Properties

Label 3.67.b.a.2.1
Level 3
Weight 67
Character 3.2
Self dual yes
Analytic conductor 82.760
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 67 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(82.7604085389\)
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 2.1
Character \(\chi\) \(=\) 3.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.55906e15 q^{3} +7.37870e19 q^{4} -1.54595e28 q^{7} +3.09032e31 q^{9} +O(q^{10})\) \(q-5.55906e15 q^{3} +7.37870e19 q^{4} -1.54595e28 q^{7} +3.09032e31 q^{9} -4.10186e35 q^{12} -1.09938e37 q^{13} +5.44452e39 q^{16} -8.32798e41 q^{19} +8.59401e43 q^{21} +1.35525e46 q^{25} -1.71793e47 q^{27} -1.14071e48 q^{28} -2.08297e49 q^{31} +2.28025e51 q^{36} -7.41794e51 q^{37} +6.11153e52 q^{39} -1.24524e54 q^{43} -3.02664e55 q^{48} +1.79227e56 q^{49} -8.11200e56 q^{52} +4.62958e57 q^{57} +7.67187e58 q^{61} -4.77746e59 q^{63} +4.01735e59 q^{64} +2.36444e59 q^{67} +5.64366e61 q^{73} -7.53393e61 q^{75} -6.14497e61 q^{76} +6.24528e62 q^{79} +9.55005e62 q^{81} +6.34126e63 q^{84} +1.69958e65 q^{91} +1.15793e65 q^{93} +2.44251e65 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −5.55906e15 −1.00000
\(4\) 7.37870e19 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.54595e28 −1.99967 −0.999837 0.0180798i \(-0.994245\pi\)
−0.999837 + 0.0180798i \(0.994245\pi\)
\(8\) 0 0
\(9\) 3.09032e31 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −4.10186e35 −1.00000
\(13\) −1.09938e37 −1.90993 −0.954966 0.296716i \(-0.904108\pi\)
−0.954966 + 0.296716i \(0.904108\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.44452e39 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −8.32798e41 −0.526831 −0.263415 0.964683i \(-0.584849\pi\)
−0.263415 + 0.964683i \(0.584849\pi\)
\(20\) 0 0
\(21\) 8.59401e43 1.99967
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.35525e46 1.00000
\(26\) 0 0
\(27\) −1.71793e47 −1.00000
\(28\) −1.14071e48 −1.99967
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.08297e49 −1.26983 −0.634917 0.772581i \(-0.718966\pi\)
−0.634917 + 0.772581i \(0.718966\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.28025e51 1.00000
\(37\) −7.41794e51 −1.31712 −0.658562 0.752527i \(-0.728835\pi\)
−0.658562 + 0.752527i \(0.728835\pi\)
\(38\) 0 0
\(39\) 6.11153e52 1.90993
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.24524e54 −1.55165 −0.775825 0.630948i \(-0.782666\pi\)
−0.775825 + 0.630948i \(0.782666\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −3.02664e55 −1.00000
\(49\) 1.79227e56 2.99869
\(50\) 0 0
\(51\) 0 0
\(52\) −8.11200e56 −1.90993
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.62958e57 0.526831
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 7.67187e58 0.931143 0.465572 0.885010i \(-0.345849\pi\)
0.465572 + 0.885010i \(0.345849\pi\)
\(62\) 0 0
\(63\) −4.77746e59 −1.99967
\(64\) 4.01735e59 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.36444e59 0.129795 0.0648977 0.997892i \(-0.479328\pi\)
0.0648977 + 0.997892i \(0.479328\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.64366e61 1.82771 0.913853 0.406046i \(-0.133093\pi\)
0.913853 + 0.406046i \(0.133093\pi\)
\(74\) 0 0
\(75\) −7.53393e61 −1.00000
\(76\) −6.14497e61 −0.526831
\(77\) 0 0
\(78\) 0 0
\(79\) 6.24528e62 1.49231 0.746153 0.665775i \(-0.231899\pi\)
0.746153 + 0.665775i \(0.231899\pi\)
\(80\) 0 0
\(81\) 9.55005e62 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 6.34126e63 1.99967
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.69958e65 3.81924
\(92\) 0 0
\(93\) 1.15793e65 1.26983
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.44251e65 0.667374 0.333687 0.942684i \(-0.391707\pi\)
0.333687 + 0.942684i \(0.391707\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e66 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.68709e66 0.636078 0.318039 0.948078i \(-0.396976\pi\)
0.318039 + 0.948078i \(0.396976\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.26760e67 −1.00000
\(109\) −3.43242e67 −1.99768 −0.998840 0.0481558i \(-0.984666\pi\)
−0.998840 + 0.0481558i \(0.984666\pi\)
\(110\) 0 0
\(111\) 4.12368e67 1.31712
\(112\) −8.41693e67 −1.99967
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.39744e68 −1.90993
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.39408e68 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.53696e69 −1.26983
\(125\) 0 0
\(126\) 0 0
\(127\) 5.20818e69 1.95507 0.977537 0.210762i \(-0.0675946\pi\)
0.977537 + 0.210762i \(0.0675946\pi\)
\(128\) 0 0
\(129\) 6.92236e69 1.55165
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.28746e70 1.05349
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −5.27590e70 −1.00649 −0.503244 0.864145i \(-0.667860\pi\)
−0.503244 + 0.864145i \(0.667860\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.68253e71 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −9.96332e71 −2.99869
\(148\) −5.47347e71 −1.31712
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 3.28971e71 0.408243 0.204121 0.978946i \(-0.434566\pi\)
0.204121 + 0.978946i \(0.434566\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 4.50951e72 1.90993
\(157\) −8.54429e71 −0.293081 −0.146540 0.989205i \(-0.546814\pi\)
−0.146540 + 0.989205i \(0.546814\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.15465e73 1.14884 0.574419 0.818561i \(-0.305228\pi\)
0.574419 + 0.818561i \(0.305228\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.77309e73 2.64784
\(170\) 0 0
\(171\) −2.57361e73 −0.526831
\(172\) −9.18825e73 −1.55165
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −2.09515e74 −1.99967
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −5.80266e73 −0.182068 −0.0910341 0.995848i \(-0.529017\pi\)
−0.0910341 + 0.995848i \(0.529017\pi\)
\(182\) 0 0
\(183\) −4.26484e74 −0.931143
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.65582e75 1.99967
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −2.23327e75 −1.00000
\(193\) −2.25295e74 −0.0849887 −0.0424943 0.999097i \(-0.513530\pi\)
−0.0424943 + 0.999097i \(0.513530\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.32246e76 2.99869
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.88546e75 −0.533692 −0.266846 0.963739i \(-0.585982\pi\)
−0.266846 + 0.963739i \(0.585982\pi\)
\(200\) 0 0
\(201\) −1.31440e75 −0.129795
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −5.98560e76 −1.90993
\(209\) 0 0
\(210\) 0 0
\(211\) −7.03627e76 −1.39966 −0.699829 0.714311i \(-0.746740\pi\)
−0.699829 + 0.714311i \(0.746740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.22016e77 2.53925
\(218\) 0 0
\(219\) −3.13734e77 −1.82771
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −6.59547e76 −0.211439 −0.105720 0.994396i \(-0.533715\pi\)
−0.105720 + 0.994396i \(0.533715\pi\)
\(224\) 0 0
\(225\) 4.18816e77 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 3.41602e77 0.526831
\(229\) 1.42980e78 1.90855 0.954274 0.298932i \(-0.0966305\pi\)
0.954274 + 0.298932i \(0.0966305\pi\)
\(230\) 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
\(237\) −3.47179e78 −1.49231
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −6.92414e78 −1.71318 −0.856591 0.515995i \(-0.827422\pi\)
−0.856591 + 0.515995i \(0.827422\pi\)
\(242\) 0 0
\(243\) −5.30893e78 −1.00000
\(244\) 5.66084e78 0.931143
\(245\) 0 0
\(246\) 0 0
\(247\) 9.15563e78 1.00621
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −3.52514e79 −1.99967
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.96428e79 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.14677e80 2.63382
\(260\) 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
\(268\) 1.74465e79 0.129795
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 1.63267e80 0.841223 0.420612 0.907241i \(-0.361816\pi\)
0.420612 + 0.907241i \(0.361816\pi\)
\(272\) 0 0
\(273\) −9.44809e80 −3.81924
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.91183e80 1.72886 0.864432 0.502749i \(-0.167678\pi\)
0.864432 + 0.502749i \(0.167678\pi\)
\(278\) 0 0
\(279\) −6.43703e80 −1.26983
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −4.86375e80 −0.599818 −0.299909 0.953968i \(-0.596956\pi\)
−0.299909 + 0.953968i \(0.596956\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.62042e81 1.00000
\(290\) 0 0
\(291\) −1.35781e81 −0.667374
\(292\) 4.16428e81 1.82771
\(293\) 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
\(300\) −5.55906e81 −1.00000
\(301\) 1.92507e82 3.10279
\(302\) 0 0
\(303\) 0 0
\(304\) −4.53419e81 −0.526831
\(305\) 0 0
\(306\) 0 0
\(307\) −2.36245e82 −1.98516 −0.992580 0.121596i \(-0.961199\pi\)
−0.992580 + 0.121596i \(0.961199\pi\)
\(308\) 0 0
\(309\) −9.37864e81 −0.636078
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −4.30323e82 −1.90911 −0.954556 0.298031i \(-0.903670\pi\)
−0.954556 + 0.298031i \(0.903670\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 4.60820e82 1.49231
\(317\) 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
\(324\) 7.04669e82 1.00000
\(325\) −1.48994e83 −1.90993
\(326\) 0 0
\(327\) 1.90810e83 1.99768
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.87323e82 0.481767 0.240884 0.970554i \(-0.422563\pi\)
0.240884 + 0.970554i \(0.422563\pi\)
\(332\) 0 0
\(333\) −2.29238e83 −1.31712
\(334\) 0 0
\(335\) 0 0
\(336\) 4.67902e83 1.99967
\(337\) 4.73696e83 1.83533 0.917663 0.397360i \(-0.130074\pi\)
0.917663 + 0.397360i \(0.130074\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.84676e84 −3.99673
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.59971e84 −1.95344 −0.976722 0.214509i \(-0.931185\pi\)
−0.976722 + 0.214509i \(0.931185\pi\)
\(350\) 0 0
\(351\) 1.88865e84 1.90993
\(352\) 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
\(361\) −1.80528e84 −0.722449
\(362\) 0 0
\(363\) −2.99860e84 −1.00000
\(364\) 1.25407e85 3.81924
\(365\) 0 0
\(366\) 0 0
\(367\) −7.00892e84 −1.62806 −0.814031 0.580821i \(-0.802732\pi\)
−0.814031 + 0.580821i \(0.802732\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 8.54405e84 1.26983
\(373\) 1.15803e85 1.57518 0.787589 0.616201i \(-0.211329\pi\)
0.787589 + 0.616201i \(0.211329\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.14770e85 −0.922004 −0.461002 0.887399i \(-0.652510\pi\)
−0.461002 + 0.887399i \(0.652510\pi\)
\(380\) 0 0
\(381\) −2.89526e85 −1.95507
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.84818e85 −1.55165
\(388\) 1.80226e85 0.667374
\(389\) 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
\(397\) 9.84148e84 0.170991 0.0854956 0.996339i \(-0.472753\pi\)
0.0854956 + 0.996339i \(0.472753\pi\)
\(398\) 0 0
\(399\) −7.15708e85 −1.05349
\(400\) 7.37870e85 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 2.28998e86 2.42529
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.43225e86 −0.931428 −0.465714 0.884935i \(-0.654202\pi\)
−0.465714 + 0.884935i \(0.654202\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.24485e86 0.636078
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.93291e86 1.00649
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −7.48802e86 −1.87524 −0.937622 0.347656i \(-0.886978\pi\)
−0.937622 + 0.347656i \(0.886978\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.18603e87 −1.86198
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −9.35327e86 −1.00000
\(433\) 2.01798e87 1.99901 0.999507 0.0313819i \(-0.00999081\pi\)
0.999507 + 0.0313819i \(0.00999081\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.53268e87 −1.99768
\(437\) 0 0
\(438\) 0 0
\(439\) 1.23295e87 0.775559 0.387780 0.921752i \(-0.373242\pi\)
0.387780 + 0.921752i \(0.373242\pi\)
\(440\) 0 0
\(441\) 5.53867e87 2.99869
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 3.04274e87 1.31712
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −6.21060e87 −1.99967
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.82877e87 −0.408243
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.18784e88 1.98392 0.991960 0.126549i \(-0.0403900\pi\)
0.991960 + 0.126549i \(0.0403900\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.62965e88 −1.76979 −0.884893 0.465795i \(-0.845769\pi\)
−0.884893 + 0.465795i \(0.845769\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −2.50686e88 −1.90993
\(469\) −3.65529e87 −0.259548
\(470\) 0 0
\(471\) 4.74982e87 0.293081
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.12865e88 −0.526831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 8.15514e88 2.51561
\(482\) 0 0
\(483\) 0 0
\(484\) 3.98013e88 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 4.44721e88 0.911238 0.455619 0.890175i \(-0.349418\pi\)
0.455619 + 0.890175i \(0.349418\pi\)
\(488\) 0 0
\(489\) −6.41876e88 −1.14884
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.13408e89 −1.26983
\(497\) 0 0
\(498\) 0 0
\(499\) 1.65142e89 1.51545 0.757723 0.652576i \(-0.226312\pi\)
0.757723 + 0.652576i \(0.226312\pi\)
\(500\) 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
\(507\) −4.87701e89 −2.64784
\(508\) 3.84296e89 1.95507
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −8.72479e89 −3.65481
\(512\) 0 0
\(513\) 1.43069e89 0.526831
\(514\) 0 0
\(515\) 0 0
\(516\) 5.10780e89 1.55165
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −5.86692e89 −1.14250 −0.571248 0.820778i \(-0.693540\pi\)
−0.571248 + 0.820778i \(0.693540\pi\)
\(524\) 0 0
\(525\) 1.16471e90 1.99967
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.48234e89 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 9.49979e89 1.05349
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.83632e90 −1.80820 −0.904099 0.427323i \(-0.859457\pi\)
−0.904099 + 0.427323i \(0.859457\pi\)
\(542\) 0 0
\(543\) 3.22573e89 0.182068
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7.17956e89 −0.318065 −0.159033 0.987273i \(-0.550837\pi\)
−0.159033 + 0.987273i \(0.550837\pi\)
\(548\) 0 0
\(549\) 2.37085e90 0.931143
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −9.65486e90 −2.98412
\(554\) 0 0
\(555\) 0 0
\(556\) −3.89293e90 −1.00649
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.36899e91 2.96355
\(560\) 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
\(567\) −1.47639e91 −1.99967
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.59715e91 1.71536 0.857679 0.514186i \(-0.171906\pi\)
0.857679 + 0.514186i \(0.171906\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.24149e91 1.00000
\(577\) 2.35499e91 1.79138 0.895690 0.444679i \(-0.146682\pi\)
0.895690 + 0.444679i \(0.146682\pi\)
\(578\) 0 0
\(579\) 1.25243e90 0.0849887
\(580\) 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 \(\pi\)
\(588\) −7.35163e91 −2.99869
\(589\) 1.73469e91 0.668987
\(590\) 0 0
\(591\) 0 0
\(592\) −4.03871e91 −1.31712
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.15995e91 0.533692
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −6.95992e91 −1.37958 −0.689790 0.724009i \(-0.742297\pi\)
−0.689790 + 0.724009i \(0.742297\pi\)
\(602\) 0 0
\(603\) 7.30685e90 0.129795
\(604\) 2.42738e91 0.408243
\(605\) 0 0
\(606\) 0 0
\(607\) 1.08037e92 1.54293 0.771466 0.636271i \(-0.219524\pi\)
0.771466 + 0.636271i \(0.219524\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.27210e92 −1.31318 −0.656592 0.754246i \(-0.728003\pi\)
−0.656592 + 0.754246i \(0.728003\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.39663e92 1.04542 0.522710 0.852511i \(-0.324921\pi\)
0.522710 + 0.852511i \(0.324921\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 3.32743e92 1.90993
\(625\) 1.83671e92 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −6.30457e91 −0.293081
\(629\) 0 0
\(630\) 0 0
\(631\) −1.27980e92 −0.508361 −0.254181 0.967157i \(-0.581806\pi\)
−0.254181 + 0.967157i \(0.581806\pi\)
\(632\) 0 0
\(633\) 3.91151e92 1.39966
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.97038e93 −5.72730
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 8.99501e92 1.91884 0.959422 0.281974i \(-0.0909892\pi\)
0.959422 + 0.281974i \(0.0909892\pi\)
\(644\) 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
\(651\) −1.79010e93 −2.53925
\(652\) 8.51980e92 1.14884
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.74407e93 1.82771
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.13909e92 0.183477 0.0917384 0.995783i \(-0.470758\pi\)
0.0917384 + 0.995783i \(0.470758\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.66646e92 0.211439
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.25672e93 1.06901 0.534504 0.845166i \(-0.320499\pi\)
0.534504 + 0.845166i \(0.320499\pi\)
\(674\) 0 0
\(675\) −2.32822e93 −1.00000
\(676\) 6.47340e93 2.64784
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −3.77599e93 −1.33453
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −1.89899e93 −0.526831
\(685\) 0 0
\(686\) 0 0
\(687\) −7.94832e93 −1.90855
\(688\) −6.77973e93 −1.55165
\(689\) 0 0
\(690\) 0 0
\(691\) 7.98537e93 1.58314 0.791570 0.611078i \(-0.209264\pi\)
0.791570 + 0.611078i \(0.209264\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.54595e94 −1.99967
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 6.17765e93 0.693901
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.32423e94 1.97222 0.986108 0.166108i \(-0.0531201\pi\)
0.986108 + 0.166108i \(0.0531201\pi\)
\(710\) 0 0
\(711\) 1.92999e94 1.49231
\(712\) 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
\(721\) −2.60815e94 −1.27195
\(722\) 0 0
\(723\) 3.84917e94 1.71318
\(724\) −4.28161e93 −0.182068
\(725\) 0 0
\(726\) 0 0
\(727\) −5.32643e94 −1.97606 −0.988031 0.154253i \(-0.950703\pi\)
−0.988031 + 0.154253i \(0.950703\pi\)
\(728\) 0 0
\(729\) 2.95127e94 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −3.14690e94 −0.931143
\(733\) 4.56664e94 1.29171 0.645856 0.763459i \(-0.276501\pi\)
0.645856 + 0.763459i \(0.276501\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −6.00129e94 −1.29711 −0.648557 0.761166i \(-0.724627\pi\)
−0.648557 + 0.761166i \(0.724627\pi\)
\(740\) 0 0
\(741\) −5.08967e94 −1.00621
\(742\) 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
\(751\) −1.13008e95 −1.43545 −0.717727 0.696324i \(-0.754818\pi\)
−0.717727 + 0.696324i \(0.754818\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 1.95965e95 1.99967
\(757\) −9.94070e94 −0.971075 −0.485538 0.874216i \(-0.661376\pi\)
−0.485538 + 0.874216i \(0.661376\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 5.30633e95 3.99471
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.64786e95 −1.00000
\(769\) 2.49117e95 1.44822 0.724108 0.689686i \(-0.242252\pi\)
0.724108 + 0.689686i \(0.242252\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.66239e94 −0.0849887
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −2.82295e95 −1.26983
\(776\) 0 0
\(777\) −6.37498e95 −2.63382
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.75803e95 2.99869
\(785\) 0 0
\(786\) 0 0
\(787\) −4.37843e95 −1.18618 −0.593092 0.805135i \(-0.702093\pi\)
−0.593092 + 0.805135i \(0.702093\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −8.43431e95 −1.77842
\(794\) 0 0
\(795\) 0 0
\(796\) −2.86696e95 −0.533692
\(797\) 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
\(804\) −9.69859e94 −0.129795
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1.78605e96 1.79559 0.897793 0.440418i \(-0.145170\pi\)
0.897793 + 0.440418i \(0.145170\pi\)
\(812\) 0 0
\(813\) −9.07610e95 −0.841223
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.03703e96 0.817457
\(818\) 0 0
\(819\) 5.25225e96 3.81924
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −2.79620e96 −1.73131 −0.865653 0.500644i \(-0.833097\pi\)
−0.865653 + 0.500644i \(0.833097\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 7.70745e95 0.375501 0.187751 0.982217i \(-0.439880\pi\)
0.187751 + 0.982217i \(0.439880\pi\)
\(830\) 0 0
\(831\) −3.84233e96 −1.72886
\(832\) −4.41659e96 −1.90993
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.57838e96 1.26983
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.29813e96 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −5.19185e96 −1.39966
\(845\) 0 0
\(846\) 0 0
\(847\) −8.33895e96 −1.99967
\(848\) 0 0
\(849\) 2.70379e96 0.599818
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 7.73144e94 0.0146873 0.00734366 0.999973i \(-0.497662\pi\)
0.00734366 + 0.999973i \(0.497662\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 7.73990e96 1.16670 0.583352 0.812219i \(-0.301741\pi\)
0.583352 + 0.812219i \(0.301741\pi\)
\(860\) 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
\(867\) −9.00804e96 −1.00000
\(868\) 2.37606e97 2.53925
\(869\) 0 0
\(870\) 0 0
\(871\) −2.59942e96 −0.247900
\(872\) 0 0
\(873\) 7.54813e96 0.667374
\(874\) 0 0
\(875\) 0 0
\(876\) −2.31495e97 −1.82771
\(877\) 2.32641e97 1.76888 0.884442 0.466650i \(-0.154539\pi\)
0.884442 + 0.466650i \(0.154539\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.86937e97 1.74214 0.871070 0.491158i \(-0.163426\pi\)
0.871070 + 0.491158i \(0.163426\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −8.05157e97 −3.90951
\(890\) 0 0
\(891\) 0 0
\(892\) −4.86660e96 −0.211439
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 3.09032e97 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −1.07016e98 −3.10279
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.50162e97 1.37864 0.689319 0.724458i \(-0.257910\pi\)
0.689319 + 0.724458i \(0.257910\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 2.52058e97 0.526831
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.05500e98 1.90855
\(917\) 0 0
\(918\) 0 0
\(919\) −5.71068e97 −0.927419 −0.463709 0.885987i \(-0.653482\pi\)
−0.463709 + 0.885987i \(0.653482\pi\)
\(920\) 0 0
\(921\) 1.31330e98 1.98516
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00532e98 −1.31712
\(926\) 0 0
\(927\) 5.21365e97 0.636078
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.49260e98 −1.57980
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.82408e98 1.56184 0.780921 0.624630i \(-0.214750\pi\)
0.780921 + 0.624630i \(0.214750\pi\)
\(938\) 0 0
\(939\) 2.39219e98 1.90911
\(940\) 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 \(\pi\)
\(948\) −2.56173e98 −1.49231
\(949\) −6.20453e98 −3.49079
\(950\) 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
\(961\) 1.64802e98 0.612477
\(962\) 0 0
\(963\) 0 0
\(964\) −5.10912e98 −1.71318
\(965\) 0 0
\(966\) 0 0
\(967\) 6.12966e98 1.85508 0.927540 0.373724i \(-0.121919\pi\)
0.927540 + 0.373724i \(0.121919\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −3.91730e98 −1.00000
\(973\) 8.15626e98 2.01265
\(974\) 0 0
\(975\) 8.28266e98 1.90993
\(976\) 4.17696e98 0.931143
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.06073e99 −1.99768
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 6.75566e98 1.00621
\(989\) 0 0
\(990\) 0 0
\(991\) −1.02882e99 −1.38646 −0.693231 0.720716i \(-0.743813\pi\)
−0.693231 + 0.720716i \(0.743813\pi\)
\(992\) 0 0
\(993\) −3.82087e98 −0.481767
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.68647e98 0.959187 0.479593 0.877491i \(-0.340784\pi\)
0.479593 + 0.877491i \(0.340784\pi\)
\(998\) 0 0
\(999\) 1.27435e99 1.31712
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 3.67.b.a.2.1 1
3.2 odd 2 CM 3.67.b.a.2.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.67.b.a.2.1 1 1.1 even 1 trivial
3.67.b.a.2.1 1 3.2 odd 2 CM