} +1.77355e106 q^{73} -7.55948e106 q^{75} -2.76032e107 q^{76} -2.15665e108 q^{79} +6.07640e108 q^{81} -9.17861e109 q^{84} -1.42945e112 q^{91} -1.60281e112 q^{93} +1.95362e113 q^{97} +O(q^{100})\)

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 \(\pi\)
\(3\) −1.57004e27 −1.00000
\(4\) 2.07692e34 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 2.81479e48 1.90046 0.950228 0.311556i \(-0.100850\pi\)
0.950228 + 0.311556i \(0.100850\pi\)
\(8\) 0 0
\(9\) 2.46503e54 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −3.26085e61 −1.00000
\(13\) −5.07835e63 −1.62537 −0.812684 0.582704i \(-0.801995\pi\)
−0.812684 + 0.582704i \(0.801995\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.31359e68 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −1.32905e73 −1.71627 −0.858133 0.513428i \(-0.828375\pi\)
−0.858133 + 0.513428i \(0.828375\pi\)
\(20\) 0 0
\(21\) −4.41934e75 −1.90046
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 4.81482e79 1.00000
\(26\) 0 0
\(27\) −3.87021e81 −1.00000
\(28\) 5.84609e82 1.90046
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.02087e85 1.00312 0.501562 0.865122i \(-0.332759\pi\)
0.501562 + 0.865122i \(0.332759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.11968e88 1.00000
\(37\) 4.87661e89 1.99811 0.999056 0.0434372i \(-0.0138308\pi\)
0.999056 + 0.0434372i \(0.0138308\pi\)
\(38\) 0 0
\(39\) 7.97323e90 1.62537
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.10341e93 −1.64143 −0.820713 0.571341i \(-0.806423\pi\)
−0.820713 + 0.571341i \(0.806423\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −6.77252e95 −1.00000
\(49\) 5.72935e96 2.61173
\(50\) 0 0
\(51\) 0 0
\(52\) −1.05473e98 −1.62537
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.08666e100 1.71627
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −6.07714e101 −1.04688 −0.523442 0.852061i \(-0.675352\pi\)
−0.523442 + 0.852061i \(0.675352\pi\)
\(62\) 0 0
\(63\) 6.93856e102 1.90046
\(64\) 8.95898e102 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.65321e104 1.35539 0.677695 0.735343i \(-0.262979\pi\)
0.677695 + 0.735343i \(0.262979\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.77355e106 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(74\) 0 0
\(75\) −7.55948e106 −1.00000
\(76\) −2.76032e107 −1.71627
\(77\) 0 0
\(78\) 0 0
\(79\) −2.15665e108 −1.47583 −0.737914 0.674895i \(-0.764189\pi\)
−0.737914 + 0.674895i \(0.764189\pi\)
\(80\) 0 0
\(81\) 6.07640e108 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −9.17861e109 −1.90046
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.42945e112 −3.08894
\(92\) 0 0
\(93\) −1.60281e112 −1.00312
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.95362e113 1.10880 0.554398 0.832252i \(-0.312949\pi\)
0.554398 + 0.832252i \(0.312949\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e114 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 8.23466e114 1.52730 0.763650 0.645631i \(-0.223406\pi\)
0.763650 + 0.645631i \(0.223406\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −8.03811e115 −1.00000
\(109\) 1.61902e115 0.119108 0.0595540 0.998225i \(-0.481032\pi\)
0.0595540 + 0.998225i \(0.481032\pi\)
\(110\) 0 0
\(111\) −7.65649e116 −1.99811
\(112\) 1.21419e117 1.90046
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.25183e118 −1.62537
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.23319e118 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 2.12027e119 1.00312
\(125\) 0 0
\(126\) 0 0
\(127\) −5.61909e119 −0.680540 −0.340270 0.940328i \(-0.610518\pi\)
−0.340270 + 0.940328i \(0.610518\pi\)
\(128\) 0 0
\(129\) 3.30244e120 1.64143
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −3.74098e121 −3.26169
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 2.78229e122 1.96136 0.980682 0.195607i \(-0.0626676\pi\)
0.980682 + 0.195607i \(0.0626676\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.06332e123 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −8.99532e123 −2.61173
\(148\) 1.01283e124 1.99811
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −3.11069e124 −1.95511 −0.977555 0.210679i \(-0.932433\pi\)
−0.977555 + 0.210679i \(0.932433\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.65598e125 1.62537
\(157\) −9.25769e123 −0.0631281 −0.0315640 0.999502i \(-0.510049\pi\)
−0.0315640 + 0.999502i \(0.510049\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.94413e125 −0.397555 −0.198777 0.980045i \(-0.563697\pi\)
−0.198777 + 0.980045i \(0.563697\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.60276e127 1.64182
\(170\) 0 0
\(171\) −3.27614e127 −1.71627
\(172\) −4.36860e127 −1.64143
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 1.35527e128 1.90046
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.52785e128 −0.313618 −0.156809 0.987629i \(-0.550121\pi\)
−0.156809 + 0.987629i \(0.550121\pi\)
\(182\) 0 0
\(183\) 9.54136e128 1.04688
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.08938e130 −1.90046
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.40660e130 −1.00000
\(193\) 1.32906e130 0.702715 0.351358 0.936241i \(-0.385720\pi\)
0.351358 + 0.936241i \(0.385720\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.18994e131 2.61173
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 2.26212e130 0.208877 0.104439 0.994531i \(-0.466695\pi\)
0.104439 + 0.994531i \(0.466695\pi\)
\(200\) 0 0
\(201\) −2.59561e131 −1.35539
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −2.19059e132 −1.62537
\(209\) 0 0
\(210\) 0 0
\(211\) 5.29886e132 1.73813 0.869065 0.494698i \(-0.164721\pi\)
0.869065 + 0.494698i \(0.164721\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.87354e133 1.90639
\(218\) 0 0
\(219\) −2.78455e133 −1.09507
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.35190e134 −1.89482 −0.947412 0.320018i \(-0.896311\pi\)
−0.947412 + 0.320018i \(0.896311\pi\)
\(224\) 0 0
\(225\) 1.18687e134 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 4.33382e134 1.71627
\(229\) −6.20950e133 −0.191617 −0.0958085 0.995400i \(-0.530544\pi\)
−0.0958085 + 0.995400i \(0.530544\pi\)
\(230\) 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
\(237\) 3.38604e135 1.47583
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.18747e136 1.99366 0.996828 0.0795920i \(-0.0253617\pi\)
0.996828 + 0.0795920i \(0.0253617\pi\)
\(242\) 0 0
\(243\) −9.54020e135 −1.00000
\(244\) −1.26217e136 −1.04688
\(245\) 0 0
\(246\) 0 0
\(247\) 6.74936e136 2.78956
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.44108e137 1.90046
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.86071e137 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.37266e138 3.79732
\(260\) 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
\(268\) 3.43358e138 1.35539
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 9.06662e138 1.89756 0.948780 0.315936i \(-0.102319\pi\)
0.948780 + 0.315936i \(0.102319\pi\)
\(272\) 0 0
\(273\) 2.24430e139 3.08894
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.98291e138 0.179182 0.0895909 0.995979i \(-0.471444\pi\)
0.0895909 + 0.995979i \(0.471444\pi\)
\(278\) 0 0
\(279\) 2.51649e139 1.00312
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.66704e139 −0.649365 −0.324682 0.945823i \(-0.605257\pi\)
−0.324682 + 0.945823i \(0.605257\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.86714e140 1.00000
\(290\) 0 0
\(291\) −3.06727e140 −1.10880
\(292\) 3.68352e140 1.09507
\(293\) 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
\(300\) −1.57004e141 −1.00000
\(301\) −5.92065e141 −3.11946
\(302\) 0 0
\(303\) 0 0
\(304\) −5.73296e141 −1.71627
\(305\) 0 0
\(306\) 0 0
\(307\) 9.33492e141 1.59670 0.798349 0.602194i \(-0.205707\pi\)
0.798349 + 0.602194i \(0.205707\pi\)
\(308\) 0 0
\(309\) −1.29288e142 −1.52730
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.61761e142 −0.918008 −0.459004 0.888434i \(-0.651794\pi\)
−0.459004 + 0.888434i \(0.651794\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4.47919e142 −1.47583
\(317\) 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
\(324\) 1.26202e143 1.00000
\(325\) −2.44514e143 −1.62537
\(326\) 0 0
\(327\) −2.54193e142 −0.119108
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.53515e143 1.99993 0.999966 0.00819470i \(-0.00260848\pi\)
0.999966 + 0.00819470i \(0.00260848\pi\)
\(332\) 0 0
\(333\) 1.20210e144 1.99811
\(334\) 0 0
\(335\) 0 0
\(336\) −1.90632e144 −1.90046
\(337\) 2.35498e144 1.98190 0.990951 0.134225i \(-0.0428545\pi\)
0.990951 + 0.134225i \(0.0428545\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 9.95212e144 3.06303
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.71210e145 −1.96101 −0.980505 0.196494i \(-0.937044\pi\)
−0.980505 + 0.196494i \(0.937044\pi\)
\(350\) 0 0
\(351\) 1.96543e145 1.62537
\(352\) 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
\(361\) 1.16669e146 1.94557
\(362\) 0 0
\(363\) −8.21633e145 −1.00000
\(364\) −2.96885e146 −3.08894
\(365\) 0 0
\(366\) 0 0
\(367\) −2.45586e146 −1.60043 −0.800216 0.599712i \(-0.795282\pi\)
−0.800216 + 0.599712i \(0.795282\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −3.32891e146 −1.00312
\(373\) −1.06738e146 −0.276004 −0.138002 0.990432i \(-0.544068\pi\)
−0.138002 + 0.990432i \(0.544068\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.61162e145 −0.0584323 −0.0292162 0.999573i \(-0.509301\pi\)
−0.0292162 + 0.999573i \(0.509301\pi\)
\(380\) 0 0
\(381\) 8.82221e146 0.680540
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.18497e147 −1.64143
\(388\) 4.05751e147 1.10880
\(389\) 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
\(397\) 2.59878e148 1.92181 0.960905 0.276879i \(-0.0893000\pi\)
0.960905 + 0.276879i \(0.0893000\pi\)
\(398\) 0 0
\(399\) 5.87351e148 3.26169
\(400\) 2.07692e148 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −5.18435e148 −1.63045
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −8.23029e148 −1.11477 −0.557387 0.830253i \(-0.688196\pi\)
−0.557387 + 0.830253i \(0.688196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.71027e149 1.52730
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.36831e149 −1.96136
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 7.71586e148 0.201050 0.100525 0.994935i \(-0.467948\pi\)
0.100525 + 0.994935i \(0.467948\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.71059e150 −1.98956
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.66945e150 −1.00000
\(433\) −2.64686e150 −1.38970 −0.694851 0.719154i \(-0.744530\pi\)
−0.694851 + 0.719154i \(0.744530\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.36257e149 0.119108
\(437\) 0 0
\(438\) 0 0
\(439\) 1.40354e150 0.336317 0.168158 0.985760i \(-0.446218\pi\)
0.168158 + 0.985760i \(0.446218\pi\)
\(440\) 0 0
\(441\) 1.41230e151 2.61173
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −1.59019e151 −1.99811
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 2.52176e151 1.90046
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.88392e151 1.95511
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.71277e151 0.415408 0.207704 0.978192i \(-0.433401\pi\)
0.207704 + 0.978192i \(0.433401\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −2.12657e151 −0.245224 −0.122612 0.992455i \(-0.539127\pi\)
−0.122612 + 0.992455i \(0.539127\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −2.59995e152 −1.62537
\(469\) 4.65344e152 2.57586
\(470\) 0 0
\(471\) 1.45350e151 0.0631281
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.39912e152 −1.71627
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −2.47652e153 −3.24767
\(482\) 0 0
\(483\) 0 0
\(484\) 1.08689e153 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2.75904e153 −1.78487 −0.892436 0.451174i \(-0.851005\pi\)
−0.892436 + 0.451174i \(0.851005\pi\)
\(488\) 0 0
\(489\) 7.76250e152 0.397555
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 4.40363e153 1.00312
\(497\) 0 0
\(498\) 0 0
\(499\) 1.23358e154 1.99267 0.996334 0.0855536i \(-0.0272659\pi\)
0.996334 + 0.0855536i \(0.0272659\pi\)
\(500\) 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
\(507\) −2.51640e154 −1.64182
\(508\) −1.16704e154 −0.680540
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 4.99217e154 2.08112
\(512\) 0 0
\(513\) 5.14368e154 1.71627
\(514\) 0 0
\(515\) 0 0
\(516\) 6.85889e154 1.64143
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −6.64471e154 −0.737700 −0.368850 0.929489i \(-0.620248\pi\)
−0.368850 + 0.929489i \(0.620248\pi\)
\(524\) 0 0
\(525\) −2.12784e155 −1.90046
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.72573e155 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −7.76972e155 −3.26169
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.90983e155 −0.469487 −0.234744 0.972057i \(-0.575425\pi\)
−0.234744 + 0.972057i \(0.575425\pi\)
\(542\) 0 0
\(543\) 2.39878e155 0.313618
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.77139e156 1.52418 0.762089 0.647472i \(-0.224174\pi\)
0.762089 + 0.647472i \(0.224174\pi\)
\(548\) 0 0
\(549\) −1.49803e156 −1.04688
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.07052e156 −2.80475
\(554\) 0 0
\(555\) 0 0
\(556\) 5.77859e156 1.96136
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.06818e157 2.66792
\(560\) 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
\(567\) 1.71038e157 1.90046
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −9.27389e156 −0.690246 −0.345123 0.938557i \(-0.612163\pi\)
−0.345123 + 0.938557i \(0.612163\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 2.20842e157 1.00000
\(577\) −1.75207e157 −0.718672 −0.359336 0.933208i \(-0.616997\pi\)
−0.359336 + 0.933208i \(0.616997\pi\)
\(578\) 0 0
\(579\) −2.08669e157 −0.702715
\(580\) 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 \(\pi\)
\(588\) −1.86826e158 −2.61173
\(589\) −1.35679e158 −1.72163
\(590\) 0 0
\(591\) 0 0
\(592\) 2.10357e158 1.99811
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.55163e157 −0.208877
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −3.14582e158 −1.26441 −0.632206 0.774801i \(-0.717850\pi\)
−0.632206 + 0.774801i \(0.717850\pi\)
\(602\) 0 0
\(603\) 4.07522e158 1.35539
\(604\) −6.46066e158 −1.95511
\(605\) 0 0
\(606\) 0 0
\(607\) 7.59666e158 1.73327 0.866637 0.498939i \(-0.166277\pi\)
0.866637 + 0.498939i \(0.166277\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.70202e158 0.742645 0.371323 0.928504i \(-0.378904\pi\)
0.371323 + 0.928504i \(0.378904\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.23060e159 0.919922 0.459961 0.887939i \(-0.347863\pi\)
0.459961 + 0.887939i \(0.347863\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 3.43933e159 1.62537
\(625\) 2.31825e159 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.92275e158 −0.0631281
\(629\) 0 0
\(630\) 0 0
\(631\) 5.13991e159 1.28612 0.643060 0.765816i \(-0.277664\pi\)
0.643060 + 0.765816i \(0.277664\pi\)
\(632\) 0 0
\(633\) −8.31943e159 −1.73813
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.90957e160 −4.24503
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 2.33212e160 1.99400 0.997002 0.0773787i \(-0.0246551\pi\)
0.997002 + 0.0773787i \(0.0246551\pi\)
\(644\) 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
\(651\) −4.51158e160 −1.90639
\(652\) −1.02686e160 −0.397555
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.37186e160 1.09507
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −9.58949e160 −1.69947 −0.849733 0.527213i \(-0.823237\pi\)
−0.849733 + 0.527213i \(0.823237\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 2.12255e161 1.89482
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.60304e161 1.65433 0.827165 0.561959i \(-0.189952\pi\)
0.827165 + 0.561959i \(0.189952\pi\)
\(674\) 0 0
\(675\) −1.86344e161 −1.00000
\(676\) 3.32880e161 1.64182
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 5.49903e161 2.10722
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −6.80428e161 −1.71627
\(685\) 0 0
\(686\) 0 0
\(687\) 9.74918e160 0.191617
\(688\) −9.07323e161 −1.64143
\(689\) 0 0
\(690\) 0 0
\(691\) −1.92008e161 −0.271063 −0.135532 0.990773i \(-0.543274\pi\)
−0.135532 + 0.990773i \(0.543274\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.81479e162 1.90046
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −6.48124e162 −3.42929
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.89302e161 0.289877 0.144938 0.989441i \(-0.453702\pi\)
0.144938 + 0.989441i \(0.453702\pi\)
\(710\) 0 0
\(711\) −5.31622e162 −1.47583
\(712\) 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
\(721\) 2.31789e163 2.90256
\(722\) 0 0
\(723\) −1.86437e163 −1.99366
\(724\) −3.17321e162 −0.313618
\(725\) 0 0
\(726\) 0 0
\(727\) 1.02175e163 0.797779 0.398889 0.916999i \(-0.369396\pi\)
0.398889 + 0.916999i \(0.369396\pi\)
\(728\) 0 0
\(729\) 1.49785e163 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 1.98166e163 1.04688
\(733\) 2.44907e163 1.19695 0.598474 0.801142i \(-0.295774\pi\)
0.598474 + 0.801142i \(0.295774\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.70004e163 1.75043 0.875215 0.483734i \(-0.160720\pi\)
0.875215 + 0.483734i \(0.160720\pi\)
\(740\) 0 0
\(741\) −1.05968e164 −2.78956
\(742\) 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
\(751\) 8.18904e163 1.00405 0.502027 0.864852i \(-0.332588\pi\)
0.502027 + 0.864852i \(0.332588\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −2.26256e164 −1.90046
\(757\) −2.31483e164 −1.80324 −0.901622 0.432524i \(-0.857623\pi\)
−0.901622 + 0.432524i \(0.857623\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 4.55719e163 0.226360
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −2.92139e164 −1.00000
\(769\) 3.55595e164 1.13020 0.565098 0.825024i \(-0.308838\pi\)
0.565098 + 0.825024i \(0.308838\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.76036e164 0.702715
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 4.91532e164 1.00312
\(776\) 0 0
\(777\) −2.15514e165 −3.79732
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.47141e165 2.61173
\(785\) 0 0
\(786\) 0 0
\(787\) 1.70580e165 1.45000 0.725000 0.688749i \(-0.241840\pi\)
0.725000 + 0.688749i \(0.241840\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.08618e165 1.70157
\(794\) 0 0
\(795\) 0 0
\(796\) 4.69824e164 0.208877
\(797\) 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
\(804\) −5.39087e165 −1.35539
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −6.22166e165 −0.954372 −0.477186 0.878802i \(-0.658343\pi\)
−0.477186 + 0.878802i \(0.658343\pi\)
\(812\) 0 0
\(813\) −1.42350e166 −1.89756
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.79552e166 2.81712
\(818\) 0 0
\(819\) −3.52364e166 −3.08894
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 2.61019e166 1.73333 0.866667 0.498888i \(-0.166258\pi\)
0.866667 + 0.498888i \(0.166258\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.10508e166 −0.485047 −0.242523 0.970146i \(-0.577975\pi\)
−0.242523 + 0.970146i \(0.577975\pi\)
\(830\) 0 0
\(831\) −4.68330e165 −0.179182
\(832\) −4.54969e166 −1.62537
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.95099e166 −1.00312
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.16859e166 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.10053e167 1.73813
\(845\) 0 0
\(846\) 0 0
\(847\) 1.47303e167 1.90046
\(848\) 0 0
\(849\) 5.75741e166 0.649365
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.27787e167 1.10254 0.551268 0.834328i \(-0.314144\pi\)
0.551268 + 0.834328i \(0.314144\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −3.23272e167 −1.87050 −0.935249 0.353989i \(-0.884825\pi\)
−0.935249 + 0.353989i \(0.884825\pi\)
\(860\) 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
\(867\) −2.93149e167 −1.00000
\(868\) 5.96811e167 1.90639
\(869\) 0 0
\(870\) 0 0
\(871\) −8.39558e167 −2.20301
\(872\) 0 0
\(873\) 4.81574e167 1.10880
\(874\) 0 0
\(875\) 0 0
\(876\) −5.78328e167 −1.09507
\(877\) −9.37020e167 −1.66254 −0.831270 0.555868i \(-0.812386\pi\)
−0.831270 + 0.555868i \(0.812386\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.58049e168 1.90121 0.950605 0.310402i \(-0.100464\pi\)
0.950605 + 0.310402i \(0.100464\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.58166e168 −1.29334
\(890\) 0 0
\(891\) 0 0
\(892\) −2.80779e168 −1.89482
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.46503e168 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 9.29567e168 3.11946
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.38243e168 −0.882295 −0.441147 0.897435i \(-0.645428\pi\)
−0.441147 + 0.897435i \(0.645428\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 9.00099e168 1.71627
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.28966e168 −0.191617
\(917\) 0 0
\(918\) 0 0
\(919\) 4.94061e168 0.609251 0.304626 0.952472i \(-0.401469\pi\)
0.304626 + 0.952472i \(0.401469\pi\)
\(920\) 0 0
\(921\) −1.46562e169 −1.59670
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.34800e169 1.99811
\(926\) 0 0
\(927\) 2.02987e169 1.52730
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −7.61456e169 −4.48243
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.73182e169 −1.93140 −0.965698 0.259668i \(-0.916387\pi\)
−0.965698 + 0.259668i \(0.916387\pi\)
\(938\) 0 0
\(939\) 2.53972e169 0.918008
\(940\) 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 \(\pi\)
\(948\) 7.03252e169 1.47583
\(949\) −9.00671e169 −1.77989
\(950\) 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
\(961\) 6.48236e167 0.00625893
\(962\) 0 0
\(963\) 0 0
\(964\) 2.46627e170 1.99366
\(965\) 0 0
\(966\) 0 0
\(967\) −2.74720e170 −1.86029 −0.930147 0.367186i \(-0.880321\pi\)
−0.930147 + 0.367186i \(0.880321\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −1.98142e170 −1.00000
\(973\) 7.83156e170 3.72749
\(974\) 0 0
\(975\) 3.83897e170 1.62537
\(976\) −2.62143e170 −1.04688
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.99093e169 0.119108
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.40179e171 2.78956
\(989\) 0 0
\(990\) 0 0
\(991\) −1.15813e171 −1.93892 −0.969461 0.245244i \(-0.921132\pi\)
−0.969461 + 0.245244i \(0.921132\pi\)
\(992\) 0 0
\(993\) −1.34005e171 −1.99993
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.26565e171 −1.50207 −0.751036 0.660262i \(-0.770445\pi\)
−0.751036 + 0.660262i \(0.770445\pi\)
\(998\) 0 0
\(999\) −1.88735e171 −1.99811
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 3.115.b.a.2.1 1
3.2 odd 2 CM 3.115.b.a.2.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.115.b.a.2.1 1 1.1 even 1 trivial
3.115.b.a.2.1 1 3.2 odd 2 CM