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Properties

Label 12.33.c.a.5.1
Level $12$
Weight $33$
Character 12.5
Self dual yes
Analytic conductor $77.840$
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] = [12,33,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, 33); 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, 33, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 33 \)
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: \(77.8399861707\)
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)
 
\(f(q)\) \(=\) \(q+4.30467e7 q^{3} -4.39646e13 q^{7} +1.85302e15 q^{9} -1.20876e18 q^{13} +5.03460e20 q^{19} -1.89253e21 q^{21} +2.32831e22 q^{25} +7.97664e22 q^{27} +1.00430e24 q^{31} +2.11772e25 q^{37} -5.20331e25 q^{39} -1.48214e26 q^{43} +8.28455e26 q^{49} +2.16723e28 q^{57} -3.95411e28 q^{61} -8.14672e28 q^{63} +2.82906e29 q^{67} +5.25728e29 q^{73} +1.00226e30 q^{75} +2.56575e30 q^{79} +3.43368e30 q^{81} +5.31425e31 q^{91} +4.32320e31 q^{93} +1.21183e32 q^{97} +O(q^{100})\)

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
\(3\) 4.30467e7 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −4.39646e13 −1.32292 −0.661461 − 0.749980i \(-0.730063\pi\)
−0.661461 + 0.749980i \(0.730063\pi\)
\(8\) 0 0
\(9\) 1.85302e15 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.20876e18 −1.81654 −0.908272 − 0.418380i \(-0.862598\pi\)
−0.908272 + 0.418380i \(0.862598\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 5.03460e20 1.74545 0.872725 − 0.488213i \(-0.162351\pi\)
0.872725 + 0.488213i \(0.162351\pi\)
\(20\) 0 0
\(21\) −1.89253e21 −1.32292
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.32831e22 1.00000
\(26\) 0 0
\(27\) 7.97664e22 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.00430e24 1.38063 0.690317 − 0.723507i \(-0.257471\pi\)
0.690317 + 0.723507i \(0.257471\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.11772e25 1.71649 0.858243 − 0.513244i \(-0.171556\pi\)
0.858243 + 0.513244i \(0.171556\pi\)
\(38\) 0 0
\(39\) −5.20331e25 −1.81654
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.48214e26 −1.08491 −0.542454 − 0.840085i \(-0.682505\pi\)
−0.542454 + 0.840085i \(0.682505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 8.28455e26 0.750121
\(50\) 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
\(57\) 2.16723e28 1.74545
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −3.95411e28 −1.07590 −0.537949 − 0.842978i \(-0.680801\pi\)
−0.537949 + 0.842978i \(0.680801\pi\)
\(62\) 0 0
\(63\) −8.14672e28 −1.32292
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.82906e29 1.71571 0.857857 − 0.513888i \(-0.171795\pi\)
0.857857 + 0.513888i \(0.171795\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.25728e29 0.808342 0.404171 − 0.914684i \(-0.367560\pi\)
0.404171 + 0.914684i \(0.367560\pi\)
\(74\) 0 0
\(75\) 1.00226e30 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.56575e30 1.11476 0.557380 − 0.830257i \(-0.311807\pi\)
0.557380 + 0.830257i \(0.311807\pi\)
\(80\) 0 0
\(81\) 3.43368e30 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
\(91\) 5.31425e31 2.40315
\(92\) 0 0
\(93\) 4.32320e31 1.38063
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.21183e32 1.97285 0.986426 − 0.164204i \(-0.0525057\pi\)
0.986426 + 0.164204i \(0.0525057\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.24734e32 1.40047 0.700234 − 0.713914i \(-0.253079\pi\)
0.700234 + 0.713914i \(0.253079\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −7.92013e32 −1.99484 −0.997420 − 0.0717807i \(-0.977132\pi\)
−0.997420 + 0.0717807i \(0.977132\pi\)
\(110\) 0 0
\(111\) 9.11608e32 1.71649
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.23985e33 −1.81654
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.11138e33 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.52990e33 −1.86245 −0.931224 − 0.364447i \(-0.881258\pi\)
−0.931224 + 0.364447i \(0.881258\pi\)
\(128\) 0 0
\(129\) −6.38011e33 −1.08491
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −2.21344e34 −2.30909
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.15995e34 0.597312 0.298656 − 0.954361i \(-0.403462\pi\)
0.298656 + 0.954361i \(0.403462\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.56623e34 0.750121
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 7.24857e34 0.992250 0.496125 − 0.868251i \(-0.334756\pi\)
0.496125 + 0.868251i \(0.334756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.70606e35 1.98583 0.992917 − 0.118812i \(-0.0379084\pi\)
0.992917 + 0.118812i \(0.0379084\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.55609e35 −1.43209 −0.716046 − 0.698053i \(-0.754050\pi\)
−0.716046 + 0.698053i \(0.754050\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.01832e36 2.29983
\(170\) 0 0
\(171\) 9.32921e35 1.74545
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.02363e36 −1.32292
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.22411e36 −0.922493 −0.461247 − 0.887272i \(-0.652598\pi\)
−0.461247 + 0.887272i \(0.652598\pi\)
\(182\) 0 0
\(183\) −1.70211e36 −1.07590
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.50690e36 −1.32292
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −4.96837e36 −1.34059 −0.670296 − 0.742094i \(-0.733833\pi\)
−0.670296 + 0.742094i \(0.733833\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −8.45093e36 −1.39719 −0.698594 − 0.715518i \(-0.746191\pi\)
−0.698594 + 0.715518i \(0.746191\pi\)
\(200\) 0 0
\(201\) 1.21782e37 1.71571
\(202\) 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
\(211\) 3.03417e37 1.96571 0.982856 − 0.184373i \(-0.0590256\pi\)
0.982856 + 0.184373i \(0.0590256\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.41538e37 −1.82647
\(218\) 0 0
\(219\) 2.26308e37 0.808342
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.75399e37 0.736349 0.368175 − 0.929757i \(-0.379983\pi\)
0.368175 + 0.929757i \(0.379983\pi\)
\(224\) 0 0
\(225\) 4.31440e37 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 7.03176e37 1.22941 0.614704 − 0.788758i \(-0.289275\pi\)
0.614704 + 0.788758i \(0.289275\pi\)
\(230\) 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
\(237\) 1.10447e38 1.11476
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.91462e38 −1.47846 −0.739231 − 0.673452i \(-0.764811\pi\)
−0.739231 + 0.673452i \(0.764811\pi\)
\(242\) 0 0
\(243\) 1.47809e38 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.08561e38 −3.17069
\(248\) 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
\(259\) −9.31045e38 −2.27078
\(260\) 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
\(271\) 7.33881e38 0.867195 0.433598 − 0.901107i \(-0.357244\pi\)
0.433598 + 0.901107i \(0.357244\pi\)
\(272\) 0 0
\(273\) 2.28761e39 2.40315
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.20265e39 −1.83345 −0.916725 − 0.399519i \(-0.869177\pi\)
−0.916725 + 0.399519i \(0.869177\pi\)
\(278\) 0 0
\(279\) 1.86100e39 1.38063
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −2.74806e39 −1.62347 −0.811733 − 0.584028i \(-0.801476\pi\)
−0.811733 + 0.584028i \(0.801476\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.36791e39 1.00000
\(290\) 0 0
\(291\) 5.21654e39 1.97285
\(292\) 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
\(301\) 6.51615e39 1.43525
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.07115e40 −1.72043 −0.860217 − 0.509928i \(-0.829672\pi\)
−0.860217 + 0.509928i \(0.829672\pi\)
\(308\) 0 0
\(309\) 9.67406e39 1.40047
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.47316e40 −1.73597 −0.867984 − 0.496593i \(-0.834584\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(314\) 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
\(325\) −2.81436e40 −1.81654
\(326\) 0 0
\(327\) −3.40936e40 −1.99484
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.09398e40 −1.97196 −0.985980 − 0.166863i \(-0.946636\pi\)
−0.985980 + 0.166863i \(0.946636\pi\)
\(332\) 0 0
\(333\) 3.92417e40 1.71649
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.44279e40 1.96672 0.983362 − 0.181657i \(-0.0581461\pi\)
0.983362 + 0.181657i \(0.0581461\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.21330e40 0.330570
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 9.32145e40 1.92432 0.962159 − 0.272490i \(-0.0878472\pi\)
0.962159 + 0.272490i \(0.0878472\pi\)
\(350\) 0 0
\(351\) −9.64184e40 −1.81654
\(352\) 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
\(361\) 1.70273e41 2.04659
\(362\) 0 0
\(363\) 9.08879e40 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.12066e41 −1.95801 −0.979004 − 0.203840i \(-0.934658\pi\)
−0.979004 + 0.203840i \(0.934658\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.44737e41 1.74324 0.871622 − 0.490179i \(-0.163069\pi\)
0.871622 + 0.490179i \(0.163069\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.41842e41 1.88624 0.943122 − 0.332446i \(-0.107874\pi\)
0.943122 + 0.332446i \(0.107874\pi\)
\(380\) 0 0
\(381\) −3.67184e41 −1.86245
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.74643e41 −1.08491
\(388\) 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
\(397\) 2.10085e41 0.551755 0.275877 − 0.961193i \(-0.411032\pi\)
0.275877 + 0.961193i \(0.411032\pi\)
\(398\) 0 0
\(399\) −9.52813e41 −2.30909
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.21396e42 −2.50798
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.35837e41 0.384626 0.192313 − 0.981334i \(-0.438401\pi\)
0.192313 + 0.981334i \(0.438401\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.99319e41 0.597312
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.43397e42 1.47241 0.736203 − 0.676761i \(-0.236617\pi\)
0.736203 + 0.676761i \(0.236617\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.73841e42 1.42333
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 9.88697e40 0.0647525 0.0323763 − 0.999476i \(-0.489693\pi\)
0.0323763 + 0.999476i \(0.489693\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −5.51727e41 −0.289929 −0.144965 − 0.989437i \(-0.546307\pi\)
−0.144965 + 0.989437i \(0.546307\pi\)
\(440\) 0 0
\(441\) 1.53514e42 0.750121
\(442\) 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
\(453\) 3.12027e42 0.992250
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.13601e42 0.590126 0.295063 − 0.955478i \(-0.404659\pi\)
0.295063 + 0.955478i \(0.404659\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −8.66127e42 −1.94216 −0.971082 − 0.238746i \(-0.923264\pi\)
−0.971082 + 0.238746i \(0.923264\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.24378e43 −2.26976
\(470\) 0 0
\(471\) 1.16487e43 1.98583
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.17221e43 1.74545
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −2.55981e43 −3.11807
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.62080e43 −1.61908 −0.809538 − 0.587068i \(-0.800282\pi\)
−0.809538 + 0.587068i \(0.800282\pi\)
\(488\) 0 0
\(489\) −1.53078e43 −1.43209
\(490\) 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
\(499\) 2.70243e43 1.82871 0.914356 − 0.404910i \(-0.132697\pi\)
0.914356 + 0.404910i \(0.132697\pi\)
\(500\) 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
\(507\) 4.38353e43 2.29983
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.31134e43 −1.06937
\(512\) 0 0
\(513\) 4.01592e43 1.74545
\(514\) 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
\(523\) 4.96160e43 1.58341 0.791707 − 0.610901i \(-0.209193\pi\)
0.791707 + 0.610901i \(0.209193\pi\)
\(524\) 0 0
\(525\) −4.40639e43 −1.32292
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.76089e43 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
\(541\) 7.66936e42 0.142430 0.0712152 − 0.997461i \(-0.477312\pi\)
0.0712152 + 0.997461i \(0.477312\pi\)
\(542\) 0 0
\(543\) −5.26939e43 −0.922493
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.05403e43 0.319750 0.159875 − 0.987137i \(-0.448891\pi\)
0.159875 + 0.987137i \(0.448891\pi\)
\(548\) 0 0
\(549\) −7.32704e43 −1.07590
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.12802e44 −1.47474
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.79155e44 1.97078
\(560\) 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
\(567\) −1.50960e44 −1.32292
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.82075e43 −0.142585 −0.0712925 − 0.997455i \(-0.522712\pi\)
−0.0712925 + 0.997455i \(0.522712\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.01303e44 −1.99614 −0.998068 − 0.0621333i \(-0.980210\pi\)
−0.998068 + 0.0621333i \(0.980210\pi\)
\(578\) 0 0
\(579\) −2.13872e44 −1.34059
\(580\) 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
\(589\) 5.05627e44 2.40983
\(590\) 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
\(597\) −3.63785e44 −1.39719
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 3.30557e44 1.14092 0.570460 − 0.821325i \(-0.306765\pi\)
0.570460 + 0.821325i \(0.306765\pi\)
\(602\) 0 0
\(603\) 5.24230e44 1.71571
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.69336e44 1.38186 0.690932 − 0.722920i \(-0.257201\pi\)
0.690932 + 0.722920i \(0.257201\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.75209e44 1.95008 0.975038 − 0.222040i \(-0.0712714\pi\)
0.975038 + 0.222040i \(0.0712714\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −8.68961e44 −1.87047 −0.935233 − 0.354032i \(-0.884810\pi\)
−0.935233 + 0.354032i \(0.884810\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.42101e44 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.15848e45 −1.83408 −0.917039 − 0.398798i \(-0.869428\pi\)
−0.917039 + 0.398798i \(0.869428\pi\)
\(632\) 0 0
\(633\) 1.30611e45 1.96571
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00140e45 −1.36263
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.18362e45 −1.38623 −0.693116 − 0.720826i \(-0.743763\pi\)
−0.693116 + 0.720826i \(0.743763\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.90068e45 −1.82647
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.74184e44 0.808342
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.44397e44 0.184023 0.0920114 − 0.995758i \(-0.470670\pi\)
0.0920114 + 0.995758i \(0.470670\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.18550e45 0.736349
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.53363e45 −1.43055 −0.715273 − 0.698845i \(-0.753698\pi\)
−0.715273 + 0.698845i \(0.753698\pi\)
\(674\) 0 0
\(675\) 1.85721e45 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −5.32777e45 −2.60993
\(680\) 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
\(687\) 3.02694e45 1.22941
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 9.45859e44 0.350088 0.175044 − 0.984561i \(-0.443993\pi\)
0.175044 + 0.984561i \(0.443993\pi\)
\(692\) 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
\(703\) 1.06619e46 2.99604
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.97569e45 0.975152 0.487576 − 0.873080i \(-0.337881\pi\)
0.487576 + 0.873080i \(0.337881\pi\)
\(710\) 0 0
\(711\) 4.75439e45 1.11476
\(712\) 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
\(721\) −9.88033e45 −1.85271
\(722\) 0 0
\(723\) −8.24181e45 −1.47846
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.78829e45 1.27906 0.639531 − 0.768765i \(-0.279129\pi\)
0.639531 + 0.768765i \(0.279129\pi\)
\(728\) 0 0
\(729\) 6.36269e45 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.38784e46 1.99837 0.999185 − 0.0403548i \(-0.0128488\pi\)
0.999185 + 0.0403548i \(0.0128488\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.56834e46 −1.98211 −0.991056 − 0.133448i \(-0.957395\pi\)
−0.991056 + 0.133448i \(0.957395\pi\)
\(740\) 0 0
\(741\) −2.61966e46 −3.17069
\(742\) 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
\(751\) −1.32957e46 −1.29859 −0.649297 − 0.760535i \(-0.724937\pi\)
−0.649297 + 0.760535i \(0.724937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.32391e46 −1.99842 −0.999208 − 0.0397813i \(-0.987334\pi\)
−0.999208 + 0.0397813i \(0.987334\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 3.48205e46 2.63902
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.21041e46 0.809301 0.404650 − 0.914471i \(-0.367393\pi\)
0.404650 + 0.914471i \(0.367393\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.33833e46 1.38063
\(776\) 0 0
\(777\) −4.00784e46 −2.27078
\(778\) 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
\(787\) −1.71130e46 −0.790190 −0.395095 − 0.918640i \(-0.629288\pi\)
−0.395095 + 0.918640i \(0.629288\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.77956e46 1.95442
\(794\) 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
\(811\) 6.41952e46 1.83303 0.916513 − 0.400004i \(-0.130991\pi\)
0.916513 + 0.400004i \(0.130991\pi\)
\(812\) 0 0
\(813\) 3.15912e46 0.867195
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.46196e46 −1.89365
\(818\) 0 0
\(819\) 9.84742e46 2.40315
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.53536e46 0.346586 0.173293 − 0.984870i \(-0.444559\pi\)
0.173293 + 0.984870i \(0.444559\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −2.31115e46 −0.464468 −0.232234 − 0.972660i \(-0.574603\pi\)
−0.232234 + 0.972660i \(0.574603\pi\)
\(830\) 0 0
\(831\) −9.48168e46 −1.83345
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.01098e46 1.38063
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 6.26233e46 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −9.28258e46 −1.32292
\(848\) 0 0
\(849\) −1.18295e47 −1.62347
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.23845e47 −1.57650 −0.788249 − 0.615356i \(-0.789012\pi\)
−0.788249 + 0.615356i \(0.789012\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.73106e47 −1.96980 −0.984898 − 0.173134i \(-0.944611\pi\)
−0.984898 + 0.173134i \(0.944611\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.01931e47 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.41965e47 −3.11667
\(872\) 0 0
\(873\) 2.24555e47 1.97285
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.84679e47 1.50808 0.754040 − 0.656828i \(-0.228102\pi\)
0.754040 + 0.656828i \(0.228102\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.04285e47 −0.763576 −0.381788 − 0.924250i \(-0.624692\pi\)
−0.381788 + 0.924250i \(0.624692\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 3.75013e47 2.46387
\(890\) 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
\(903\) 2.80499e47 1.43525
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.32423e47 −0.631318 −0.315659 − 0.948873i \(-0.602226\pi\)
−0.315659 + 0.948873i \(0.602226\pi\)
\(908\) 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
\(919\) 2.18285e45 0.00843289 0.00421644 − 0.999991i \(-0.498658\pi\)
0.00421644 + 0.999991i \(0.498658\pi\)
\(920\) 0 0
\(921\) −4.61097e47 −1.72043
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.93069e47 1.71649
\(926\) 0 0
\(927\) 4.16436e47 1.40047
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 4.17094e47 1.30930
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.13997e46 0.202238 0.101119 − 0.994874i \(-0.467758\pi\)
0.101119 + 0.994874i \(0.467758\pi\)
\(938\) 0 0
\(939\) −6.34149e47 −1.73597
\(940\) 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
\(949\) −6.35478e47 −1.46839
\(950\) 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
\(961\) 4.79483e47 0.906148
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.79795e47 −0.991861 −0.495931 − 0.868362i \(-0.665173\pi\)
−0.495931 + 0.868362i \(0.665173\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −5.09965e47 −0.790197
\(974\) 0 0
\(975\) −1.21149e48 −1.81654
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.46762e48 −1.99484
\(982\) 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
\(991\) 1.61267e48 1.86366 0.931829 − 0.362897i \(-0.118212\pi\)
0.931829 + 0.362897i \(0.118212\pi\)
\(992\) 0 0
\(993\) −1.76233e48 −1.97196
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 7.00436e47 0.734929 0.367465 − 0.930037i \(-0.380226\pi\)
0.367465 + 0.930037i \(0.380226\pi\)
\(998\) 0 0
\(999\) 1.68923e48 1.71649
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 12.33.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.33.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.33.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.33.c.a.5.1 ✓ 1 3.2 odd 2 CM