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Properties

Label 12.35.c.a.5.1
Level $12$
Weight $35$
Character 12.5
Self dual yes
Analytic conductor $87.871$
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] = [12,35,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, 35); 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, 35, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 35 \)
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: \(87.8707848512\)
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)
 
\(f(q)\) \(=\) \(q-1.29140e8 q^{3} +3.01393e14 q^{7} +1.66772e16 q^{9} +1.71539e19 q^{13} +2.64244e21 q^{19} -3.89219e22 q^{21} +5.82077e23 q^{25} -2.15369e24 q^{27} -4.49767e25 q^{31} -1.63401e26 q^{37} -2.21526e27 q^{39} +1.11719e28 q^{43} +3.67207e28 q^{49} -3.41245e29 q^{57} -4.46800e30 q^{61} +5.02638e30 q^{63} +1.25607e31 q^{67} +7.03386e31 q^{73} -7.51695e31 q^{75} -4.97750e31 q^{79} +2.78128e32 q^{81} +5.17007e33 q^{91} +5.80830e33 q^{93} +2.07782e33 q^{97} +O(q^{100})\)

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
\(3\) −1.29140e8 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 3.01393e14 1.29559 0.647793 − 0.761816i \(-0.275692\pi\)
0.647793 + 0.761816i \(0.275692\pi\)
\(8\) 0 0
\(9\) 1.66772e16 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.71539e19 1.98302 0.991509 − 0.130039i \(-0.0415104\pi\)
0.991509 + 0.130039i \(0.0415104\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 2.64244e21 0.482163 0.241081 − 0.970505i \(-0.422498\pi\)
0.241081 + 0.970505i \(0.422498\pi\)
\(20\) 0 0
\(21\) −3.89219e22 −1.29559
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.82077e23 1.00000
\(26\) 0 0
\(27\) −2.15369e24 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.49767e25 −1.99452 −0.997261 − 0.0739583i \(-0.976437\pi\)
−0.997261 + 0.0739583i \(0.976437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.63401e26 −0.357952 −0.178976 − 0.983853i \(-0.557279\pi\)
−0.178976 + 0.983853i \(0.557279\pi\)
\(38\) 0 0
\(39\) −2.21526e27 −1.98302
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.11719e28 1.90180 0.950900 − 0.309499i \(-0.100161\pi\)
0.950900 + 0.309499i \(0.100161\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.67207e28 0.678544
\(50\) 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
\(57\) −3.41245e29 −0.482163
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −4.46800e30 −1.99299 −0.996497 − 0.0836236i \(-0.973351\pi\)
−0.996497 + 0.0836236i \(0.973351\pi\)
\(62\) 0 0
\(63\) 5.02638e30 1.29559
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.25607e31 1.13696 0.568478 − 0.822699i \(-0.307533\pi\)
0.568478 + 0.822699i \(0.307533\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 7.03386e31 1.48151 0.740756 − 0.671774i \(-0.234467\pi\)
0.740756 + 0.671774i \(0.234467\pi\)
\(74\) 0 0
\(75\) −7.51695e31 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.97750e31 −0.273748 −0.136874 − 0.990588i \(-0.543706\pi\)
−0.136874 + 0.990588i \(0.543706\pi\)
\(80\) 0 0
\(81\) 2.78128e32 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
\(91\) 5.17007e33 2.56917
\(92\) 0 0
\(93\) 5.80830e33 1.99452
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.07782e33 0.348730 0.174365 − 0.984681i \(-0.444213\pi\)
0.174365 + 0.984681i \(0.444213\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.97363e34 1.79909 0.899547 − 0.436824i \(-0.143897\pi\)
0.899547 + 0.436824i \(0.143897\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 8.59301e34 1.98561 0.992807 − 0.119729i \(-0.0382026\pi\)
0.992807 + 0.119729i \(0.0382026\pi\)
\(110\) 0 0
\(111\) 2.11016e34 0.357952
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.86079e35 1.98302
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.55477e35 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.69608e35 −1.66699 −0.833495 − 0.552528i \(-0.813663\pi\)
−0.833495 + 0.552528i \(0.813663\pi\)
\(128\) 0 0
\(129\) −1.44275e36 −1.90180
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 7.96412e35 0.624683
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −5.26365e36 −1.95000 −0.975002 − 0.222194i \(-0.928678\pi\)
−0.975002 + 0.222194i \(0.928678\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4.74212e36 −0.678544
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.65176e37 −1.49740 −0.748701 − 0.662908i \(-0.769322\pi\)
−0.748701 + 0.662908i \(0.769322\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.12546e37 −0.526063 −0.263031 − 0.964787i \(-0.584722\pi\)
−0.263031 + 0.964787i \(0.584722\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.29593e37 0.320178 0.160089 − 0.987103i \(-0.448822\pi\)
0.160089 + 0.987103i \(0.448822\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.19428e38 2.93236
\(170\) 0 0
\(171\) 4.40684e37 0.482163
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 1.75434e38 1.29559
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −4.04269e38 −1.68320 −0.841599 − 0.540104i \(-0.818385\pi\)
−0.841599 + 0.540104i \(0.818385\pi\)
\(182\) 0 0
\(183\) 5.76999e38 1.99299
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.49108e38 −1.29559
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.10139e39 1.53981 0.769907 − 0.638156i \(-0.220302\pi\)
0.769907 + 0.638156i \(0.220302\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.21125e39 −1.00631 −0.503155 − 0.864196i \(-0.667828\pi\)
−0.503155 + 0.864196i \(0.667828\pi\)
\(200\) 0 0
\(201\) −1.62210e39 −1.13696
\(202\) 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
\(211\) −3.62250e39 −1.11226 −0.556131 − 0.831095i \(-0.687715\pi\)
−0.556131 + 0.831095i \(0.687715\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.35557e40 −2.58408
\(218\) 0 0
\(219\) −9.08354e39 −1.48151
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.47728e40 1.77125 0.885623 − 0.464405i \(-0.153732\pi\)
0.885623 + 0.464405i \(0.153732\pi\)
\(224\) 0 0
\(225\) 9.70740e39 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 2.15430e40 1.64476 0.822382 − 0.568936i \(-0.192645\pi\)
0.822382 + 0.568936i \(0.192645\pi\)
\(230\) 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
\(237\) 6.42795e39 0.273748
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 6.12158e40 1.96143 0.980717 − 0.195434i \(-0.0626114\pi\)
0.980717 + 0.195434i \(0.0626114\pi\)
\(242\) 0 0
\(243\) −3.59175e40 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.53282e40 0.956137
\(248\) 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
\(259\) −4.92479e40 −0.463758
\(260\) 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
\(271\) 4.58631e41 1.99979 0.999897 − 0.0143618i \(-0.00457165\pi\)
0.999897 + 0.0143618i \(0.00457165\pi\)
\(272\) 0 0
\(273\) −6.67664e41 −2.56917
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.93737e41 −1.18318 −0.591589 − 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(278\) 0 0
\(279\) −7.50085e41 −1.99452
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.00546e41 −0.627396 −0.313698 − 0.949523i \(-0.601568\pi\)
−0.313698 + 0.949523i \(0.601568\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.84326e41 1.00000
\(290\) 0 0
\(291\) −2.68331e41 −0.348730
\(292\) 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
\(301\) 3.36714e42 2.46395
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.50108e42 1.83168 0.915839 − 0.401545i \(-0.131527\pi\)
0.915839 + 0.401545i \(0.131527\pi\)
\(308\) 0 0
\(309\) −3.84015e42 −1.79909
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.61524e42 1.36108 0.680539 − 0.732712i \(-0.261746\pi\)
0.680539 + 0.732712i \(0.261746\pi\)
\(314\) 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
\(325\) 9.98490e42 1.98302
\(326\) 0 0
\(327\) −1.10970e43 −1.98561
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.49003e42 −0.798912 −0.399456 − 0.916752i \(-0.630801\pi\)
−0.399456 + 0.916752i \(0.630801\pi\)
\(332\) 0 0
\(333\) −2.72507e42 −0.357952
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.54856e43 1.66042 0.830211 − 0.557449i \(-0.188220\pi\)
0.830211 + 0.557449i \(0.188220\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5.24310e42 −0.416474
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.97983e43 −1.76262 −0.881310 − 0.472539i \(-0.843338\pi\)
−0.881310 + 0.472539i \(0.843338\pi\)
\(350\) 0 0
\(351\) −3.69443e43 −1.98302
\(352\) 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
\(361\) −2.30522e43 −0.767519
\(362\) 0 0
\(363\) −3.29923e43 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.27668e43 1.83067 0.915335 − 0.402694i \(-0.131926\pi\)
0.915335 + 0.402694i \(0.131926\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.03531e44 1.97706 0.988529 − 0.151034i \(-0.0482604\pi\)
0.988529 + 0.151034i \(0.0482604\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00253e44 −1.45959 −0.729794 − 0.683667i \(-0.760384\pi\)
−0.729794 + 0.683667i \(0.760384\pi\)
\(380\) 0 0
\(381\) 1.25215e44 1.66699
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.86316e44 1.90180
\(388\) 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
\(397\) 2.96656e44 1.96252 0.981260 − 0.192690i \(-0.0617211\pi\)
0.981260 + 0.192690i \(0.0617211\pi\)
\(398\) 0 0
\(399\) −1.02849e44 −0.624683
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −7.71527e44 −3.95517
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3.90645e44 1.55771 0.778854 − 0.627205i \(-0.215801\pi\)
0.778854 + 0.627205i \(0.215801\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.79749e44 1.95000
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −7.58980e44 −1.85112 −0.925560 − 0.378600i \(-0.876405\pi\)
−0.925560 + 0.378600i \(0.876405\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.34662e45 −2.58210
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 8.42635e43 0.127452 0.0637258 − 0.997967i \(-0.479702\pi\)
0.0637258 + 0.997967i \(0.479702\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.61773e45 −1.93646 −0.968230 − 0.250062i \(-0.919549\pi\)
−0.968230 + 0.250062i \(0.919549\pi\)
\(440\) 0 0
\(441\) 6.12398e44 0.678544
\(442\) 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
\(453\) 2.13308e45 1.49740
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.68288e44 0.343552 0.171776 − 0.985136i \(-0.445049\pi\)
0.171776 + 0.985136i \(0.445049\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 3.08933e45 1.49619 0.748097 − 0.663589i \(-0.230968\pi\)
0.748097 + 0.663589i \(0.230968\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 3.78572e45 1.47302
\(470\) 0 0
\(471\) 1.45343e45 0.526063
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.53810e45 0.482163
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −2.80297e45 −0.709826
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8.69183e45 −1.78287 −0.891434 − 0.453151i \(-0.850300\pi\)
−0.891434 + 0.453151i \(0.850300\pi\)
\(488\) 0 0
\(489\) −1.67356e45 −0.320178
\(490\) 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
\(499\) 6.32099e45 0.857188 0.428594 − 0.903497i \(-0.359009\pi\)
0.428594 + 0.903497i \(0.359009\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.83369e46 −2.93236
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 2.11996e46 1.91943
\(512\) 0 0
\(513\) −5.69100e45 −0.482163
\(514\) 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
\(523\) −3.12578e46 −1.90735 −0.953675 − 0.300840i \(-0.902733\pi\)
−0.953675 + 0.300840i \(0.902733\pi\)
\(524\) 0 0
\(525\) −2.26555e46 −1.29559
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.98951e46 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
\(541\) 2.10822e46 0.723706 0.361853 − 0.932235i \(-0.382144\pi\)
0.361853 + 0.932235i \(0.382144\pi\)
\(542\) 0 0
\(543\) 5.22074e46 1.68320
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.67035e46 1.89830 0.949149 − 0.314827i \(-0.101947\pi\)
0.949149 + 0.314827i \(0.101947\pi\)
\(548\) 0 0
\(549\) −7.45137e46 −1.99299
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.50018e46 −0.354664
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.91643e47 3.77130
\(560\) 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
\(567\) 8.38259e46 1.29559
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −8.37092e46 −1.14805 −0.574023 − 0.818839i \(-0.694618\pi\)
−0.574023 + 0.818839i \(0.694618\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.50904e47 −1.73266 −0.866330 − 0.499472i \(-0.833527\pi\)
−0.866330 + 0.499472i \(0.833527\pi\)
\(578\) 0 0
\(579\) −1.42234e47 −1.53981
\(580\) 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
\(589\) −1.18848e47 −0.961684
\(590\) 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
\(597\) 1.56421e47 1.00631
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −3.44125e47 −1.97629 −0.988145 − 0.153524i \(-0.950938\pi\)
−0.988145 + 0.153524i \(0.950938\pi\)
\(602\) 0 0
\(603\) 2.09478e47 1.13696
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.01232e46 0.146115 0.0730573 − 0.997328i \(-0.476724\pi\)
0.0730573 + 0.997328i \(0.476724\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.93636e47 −1.61535 −0.807675 − 0.589629i \(-0.799274\pi\)
−0.807675 + 0.589629i \(0.799274\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.07573e47 −0.374078 −0.187039 − 0.982352i \(-0.559889\pi\)
−0.187039 + 0.982352i \(0.559889\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.38813e47 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.21647e47 −0.305213 −0.152607 − 0.988287i \(-0.548767\pi\)
−0.152607 + 0.988287i \(0.548767\pi\)
\(632\) 0 0
\(633\) 4.67811e47 1.11226
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.29905e47 1.34556
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −9.53364e47 −1.73649 −0.868245 − 0.496135i \(-0.834752\pi\)
−0.868245 + 0.496135i \(0.834752\pi\)
\(644\) 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
\(651\) 1.75058e48 2.58408
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.17305e48 1.48151
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.75572e48 2.00000 1.00000 0.000274157i \(-8.72669e-5\pi\)
1.00000 0.000274157i \(8.72669e-5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.90776e48 −1.77125
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.31942e48 −1.94591 −0.972956 − 0.230989i \(-0.925804\pi\)
−0.972956 + 0.230989i \(0.925804\pi\)
\(674\) 0 0
\(675\) −1.25361e48 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 6.26241e47 0.451810
\(680\) 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
\(687\) −2.78207e48 −1.64476
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.72909e48 −0.926168 −0.463084 − 0.886314i \(-0.653257\pi\)
−0.463084 + 0.886314i \(0.653257\pi\)
\(692\) 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
\(703\) −4.31777e47 −0.172591
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.94118e48 0.671553 0.335777 − 0.941942i \(-0.391001\pi\)
0.335777 + 0.941942i \(0.391001\pi\)
\(710\) 0 0
\(711\) −8.30107e47 −0.273748
\(712\) 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
\(721\) 8.96231e48 2.33088
\(722\) 0 0
\(723\) −7.90542e48 −1.96143
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −4.54443e48 −1.02658 −0.513292 − 0.858214i \(-0.671574\pi\)
−0.513292 + 0.858214i \(0.671574\pi\)
\(728\) 0 0
\(729\) 4.63840e48 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.88388e48 1.35228 0.676139 − 0.736774i \(-0.263652\pi\)
0.676139 + 0.736774i \(0.263652\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 8.70467e48 1.48866 0.744332 − 0.667809i \(-0.232768\pi\)
0.744332 + 0.667809i \(0.232768\pi\)
\(740\) 0 0
\(741\) −5.85369e48 −0.956137
\(742\) 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
\(751\) −9.77451e47 −0.127121 −0.0635603 − 0.997978i \(-0.520246\pi\)
−0.0635603 + 0.997978i \(0.520246\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.15388e48 1.03986 0.519931 − 0.854208i \(-0.325958\pi\)
0.519931 + 0.854208i \(0.325958\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 2.58987e49 2.57253
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −7.76771e48 −0.675375 −0.337688 − 0.941258i \(-0.609645\pi\)
−0.337688 + 0.941258i \(0.609645\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −2.61799e49 −1.99452
\(776\) 0 0
\(777\) 6.35988e48 0.463758
\(778\) 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
\(787\) −1.72247e49 −1.01060 −0.505302 − 0.862943i \(-0.668619\pi\)
−0.505302 + 0.862943i \(0.668619\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.66438e49 −3.95214
\(794\) 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
\(811\) 5.67484e49 1.99802 0.999008 − 0.0445279i \(-0.0141784\pi\)
0.999008 + 0.0445279i \(0.0141784\pi\)
\(812\) 0 0
\(813\) −5.92276e49 −1.99979
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.95211e49 0.916977
\(818\) 0 0
\(819\) 8.62222e49 2.56917
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −4.68914e49 −1.28616 −0.643081 − 0.765798i \(-0.722344\pi\)
−0.643081 + 0.765798i \(0.722344\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −8.24152e49 −1.99793 −0.998965 − 0.0454865i \(-0.985516\pi\)
−0.998965 + 0.0454865i \(0.985516\pi\)
\(830\) 0 0
\(831\) 5.08472e49 1.18318
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.68661e49 1.99452
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.26662e49 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.69989e49 1.29559
\(848\) 0 0
\(849\) 3.88126e49 0.627396
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.32424e49 −1.24226 −0.621128 − 0.783709i \(-0.713325\pi\)
−0.621128 + 0.783709i \(0.713325\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.07974e50 −1.43033 −0.715166 − 0.698955i \(-0.753649\pi\)
−0.715166 + 0.698955i \(0.753649\pi\)
\(860\) 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
\(867\) −8.83740e49 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.15466e50 2.25460
\(872\) 0 0
\(873\) 3.46522e49 0.348730
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.74112e49 0.720793 0.360397 − 0.932799i \(-0.382641\pi\)
0.360397 + 0.932799i \(0.382641\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.92779e49 0.242777 0.121389 − 0.992605i \(-0.461265\pi\)
0.121389 + 0.992605i \(0.461265\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2.92233e50 −2.15973
\(890\) 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
\(903\) −4.34833e50 −2.46395
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.44355e50 −1.81002 −0.905008 − 0.425394i \(-0.860135\pi\)
−0.905008 + 0.425394i \(0.860135\pi\)
\(908\) 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
\(919\) 4.20589e50 1.76805 0.884024 − 0.467441i \(-0.154824\pi\)
0.884024 + 0.467441i \(0.154824\pi\)
\(920\) 0 0
\(921\) −4.52130e50 −1.83168
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −9.51119e49 −0.357952
\(926\) 0 0
\(927\) 4.95918e50 1.79909
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 9.70322e49 0.327168
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.63389e50 1.40079 0.700395 − 0.713756i \(-0.253007\pi\)
0.700395 + 0.713756i \(0.253007\pi\)
\(938\) 0 0
\(939\) −4.66872e50 −1.36108
\(940\) 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
\(949\) 1.20658e51 2.93787
\(950\) 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
\(961\) 1.51440e51 2.97812
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.53150e50 −0.270935 −0.135468 − 0.990782i \(-0.543254\pi\)
−0.135468 + 0.990782i \(0.543254\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −1.58643e51 −2.52640
\(974\) 0 0
\(975\) −1.28945e51 −1.98302
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.43307e51 1.98561
\(982\) 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
\(991\) −6.59319e50 −0.768853 −0.384426 − 0.923156i \(-0.625601\pi\)
−0.384426 + 0.923156i \(0.625601\pi\)
\(992\) 0 0
\(993\) 7.08984e50 0.798912
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.21601e51 −1.27973 −0.639866 − 0.768487i \(-0.721010\pi\)
−0.639866 + 0.768487i \(0.721010\pi\)
\(998\) 0 0
\(999\) 3.51916e50 0.357952
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 12.35.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.35.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.35.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.35.c.a.5.1 ✓ 1 3.2 odd 2 CM