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Properties

Label 12.39.c.a.5.1
Level $12$
Weight $39$
Character 12.5
Self dual yes
Analytic conductor $109.756$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,39,Mod(5,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.5"); S:= CuspForms(chi, 39); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 39, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 39 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.756162840\)
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 5.1
Character \(\chi\) \(=\) 12.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.16226e9 q^{3} -9.25415e15 q^{7} +1.35085e18 q^{9} +9.09323e20 q^{13} -3.89311e24 q^{19} +1.07557e25 q^{21} +3.63798e26 q^{25} -1.57004e27 q^{27} -7.55182e27 q^{31} -6.40543e29 q^{37} -1.05687e30 q^{39} -2.12787e31 q^{43} -4.42955e31 q^{49} +4.52481e33 q^{57} +3.04644e33 q^{61} -1.25010e34 q^{63} -2.43668e34 q^{67} -9.71243e34 q^{73} -4.22828e35 q^{75} +1.58081e36 q^{79} +1.82480e36 q^{81} -8.41501e36 q^{91} +8.77719e36 q^{93} -2.18224e37 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) −1.16226e9 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −9.25415e15 −0.811847 −0.405923 − 0.913907i \(-0.633050\pi\)
−0.405923 + 0.913907i \(0.633050\pi\)
\(8\) 0 0
\(9\) 1.35085e18 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 9.09323e20 0.622006 0.311003 − 0.950409i \(-0.399335\pi\)
0.311003 + 0.950409i \(0.399335\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −3.89311e24 −1.96779 −0.983893 − 0.178759i \(-0.942792\pi\)
−0.983893 + 0.178759i \(0.942792\pi\)
\(20\) 0 0
\(21\) 1.07557e25 0.811847
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.63798e26 1.00000
\(26\) 0 0
\(27\) −1.57004e27 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.55182e27 −0.348481 −0.174241 − 0.984703i \(-0.555747\pi\)
−0.174241 + 0.984703i \(0.555747\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.40543e29 −1.02498 −0.512490 − 0.858693i \(-0.671277\pi\)
−0.512490 + 0.858693i \(0.671277\pi\)
\(38\) 0 0
\(39\) −1.05687e30 −0.622006
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.12787e31 −1.95905 −0.979524 − 0.201328i \(-0.935474\pi\)
−0.979524 + 0.201328i \(0.935474\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −4.42955e31 −0.340905
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.52481e33 1.96779
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 3.04644e33 0.365196 0.182598 − 0.983188i \(-0.441549\pi\)
0.182598 + 0.983188i \(0.441549\pi\)
\(62\) 0 0
\(63\) −1.25010e34 −0.811847
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.43668e34 −0.491334 −0.245667 − 0.969354i \(-0.579007\pi\)
−0.245667 + 0.969354i \(0.579007\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9.71243e34 −0.383878 −0.191939 − 0.981407i \(-0.561478\pi\)
−0.191939 + 0.981407i \(0.561478\pi\)
\(74\) 0 0
\(75\) −4.22828e35 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.58081e36 1.39304 0.696522 − 0.717535i \(-0.254730\pi\)
0.696522 + 0.717535i \(0.254730\pi\)
\(80\) 0 0
\(81\) 1.82480e36 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
\(91\) −8.41501e36 −0.504973
\(92\) 0 0
\(93\) 8.77719e36 0.348481
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.18224e37 −0.389259 −0.194630 − 0.980877i \(-0.562350\pi\)
−0.194630 + 0.980877i \(0.562350\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 3.41148e38 1.94552 0.972758 − 0.231821i \(-0.0744685\pi\)
0.972758 + 0.231821i \(0.0744685\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 9.00598e38 1.75157 0.875785 − 0.482701i \(-0.160344\pi\)
0.875785 + 0.482701i \(0.160344\pi\)
\(110\) 0 0
\(111\) 7.44479e38 1.02498
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.22836e39 0.622006
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.74043e39 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.50571e40 1.60499 0.802494 − 0.596661i \(-0.203506\pi\)
0.802494 + 0.596661i \(0.203506\pi\)
\(128\) 0 0
\(129\) 2.47314e40 1.95905
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 3.60274e40 1.59754
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.04082e41 1.99569 0.997847 − 0.0655783i \(-0.0208892\pi\)
0.997847 + 0.0655783i \(0.0208892\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.14829e40 0.340905
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −5.01769e41 −1.99500 −0.997498 − 0.0706973i \(-0.977478\pi\)
−0.997498 + 0.0706973i \(0.977478\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.07786e41 1.72143 0.860716 − 0.509085i \(-0.170016\pi\)
0.860716 + 0.509085i \(0.170016\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.93016e42 −1.79485 −0.897426 − 0.441164i \(-0.854566\pi\)
−0.897426 + 0.441164i \(0.854566\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.31034e42 −0.613109
\(170\) 0 0
\(171\) −5.25901e42 −1.96779
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −3.36664e42 −0.811847
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.40227e43 −1.78213 −0.891064 − 0.453878i \(-0.850040\pi\)
−0.891064 + 0.453878i \(0.850040\pi\)
\(182\) 0 0
\(183\) −3.54077e42 −0.365196
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.45294e43 0.811847
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 4.89985e43 1.83905 0.919525 − 0.393031i \(-0.128573\pi\)
0.919525 + 0.393031i \(0.128573\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.32417e42 −0.0697387 −0.0348694 − 0.999392i \(-0.511102\pi\)
−0.0348694 + 0.999392i \(0.511102\pi\)
\(200\) 0 0
\(201\) 2.83206e43 0.491334
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.85695e44 1.97032 0.985159 − 0.171642i \(-0.0549071\pi\)
0.985159 + 0.171642i \(0.0549071\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.98857e43 0.282913
\(218\) 0 0
\(219\) 1.12884e44 0.383878
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.90163e44 1.18181 0.590906 − 0.806740i \(-0.298770\pi\)
0.590906 + 0.806740i \(0.298770\pi\)
\(224\) 0 0
\(225\) 4.91437e44 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.21105e45 −1.76314 −0.881572 − 0.472049i \(-0.843515\pi\)
−0.881572 + 0.472049i \(0.843515\pi\)
\(230\) 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
\(237\) −1.83732e45 −1.39304
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.72868e45 −0.953652 −0.476826 − 0.878998i \(-0.658213\pi\)
−0.476826 + 0.878998i \(0.658213\pi\)
\(242\) 0 0
\(243\) −2.12090e45 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.54009e45 −1.22397
\(248\) 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
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 5.92768e45 0.832127
\(260\) 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
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.98661e46 −1.17950 −0.589748 − 0.807587i \(-0.700773\pi\)
−0.589748 + 0.807587i \(0.700773\pi\)
\(272\) 0 0
\(273\) 9.78044e45 0.504973
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.49695e46 1.76118 0.880588 − 0.473883i \(-0.157148\pi\)
0.880588 + 0.473883i \(0.157148\pi\)
\(278\) 0 0
\(279\) −1.02014e46 −0.348481
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 6.18276e46 1.61154 0.805769 − 0.592230i \(-0.201752\pi\)
0.805769 + 0.592230i \(0.201752\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.71556e46 1.00000
\(290\) 0 0
\(291\) 2.53633e46 0.389259
\(292\) 0 0
\(293\) 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
\(300\) 0 0
\(301\) 1.96917e47 1.59045
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.42681e47 −1.34712 −0.673560 − 0.739133i \(-0.735236\pi\)
−0.673560 + 0.739133i \(0.735236\pi\)
\(308\) 0 0
\(309\) −3.96503e47 −1.94552
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −4.86224e47 −1.86850 −0.934252 − 0.356614i \(-0.883931\pi\)
−0.934252 + 0.356614i \(0.883931\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(324\) 0 0
\(325\) 3.30810e47 0.622006
\(326\) 0 0
\(327\) −1.04673e48 −1.75157
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.50578e48 1.99999 0.999996 − 0.00273159i \(-0.000869494\pi\)
0.999996 + 0.00273159i \(0.000869494\pi\)
\(332\) 0 0
\(333\) −8.65279e47 −1.02498
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.14041e48 −1.07670 −0.538348 − 0.842722i \(-0.680952\pi\)
−0.538348 + 0.842722i \(0.680952\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.61235e48 1.08861
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 1.81969e48 0.883721 0.441861 − 0.897084i \(-0.354319\pi\)
0.441861 + 0.897084i \(0.354319\pi\)
\(350\) 0 0
\(351\) −1.42768e48 −0.622006
\(352\) 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
\(361\) 1.12421e49 2.87218
\(362\) 0 0
\(363\) −4.34736e48 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.20390e48 1.34559 0.672794 − 0.739830i \(-0.265094\pi\)
0.672794 + 0.739830i \(0.265094\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.22696e49 1.68408 0.842038 − 0.539419i \(-0.181356\pi\)
0.842038 + 0.539419i \(0.181356\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.92190e47 0.0194799 0.00973995 − 0.999953i \(-0.496900\pi\)
0.00973995 + 0.999953i \(0.496900\pi\)
\(380\) 0 0
\(381\) −1.75003e49 −1.60499
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.87444e49 −1.95905
\(388\) 0 0
\(389\) 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
\(397\) −1.98669e49 −0.833894 −0.416947 − 0.908931i \(-0.636900\pi\)
−0.416947 + 0.908931i \(0.636900\pi\)
\(398\) 0 0
\(399\) −4.18733e49 −1.59754
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.86704e48 −0.216757
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.30406e49 1.50272 0.751359 − 0.659894i \(-0.229399\pi\)
0.751359 + 0.659894i \(0.229399\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.20970e50 −1.99569
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.28237e50 1.76463 0.882317 − 0.470656i \(-0.155983\pi\)
0.882317 + 0.470656i \(0.155983\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.81923e49 −0.296483
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 6.27919e49 0.506563 0.253281 − 0.967393i \(-0.418490\pi\)
0.253281 + 0.967393i \(0.418490\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.87481e50 1.78560 0.892798 − 0.450458i \(-0.148739\pi\)
0.892798 + 0.450458i \(0.148739\pi\)
\(440\) 0 0
\(441\) −5.98366e49 −0.340905
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 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
\(453\) 5.83187e50 1.99500
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.20988e50 1.79753 0.898763 − 0.438435i \(-0.144467\pi\)
0.898763 + 0.438435i \(0.144467\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −7.84142e50 −1.77156 −0.885780 − 0.464106i \(-0.846376\pi\)
−0.885780 + 0.464106i \(0.846376\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 2.25494e50 0.398888
\(470\) 0 0
\(471\) −1.05508e51 −1.72143
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.41630e51 −1.96779
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −5.82460e50 −0.637544
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.28440e51 −1.97570 −0.987852 − 0.155395i \(-0.950335\pi\)
−0.987852 + 0.155395i \(0.950335\pi\)
\(488\) 0 0
\(489\) 2.24335e51 1.79485
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.74462e51 −0.950144 −0.475072 − 0.879947i \(-0.657578\pi\)
−0.475072 + 0.879947i \(0.657578\pi\)
\(500\) 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
\(507\) 1.52296e51 0.613109
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 8.98803e50 0.311650
\(512\) 0 0
\(513\) 6.11234e51 1.96779
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −8.26563e51 −1.84393 −0.921964 − 0.387276i \(-0.873416\pi\)
−0.921964 + 0.387276i \(0.873416\pi\)
\(524\) 0 0
\(525\) 3.91292e51 0.811847
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 5.56747e51 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
\(541\) 1.40488e52 1.64775 0.823874 − 0.566773i \(-0.191808\pi\)
0.823874 + 0.566773i \(0.191808\pi\)
\(542\) 0 0
\(543\) 1.62980e52 1.78213
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.98959e51 −0.569689 −0.284845 − 0.958574i \(-0.591942\pi\)
−0.284845 + 0.958574i \(0.591942\pi\)
\(548\) 0 0
\(549\) 4.11529e51 0.365196
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.46291e52 −1.13094
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.93492e52 −1.21854
\(560\) 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
\(567\) −1.68870e52 −0.811847
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −4.36784e52 −1.83730 −0.918652 − 0.395068i \(-0.870721\pi\)
−0.918652 + 0.395068i \(0.870721\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.08738e51 0.244425 0.122212 − 0.992504i \(-0.461001\pi\)
0.122212 + 0.992504i \(0.461001\pi\)
\(578\) 0 0
\(579\) −5.69490e52 −1.83905
\(580\) 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
\(589\) 2.94000e52 0.685737
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.86355e51 0.0697387
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 9.18903e52 1.46102 0.730508 − 0.682904i \(-0.239283\pi\)
0.730508 + 0.682904i \(0.239283\pi\)
\(602\) 0 0
\(603\) −3.29159e52 −0.491334
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.49623e53 1.96976 0.984879 − 0.173246i \(-0.0554257\pi\)
0.984879 + 0.173246i \(0.0554257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.31618e52 −0.252943 −0.126471 − 0.991970i \(-0.540365\pi\)
−0.126471 + 0.991970i \(0.540365\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −3.49603e52 −0.317288 −0.158644 − 0.987336i \(-0.550712\pi\)
−0.158644 + 0.987336i \(0.550712\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.32349e53 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.04063e53 1.91605 0.958024 − 0.286690i \(-0.0925548\pi\)
0.958024 + 0.286690i \(0.0925548\pi\)
\(632\) 0 0
\(633\) −3.32053e53 −1.97032
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.02789e52 −0.212045
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 4.53831e53 1.99933 0.999667 − 0.0258158i \(-0.00821835\pi\)
0.999667 + 0.0258158i \(0.00821835\pi\)
\(644\) 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
\(651\) −8.12255e52 −0.282913
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.31201e53 −0.383878
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −7.54005e53 −1.96583 −0.982917 − 0.184051i \(-0.941079\pi\)
−0.982917 + 0.184051i \(0.941079\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.69698e53 −1.18181
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.13999e53 −1.32255 −0.661274 − 0.750144i \(-0.729984\pi\)
−0.661274 + 0.750144i \(0.729984\pi\)
\(674\) 0 0
\(675\) −5.71178e53 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 2.01947e53 0.316019
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.40756e54 1.76314
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.58278e54 −1.77557 −0.887787 − 0.460255i \(-0.847758\pi\)
−0.887787 + 0.460255i \(0.847758\pi\)
\(692\) 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
\(703\) 2.49370e54 2.01694
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.58421e54 1.77848 0.889240 − 0.457441i \(-0.151234\pi\)
0.889240 + 0.457441i \(0.151234\pi\)
\(710\) 0 0
\(711\) 2.13544e54 1.39304
\(712\) 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
\(721\) −3.15703e54 −1.57946
\(722\) 0 0
\(723\) 2.00917e54 0.953652
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.37992e53 −0.272685 −0.136342 − 0.990662i \(-0.543535\pi\)
−0.136342 + 0.990662i \(0.543535\pi\)
\(728\) 0 0
\(729\) 2.46503e54 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.04893e54 −1.48035 −0.740176 − 0.672413i \(-0.765258\pi\)
−0.740176 + 0.672413i \(0.765258\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.22170e54 −0.695731 −0.347866 − 0.937544i \(-0.613093\pi\)
−0.347866 + 0.937544i \(0.613093\pi\)
\(740\) 0 0
\(741\) 4.11451e54 1.22397
\(742\) 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
\(751\) 8.15263e54 1.87992 0.939959 − 0.341287i \(-0.110863\pi\)
0.939959 + 0.341287i \(0.110863\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −9.97715e54 −1.97781 −0.988906 − 0.148545i \(-0.952541\pi\)
−0.988906 + 0.148545i \(0.952541\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −8.33428e54 −1.42201
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.70493e54 −0.397700 −0.198850 − 0.980030i \(-0.563721\pi\)
−0.198850 + 0.980030i \(0.563721\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −2.74734e54 −0.348481
\(776\) 0 0
\(777\) −6.88952e54 −0.832127
\(778\) 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
\(787\) −1.48010e55 −1.40208 −0.701038 − 0.713123i \(-0.747280\pi\)
−0.701038 + 0.713123i \(0.747280\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.77020e54 0.227154
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 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
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 2.88289e55 1.54324 0.771618 − 0.636086i \(-0.219448\pi\)
0.771618 + 0.636086i \(0.219448\pi\)
\(812\) 0 0
\(813\) 2.30896e55 1.17950
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.28404e55 3.85499
\(818\) 0 0
\(819\) −1.13674e55 −0.504973
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −3.17302e55 −1.28492 −0.642460 − 0.766320i \(-0.722086\pi\)
−0.642460 + 0.766320i \(0.722086\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 4.66258e55 1.64472 0.822358 − 0.568971i \(-0.192658\pi\)
0.822358 + 0.568971i \(0.192658\pi\)
\(830\) 0 0
\(831\) −5.22663e55 −1.76118
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.18567e55 0.348481
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.72498e55 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.46146e55 −0.811847
\(848\) 0 0
\(849\) −7.18599e55 −1.61154
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.88591e55 −0.386803 −0.193401 − 0.981120i \(-0.561952\pi\)
−0.193401 + 0.981120i \(0.561952\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 4.36891e55 0.784337 0.392168 − 0.919893i \(-0.371725\pi\)
0.392168 + 0.919893i \(0.371725\pi\)
\(860\) 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
\(867\) −6.64298e55 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −2.21573e55 −0.305613
\(872\) 0 0
\(873\) −2.94788e55 −0.389259
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.30846e55 0.642652 0.321326 − 0.946969i \(-0.395871\pi\)
0.321326 + 0.946969i \(0.395871\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −7.64052e55 −0.812586 −0.406293 − 0.913743i \(-0.633179\pi\)
−0.406293 + 0.913743i \(0.633179\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.39341e56 −1.30300
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −2.28869e56 −1.59045
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.44200e56 1.56030 0.780150 − 0.625593i \(-0.215143\pi\)
0.780150 + 0.625593i \(0.215143\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.13863e55 −0.205998 −0.102999 − 0.994681i \(-0.532844\pi\)
−0.102999 + 0.994681i \(0.532844\pi\)
\(920\) 0 0
\(921\) 2.82058e56 1.34712
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.33028e56 −1.02498
\(926\) 0 0
\(927\) 4.60840e56 1.94552
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.72447e56 0.670828
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.33315e56 1.14763 0.573816 − 0.818984i \(-0.305462\pi\)
0.573816 + 0.818984i \(0.305462\pi\)
\(938\) 0 0
\(939\) 5.65119e56 1.86850
\(940\) 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 \(\pi\)
\(948\) 0 0
\(949\) −8.83174e55 −0.238775
\(950\) 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
\(961\) −4.12588e56 −0.878561
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.04885e57 −1.98431 −0.992157 − 0.125000i \(-0.960107\pi\)
−0.992157 + 0.125000i \(0.960107\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −9.63190e56 −1.62020
\(974\) 0 0
\(975\) −3.84487e56 −0.622006
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.21657e57 1.75157
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 7.18963e56 0.853703 0.426852 − 0.904322i \(-0.359623\pi\)
0.426852 + 0.904322i \(0.359623\pi\)
\(992\) 0 0
\(993\) −1.75010e57 −1.99999
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.83471e57 −1.94249 −0.971245 − 0.238081i \(-0.923482\pi\)
−0.971245 + 0.238081i \(0.923482\pi\)
\(998\) 0 0
\(999\) 1.00568e57 1.02498
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 12.39.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.39.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.39.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.39.c.a.5.1 ✓ 1 3.2 odd 2 CM