Properties

Label 83.13.b.a.82.1
Level $83$
Weight $13$
Character 83.82
Self dual yes
Analytic conductor $75.861$
Analytic rank $0$
Dimension $1$
CM discriminant -83
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(75.8614868339\)
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 82.1
Character \(\chi\) \(=\) 83.82

$q$-expansion

\(f(q)\) \(=\) \(q-617.000 q^{3} +4096.00 q^{4} -231577. q^{7} -150752. q^{9} +O(q^{10})\) \(q-617.000 q^{3} +4096.00 q^{4} -231577. q^{7} -150752. q^{9} -3.19855e6 q^{11} -2.52723e6 q^{12} +1.67772e7 q^{16} -3.06267e7 q^{17} +1.42883e8 q^{21} -2.31011e8 q^{23} +2.44141e8 q^{25} +4.20913e8 q^{27} -9.48539e8 q^{28} +9.91524e8 q^{29} -1.01625e8 q^{31} +1.97351e9 q^{33} -6.17480e8 q^{36} -5.01223e9 q^{37} -9.47479e9 q^{41} -1.31013e10 q^{44} -1.03515e10 q^{48} +3.97866e10 q^{49} +1.88967e10 q^{51} -4.86714e10 q^{59} +8.64340e10 q^{61} +3.49107e10 q^{63} +6.87195e10 q^{64} -1.25447e11 q^{68} +1.42534e11 q^{69} -1.50635e11 q^{75} +7.40711e11 q^{77} -1.79588e11 q^{81} +3.26940e11 q^{83} +5.85249e11 q^{84} -6.11770e11 q^{87} -9.46223e11 q^{92} +6.27024e10 q^{93} +4.82188e11 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/83\mathbb{Z}\right)^\times\).

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

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 \(\pi\)
\(3\) −617.000 −0.846365 −0.423182 0.906044i \(-0.639087\pi\)
−0.423182 + 0.906044i \(0.639087\pi\)
\(4\) 4096.00 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −231577. −1.96837 −0.984186 0.177138i \(-0.943316\pi\)
−0.984186 + 0.177138i \(0.943316\pi\)
\(8\) 0 0
\(9\) −150752. −0.283666
\(10\) 0 0
\(11\) −3.19855e6 −1.80550 −0.902750 0.430166i \(-0.858455\pi\)
−0.902750 + 0.430166i \(0.858455\pi\)
\(12\) −2.52723e6 −0.846365
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.67772e7 1.00000
\(17\) −3.06267e7 −1.26884 −0.634420 0.772988i \(-0.718761\pi\)
−0.634420 + 0.772988i \(0.718761\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 1.42883e8 1.66596
\(22\) 0 0
\(23\) −2.31011e8 −1.56051 −0.780255 0.625462i \(-0.784911\pi\)
−0.780255 + 0.625462i \(0.784911\pi\)
\(24\) 0 0
\(25\) 2.44141e8 1.00000
\(26\) 0 0
\(27\) 4.20913e8 1.08645
\(28\) −9.48539e8 −1.96837
\(29\) 9.91524e8 1.66692 0.833461 0.552579i \(-0.186356\pi\)
0.833461 + 0.552579i \(0.186356\pi\)
\(30\) 0 0
\(31\) −1.01625e8 −0.114506 −0.0572531 0.998360i \(-0.518234\pi\)
−0.0572531 + 0.998360i \(0.518234\pi\)
\(32\) 0 0
\(33\) 1.97351e9 1.52811
\(34\) 0 0
\(35\) 0 0
\(36\) −6.17480e8 −0.283666
\(37\) −5.01223e9 −1.95353 −0.976766 0.214309i \(-0.931250\pi\)
−0.976766 + 0.214309i \(0.931250\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.47479e9 −1.99465 −0.997324 0.0731073i \(-0.976708\pi\)
−0.997324 + 0.0731073i \(0.976708\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −1.31013e10 −1.80550
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.03515e10 −0.846365
\(49\) 3.97866e10 2.87449
\(50\) 0 0
\(51\) 1.88967e10 1.07390
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.86714e10 −1.15388 −0.576942 0.816785i \(-0.695754\pi\)
−0.576942 + 0.816785i \(0.695754\pi\)
\(60\) 0 0
\(61\) 8.64340e10 1.67767 0.838833 0.544388i \(-0.183238\pi\)
0.838833 + 0.544388i \(0.183238\pi\)
\(62\) 0 0
\(63\) 3.49107e10 0.558361
\(64\) 6.87195e10 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.25447e11 −1.26884
\(69\) 1.42534e11 1.32076
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1.50635e11 −0.846365
\(76\) 0 0
\(77\) 7.40711e11 3.55390
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.79588e11 −0.635867
\(82\) 0 0
\(83\) 3.26940e11 1.00000
\(84\) 5.85249e11 1.66596
\(85\) 0 0
\(86\) 0 0
\(87\) −6.11770e11 −1.41082
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.46223e11 −1.56051
\(93\) 6.27024e10 0.0969141
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 4.82188e11 0.512160
\(100\) 1.00000e12 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.72406e12 1.08645
\(109\) 1.15371e12 0.687917 0.343959 0.938985i \(-0.388232\pi\)
0.343959 + 0.938985i \(0.388232\pi\)
\(110\) 0 0
\(111\) 3.09254e12 1.65340
\(112\) −3.88522e12 −1.96837
\(113\) −2.02812e12 −0.974145 −0.487073 0.873362i \(-0.661935\pi\)
−0.487073 + 0.873362i \(0.661935\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.06128e12 1.66692
\(117\) 0 0
\(118\) 0 0
\(119\) 7.09245e12 2.49755
\(120\) 0 0
\(121\) 7.09231e12 2.25983
\(122\) 0 0
\(123\) 5.84594e12 1.68820
\(124\) −4.16255e11 −0.114506
\(125\) 0 0
\(126\) 0 0
\(127\) −6.15236e12 −1.46629 −0.733144 0.680074i \(-0.761948\pi\)
−0.733144 + 0.680074i \(0.761948\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00340e12 0.396405 0.198202 0.980161i \(-0.436490\pi\)
0.198202 + 0.980161i \(0.436490\pi\)
\(132\) 8.08349e12 1.52811
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.52920e12 −0.283666
\(145\) 0 0
\(146\) 0 0
\(147\) −2.45483e13 −2.43287
\(148\) −2.05301e13 −1.95353
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.47487e13 −1.24421 −0.622104 0.782934i \(-0.713722\pi\)
−0.622104 + 0.782934i \(0.713722\pi\)
\(152\) 0 0
\(153\) 4.61704e12 0.359928
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.34969e13 3.07166
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −3.88087e13 −1.99465
\(165\) 0 0
\(166\) 0 0
\(167\) 2.84875e13 1.31327 0.656637 0.754207i \(-0.271978\pi\)
0.656637 + 0.754207i \(0.271978\pi\)
\(168\) 0 0
\(169\) 2.32981e13 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.25936e12 −0.345386 −0.172693 0.984976i \(-0.555247\pi\)
−0.172693 + 0.984976i \(0.555247\pi\)
\(174\) 0 0
\(175\) −5.65374e13 −1.96837
\(176\) −5.36628e13 −1.80550
\(177\) 3.00303e13 0.976607
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −5.33298e13 −1.41992
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.79612e13 2.29089
\(188\) 0 0
\(189\) −9.74738e13 −2.13854
\(190\) 0 0
\(191\) −3.84442e13 −0.791828 −0.395914 0.918288i \(-0.629572\pi\)
−0.395914 + 0.918288i \(0.629572\pi\)
\(192\) −4.23999e13 −0.846365
\(193\) −5.84052e13 −1.13008 −0.565038 0.825065i \(-0.691138\pi\)
−0.565038 + 0.825065i \(0.691138\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.62966e14 2.87449
\(197\) 1.16897e14 1.99988 0.999941 0.0108496i \(-0.00345360\pi\)
0.999941 + 0.0108496i \(0.00345360\pi\)
\(198\) 0 0
\(199\) −1.23113e14 −1.98237 −0.991186 0.132476i \(-0.957707\pi\)
−0.991186 + 0.132476i \(0.957707\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.29614e14 −3.28112
\(204\) 7.74009e13 1.07390
\(205\) 0 0
\(206\) 0 0
\(207\) 3.48254e13 0.442664
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.35339e13 0.225391
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −3.68047e13 −0.283666
\(226\) 0 0
\(227\) 1.68465e13 0.123127 0.0615635 0.998103i \(-0.480391\pi\)
0.0615635 + 0.998103i \(0.480391\pi\)
\(228\) 0 0
\(229\) 1.66485e14 1.15442 0.577208 0.816597i \(-0.304142\pi\)
0.577208 + 0.816597i \(0.304142\pi\)
\(230\) 0 0
\(231\) −4.57019e14 −3.00789
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.99358e14 −1.15388
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −2.40833e14 −1.22917 −0.614587 0.788849i \(-0.710677\pi\)
−0.614587 + 0.788849i \(0.710677\pi\)
\(242\) 0 0
\(243\) −1.12885e14 −0.548275
\(244\) 3.54034e14 1.67767
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −2.01722e14 −0.846365
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.42994e14 0.558361
\(253\) 7.38902e14 2.81750
\(254\) 0 0
\(255\) 0 0
\(256\) 2.81475e14 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.16072e15 3.84528
\(260\) 0 0
\(261\) −1.49474e14 −0.472850
\(262\) 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
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −5.13831e14 −1.26884
\(273\) 0 0
\(274\) 0 0
\(275\) −7.80897e14 −1.80550
\(276\) 5.83819e14 1.32076
\(277\) 4.83137e14 1.06953 0.534764 0.845002i \(-0.320401\pi\)
0.534764 + 0.845002i \(0.320401\pi\)
\(278\) 0 0
\(279\) 1.53201e13 0.0324816
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.19414e15 3.92621
\(288\) 0 0
\(289\) 3.55375e14 0.609957
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.26021e15 −1.99177 −0.995883 0.0906530i \(-0.971105\pi\)
−0.995883 + 0.0906530i \(0.971105\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.34631e15 −1.96159
\(298\) 0 0
\(299\) 0 0
\(300\) −6.17000e14 −0.846365
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 3.03395e15 3.55390
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −3.61810e14 −0.384782 −0.192391 0.981318i \(-0.561624\pi\)
−0.192391 + 0.981318i \(0.561624\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.97665e15 1.94793 0.973965 0.226697i \(-0.0727927\pi\)
0.973965 + 0.226697i \(0.0727927\pi\)
\(318\) 0 0
\(319\) −3.17144e15 −3.00963
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −7.35591e14 −0.635867
\(325\) 0 0
\(326\) 0 0
\(327\) −7.11836e14 −0.582229
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.33915e15 1.00000
\(333\) 7.55603e14 0.554152
\(334\) 0 0
\(335\) 0 0
\(336\) 2.39718e15 1.66596
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 1.25135e15 0.824482
\(340\) 0 0
\(341\) 3.25052e14 0.206741
\(342\) 0 0
\(343\) −6.00834e15 −3.68969
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −2.50581e15 −1.41082
\(349\) 3.61078e15 1.99824 0.999121 0.0419086i \(-0.0133438\pi\)
0.999121 + 0.0419086i \(0.0133438\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.11221e15 −0.574827 −0.287413 0.957807i \(-0.592795\pi\)
−0.287413 + 0.957807i \(0.592795\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.37604e15 −2.11384
\(358\) 0 0
\(359\) −3.67844e15 −1.71829 −0.859145 0.511732i \(-0.829004\pi\)
−0.859145 + 0.511732i \(0.829004\pi\)
\(360\) 0 0
\(361\) 2.21331e15 1.00000
\(362\) 0 0
\(363\) −4.37596e15 −1.91264
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −3.87573e15 −1.56051
\(369\) 1.42834e15 0.565815
\(370\) 0 0
\(371\) 0 0
\(372\) 2.56829e14 0.0969141
\(373\) −2.31439e12 −0.000859377 0 −0.000429689 1.00000i \(-0.500137\pi\)
−0.000429689 1.00000i \(0.500137\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 3.79600e15 1.24101
\(382\) 0 0
\(383\) −3.43347e15 −1.08778 −0.543890 0.839157i \(-0.683049\pi\)
−0.543890 + 0.839157i \(0.683049\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 7.07513e15 1.98004
\(392\) 0 0
\(393\) −1.23609e15 −0.335503
\(394\) 0 0
\(395\) 0 0
\(396\) 1.97504e15 0.512160
\(397\) −7.34095e15 −1.87503 −0.937517 0.347939i \(-0.886882\pi\)
−0.937517 + 0.347939i \(0.886882\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.09600e15 1.00000
\(401\) −7.69827e15 −1.85151 −0.925756 0.378120i \(-0.876571\pi\)
−0.925756 + 0.378120i \(0.876571\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.60319e16 3.52710
\(408\) 0 0
\(409\) −9.35838e15 −1.99922 −0.999611 0.0278991i \(-0.991118\pi\)
−0.999611 + 0.0278991i \(0.991118\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.12712e16 2.27127
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.37973e15 0.809400 0.404700 0.914450i \(-0.367376\pi\)
0.404700 + 0.914450i \(0.367376\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.47723e15 −1.26884
\(426\) 0 0
\(427\) −2.00161e16 −3.30227
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.48443e15 −1.01160 −0.505800 0.862651i \(-0.668802\pi\)
−0.505800 + 0.862651i \(0.668802\pi\)
\(432\) 7.06175e15 1.08645
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.72558e15 0.687917
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −5.99791e15 −0.815396
\(442\) 0 0
\(443\) −1.04815e16 −1.38675 −0.693377 0.720575i \(-0.743878\pi\)
−0.693377 + 0.720575i \(0.743878\pi\)
\(444\) 1.26671e16 1.65340
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.59139e16 −1.96837
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 3.03056e16 3.60134
\(452\) −8.30719e15 −0.974145
\(453\) 9.09997e15 1.05305
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.28912e16 −1.37853
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.81377e16 1.84118 0.920590 0.390531i \(-0.127709\pi\)
0.920590 + 0.390531i \(0.127709\pi\)
\(464\) 1.66350e16 1.66692
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 2.90507e16 2.49755
\(477\) 0 0
\(478\) 0 0
\(479\) 3.28278e15 0.271787 0.135894 0.990723i \(-0.456609\pi\)
0.135894 + 0.990723i \(0.456609\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −3.30076e16 −2.59975
\(484\) 2.90501e16 2.25983
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 2.39450e16 1.68820
\(493\) −3.03671e16 −2.11506
\(494\) 0 0
\(495\) 0 0
\(496\) −1.70498e15 −0.114506
\(497\) 0 0
\(498\) 0 0
\(499\) 3.04651e16 1.97333 0.986664 0.162773i \(-0.0520438\pi\)
0.986664 + 0.162773i \(0.0520438\pi\)
\(500\) 0 0
\(501\) −1.75768e16 −1.11151
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.43749e16 −0.846365
\(508\) −2.52001e16 −1.46629
\(509\) 4.13069e15 0.237529 0.118764 0.992922i \(-0.462107\pi\)
0.118764 + 0.992922i \(0.462107\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 5.71303e15 0.292322
\(520\) 0 0
\(521\) 2.81773e16 1.40888 0.704440 0.709764i \(-0.251198\pi\)
0.704440 + 0.709764i \(0.251198\pi\)
\(522\) 0 0
\(523\) 3.20950e16 1.56829 0.784147 0.620576i \(-0.213101\pi\)
0.784147 + 0.620576i \(0.213101\pi\)
\(524\) 8.20591e15 0.396405
\(525\) 3.48835e16 1.66596
\(526\) 0 0
\(527\) 3.11243e15 0.145290
\(528\) 3.31100e16 1.52811
\(529\) 3.14517e16 1.43519
\(530\) 0 0
\(531\) 7.33732e15 0.327318
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.27260e17 −5.18989
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.18837e16 −1.93690 −0.968449 0.249212i \(-0.919829\pi\)
−0.968449 + 0.249212i \(0.919829\pi\)
\(548\) 0 0
\(549\) −1.30301e16 −0.475898
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.38611e16 −0.799023 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −6.04421e16 −1.93893
\(562\) 0 0
\(563\) 5.02440e15 0.157773 0.0788867 0.996884i \(-0.474863\pi\)
0.0788867 + 0.996884i \(0.474863\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.15884e16 1.25162
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 2.37201e16 0.670175
\(574\) 0 0
\(575\) −5.63993e16 −1.56051
\(576\) −1.03596e16 −0.283666
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 3.60360e16 0.956456
\(580\) 0 0
\(581\) −7.57119e16 −1.96837
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −1.00550e17 −2.43287
\(589\) 0 0
\(590\) 0 0
\(591\) −7.21252e16 −1.69263
\(592\) −8.40912e16 −1.95353
\(593\) 1.08476e16 0.249464 0.124732 0.992190i \(-0.460193\pi\)
0.124732 + 0.992190i \(0.460193\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.59607e16 1.67781
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −6.04108e16 −1.24421
\(605\) 0 0
\(606\) 0 0
\(607\) 7.16482e16 1.43243 0.716214 0.697881i \(-0.245873\pi\)
0.716214 + 0.697881i \(0.245873\pi\)
\(608\) 0 0
\(609\) 1.41672e17 2.77703
\(610\) 0 0
\(611\) 0 0
\(612\) 1.89114e16 0.359928
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.27774e16 −1.50038 −0.750189 0.661223i \(-0.770038\pi\)
−0.750189 + 0.661223i \(0.770038\pi\)
\(618\) 0 0
\(619\) 4.04116e15 0.0718393 0.0359196 0.999355i \(-0.488564\pi\)
0.0359196 + 0.999355i \(0.488564\pi\)
\(620\) 0 0
\(621\) −9.72357e16 −1.69542
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.96046e16 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.53508e17 2.47872
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 2.19123e17 3.07166
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1.55678e17 2.08334
\(650\) 0 0
\(651\) −1.45204e16 −0.190763
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.58961e17 −1.99465
\(657\) 0 0
\(658\) 0 0
\(659\) −5.75231e16 −0.702311 −0.351156 0.936317i \(-0.614211\pi\)
−0.351156 + 0.936317i \(0.614211\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.29053e17 −2.60125
\(668\) 1.16685e17 1.31327
\(669\) 0 0
\(670\) 0 0
\(671\) −2.76464e17 −3.02903
\(672\) 0 0
\(673\) −4.68025e16 −0.503707 −0.251854 0.967765i \(-0.581040\pi\)
−0.251854 + 0.967765i \(0.581040\pi\)
\(674\) 0 0
\(675\) 1.02762e17 1.08645
\(676\) 9.54290e16 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.03943e16 −0.104210
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.02721e17 −0.977058
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.90288e16 0.634107 0.317053 0.948408i \(-0.397307\pi\)
0.317053 + 0.948408i \(0.397307\pi\)
\(692\) −3.79263e16 −0.345386
\(693\) −1.11664e17 −1.00812
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.90182e17 2.53089
\(698\) 0 0
\(699\) 0 0
\(700\) −2.31577e17 −1.96837
\(701\) 2.35083e17 1.98113 0.990566 0.137037i \(-0.0437578\pi\)
0.990566 + 0.137037i \(0.0437578\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.19803e17 −1.80550
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 1.23004e17 0.976607
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.34765e16 0.178688
\(714\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 1.48594e17 1.04033
\(724\) 0 0
\(725\) 2.42071e17 1.66692
\(726\) 0 0
\(727\) 2.54140e17 1.72134 0.860670 0.509163i \(-0.170045\pi\)
0.860670 + 0.509163i \(0.170045\pi\)
\(728\) 0 0
\(729\) 1.65090e17 1.09991
\(730\) 0 0
\(731\) 0 0
\(732\) −2.18439e17 −1.41992
\(733\) 2.92603e17 1.88649 0.943247 0.332092i \(-0.107754\pi\)
0.943247 + 0.332092i \(0.107754\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 3.25531e17 1.99860 0.999299 0.0374245i \(-0.0119154\pi\)
0.999299 + 0.0374245i \(0.0119154\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.92869e16 −0.283666
\(748\) 4.01249e17 2.29089
\(749\) 0 0
\(750\) 0 0
\(751\) 2.29571e17 1.27961 0.639805 0.768537i \(-0.279015\pi\)
0.639805 + 0.768537i \(0.279015\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −3.99253e17 −2.13854
\(757\) 2.88508e17 1.53314 0.766572 0.642158i \(-0.221961\pi\)
0.766572 + 0.642158i \(0.221961\pi\)
\(758\) 0 0
\(759\) −4.55903e17 −2.38463
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −2.67172e17 −1.35408
\(764\) −1.57467e17 −0.791828
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.73670e17 −0.846365
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.39228e17 −1.13008
\(773\) −2.81962e17 −1.32164 −0.660822 0.750543i \(-0.729792\pi\)
−0.660822 + 0.750543i \(0.729792\pi\)
\(774\) 0 0
\(775\) −2.48107e16 −0.114506
\(776\) 0 0
\(777\) −7.16162e17 −3.25451
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 4.17345e17 1.81103
\(784\) 6.67509e17 2.87449
\(785\) 0 0
\(786\) 0 0
\(787\) 4.67727e17 1.96854 0.984269 0.176677i \(-0.0565349\pi\)
0.984269 + 0.176677i \(0.0565349\pi\)
\(788\) 4.78808e17 1.99988
\(789\) 0 0
\(790\) 0 0
\(791\) 4.69667e17 1.91748
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −5.04271e17 −1.98237
\(797\) 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
\(811\) −5.47610e17 −1.92463 −0.962313 0.271945i \(-0.912333\pi\)
−0.962313 + 0.271945i \(0.912333\pi\)
\(812\) −9.40499e17 −3.28112
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 3.17034e17 1.07390
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 4.81813e17 1.52811
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 1.42645e17 0.442664
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −2.98096e17 −0.905211
\(832\) 0 0
\(833\) −1.21853e18 −3.64727
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.27752e16 −0.124405
\(838\) 0 0
\(839\) −5.99895e17 −1.71990 −0.859951 0.510377i \(-0.829506\pi\)
−0.859951 + 0.510377i \(0.829506\pi\)
\(840\) 0 0
\(841\) 6.29304e17 1.77863
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.64242e18 −4.44819
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.15788e18 3.04851
\(852\) 0 0
\(853\) −7.46510e17 −1.93794 −0.968972 0.247169i \(-0.920500\pi\)
−0.968972 + 0.247169i \(0.920500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.91177e17 −1.99705 −0.998524 0.0543084i \(-0.982705\pi\)
−0.998524 + 0.0543084i \(0.982705\pi\)
\(858\) 0 0
\(859\) 7.95514e17 1.98011 0.990054 0.140689i \(-0.0449318\pi\)
0.990054 + 0.140689i \(0.0449318\pi\)
\(860\) 0 0
\(861\) −1.35379e18 −3.32301
\(862\) 0 0
\(863\) 2.25913e17 0.546860 0.273430 0.961892i \(-0.411842\pi\)
0.273430 + 0.961892i \(0.411842\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.19266e17 −0.516247
\(868\) 9.63950e16 0.225391
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 7.77551e17 1.68576
\(880\) 0 0
\(881\) 2.39826e15 0.00512910 0.00256455 0.999997i \(-0.499184\pi\)
0.00256455 + 0.999997i \(0.499184\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.42474e18 2.88620
\(890\) 0 0
\(891\) 5.74420e17 1.14806
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.00763e17 −0.190873
\(900\) −1.50752e17 −0.283666
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.96293e17 1.25069 0.625343 0.780350i \(-0.284959\pi\)
0.625343 + 0.780350i \(0.284959\pi\)
\(908\) 6.90031e16 0.123127
\(909\) 0 0
\(910\) 0 0
\(911\) 1.14324e18 1.99999 0.999995 0.00331330i \(-0.00105466\pi\)
0.999995 + 0.00331330i \(0.00105466\pi\)
\(912\) 0 0
\(913\) −1.04574e18 −1.80550
\(914\) 0 0
\(915\) 0 0
\(916\) 6.81923e17 1.15442
\(917\) −4.63940e17 −0.780272
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) −1.87195e18 −3.00789
\(925\) −1.22369e18 −1.95353
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.28218e18 1.99460 0.997298 0.0734666i \(-0.0234062\pi\)
0.997298 + 0.0734666i \(0.0234062\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 2.23237e17 0.325666
\(940\) 0 0
\(941\) 1.17140e18 1.68720 0.843601 0.536971i \(-0.180431\pi\)
0.843601 + 0.536971i \(0.180431\pi\)
\(942\) 0 0
\(943\) 2.18878e18 3.11267
\(944\) −8.16571e17 −1.15388
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.21959e18 −1.64866
\(952\) 0 0
\(953\) −1.59139e17 −0.212431 −0.106216 0.994343i \(-0.533873\pi\)
−0.106216 + 0.994343i \(0.533873\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.95678e18 2.54724
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −7.77335e17 −0.986888
\(962\) 0 0
\(963\) 0 0
\(964\) −9.86451e17 −1.22917
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −4.62377e17 −0.548275
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.45012e18 1.67767
\(977\) 1.61226e18 1.85382 0.926910 0.375284i \(-0.122455\pi\)
0.926910 + 0.375284i \(0.122455\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.73923e17 −0.195139
\(982\) 0 0
\(983\) −1.66457e18 −1.84493 −0.922467 0.386076i \(-0.873830\pi\)
−0.922467 + 0.386076i \(0.873830\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.81772e18 −1.91904 −0.959520 0.281640i \(-0.909122\pi\)
−0.959520 + 0.281640i \(0.909122\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) −8.26254e17 −0.846365
\(997\) 1.04955e18 1.06864 0.534320 0.845282i \(-0.320568\pi\)
0.534320 + 0.845282i \(0.320568\pi\)
\(998\) 0 0
\(999\) −2.10971e18 −2.12242
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 83.13.b.a.82.1 1
83.82 odd 2 CM 83.13.b.a.82.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.13.b.a.82.1 1 1.1 even 1 trivial
83.13.b.a.82.1 1 83.82 odd 2 CM