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Properties

Label 12.41.c.a.5.1
Level $12$
Weight $41$
Character 12.5
Self dual yes
Analytic conductor $121.611$
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] = [12,41,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, 41); 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, 41, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 41 \)
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: \(121.610742236\)
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)
 
\(f(q)\) \(=\) \(q+3.48678e9 q^{3} -1.53604e17 q^{7} +1.21577e19 q^{9} +9.24985e21 q^{13} -6.04113e25 q^{19} -5.35583e26 q^{21} +9.09495e27 q^{25} +4.23912e28 q^{27} -7.13361e29 q^{31} +2.88514e31 q^{37} +3.22522e31 q^{39} +4.15871e32 q^{43} +1.72273e34 q^{49} -2.10641e35 q^{57} +9.08240e35 q^{61} -1.86746e36 q^{63} -4.15087e36 q^{67} +3.66398e37 q^{73} +3.17121e37 q^{75} -1.68650e38 q^{79} +1.47809e38 q^{81} -1.42081e39 q^{91} -2.48733e39 q^{93} +1.06455e40 q^{97} +O(q^{100})\)

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
\(3\) 3.48678e9 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.53604e17 −1.92505 −0.962523 − 0.271202i \(-0.912579\pi\)
−0.962523 + 0.271202i \(0.912579\pi\)
\(8\) 0 0
\(9\) 1.21577e19 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 9.24985e21 0.486707 0.243353 − 0.969938i \(-0.421752\pi\)
0.243353 + 0.969938i \(0.421752\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −6.04113e25 −1.60711 −0.803556 − 0.595229i \(-0.797061\pi\)
−0.803556 + 0.595229i \(0.797061\pi\)
\(20\) 0 0
\(21\) −5.35583e26 −1.92505
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.09495e27 1.00000
\(26\) 0 0
\(27\) 4.23912e28 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.13361e29 −1.06188 −0.530940 − 0.847410i \(-0.678161\pi\)
−0.530940 + 0.847410i \(0.678161\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.88514e31 1.24776 0.623882 − 0.781519i \(-0.285555\pi\)
0.623882 + 0.781519i \(0.285555\pi\)
\(38\) 0 0
\(39\) 3.22522e31 0.486707
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 4.15871e32 0.890409 0.445204 − 0.895429i \(-0.353131\pi\)
0.445204 + 0.895429i \(0.353131\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.72273e34 2.70580
\(50\) 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
\(57\) −2.10641e35 −1.60711
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 9.08240e35 1.78486 0.892429 − 0.451187i \(-0.148999\pi\)
0.892429 + 0.451187i \(0.148999\pi\)
\(62\) 0 0
\(63\) −1.86746e36 −1.92505
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.15087e36 −1.24923 −0.624615 − 0.780933i \(-0.714744\pi\)
−0.624615 + 0.780933i \(0.714744\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 3.66398e37 1.98379 0.991897 − 0.127047i \(-0.0405499\pi\)
0.991897 + 0.127047i \(0.0405499\pi\)
\(74\) 0 0
\(75\) 3.17121e37 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.68650e38 −1.88125 −0.940623 − 0.339453i \(-0.889758\pi\)
−0.940623 + 0.339453i \(0.889758\pi\)
\(80\) 0 0
\(81\) 1.47809e38 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
\(91\) −1.42081e39 −0.936933
\(92\) 0 0
\(93\) −2.48733e39 −1.06188
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.06455e40 1.95764 0.978818 − 0.204731i \(-0.0656321\pi\)
0.978818 + 0.204731i \(0.0656321\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 3.02750e40 1.67625 0.838126 − 0.545476i \(-0.183651\pi\)
0.838126 + 0.545476i \(0.183651\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −8.60470e40 −1.53534 −0.767672 − 0.640842i \(-0.778585\pi\)
−0.767672 + 0.640842i \(0.778585\pi\)
\(110\) 0 0
\(111\) 1.00599e41 1.24776
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.12457e41 0.486707
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.52593e41 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.26260e42 1.89904 0.949518 − 0.313711i \(-0.101572\pi\)
0.949518 + 0.313711i \(0.101572\pi\)
\(128\) 0 0
\(129\) 1.45005e42 0.890409
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 9.27940e42 3.09376
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.97090e41 −0.0271875 −0.0135937 − 0.999908i \(-0.504327\pi\)
−0.0135937 + 0.999908i \(0.504327\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00679e43 2.70580
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 7.34777e43 1.93471 0.967356 − 0.253420i \(-0.0815553\pi\)
0.967356 + 0.253420i \(0.0815553\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.45592e43 0.296634 0.148317 − 0.988940i \(-0.452614\pi\)
0.148317 + 0.988940i \(0.452614\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.44884e44 1.96753 0.983764 − 0.179468i \(-0.0574376\pi\)
0.983764 + 0.179468i \(0.0574376\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −2.75629e44 −0.763116
\(170\) 0 0
\(171\) −7.34460e44 −1.60711
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.39702e45 −1.92505
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.55825e45 −1.09412 −0.547060 − 0.837094i \(-0.684253\pi\)
−0.547060 + 0.837094i \(0.684253\pi\)
\(182\) 0 0
\(183\) 3.16684e45 1.78486
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.51144e45 −1.92505
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −9.93911e45 −1.93286 −0.966432 − 0.256923i \(-0.917291\pi\)
−0.966432 + 0.256923i \(0.917291\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 3.97898e45 0.419479 0.209739 − 0.977757i \(-0.432738\pi\)
0.209739 + 0.977757i \(0.432738\pi\)
\(200\) 0 0
\(201\) −1.44732e46 −1.24923
\(202\) 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
\(211\) −1.40567e46 −0.459446 −0.229723 − 0.973256i \(-0.573782\pi\)
−0.229723 + 0.973256i \(0.573782\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.09575e47 2.04417
\(218\) 0 0
\(219\) 1.27755e47 1.98379
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.45259e46 −0.157053 −0.0785263 − 0.996912i \(-0.525021\pi\)
−0.0785263 + 0.996912i \(0.525021\pi\)
\(224\) 0 0
\(225\) 1.10573e47 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.32517e47 0.842484 0.421242 − 0.906948i \(-0.361594\pi\)
0.421242 + 0.906948i \(0.361594\pi\)
\(230\) 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
\(237\) −5.88048e47 −1.88125
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 8.65388e47 1.98093 0.990467 − 0.137749i \(-0.0439866\pi\)
0.990467 + 0.137749i \(0.0439866\pi\)
\(242\) 0 0
\(243\) 5.15378e47 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.58795e47 −0.782193
\(248\) 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
\(259\) −4.43168e48 −2.40200
\(260\) 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
\(271\) −9.00047e48 −1.97187 −0.985937 − 0.167115i \(-0.946555\pi\)
−0.985937 + 0.167115i \(0.946555\pi\)
\(272\) 0 0
\(273\) −4.95406e48 −0.936933
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.79572e48 −0.536659 −0.268330 − 0.963327i \(-0.586472\pi\)
−0.268330 + 0.963327i \(0.586472\pi\)
\(278\) 0 0
\(279\) −8.67280e48 −1.06188
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.17146e49 1.99997 0.999983 − 0.00579524i \(-0.00184469\pi\)
0.999983 + 0.00579524i \(0.00184469\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.65180e49 1.00000
\(290\) 0 0
\(291\) 3.71186e49 1.95764
\(292\) 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
\(301\) −6.38793e49 −1.71408
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.09861e50 1.98644 0.993219 − 0.116259i \(-0.0370901\pi\)
0.993219 + 0.116259i \(0.0370901\pi\)
\(308\) 0 0
\(309\) 1.05562e50 1.67625
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −2.20556e49 −0.270790 −0.135395 − 0.990792i \(-0.543230\pi\)
−0.135395 + 0.990792i \(0.543230\pi\)
\(314\) 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
\(325\) 8.41269e49 0.486707
\(326\) 0 0
\(327\) −3.00027e50 −1.53534
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.71406e50 1.08908 0.544539 − 0.838736i \(-0.316705\pi\)
0.544539 + 0.838736i \(0.316705\pi\)
\(332\) 0 0
\(333\) 3.50766e50 1.24776
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.95355e50 −1.94809 −0.974044 − 0.226360i \(-0.927317\pi\)
−0.974044 + 0.226360i \(0.927317\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.66821e51 −3.28374
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.35259e51 −1.88217 −0.941084 − 0.338173i \(-0.890191\pi\)
−0.941084 + 0.338173i \(0.890191\pi\)
\(350\) 0 0
\(351\) 3.92112e50 0.486707
\(352\) 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
\(361\) 2.23652e51 1.58281
\(362\) 0 0
\(363\) 1.57809e51 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.98249e51 −1.00899 −0.504497 − 0.863413i \(-0.668322\pi\)
−0.504497 + 0.863413i \(0.668322\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.24742e51 1.19498 0.597491 − 0.801875i \(-0.296164\pi\)
0.597491 + 0.801875i \(0.296164\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.07414e51 −0.822127 −0.411063 − 0.911607i \(-0.634842\pi\)
−0.411063 + 0.911607i \(0.634842\pi\)
\(380\) 0 0
\(381\) 7.88920e51 1.89904
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.05602e51 0.890409
\(388\) 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
\(397\) −1.88987e52 −1.99812 −0.999062 − 0.0433074i \(-0.986211\pi\)
−0.999062 + 0.0433074i \(0.986211\pi\)
\(398\) 0 0
\(399\) 3.23553e52 3.09376
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.59848e51 −0.516824
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.15529e51 0.300460 0.150230 − 0.988651i \(-0.451999\pi\)
0.150230 + 0.988651i \(0.451999\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.87211e50 −0.0271875
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −4.90206e52 −1.60227 −0.801137 − 0.598481i \(-0.795771\pi\)
−0.801137 + 0.598481i \(0.795771\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.39509e53 −3.43593
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −3.70328e52 −0.689968 −0.344984 − 0.938609i \(-0.612116\pi\)
−0.344984 + 0.938609i \(0.612116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.68215e52 −1.08691 −0.543453 − 0.839439i \(-0.682884\pi\)
−0.543453 + 0.839439i \(0.682884\pi\)
\(440\) 0 0
\(441\) 2.09444e53 2.70580
\(442\) 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
\(453\) 2.56201e53 1.93471
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.15705e53 −1.99967 −0.999833 − 0.0182964i \(-0.994176\pi\)
−0.999833 + 0.0182964i \(0.994176\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 3.62785e53 1.77023 0.885115 − 0.465373i \(-0.154080\pi\)
0.885115 + 0.465373i \(0.154080\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 6.37588e53 2.40483
\(470\) 0 0
\(471\) 8.56325e52 0.296634
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.49437e53 −1.60711
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 2.66871e53 0.607295
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.12618e54 −2.00000 −0.999999 − 0.00110815i \(-0.999647\pi\)
−0.999999 + 0.00110815i \(0.999647\pi\)
\(488\) 0 0
\(489\) 1.20254e54 1.96753
\(490\) 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
\(499\) 1.58926e54 1.73454 0.867272 − 0.497835i \(-0.165872\pi\)
0.867272 + 0.497835i \(0.165872\pi\)
\(500\) 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
\(507\) −9.61059e53 −0.763116
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −5.62802e54 −3.81889
\(512\) 0 0
\(513\) −2.56090e54 −1.60711
\(514\) 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
\(523\) 3.43300e54 1.46433 0.732166 − 0.681126i \(-0.238509\pi\)
0.732166 + 0.681126i \(0.238509\pi\)
\(524\) 0 0
\(525\) −4.87110e54 −1.92505
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 2.94519e54 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
\(541\) 8.80328e54 1.90853 0.954265 − 0.298962i \(-0.0966404\pi\)
0.954265 + 0.298962i \(0.0966404\pi\)
\(542\) 0 0
\(543\) −5.43327e54 −1.09412
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.12907e55 −1.96325 −0.981626 − 0.190817i \(-0.938886\pi\)
−0.981626 + 0.190817i \(0.938886\pi\)
\(548\) 0 0
\(549\) 1.10421e55 1.78486
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.59053e55 3.62148
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 3.84674e54 0.433368
\(560\) 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
\(567\) −2.27040e55 −1.92505
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.25823e55 0.926910 0.463455 − 0.886120i \(-0.346610\pi\)
0.463455 + 0.886120i \(0.346610\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.17525e55 1.30015 0.650074 − 0.759871i \(-0.274738\pi\)
0.650074 + 0.759871i \(0.274738\pi\)
\(578\) 0 0
\(579\) −3.46555e55 −1.93286
\(580\) 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
\(589\) 4.30950e55 1.70656
\(590\) 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
\(597\) 1.38738e55 0.419479
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −7.05888e55 −1.86744 −0.933719 − 0.358007i \(-0.883456\pi\)
−0.933719 + 0.358007i \(0.883456\pi\)
\(602\) 0 0
\(603\) −5.04648e55 −1.24923
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.90439e55 −1.06368 −0.531840 − 0.846845i \(-0.678499\pi\)
−0.531840 + 0.846845i \(0.678499\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.10120e55 −0.552485 −0.276242 − 0.961088i \(-0.589089\pi\)
−0.276242 + 0.961088i \(0.589089\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.28913e56 −1.89010 −0.945050 − 0.326926i \(-0.893987\pi\)
−0.945050 + 0.326926i \(0.893987\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8.27181e55 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 5.39281e55 0.538552 0.269276 − 0.963063i \(-0.413216\pi\)
0.269276 + 0.963063i \(0.413216\pi\)
\(632\) 0 0
\(633\) −4.90127e55 −0.459446
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.59350e56 1.31693
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 6.39436e55 0.438104 0.219052 − 0.975713i \(-0.429704\pi\)
0.219052 + 0.975713i \(0.429704\pi\)
\(644\) 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
\(651\) 3.82064e56 2.04417
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.45455e56 1.98379
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.38921e56 −0.547946 −0.273973 − 0.961737i \(-0.588338\pi\)
−0.273973 + 0.961737i \(0.588338\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.06486e55 −0.157053
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.31401e56 −0.361657 −0.180829 − 0.983515i \(-0.557878\pi\)
−0.180829 + 0.983515i \(0.557878\pi\)
\(674\) 0 0
\(675\) 3.85545e56 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.63519e57 −3.76854
\(680\) 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
\(687\) 4.62058e56 0.842484
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.21361e57 1.97023 0.985114 − 0.171903i \(-0.0549915\pi\)
0.985114 + 0.171903i \(0.0549915\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −1.74295e57 −2.00530
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.97648e56 −0.483056 −0.241528 − 0.970394i \(-0.577649\pi\)
−0.241528 + 0.970394i \(0.577649\pi\)
\(710\) 0 0
\(711\) −2.05040e57 −1.88125
\(712\) 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
\(721\) −4.65035e57 −3.22686
\(722\) 0 0
\(723\) 3.01742e57 1.98093
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.02579e57 1.77890 0.889448 − 0.457037i \(-0.151089\pi\)
0.889448 + 0.457037i \(0.151089\pi\)
\(728\) 0 0
\(729\) 1.79701e57 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.00457e57 −1.99745 −0.998727 − 0.0504358i \(-0.983939\pi\)
−0.998727 + 0.0504358i \(0.983939\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 3.84654e57 1.62998 0.814988 − 0.579478i \(-0.196744\pi\)
0.814988 + 0.579478i \(0.196744\pi\)
\(740\) 0 0
\(741\) −1.94840e57 −0.782193
\(742\) 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
\(751\) 6.23277e57 1.91374 0.956868 − 0.290522i \(-0.0938288\pi\)
0.956868 + 0.290522i \(0.0938288\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.12530e57 1.34215 0.671075 − 0.741389i \(-0.265833\pi\)
0.671075 + 0.741389i \(0.265833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.32171e58 2.95561
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.33579e57 0.255394 0.127697 − 0.991813i \(-0.459241\pi\)
0.127697 + 0.991813i \(0.459241\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −6.48798e57 −1.06188
\(776\) 0 0
\(777\) −1.54523e58 −2.40200
\(778\) 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
\(787\) −1.30200e58 −1.56717 −0.783584 − 0.621286i \(-0.786610\pi\)
−0.783584 + 0.621286i \(0.786610\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.40108e57 0.868703
\(794\) 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
\(811\) 1.49082e58 0.984029 0.492014 − 0.870587i \(-0.336261\pi\)
0.492014 + 0.870587i \(0.336261\pi\)
\(812\) 0 0
\(813\) −3.13827e58 −1.97187
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.51233e58 −1.43099
\(818\) 0 0
\(819\) −1.72737e58 −0.936933
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 7.07628e57 0.348184 0.174092 − 0.984729i \(-0.444301\pi\)
0.174092 + 0.984729i \(0.444301\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 3.63798e58 1.54799 0.773997 − 0.633189i \(-0.218254\pi\)
0.773997 + 0.633189i \(0.218254\pi\)
\(830\) 0 0
\(831\) −1.32348e58 −0.536659
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.02402e58 −1.06188
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.13271e58 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.95199e58 −1.92505
\(848\) 0 0
\(849\) 7.57141e58 1.99997
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.58796e57 0.0862715 0.0431358 − 0.999069i \(-0.486265\pi\)
0.0431358 + 0.999069i \(0.486265\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 8.06755e58 1.68608 0.843041 − 0.537850i \(-0.180763\pi\)
0.843041 + 0.537850i \(0.180763\pi\)
\(860\) 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
\(867\) 5.75946e58 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.83949e58 −0.608009
\(872\) 0 0
\(873\) 1.29425e59 1.95764
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.13109e59 1.56136 0.780681 − 0.624929i \(-0.214872\pi\)
0.780681 + 0.624929i \(0.214872\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.05498e59 1.27066 0.635329 − 0.772241i \(-0.280864\pi\)
0.635329 + 0.772241i \(0.280864\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −3.47544e59 −3.65573
\(890\) 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
\(903\) −2.22733e59 −1.71408
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −2.02499e59 −1.42652 −0.713259 − 0.700900i \(-0.752782\pi\)
−0.713259 + 0.700900i \(0.752782\pi\)
\(908\) 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
\(919\) −3.41898e59 −1.85177 −0.925884 − 0.377809i \(-0.876678\pi\)
−0.925884 + 0.377809i \(0.876678\pi\)
\(920\) 0 0
\(921\) 3.83061e59 1.98644
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.62402e59 1.24776
\(926\) 0 0
\(927\) 3.68073e59 1.67625
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.04072e60 −4.34852
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.43121e59 0.525910 0.262955 − 0.964808i \(-0.415303\pi\)
0.262955 + 0.964808i \(0.415303\pi\)
\(938\) 0 0
\(939\) −7.69031e58 −0.270790
\(940\) 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
\(949\) 3.38913e59 0.965526
\(950\) 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
\(961\) 5.75810e58 0.127588
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.82286e59 1.72616 0.863078 − 0.505071i \(-0.168534\pi\)
0.863078 + 0.505071i \(0.168534\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 3.02738e58 0.0523372
\(974\) 0 0
\(975\) 2.93332e59 0.486707
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.04613e60 −1.53534
\(982\) 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
\(991\) 1.49253e60 1.78834 0.894171 − 0.447725i \(-0.147766\pi\)
0.894171 + 0.447725i \(0.147766\pi\)
\(992\) 0 0
\(993\) 9.46332e59 1.08908
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.87768e60 1.99397 0.996987 − 0.0775748i \(-0.0247177\pi\)
0.996987 + 0.0775748i \(0.0247177\pi\)
\(998\) 0 0
\(999\) 1.22304e60 1.24776
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 12.41.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.41.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.41.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.41.c.a.5.1 ✓ 1 3.2 odd 2 CM