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Properties

Label 12.51.c.a.5.1
Level $12$
Weight $51$
Character 12.5
Self dual yes
Analytic conductor $190.003$
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] = [12,51,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, 51); 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, 51, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 51 \)
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: \(190.002544227\)
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)
 
\(f(q)\) \(=\) \(q-8.47289e11 q^{3} -1.14751e21 q^{7} +7.17898e23 q^{9} -1.20408e27 q^{13} -2.14825e30 q^{19} +9.72275e32 q^{21} +8.88178e34 q^{25} -6.08267e35 q^{27} -3.41461e37 q^{31} +1.39319e39 q^{37} +1.02020e39 q^{39} -2.51323e40 q^{43} -4.81678e41 q^{49} +1.82019e42 q^{57} -4.74733e44 q^{61} -8.23797e44 q^{63} -2.94809e45 q^{67} -7.56087e46 q^{73} -7.52543e46 q^{75} -1.90463e47 q^{79} +5.15378e47 q^{81} +1.38170e48 q^{91} +2.89316e49 q^{93} +2.38057e49 q^{97} +O(q^{100})\)

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
\(3\) −8.47289e11 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.14751e21 −0.855671 −0.427835 − 0.903857i \(-0.640724\pi\)
−0.427835 + 0.903857i \(0.640724\pi\)
\(8\) 0 0
\(9\) 7.17898e23 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.20408e27 −0.170636 −0.0853180 − 0.996354i \(-0.527191\pi\)
−0.0853180 + 0.996354i \(0.527191\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.14825e30 −0.0230805 −0.0115403 − 0.999933i \(-0.503673\pi\)
−0.0115403 + 0.999933i \(0.503673\pi\)
\(20\) 0 0
\(21\) 9.72275e32 0.855671
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 8.88178e34 1.00000
\(26\) 0 0
\(27\) −6.08267e35 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −3.41461e37 −1.77541 −0.887705 − 0.460413i \(-0.847701\pi\)
−0.887705 + 0.460413i \(0.847701\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.39319e39 0.868897 0.434448 − 0.900697i \(-0.356943\pi\)
0.434448 + 0.900697i \(0.356943\pi\)
\(38\) 0 0
\(39\) 1.02020e39 0.170636
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.51323e40 −0.366034 −0.183017 − 0.983110i \(-0.558586\pi\)
−0.183017 + 0.983110i \(0.558586\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −4.81678e41 −0.267827
\(50\) 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
\(57\) 1.82019e42 0.0230805
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −4.74733e44 −1.10460 −0.552298 − 0.833647i \(-0.686249\pi\)
−0.552298 + 0.833647i \(0.686249\pi\)
\(62\) 0 0
\(63\) −8.23797e44 −0.855671
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.94809e45 −0.657158 −0.328579 − 0.944476i \(-0.606570\pi\)
−0.328579 + 0.944476i \(0.606570\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −7.56087e46 −1.97470 −0.987348 − 0.158567i \(-0.949313\pi\)
−0.987348 + 0.158567i \(0.949313\pi\)
\(74\) 0 0
\(75\) −7.52543e46 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.90463e47 −0.690452 −0.345226 − 0.938520i \(-0.612198\pi\)
−0.345226 + 0.938520i \(0.612198\pi\)
\(80\) 0 0
\(81\) 5.15378e47 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
\(91\) 1.38170e48 0.146008
\(92\) 0 0
\(93\) 2.89316e49 1.77541
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.38057e49 0.509785 0.254892 − 0.966969i \(-0.417960\pi\)
0.254892 + 0.966969i \(0.417960\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.76497e50 −1.32057 −0.660283 − 0.751017i \(-0.729564\pi\)
−0.660283 + 0.751017i \(0.729564\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.44862e50 −0.167993 −0.0839966 − 0.996466i \(-0.526768\pi\)
−0.0839966 + 0.996466i \(0.526768\pi\)
\(110\) 0 0
\(111\) −1.18044e51 −0.868897
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.64405e50 −0.170636
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.17391e52 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.05760e52 0.776762 0.388381 − 0.921499i \(-0.373034\pi\)
0.388381 + 0.921499i \(0.373034\pi\)
\(128\) 0 0
\(129\) 2.12943e52 0.366034
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 2.46515e51 0.0197493
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −6.90057e53 −1.83449 −0.917244 − 0.398327i \(-0.869591\pi\)
−0.917244 + 0.398327i \(0.869591\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.08121e53 0.267827
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 1.87722e54 0.629640 0.314820 − 0.949151i \(-0.398056\pi\)
0.314820 + 0.949151i \(0.398056\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.24035e54 0.410299 0.205150 − 0.978731i \(-0.434232\pi\)
0.205150 + 0.978731i \(0.434232\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.93167e55 1.94934 0.974670 − 0.223649i \(-0.0717969\pi\)
0.974670 + 0.223649i \(0.0717969\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −4.83431e55 −0.970883
\(170\) 0 0
\(171\) −1.54223e54 −0.0230805
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.01920e56 −0.855671
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 4.97159e56 1.79693 0.898464 − 0.439047i \(-0.144684\pi\)
0.898464 + 0.439047i \(0.144684\pi\)
\(182\) 0 0
\(183\) 4.02236e56 1.10460
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 6.97994e56 0.855671
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 2.46699e57 1.79157 0.895785 − 0.444487i \(-0.146614\pi\)
0.895785 + 0.444487i \(0.146614\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 5.87162e57 1.98349 0.991747 − 0.128210i \(-0.0409230\pi\)
0.991747 + 0.128210i \(0.0409230\pi\)
\(200\) 0 0
\(201\) 2.49788e57 0.657158
\(202\) 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
\(211\) 1.61401e58 1.26138 0.630689 − 0.776036i \(-0.282773\pi\)
0.630689 + 0.776036i \(0.282773\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.91831e58 1.51917
\(218\) 0 0
\(219\) 6.40624e58 1.97470
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.80957e58 0.942942 0.471471 − 0.881882i \(-0.343723\pi\)
0.471471 + 0.881882i \(0.343723\pi\)
\(224\) 0 0
\(225\) 6.37622e58 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.95866e59 1.97730 0.988648 − 0.150248i \(-0.0480072\pi\)
0.988648 + 0.150248i \(0.0480072\pi\)
\(230\) 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
\(237\) 1.61377e59 0.690452
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.22088e59 0.343754 0.171877 − 0.985118i \(-0.445017\pi\)
0.171877 + 0.985118i \(0.445017\pi\)
\(242\) 0 0
\(243\) −4.36674e59 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.58666e57 0.00393837
\(248\) 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
\(259\) −1.59871e60 −0.743490
\(260\) 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
\(271\) −1.11938e61 −1.67782 −0.838910 − 0.544271i \(-0.816806\pi\)
−0.838910 + 0.544271i \(0.816806\pi\)
\(272\) 0 0
\(273\) −1.17069e60 −0.146008
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.54229e61 −1.33712 −0.668561 − 0.743657i \(-0.733089\pi\)
−0.668561 + 0.743657i \(0.733089\pi\)
\(278\) 0 0
\(279\) −2.45134e61 −1.77541
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.94166e61 −1.99995 −0.999974 − 0.00724403i \(-0.997694\pi\)
−0.999974 + 0.00724403i \(0.997694\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.33001e61 1.00000
\(290\) 0 0
\(291\) −2.01703e61 −0.509785
\(292\) 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
\(301\) 2.88396e61 0.313204
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.37795e61 −0.290276 −0.145138 − 0.989411i \(-0.546363\pi\)
−0.145138 + 0.989411i \(0.546363\pi\)
\(308\) 0 0
\(309\) 2.34273e62 1.32057
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.60968e62 1.06654 0.533272 − 0.845944i \(-0.320962\pi\)
0.533272 + 0.845944i \(0.320962\pi\)
\(314\) 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
\(325\) −1.06944e62 −0.170636
\(326\) 0 0
\(327\) 1.22740e62 0.167993
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.87530e63 1.89396 0.946978 − 0.321298i \(-0.104119\pi\)
0.946978 + 0.321298i \(0.104119\pi\)
\(332\) 0 0
\(333\) 1.00017e63 0.868897
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.72315e63 1.75519 0.877595 − 0.479403i \(-0.159147\pi\)
0.877595 + 0.479403i \(0.159147\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.61649e63 1.08484
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −6.97970e63 −1.87586 −0.937932 − 0.346821i \(-0.887261\pi\)
−0.937932 + 0.346821i \(0.887261\pi\)
\(350\) 0 0
\(351\) 7.32401e62 0.170636
\(352\) 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
\(361\) −8.65862e63 −0.999467
\(362\) 0 0
\(363\) −9.94639e63 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.29337e64 −0.988718 −0.494359 − 0.869258i \(-0.664597\pi\)
−0.494359 + 0.869258i \(0.664597\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.60245e64 −1.83602 −0.918011 − 0.396556i \(-0.870205\pi\)
−0.918011 + 0.396556i \(0.870205\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.53250e64 1.20810 0.604049 − 0.796947i \(-0.293553\pi\)
0.604049 + 0.796947i \(0.293553\pi\)
\(380\) 0 0
\(381\) −2.59067e64 −0.776762
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.80424e64 −0.366034
\(388\) 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
\(397\) −1.24577e65 −1.33560 −0.667798 − 0.744342i \(-0.732763\pi\)
−0.667798 + 0.744342i \(0.732763\pi\)
\(398\) 0 0
\(399\) −2.08869e63 −0.0197493
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 4.11145e64 0.302949
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.96419e64 0.405564 0.202782 − 0.979224i \(-0.435002\pi\)
0.202782 + 0.979224i \(0.435002\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.84677e65 1.83449
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 8.09138e65 1.99972 0.999862 − 0.0166060i \(-0.00528610\pi\)
0.999862 + 0.0166060i \(0.00528610\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.44763e65 0.945171
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 1.20899e66 1.47987 0.739937 − 0.672677i \(-0.234855\pi\)
0.739937 + 0.672677i \(0.234855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.02306e66 −0.887745 −0.443872 − 0.896090i \(-0.646396\pi\)
−0.443872 + 0.896090i \(0.646396\pi\)
\(440\) 0 0
\(441\) −3.45796e65 −0.267827
\(442\) 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
\(453\) −1.59055e66 −0.629640
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.55124e66 −1.44619 −0.723093 − 0.690751i \(-0.757280\pi\)
−0.723093 + 0.690751i \(0.757280\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −4.96054e66 −1.13764 −0.568819 − 0.822463i \(-0.692600\pi\)
−0.568819 + 0.822463i \(0.692600\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 3.38297e66 0.562311
\(470\) 0 0
\(471\) −2.74551e66 −0.410299
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.90803e65 −0.0230805
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.67751e66 −0.148265
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.17839e67 −1.41225 −0.706127 − 0.708086i \(-0.749559\pi\)
−0.706127 + 0.708086i \(0.749559\pi\)
\(488\) 0 0
\(489\) −3.33126e67 −1.94934
\(490\) 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
\(499\) 3.43730e67 1.21256 0.606281 − 0.795250i \(-0.292660\pi\)
0.606281 + 0.795250i \(0.292660\pi\)
\(500\) 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
\(507\) 4.09606e67 0.970883
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 8.67620e67 1.68969
\(512\) 0 0
\(513\) 1.30671e66 0.0230805
\(514\) 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
\(523\) 1.47799e68 1.61112 0.805562 − 0.592511i \(-0.201863\pi\)
0.805562 + 0.592511i \(0.201863\pi\)
\(524\) 0 0
\(525\) 8.63554e67 0.855671
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.22009e68 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
\(541\) 1.58382e68 0.740923 0.370462 − 0.928848i \(-0.379200\pi\)
0.370462 + 0.928848i \(0.379200\pi\)
\(542\) 0 0
\(543\) −4.21237e68 −1.79693
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.92202e68 −1.03753 −0.518765 − 0.854917i \(-0.673608\pi\)
−0.518765 + 0.854917i \(0.673608\pi\)
\(548\) 0 0
\(549\) −3.40810e68 −1.10460
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.18559e68 0.590800
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 3.02612e67 0.0624585
\(560\) 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
\(567\) −5.91403e68 −0.855671
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 3.42981e68 0.416262 0.208131 − 0.978101i \(-0.433262\pi\)
0.208131 + 0.978101i \(0.433262\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.01076e69 −0.944608 −0.472304 − 0.881436i \(-0.656578\pi\)
−0.472304 + 0.881436i \(0.656578\pi\)
\(578\) 0 0
\(579\) −2.09025e69 −1.79157
\(580\) 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
\(589\) 7.33544e67 0.0409773
\(590\) 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
\(597\) −4.97496e69 −1.98349
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 5.61227e69 1.89355 0.946775 − 0.321896i \(-0.104320\pi\)
0.946775 + 0.321896i \(0.104320\pi\)
\(602\) 0 0
\(603\) −2.11642e69 −0.657158
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.48964e69 0.918467 0.459233 − 0.888316i \(-0.348124\pi\)
0.459233 + 0.888316i \(0.348124\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.15912e69 1.47349 0.736746 − 0.676170i \(-0.236362\pi\)
0.736746 + 0.676170i \(0.236362\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.15616e70 1.86531 0.932657 − 0.360764i \(-0.117484\pi\)
0.932657 + 0.360764i \(0.117484\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.88861e69 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.00069e70 1.99731 0.998654 − 0.0518742i \(-0.0165195\pi\)
0.998654 + 0.0518742i \(0.0165195\pi\)
\(632\) 0 0
\(633\) −1.36753e70 −1.26138
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.79978e68 0.0457010
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −3.73411e69 −0.232762 −0.116381 − 0.993205i \(-0.537129\pi\)
−0.116381 + 0.993205i \(0.537129\pi\)
\(644\) 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
\(651\) −3.31994e70 −1.51917
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.42793e70 −1.97470
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 4.72668e70 1.47747 0.738737 − 0.673994i \(-0.235422\pi\)
0.738737 + 0.673994i \(0.235422\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −4.07510e70 −0.942942
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.82908e70 1.16205 0.581023 − 0.813887i \(-0.302653\pi\)
0.581023 + 0.813887i \(0.302653\pi\)
\(674\) 0 0
\(675\) −5.40249e70 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2.73173e70 −0.436208
\(680\) 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
\(687\) −1.65955e71 −1.97730
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.15868e70 −0.428553 −0.214277 − 0.976773i \(-0.568739\pi\)
−0.214277 + 0.976773i \(0.568739\pi\)
\(692\) 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
\(703\) −2.99293e69 −0.0200546
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.82895e71 1.53274 0.766370 − 0.642399i \(-0.222061\pi\)
0.766370 + 0.642399i \(0.222061\pi\)
\(710\) 0 0
\(711\) −1.36733e71 −0.690452
\(712\) 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
\(721\) 3.17284e71 1.12997
\(722\) 0 0
\(723\) −1.03444e71 −0.343754
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 5.72782e71 1.65817 0.829086 − 0.559122i \(-0.188862\pi\)
0.829086 + 0.559122i \(0.188862\pi\)
\(728\) 0 0
\(729\) 3.69988e71 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5.60942e71 1.32226 0.661132 − 0.750270i \(-0.270076\pi\)
0.661132 + 0.750270i \(0.270076\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.44919e71 1.43218 0.716091 − 0.698007i \(-0.245930\pi\)
0.716091 + 0.698007i \(0.245930\pi\)
\(740\) 0 0
\(741\) −2.19165e69 −0.00393837
\(742\) 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
\(751\) −1.45163e72 −1.86576 −0.932881 − 0.360184i \(-0.882714\pi\)
−0.932881 + 0.360184i \(0.882714\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.64109e72 −1.72877 −0.864383 − 0.502834i \(-0.832291\pi\)
−0.864383 + 0.502834i \(0.832291\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.66231e71 0.143747
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.48559e71 0.461096 0.230548 − 0.973061i \(-0.425948\pi\)
0.230548 + 0.973061i \(0.425948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −3.03278e72 −1.77541
\(776\) 0 0
\(777\) 1.35457e72 0.743490
\(778\) 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
\(787\) −5.00954e72 −1.99724 −0.998618 − 0.0525566i \(-0.983263\pi\)
−0.998618 + 0.0525566i \(0.983263\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5.71616e71 0.188484
\(794\) 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
\(811\) −1.02992e73 −1.93767 −0.968837 − 0.247698i \(-0.920326\pi\)
−0.968837 + 0.247698i \(0.920326\pi\)
\(812\) 0 0
\(813\) 9.48438e72 1.67782
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.39905e70 0.00844824
\(818\) 0 0
\(819\) 9.91916e71 0.146008
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −1.51154e73 −1.96981 −0.984905 − 0.173094i \(-0.944624\pi\)
−0.984905 + 0.173094i \(0.944624\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.20476e73 1.30929 0.654645 − 0.755936i \(-0.272818\pi\)
0.654645 + 0.755936i \(0.272818\pi\)
\(830\) 0 0
\(831\) 1.30676e73 1.33712
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.07699e73 1.77541
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.31795e73 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.34708e73 −0.855671
\(848\) 0 0
\(849\) 3.33973e73 1.99995
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.54278e73 1.88633 0.943167 − 0.332320i \(-0.107832\pi\)
0.943167 + 0.332320i \(0.107832\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −3.39461e73 −1.51691 −0.758456 − 0.651724i \(-0.774046\pi\)
−0.758456 + 0.651724i \(0.774046\pi\)
\(860\) 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
\(867\) −2.82148e73 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 3.54972e72 0.112135
\(872\) 0 0
\(873\) 1.70900e73 0.509785
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.99572e73 −1.32925 −0.664627 − 0.747175i \(-0.731410\pi\)
−0.664627 + 0.747175i \(0.731410\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −7.95566e73 −1.78508 −0.892542 − 0.450963i \(-0.851081\pi\)
−0.892542 + 0.450963i \(0.851081\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −3.50864e73 −0.664653
\(890\) 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
\(903\) −2.44355e73 −0.313204
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.21284e73 −0.368729 −0.184364 − 0.982858i \(-0.559023\pi\)
−0.184364 + 0.982858i \(0.559023\pi\)
\(908\) 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
\(919\) −7.17900e73 −0.593166 −0.296583 − 0.955007i \(-0.595847\pi\)
−0.296583 + 0.955007i \(0.595847\pi\)
\(920\) 0 0
\(921\) 3.70939e73 0.290276
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.23740e74 0.868897
\(926\) 0 0
\(927\) −1.98497e74 −1.32057
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.03477e72 0.00618159
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.92406e74 1.99639 0.998195 − 0.0600580i \(-0.0191286\pi\)
0.998195 + 0.0600580i \(0.0191286\pi\)
\(938\) 0 0
\(939\) −2.21115e74 −1.06654
\(940\) 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
\(949\) 9.10388e73 0.336954
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.96054e74 2.15208
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.81862e74 1.57774 0.788869 − 0.614562i \(-0.210667\pi\)
0.788869 + 0.614562i \(0.210667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 7.91849e74 1.56972
\(974\) 0 0
\(975\) 9.06121e73 0.170636
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.03996e74 −0.167993
\(982\) 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
\(991\) −8.74687e74 −1.09651 −0.548253 − 0.836312i \(-0.684707\pi\)
−0.548253 + 0.836312i \(0.684707\pi\)
\(992\) 0 0
\(993\) −1.58892e75 −1.89396
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.79802e74 0.193827 0.0969137 − 0.995293i \(-0.469103\pi\)
0.0969137 + 0.995293i \(0.469103\pi\)
\(998\) 0 0
\(999\) −8.47433e74 −0.868897
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 12.51.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.51.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.51.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.51.c.a.5.1 ✓ 1 3.2 odd 2 CM