Properties

Label 1840.2.m.e.1839.4
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM discriminant -115
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14166950625.1
Defining polynomial: \(x^{8} - 9 x^{6} + 32 x^{4} - 441 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1839.4
Root \(2.63567 + 0.230712i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.e.1839.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{5} +4.33441i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{5} +4.33441i q^{7} -3.00000 q^{9} +7.50741 q^{17} +4.79583i q^{23} +5.00000 q^{25} -8.78709 q^{29} -2.76383i q^{31} -9.69203i q^{35} -1.43686 q^{37} -12.7871 q^{41} -9.59166i q^{43} +6.70820 q^{45} -11.7871 q^{49} -13.5780 q^{53} +4.16438i q^{59} -13.0032i q^{63} -11.1575i q^{67} -16.6202i q^{71} +9.00000 q^{81} +2.48871i q^{83} -16.7871 q^{85} +4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 40q^{25} + 4q^{29} - 28q^{41} - 20q^{49} + 72q^{81} - 60q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) 4.33441i 1.63825i 0.573614 + 0.819126i \(0.305541\pi\)
−0.573614 + 0.819126i \(0.694459\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.50741 1.82082 0.910408 0.413712i \(-0.135768\pi\)
0.910408 + 0.413712i \(0.135768\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.78709 −1.63172 −0.815861 0.578249i \(-0.803736\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(30\) 0 0
\(31\) − 2.76383i − 0.496398i −0.968709 0.248199i \(-0.920161\pi\)
0.968709 0.248199i \(-0.0798387\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 9.69203i − 1.63825i
\(36\) 0 0
\(37\) −1.43686 −0.236218 −0.118109 0.993001i \(-0.537683\pi\)
−0.118109 + 0.993001i \(0.537683\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.7871 −1.99701 −0.998504 0.0546823i \(-0.982585\pi\)
−0.998504 + 0.0546823i \(0.982585\pi\)
\(42\) 0 0
\(43\) − 9.59166i − 1.46271i −0.681994 0.731357i \(-0.738887\pi\)
0.681994 0.731357i \(-0.261113\pi\)
\(44\) 0 0
\(45\) 6.70820 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −11.7871 −1.68387
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.5780 −1.86508 −0.932539 0.361070i \(-0.882412\pi\)
−0.932539 + 0.361070i \(0.882412\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.16438i 0.542156i 0.962557 + 0.271078i \(0.0873801\pi\)
−0.962557 + 0.271078i \(0.912620\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 13.0032i − 1.63825i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.1575i − 1.36311i −0.731767 0.681554i \(-0.761304\pi\)
0.731767 0.681554i \(-0.238696\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 16.6202i − 1.97246i −0.165383 0.986229i \(-0.552886\pi\)
0.165383 0.986229i \(-0.447114\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 2.48871i 0.273172i 0.990628 + 0.136586i \(0.0436129\pi\)
−0.990628 + 0.136586i \(0.956387\pi\)
\(84\) 0 0
\(85\) −16.7871 −1.82082
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.47214 0.454077 0.227038 0.973886i \(-0.427096\pi\)
0.227038 + 0.973886i \(0.427096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.787088 −0.0783182 −0.0391591 0.999233i \(-0.512468\pi\)
−0.0391591 + 0.999233i \(0.512468\pi\)
\(102\) 0 0
\(103\) 9.59166i 0.945095i 0.881305 + 0.472547i \(0.156665\pi\)
−0.881305 + 0.472547i \(0.843335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.8263i 1.91668i 0.285622 + 0.958342i \(0.407800\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.63370 −0.435902 −0.217951 0.975960i \(-0.569937\pi\)
−0.217951 + 0.975960i \(0.569937\pi\)
\(114\) 0 0
\(115\) − 10.7238i − 1.00000i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 32.5402i 2.98295i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 21.4476i − 1.87389i −0.349482 0.936943i \(-0.613642\pi\)
0.349482 0.936943i \(-0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.4164 −1.14624 −0.573121 0.819471i \(-0.694267\pi\)
−0.573121 + 0.819471i \(0.694267\pi\)
\(138\) 0 0
\(139\) − 23.5484i − 1.99735i −0.0514389 0.998676i \(-0.516381\pi\)
0.0514389 0.998676i \(-0.483619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 19.6485 1.63172
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 21.4476i 1.74538i 0.488273 + 0.872691i \(0.337627\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −22.5222 −1.82082
\(154\) 0 0
\(155\) 6.18010i 0.496398i
\(156\) 0 0
\(157\) 16.4517 1.31299 0.656494 0.754331i \(-0.272039\pi\)
0.656494 + 0.754331i \(0.272039\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −20.7871 −1.63825
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 21.6720i 1.63825i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 21.4476i − 1.60307i −0.597948 0.801535i \(-0.704017\pi\)
0.597948 0.801535i \(-0.295983\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.21291 0.236218
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 38.0868i − 2.67317i
\(204\) 0 0
\(205\) 28.5928 1.99701
\(206\) 0 0
\(207\) − 14.3875i − 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 18.0208i 1.24060i 0.784364 + 0.620301i \(0.212990\pi\)
−0.784364 + 0.620301i \(0.787010\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 21.4476i 1.46271i
\(216\) 0 0
\(217\) 11.9795 0.813225
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) 28.7750i 1.90986i 0.296826 + 0.954932i \(0.404072\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 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
\(237\) 0 0
\(238\) 0 0
\(239\) − 30.4766i − 1.97137i −0.168598 0.985685i \(-0.553924\pi\)
0.168598 0.985685i \(-0.446076\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 26.3567 1.68387
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 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
\(259\) − 6.22793i − 0.386985i
\(260\) 0 0
\(261\) 26.3613 1.63172
\(262\) 0 0
\(263\) 17.9806i 1.10873i 0.832272 + 0.554367i \(0.187040\pi\)
−0.832272 + 0.554367i \(0.812960\pi\)
\(264\) 0 0
\(265\) 30.3613 1.86508
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −24.7871 −1.51130 −0.755648 0.654978i \(-0.772678\pi\)
−0.755648 + 0.654978i \(0.772678\pi\)
\(270\) 0 0
\(271\) 22.1479i 1.34539i 0.739921 + 0.672694i \(0.234863\pi\)
−0.739921 + 0.672694i \(0.765137\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 8.29148i 0.496398i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 14.8489i − 0.882677i −0.897341 0.441338i \(-0.854504\pi\)
0.897341 0.441338i \(-0.145496\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 55.4244i − 3.27160i
\(288\) 0 0
\(289\) 39.3613 2.31537
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −25.7191 −1.50252 −0.751262 0.660004i \(-0.770555\pi\)
−0.751262 + 0.660004i \(0.770555\pi\)
\(294\) 0 0
\(295\) − 9.31183i − 0.542156i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 41.5742 2.39630
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.4476i 1.21618i 0.793867 + 0.608091i \(0.208065\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) −34.6634 −1.95929 −0.979644 0.200741i \(-0.935665\pi\)
−0.979644 + 0.200741i \(0.935665\pi\)
\(314\) 0 0
\(315\) 29.0761i 1.63825i
\(316\) 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
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.36328i 0.0749324i 0.999298 + 0.0374662i \(0.0119287\pi\)
−0.999298 + 0.0374662i \(0.988071\pi\)
\(332\) 0 0
\(333\) 4.31057 0.236218
\(334\) 0 0
\(335\) 24.9490i 1.36311i
\(336\) 0 0
\(337\) −31.3050 −1.70529 −0.852645 0.522491i \(-0.825003\pi\)
−0.852645 + 0.522491i \(0.825003\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.7492i − 1.12035i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 18.3613 0.982856 0.491428 0.870918i \(-0.336475\pi\)
0.491428 + 0.870918i \(0.336475\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 37.1640i 1.97246i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 35.3183i 1.84360i 0.387667 + 0.921799i \(0.373281\pi\)
−0.387667 + 0.921799i \(0.626719\pi\)
\(368\) 0 0
\(369\) 38.3613 1.99701
\(370\) 0 0
\(371\) − 58.8525i − 3.05547i
\(372\) 0 0
\(373\) −4.47214 −0.231558 −0.115779 0.993275i \(-0.536937\pi\)
−0.115779 + 0.993275i \(0.536937\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.643015i 0.0328565i 0.999865 + 0.0164283i \(0.00522951\pi\)
−0.999865 + 0.0164283i \(0.994770\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 28.7750i 1.46271i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 36.0043i 1.82082i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −20.1246 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −28.7871 −1.42343 −0.711715 0.702468i \(-0.752081\pi\)
−0.711715 + 0.702468i \(0.752081\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.0501 −0.888188
\(414\) 0 0
\(415\) − 5.56493i − 0.273172i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 37.5371 1.82082
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −16.7748 −0.806146 −0.403073 0.915168i \(-0.632058\pi\)
−0.403073 + 0.915168i \(0.632058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 21.4476i 1.02364i 0.859093 + 0.511819i \(0.171028\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 35.3613 1.68387
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.2129 0.529170 0.264585 0.964362i \(-0.414765\pi\)
0.264585 + 0.964362i \(0.414765\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.3960 1.18797 0.593986 0.804475i \(-0.297553\pi\)
0.593986 + 0.804475i \(0.297553\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.5177i 1.08827i 0.838997 + 0.544135i \(0.183142\pi\)
−0.838997 + 0.544135i \(0.816858\pi\)
\(468\) 0 0
\(469\) 48.3613 2.23312
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 40.7339 1.86508
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.8772i 1.43860i 0.694701 + 0.719299i \(0.255537\pi\)
−0.694701 + 0.719299i \(0.744463\pi\)
\(492\) 0 0
\(493\) −65.9683 −2.97106
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 72.0389 3.23138
\(498\) 0 0
\(499\) 42.9325i 1.92192i 0.276683 + 0.960961i \(0.410765\pi\)
−0.276683 + 0.960961i \(0.589235\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 43.9871i − 1.96129i −0.195801 0.980644i \(-0.562731\pi\)
0.195801 0.980644i \(-0.437269\pi\)
\(504\) 0 0
\(505\) 1.75998 0.0783182
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 21.4476i − 0.945095i
\(516\) 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
\(523\) − 9.59166i − 0.419414i −0.977764 0.209707i \(-0.932749\pi\)
0.977764 0.209707i \(-0.0672510\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 20.7492i − 0.903849i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 12.4931i − 0.542156i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 44.3330i − 1.91668i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.3254 −0.818844 −0.409422 0.912345i \(-0.634270\pi\)
−0.409422 + 0.912345i \(0.634270\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 45.8328i − 1.93162i −0.259248 0.965811i \(-0.583475\pi\)
0.259248 0.965811i \(-0.416525\pi\)
\(564\) 0 0
\(565\) 10.3613 0.435902
\(566\) 0 0
\(567\) 39.0097i 1.63825i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −10.7871 −0.447524
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) − 72.7621i − 2.98295i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.4476i 0.876326i 0.898896 + 0.438163i \(0.144371\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 46.3613 1.89112 0.945558 0.325455i \(-0.105517\pi\)
0.945558 + 0.325455i \(0.105517\pi\)
\(602\) 0 0
\(603\) 33.4726i 1.36311i
\(604\) 0 0
\(605\) 24.5967 1.00000
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.3050 1.26440 0.632198 0.774807i \(-0.282153\pi\)
0.632198 + 0.774807i \(0.282153\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.6782 1.99997 0.999984 0.00563284i \(-0.00179300\pi\)
0.999984 + 0.00563284i \(0.00179300\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.7871 −0.430109
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 49.8607i 1.97246i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 16.1349i 0.636300i 0.948040 + 0.318150i \(0.103062\pi\)
−0.948040 + 0.318150i \(0.896938\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 47.9583i 1.87389i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 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
\(667\) − 42.1414i − 1.63172i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.8454 0.878019 0.439009 0.898482i \(-0.355329\pi\)
0.439009 + 0.898482i \(0.355329\pi\)
\(678\) 0 0
\(679\) 19.3841i 0.743892i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 21.4476i − 0.815906i −0.913003 0.407953i \(-0.866243\pi\)
0.913003 0.407953i \(-0.133757\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 52.6559i 1.99735i
\(696\) 0 0
\(697\) −95.9980 −3.63618
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 3.41156i − 0.128305i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.2548 0.496398
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 27.6755i − 1.03212i −0.856551 0.516062i \(-0.827398\pi\)
0.856551 0.516062i \(-0.172602\pi\)
\(720\) 0 0
\(721\) −41.5742 −1.54830
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −43.9354 −1.63172
\(726\) 0 0
\(727\) − 16.6946i − 0.619169i −0.950872 0.309584i \(-0.899810\pi\)
0.950872 0.309584i \(-0.100190\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) − 72.0086i − 2.66333i
\(732\) 0 0
\(733\) −7.83054 −0.289228 −0.144614 0.989488i \(-0.546194\pi\)
−0.144614 + 0.989488i \(0.546194\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 34.6037i − 1.27292i −0.771310 0.636460i \(-0.780398\pi\)
0.771310 0.636460i \(-0.219602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.59166i 0.351884i 0.984401 + 0.175942i \(0.0562971\pi\)
−0.984401 + 0.175942i \(0.943703\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 7.46613i − 0.273172i
\(748\) 0 0
\(749\) −85.9354 −3.14001
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 47.9583i − 1.74538i
\(756\) 0 0
\(757\) −37.8602 −1.37605 −0.688026 0.725686i \(-0.741523\pi\)
−0.688026 + 0.725686i \(0.741523\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −39.9354 −1.44766 −0.723829 0.689979i \(-0.757620\pi\)
−0.723829 + 0.689979i \(0.757620\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 50.3613 1.82082
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −40.2492 −1.44766 −0.723832 0.689976i \(-0.757621\pi\)
−0.723832 + 0.689976i \(0.757621\pi\)
\(774\) 0 0
\(775\) − 13.8191i − 0.496398i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.7871 −1.31299
\(786\) 0 0
\(787\) 50.8102i 1.81119i 0.424145 + 0.905594i \(0.360575\pi\)
−0.424145 + 0.905594i \(0.639425\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 20.0843i − 0.714117i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −55.7487 −1.97472 −0.987361 0.158489i \(-0.949338\pi\)
−0.987361 + 0.158489i \(0.949338\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 46.4813 1.63825
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −55.9354 −1.96659 −0.983293 0.182032i \(-0.941733\pi\)
−0.983293 + 0.182032i \(0.941733\pi\)
\(810\) 0 0
\(811\) 56.7889i 1.99413i 0.0765723 + 0.997064i \(0.475602\pi\)
−0.0765723 + 0.997064i \(0.524398\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 54.5016i 1.89521i 0.319450 + 0.947603i \(0.396502\pi\)
−0.319450 + 0.947603i \(0.603498\pi\)
\(828\) 0 0
\(829\) −35.9354 −1.24809 −0.624045 0.781389i \(-0.714512\pi\)
−0.624045 + 0.781389i \(0.714512\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −88.4905 −3.06602
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.2129 1.66251
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −29.0689 −1.00000
\(846\) 0 0
\(847\) − 47.6785i − 1.63825i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 6.89093i − 0.236218i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) − 26.3495i − 0.899035i −0.893272 0.449517i \(-0.851596\pi\)
0.893272 0.449517i \(-0.148404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −13.4164 −0.454077
\(874\) 0 0
\(875\) − 48.4601i − 1.63825i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 47.9583i 1.60307i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.2860i 0.809983i
\(900\) 0 0
\(901\) −101.935 −3.39596
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.20268i − 0.0399344i −0.999801 0.0199672i \(-0.993644\pi\)
0.999801 0.0199672i \(-0.00635618\pi\)
\(908\) 0 0
\(909\) 2.36126 0.0783182
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 92.9627 3.06990
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.18429 −0.236218
\(926\) 0 0
\(927\) − 28.7750i − 0.945095i
\(928\) 0 0
\(929\) −31.9354 −1.04777 −0.523884 0.851790i \(-0.675517\pi\)
−0.523884 + 0.851790i \(0.675517\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 58.1378 1.89928 0.949639 0.313346i \(-0.101450\pi\)
0.949639 + 0.313346i \(0.101450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) − 61.3247i − 1.99701i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.3607 0.724333 0.362167 0.932113i \(-0.382037\pi\)
0.362167 + 0.932113i \(0.382037\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 58.1522i − 1.87783i
\(960\) 0 0
\(961\) 23.3613 0.753589
\(962\) 0 0
\(963\) − 59.4790i − 1.91668i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 102.069 3.27217
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.9159 −0.925102 −0.462551 0.886593i \(-0.653066\pi\)
−0.462551 + 0.886593i \(0.653066\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25.3634i 0.808968i 0.914545 + 0.404484i \(0.132549\pi\)
−0.914545 + 0.404484i \(0.867451\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.0000 1.46271
\(990\) 0 0
\(991\) − 55.3884i − 1.75947i −0.475465 0.879734i \(-0.657720\pi\)
0.475465 0.879734i \(-0.342280\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
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 1840.2.m.e.1839.4 yes 8
4.3 odd 2 inner 1840.2.m.e.1839.1 8
5.4 even 2 inner 1840.2.m.e.1839.5 yes 8
20.19 odd 2 inner 1840.2.m.e.1839.8 yes 8
23.22 odd 2 inner 1840.2.m.e.1839.5 yes 8
92.91 even 2 inner 1840.2.m.e.1839.8 yes 8
115.114 odd 2 CM 1840.2.m.e.1839.4 yes 8
460.459 even 2 inner 1840.2.m.e.1839.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.e.1839.1 8 4.3 odd 2 inner
1840.2.m.e.1839.1 8 460.459 even 2 inner
1840.2.m.e.1839.4 yes 8 1.1 even 1 trivial
1840.2.m.e.1839.4 yes 8 115.114 odd 2 CM
1840.2.m.e.1839.5 yes 8 5.4 even 2 inner
1840.2.m.e.1839.5 yes 8 23.22 odd 2 inner
1840.2.m.e.1839.8 yes 8 20.19 odd 2 inner
1840.2.m.e.1839.8 yes 8 92.91 even 2 inner