Properties

Label 83.7.b.a.82.1
Level $83$
Weight $7$
Character 83.82
Self dual yes
Analytic conductor $19.094$
Analytic rank $0$
Dimension $1$
CM discriminant -83
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(19.0944889404\)
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

\(f(q)\) \(=\) \(q-29.0000 q^{3} +64.0000 q^{4} -61.0000 q^{7} +112.000 q^{9} +O(q^{10})\) \(q-29.0000 q^{3} +64.0000 q^{4} -61.0000 q^{7} +112.000 q^{9} +587.000 q^{11} -1856.00 q^{12} +4096.00 q^{16} -4201.00 q^{17} +1769.00 q^{21} +8066.00 q^{23} +15625.0 q^{25} +17893.0 q^{27} -3904.00 q^{28} +46703.0 q^{29} +40907.0 q^{31} -17023.0 q^{33} +7168.00 q^{36} +10919.0 q^{37} +5042.00 q^{41} +37568.0 q^{44} -118784. q^{48} -113928. q^{49} +121829. q^{51} -188917. q^{59} +435287. q^{61} -6832.00 q^{63} +262144. q^{64} -268864. q^{68} -233914. q^{69} -453125. q^{75} -35807.0 q^{77} -600545. q^{81} -571787. q^{83} +113216. q^{84} -1.35439e6 q^{87} +516224. q^{92} -1.18630e6 q^{93} +65744.0 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −29.0000 −1.07407 −0.537037 0.843559i \(-0.680456\pi\)
−0.537037 + 0.843559i \(0.680456\pi\)
\(4\) 64.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −61.0000 −0.177843 −0.0889213 0.996039i \(-0.528342\pi\)
−0.0889213 + 0.996039i \(0.528342\pi\)
\(8\) 0 0
\(9\) 112.000 0.153635
\(10\) 0 0
\(11\) 587.000 0.441022 0.220511 0.975385i \(-0.429228\pi\)
0.220511 + 0.975385i \(0.429228\pi\)
\(12\) −1856.00 −1.07407
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) −4201.00 −0.855078 −0.427539 0.903997i \(-0.640619\pi\)
−0.427539 + 0.903997i \(0.640619\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 1769.00 0.191016
\(22\) 0 0
\(23\) 8066.00 0.662941 0.331470 0.943466i \(-0.392455\pi\)
0.331470 + 0.943466i \(0.392455\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 17893.0 0.909059
\(28\) −3904.00 −0.177843
\(29\) 46703.0 1.91492 0.957460 0.288565i \(-0.0931781\pi\)
0.957460 + 0.288565i \(0.0931781\pi\)
\(30\) 0 0
\(31\) 40907.0 1.37313 0.686566 0.727067i \(-0.259117\pi\)
0.686566 + 0.727067i \(0.259117\pi\)
\(32\) 0 0
\(33\) −17023.0 −0.473690
\(34\) 0 0
\(35\) 0 0
\(36\) 7168.00 0.153635
\(37\) 10919.0 0.215565 0.107782 0.994175i \(-0.465625\pi\)
0.107782 + 0.994175i \(0.465625\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5042.00 0.0731562 0.0365781 0.999331i \(-0.488354\pi\)
0.0365781 + 0.999331i \(0.488354\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 37568.0 0.441022
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −118784. −1.07407
\(49\) −113928. −0.968372
\(50\) 0 0
\(51\) 121829. 0.918418
\(52\) 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
\(59\) −188917. −0.919846 −0.459923 0.887959i \(-0.652123\pi\)
−0.459923 + 0.887959i \(0.652123\pi\)
\(60\) 0 0
\(61\) 435287. 1.91772 0.958862 0.283872i \(-0.0916191\pi\)
0.958862 + 0.283872i \(0.0916191\pi\)
\(62\) 0 0
\(63\) −6832.00 −0.0273229
\(64\) 262144. 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −268864. −0.855078
\(69\) −233914. −0.712047
\(70\) 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
\(75\) −453125. −1.07407
\(76\) 0 0
\(77\) −35807.0 −0.0784324
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −600545. −1.13003
\(82\) 0 0
\(83\) −571787. −1.00000
\(84\) 113216. 0.191016
\(85\) 0 0
\(86\) 0 0
\(87\) −1.35439e6 −2.05677
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 516224. 0.662941
\(93\) −1.18630e6 −1.47485
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 65744.0 0.0677564
\(100\) 1.00000e6 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 \(\pi\)
\(108\) 1.14515e6 0.909059
\(109\) 2.12318e6 1.63949 0.819743 0.572731i \(-0.194116\pi\)
0.819743 + 0.572731i \(0.194116\pi\)
\(110\) 0 0
\(111\) −316651. −0.231532
\(112\) −249856. −0.177843
\(113\) 1.46143e6 1.01284 0.506422 0.862285i \(-0.330968\pi\)
0.506422 + 0.862285i \(0.330968\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.98899e6 1.91492
\(117\) 0 0
\(118\) 0 0
\(119\) 256261. 0.152069
\(120\) 0 0
\(121\) −1.42699e6 −0.805500
\(122\) 0 0
\(123\) −146218. −0.0785752
\(124\) 2.61805e6 1.37313
\(125\) 0 0
\(126\) 0 0
\(127\) 1.49646e6 0.730556 0.365278 0.930898i \(-0.380974\pi\)
0.365278 + 0.930898i \(0.380974\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.48012e6 −1.54803 −0.774016 0.633166i \(-0.781755\pi\)
−0.774016 + 0.633166i \(0.781755\pi\)
\(132\) −1.08947e6 −0.473690
\(133\) 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
\(144\) 458752. 0.153635
\(145\) 0 0
\(146\) 0 0
\(147\) 3.30391e6 1.04010
\(148\) 698816. 0.215565
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −2.99317e6 −0.869363 −0.434681 0.900584i \(-0.643139\pi\)
−0.434681 + 0.900584i \(0.643139\pi\)
\(152\) 0 0
\(153\) −470512. −0.131370
\(154\) 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
\(161\) −492026. −0.117899
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 322688. 0.0731562
\(165\) 0 0
\(166\) 0 0
\(167\) −8.47770e6 −1.82024 −0.910120 0.414345i \(-0.864011\pi\)
−0.910120 + 0.414345i \(0.864011\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.66019e6 1.28632 0.643159 0.765733i \(-0.277623\pi\)
0.643159 + 0.765733i \(0.277623\pi\)
\(174\) 0 0
\(175\) −953125. −0.177843
\(176\) 2.40435e6 0.441022
\(177\) 5.47859e6 0.987982
\(178\) 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
\(183\) −1.26233e7 −2.05978
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.46599e6 −0.377108
\(188\) 0 0
\(189\) −1.09147e6 −0.161669
\(190\) 0 0
\(191\) 7.65887e6 1.09917 0.549584 0.835438i \(-0.314786\pi\)
0.549584 + 0.835438i \(0.314786\pi\)
\(192\) −7.60218e6 −1.07407
\(193\) −6.70521e6 −0.932697 −0.466349 0.884601i \(-0.654431\pi\)
−0.466349 + 0.884601i \(0.654431\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −7.29139e6 −0.968372
\(197\) −1.52905e7 −1.99997 −0.999985 0.00542488i \(-0.998273\pi\)
−0.999985 + 0.00542488i \(0.998273\pi\)
\(198\) 0 0
\(199\) −1.04630e6 −0.132769 −0.0663847 0.997794i \(-0.521146\pi\)
−0.0663847 + 0.997794i \(0.521146\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.84888e6 −0.340554
\(204\) 7.79706e6 0.918418
\(205\) 0 0
\(206\) 0 0
\(207\) 903392. 0.101851
\(208\) 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
\(217\) −2.49533e6 −0.244201
\(218\) 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
\(225\) 1.75000e6 0.153635
\(226\) 0 0
\(227\) −1.70438e7 −1.45710 −0.728548 0.684995i \(-0.759804\pi\)
−0.728548 + 0.684995i \(0.759804\pi\)
\(228\) 0 0
\(229\) 2.13288e7 1.77607 0.888034 0.459778i \(-0.152071\pi\)
0.888034 + 0.459778i \(0.152071\pi\)
\(230\) 0 0
\(231\) 1.03840e6 0.0842423
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.20907e7 −0.919846
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.22894e7 0.877967 0.438984 0.898495i \(-0.355339\pi\)
0.438984 + 0.898495i \(0.355339\pi\)
\(242\) 0 0
\(243\) 4.37181e6 0.304679
\(244\) 2.78584e7 1.91772
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.65818e7 1.07407
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −437248. −0.0273229
\(253\) 4.73474e6 0.292371
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −666059. −0.0383366
\(260\) 0 0
\(261\) 5.23074e6 0.294199
\(262\) 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 \(\pi\)
\(272\) −1.72073e7 −0.855078
\(273\) 0 0
\(274\) 0 0
\(275\) 9.17188e6 0.441022
\(276\) −1.49705e7 −0.712047
\(277\) 3.72370e7 1.75201 0.876003 0.482305i \(-0.160200\pi\)
0.876003 + 0.482305i \(0.160200\pi\)
\(278\) 0 0
\(279\) 4.58158e6 0.210961
\(280\) 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
\(287\) −307562. −0.0130103
\(288\) 0 0
\(289\) −6.48917e6 −0.268841
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.28261e6 −0.0907464 −0.0453732 0.998970i \(-0.514448\pi\)
−0.0453732 + 0.998970i \(0.514448\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.05032e7 0.400915
\(298\) 0 0
\(299\) 0 0
\(300\) −2.90000e7 −1.07407
\(301\) 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 \(\pi\)
\(308\) −2.29165e6 −0.0784324
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.89716e7 1.27091 0.635456 0.772137i \(-0.280812\pi\)
0.635456 + 0.772137i \(0.280812\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.32940e7 −1.98694 −0.993470 0.114093i \(-0.963604\pi\)
−0.993470 + 0.114093i \(0.963604\pi\)
\(318\) 0 0
\(319\) 2.74147e7 0.844522
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −3.84349e7 −1.13003
\(325\) 0 0
\(326\) 0 0
\(327\) −6.15723e7 −1.76093
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −3.65944e7 −1.00000
\(333\) 1.22293e6 0.0331183
\(334\) 0 0
\(335\) 0 0
\(336\) 7.24582e6 0.191016
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −4.23815e7 −1.08787
\(340\) 0 0
\(341\) 2.40124e7 0.605581
\(342\) 0 0
\(343\) 1.41262e7 0.350060
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −8.66808e7 −2.05677
\(349\) 8.49984e7 1.99956 0.999780 0.0209589i \(-0.00667191\pi\)
0.999780 + 0.0209589i \(0.00667191\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.25119e7 −1.19381 −0.596903 0.802313i \(-0.703602\pi\)
−0.596903 + 0.802313i \(0.703602\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.43157e6 −0.163334
\(358\) 0 0
\(359\) 2.45575e7 0.530763 0.265381 0.964144i \(-0.414502\pi\)
0.265381 + 0.964144i \(0.414502\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 4.13828e7 0.865166
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 3.30383e7 0.662941
\(369\) 564704. 0.0112394
\(370\) 0 0
\(371\) 0 0
\(372\) −7.59234e7 −1.47485
\(373\) −7.33750e7 −1.41391 −0.706955 0.707259i \(-0.749932\pi\)
−0.706955 + 0.707259i \(0.749932\pi\)
\(374\) 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
\(381\) −4.33973e7 −0.784672
\(382\) 0 0
\(383\) 5.36595e7 0.955102 0.477551 0.878604i \(-0.341525\pi\)
0.477551 + 0.878604i \(0.341525\pi\)
\(384\) 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
\(391\) −3.38853e7 −0.566866
\(392\) 0 0
\(393\) 1.00923e8 1.66270
\(394\) 0 0
\(395\) 0 0
\(396\) 4.20762e6 0.0677564
\(397\) −2.21190e7 −0.353504 −0.176752 0.984255i \(-0.556559\pi\)
−0.176752 + 0.984255i \(0.556559\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.40000e7 1.00000
\(401\) 2.48472e7 0.385340 0.192670 0.981264i \(-0.438285\pi\)
0.192670 + 0.981264i \(0.438285\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.40945e6 0.0950687
\(408\) 0 0
\(409\) 1.90898e6 0.0279018 0.0139509 0.999903i \(-0.495559\pi\)
0.0139509 + 0.999903i \(0.495559\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.15239e7 0.163588
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.23296e8 −1.67613 −0.838063 0.545573i \(-0.816312\pi\)
−0.838063 + 0.545573i \(0.816312\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.56406e7 −0.855078
\(426\) 0 0
\(427\) −2.65525e7 −0.341053
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.95973e7 0.994184 0.497092 0.867698i \(-0.334401\pi\)
0.497092 + 0.867698i \(0.334401\pi\)
\(432\) 7.32897e7 0.909059
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.35884e8 1.63949
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.27599e7 −0.148776
\(442\) 0 0
\(443\) 6.80814e7 0.783100 0.391550 0.920157i \(-0.371939\pi\)
0.391550 + 0.920157i \(0.371939\pi\)
\(444\) −2.02657e7 −0.231532
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.59908e7 −0.177843
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 2.95965e6 0.0322635
\(452\) 9.35316e7 1.01284
\(453\) 8.68020e7 0.933760
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −7.51685e7 −0.777316
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.94525e8 1.95989 0.979946 0.199261i \(-0.0638543\pi\)
0.979946 + 0.199261i \(0.0638543\pi\)
\(464\) 1.91295e8 1.91492
\(465\) 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
\(476\) 1.64007e7 0.152069
\(477\) 0 0
\(478\) 0 0
\(479\) −1.65650e8 −1.50724 −0.753622 0.657308i \(-0.771695\pi\)
−0.753622 + 0.657308i \(0.771695\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 1.42688e7 0.126632
\(484\) −9.13275e7 −0.805500
\(485\) 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 \(\pi\)
\(492\) −9.35795e6 −0.0785752
\(493\) −1.96199e8 −1.63741
\(494\) 0 0
\(495\) 0 0
\(496\) 1.67555e8 1.37313
\(497\) 0 0
\(498\) 0 0
\(499\) −2.47673e8 −1.99332 −0.996660 0.0816591i \(-0.973978\pi\)
−0.996660 + 0.0816591i \(0.973978\pi\)
\(500\) 0 0
\(501\) 2.45853e8 1.95507
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.39977e8 −1.07407
\(508\) 9.57734e7 0.730556
\(509\) 1.97259e8 1.49584 0.747919 0.663790i \(-0.231053\pi\)
0.747919 + 0.663790i \(0.231053\pi\)
\(510\) 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
\(519\) −1.93146e8 −1.38160
\(520\) 0 0
\(521\) −2.61107e8 −1.84632 −0.923158 0.384422i \(-0.874401\pi\)
−0.923158 + 0.384422i \(0.874401\pi\)
\(522\) 0 0
\(523\) −2.70231e8 −1.88899 −0.944496 0.328522i \(-0.893450\pi\)
−0.944496 + 0.328522i \(0.893450\pi\)
\(524\) −2.22728e8 −1.54803
\(525\) 2.76406e7 0.191016
\(526\) 0 0
\(527\) −1.71850e8 −1.17414
\(528\) −6.97262e7 −0.473690
\(529\) −8.29755e7 −0.560510
\(530\) 0 0
\(531\) −2.11587e7 −0.141321
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.68757e7 −0.427073
\(540\) 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
\(547\) 4.11134e7 0.251201 0.125600 0.992081i \(-0.459914\pi\)
0.125600 + 0.992081i \(0.459914\pi\)
\(548\) 0 0
\(549\) 4.87521e7 0.294630
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.89380e8 1.09589 0.547946 0.836514i \(-0.315410\pi\)
0.547946 + 0.836514i \(0.315410\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 7.15136e7 0.405042
\(562\) 0 0
\(563\) −2.62137e8 −1.46894 −0.734468 0.678643i \(-0.762568\pi\)
−0.734468 + 0.678643i \(0.762568\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.66332e7 0.200968
\(568\) 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
\(573\) −2.22107e8 −1.18059
\(574\) 0 0
\(575\) 1.26031e8 0.662941
\(576\) 2.93601e7 0.153635
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 1.94451e8 1.00179
\(580\) 0 0
\(581\) 3.48790e7 0.177843
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 2.11450e8 1.04010
\(589\) 0 0
\(590\) 0 0
\(591\) 4.43425e8 2.14812
\(592\) 4.47242e7 0.215565
\(593\) 3.12755e8 1.49982 0.749911 0.661539i \(-0.230096\pi\)
0.749911 + 0.661539i \(0.230096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.03428e7 0.142604
\(598\) 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
\(604\) −1.91563e8 −0.869363
\(605\) 0 0
\(606\) 0 0
\(607\) −4.14349e8 −1.85268 −0.926341 0.376687i \(-0.877063\pi\)
−0.926341 + 0.376687i \(0.877063\pi\)
\(608\) 0 0
\(609\) 8.26176e7 0.365781
\(610\) 0 0
\(611\) 0 0
\(612\) −3.01128e7 −0.131370
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.66026e8 0.706839 0.353420 0.935465i \(-0.385019\pi\)
0.353420 + 0.935465i \(0.385019\pi\)
\(618\) 0 0
\(619\) −3.41389e8 −1.43939 −0.719694 0.694291i \(-0.755718\pi\)
−0.719694 + 0.694291i \(0.755718\pi\)
\(620\) 0 0
\(621\) 1.44325e8 0.602652
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.58707e7 −0.184325
\(630\) 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 \(\pi\)
\(644\) −3.14897e7 −0.117899
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −1.10894e8 −0.405672
\(650\) 0 0
\(651\) 7.23645e7 0.262290
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.06520e7 0.0731562
\(657\) 0 0
\(658\) 0 0
\(659\) −3.26018e8 −1.13916 −0.569581 0.821935i \(-0.692894\pi\)
−0.569581 + 0.821935i \(0.692894\pi\)
\(660\) 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
\(667\) 3.76706e8 1.26948
\(668\) −5.42573e8 −1.82024
\(669\) 0 0
\(670\) 0 0
\(671\) 2.55513e8 0.845758
\(672\) 0 0
\(673\) 3.72867e8 1.22323 0.611615 0.791155i \(-0.290520\pi\)
0.611615 + 0.791155i \(0.290520\pi\)
\(674\) 0 0
\(675\) 2.79578e8 0.909059
\(676\) 3.08916e8 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4.94269e8 1.56503
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.18535e8 −1.90763
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −5.35489e8 −1.62299 −0.811497 0.584357i \(-0.801347\pi\)
−0.811497 + 0.584357i \(0.801347\pi\)
\(692\) 4.26252e8 1.28632
\(693\) −4.01038e6 −0.0120500
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.11814e7 −0.0625543
\(698\) 0 0
\(699\) 0 0
\(700\) −6.10000e7 −0.177843
\(701\) 6.87317e8 1.99528 0.997639 0.0686805i \(-0.0218789\pi\)
0.997639 + 0.0686805i \(0.0218789\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.53879e8 0.441022
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 3.50630e8 0.987982
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.29956e8 0.910306
\(714\) 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
\(723\) −3.56392e8 −0.943002
\(724\) 0 0
\(725\) 7.29734e8 1.91492
\(726\) 0 0
\(727\) 7.41230e8 1.92908 0.964539 0.263941i \(-0.0850225\pi\)
0.964539 + 0.263941i \(0.0850225\pi\)
\(728\) 0 0
\(729\) 3.11015e8 0.802784
\(730\) 0 0
\(731\) 0 0
\(732\) −8.07893e8 −2.05978
\(733\) −7.76410e8 −1.97142 −0.985710 0.168453i \(-0.946123\pi\)
−0.985710 + 0.168453i \(0.946123\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.07025e8 −1.99965 −0.999825 0.0187155i \(-0.994042\pi\)
−0.999825 + 0.0187155i \(0.994042\pi\)
\(740\) 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
\(747\) −6.40401e7 −0.153635
\(748\) −1.57823e8 −0.377108
\(749\) 0 0
\(750\) 0 0
\(751\) −7.67063e8 −1.81097 −0.905485 0.424379i \(-0.860492\pi\)
−0.905485 + 0.424379i \(0.860492\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −6.98543e7 −0.161669
\(757\) −8.15396e8 −1.87967 −0.939833 0.341634i \(-0.889020\pi\)
−0.939833 + 0.341634i \(0.889020\pi\)
\(758\) 0 0
\(759\) −1.37308e8 −0.314028
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.29514e8 −0.291571
\(764\) 4.90167e8 1.09917
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −4.86539e8 −1.07407
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.29134e8 −0.932697
\(773\) −3.80424e8 −0.823624 −0.411812 0.911269i \(-0.635104\pi\)
−0.411812 + 0.911269i \(0.635104\pi\)
\(774\) 0 0
\(775\) 6.39172e8 1.37313
\(776\) 0 0
\(777\) 1.93157e7 0.0411763
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.35657e8 1.74078
\(784\) −4.66649e8 −0.968372
\(785\) 0 0
\(786\) 0 0
\(787\) −9.71045e8 −1.99212 −0.996059 0.0886882i \(-0.971733\pi\)
−0.996059 + 0.0886882i \(0.971733\pi\)
\(788\) −9.78593e8 −1.99997
\(789\) 0 0
\(790\) 0 0
\(791\) −8.91473e7 −0.180127
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −6.69633e7 −0.132769
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1.46445e8 0.274544 0.137272 0.990533i \(-0.456167\pi\)
0.137272 + 0.990533i \(0.456167\pi\)
\(812\) −1.82329e8 −0.340554
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 4.99012e8 0.918418
\(817\) 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
\(825\) −2.65984e8 −0.473690
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 5.78171e7 0.101851
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −1.07987e9 −1.88179
\(832\) 0 0
\(833\) 4.78612e8 0.828034
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.31949e8 1.24826
\(838\) 0 0
\(839\) 3.12566e8 0.529243 0.264622 0.964352i \(-0.414753\pi\)
0.264622 + 0.964352i \(0.414753\pi\)
\(840\) 0 0
\(841\) 1.58635e9 2.66692
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.70465e7 0.143252
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.80727e7 0.142907
\(852\) 0 0
\(853\) −1.54610e8 −0.249109 −0.124555 0.992213i \(-0.539750\pi\)
−0.124555 + 0.992213i \(0.539750\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.41956e7 −0.0543284 −0.0271642 0.999631i \(-0.508648\pi\)
−0.0271642 + 0.999631i \(0.508648\pi\)
\(858\) 0 0
\(859\) 1.26452e9 1.99502 0.997510 0.0705202i \(-0.0224659\pi\)
0.997510 + 0.0705202i \(0.0224659\pi\)
\(860\) 0 0
\(861\) 8.91930e6 0.0139740
\(862\) 0 0
\(863\) 1.02573e9 1.59589 0.797944 0.602731i \(-0.205921\pi\)
0.797944 + 0.602731i \(0.205921\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.88186e8 0.288755
\(868\) −1.59701e8 −0.244201
\(869\) 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
\(879\) 6.61958e7 0.0974684
\(880\) 0 0
\(881\) −9.68275e8 −1.41603 −0.708013 0.706199i \(-0.750408\pi\)
−0.708013 + 0.706199i \(0.750408\pi\)
\(882\) 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
\(889\) −9.12840e7 −0.129924
\(890\) 0 0
\(891\) −3.52520e8 −0.498368
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.91048e9 2.62944
\(900\) 1.12000e8 0.153635
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.34527e9 −1.80297 −0.901483 0.432815i \(-0.857520\pi\)
−0.901483 + 0.432815i \(0.857520\pi\)
\(908\) −1.09080e9 −1.45710
\(909\) 0 0
\(910\) 0 0
\(911\) 1.51211e9 2.00000 0.999999 0.00165665i \(-0.000527329\pi\)
0.999999 + 0.00165665i \(0.000527329\pi\)
\(912\) 0 0
\(913\) −3.35639e8 −0.441022
\(914\) 0 0
\(915\) 0 0
\(916\) 1.36504e9 1.77607
\(917\) 2.12287e8 0.275306
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 6.64578e7 0.0842423
\(925\) 1.70609e8 0.215565
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.60245e9 −1.99865 −0.999324 0.0367582i \(-0.988297\pi\)
−0.999324 + 0.0367582i \(0.988297\pi\)
\(930\) 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
\(939\) −1.13018e9 −1.36505
\(940\) 0 0
\(941\) 1.59999e9 1.92021 0.960104 0.279642i \(-0.0902157\pi\)
0.960104 + 0.279642i \(0.0902157\pi\)
\(942\) 0 0
\(943\) 4.06688e7 0.0484982
\(944\) −7.73804e8 −0.919846
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.83553e9 2.13412
\(952\) 0 0
\(953\) −1.15720e9 −1.33700 −0.668500 0.743712i \(-0.733063\pi\)
−0.668500 + 0.743712i \(0.733063\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7.95025e8 −0.907079
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.85879e8 0.885494
\(962\) 0 0
\(963\) 0 0
\(964\) 7.86519e8 0.877967
\(965\) 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 \(\pi\)
\(972\) 2.79796e8 0.304679
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.78294e9 1.91772
\(977\) −1.83075e9 −1.96311 −0.981557 0.191168i \(-0.938773\pi\)
−0.981557 + 0.191168i \(0.938773\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.37796e8 0.251883
\(982\) 0 0
\(983\) −3.74041e8 −0.393784 −0.196892 0.980425i \(-0.563085\pi\)
−0.196892 + 0.980425i \(0.563085\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 2.76921e8 0.284534 0.142267 0.989828i \(-0.454561\pi\)
0.142267 + 0.989828i \(0.454561\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 1.06124e9 1.07407
\(997\) −1.73603e9 −1.75175 −0.875877 0.482535i \(-0.839716\pi\)
−0.875877 + 0.482535i \(0.839716\pi\)
\(998\) 0 0
\(999\) 1.95374e8 0.195961
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 83.7.b.a.82.1 1
83.82 odd 2 CM 83.7.b.a.82.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.7.b.a.82.1 1 1.1 even 1 trivial
83.7.b.a.82.1 1 83.82 odd 2 CM