Properties

Label 1200.2.o.h.1199.4
Level $1200$
Weight $2$
Character 1200.1199
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1199.4
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1199
Dual form 1200.2.o.h.1199.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} +1.73205 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +1.73205 q^{7} +3.00000 q^{9} +5.00000i q^{13} +8.66025i q^{19} +3.00000 q^{21} +5.19615 q^{27} -8.66025i q^{31} -10.0000i q^{37} +8.66025i q^{39} +12.1244 q^{43} -4.00000 q^{49} +15.0000i q^{57} -13.0000 q^{61} +5.19615 q^{63} +15.5885 q^{67} -10.0000i q^{73} +17.3205i q^{79} +9.00000 q^{81} +8.66025i q^{91} -15.0000i q^{93} -5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 12q^{21} - 16q^{49} - 52q^{61} + 36q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205 1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 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
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 8.66025i 1.98680i 0.114708 + 0.993399i \(0.463407\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) − 8.66025i − 1.55543i −0.628619 0.777714i \(-0.716379\pi\)
0.628619 0.777714i \(-0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 8.66025i 1.38675i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 12.1244 1.84895 0.924473 0.381246i \(-0.124505\pi\)
0.924473 + 0.381246i \(0.124505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15.0000i 1.98680i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 5.19615 0.654654
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.5885 1.90443 0.952217 0.305424i \(-0.0987981\pi\)
0.952217 + 0.305424i \(0.0987981\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 17.3205i 1.94871i 0.225018 + 0.974355i \(0.427756\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 8.66025i 0.907841i
\(92\) 0 0
\(93\) − 15.0000i − 1.55543i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 5.00000i − 0.507673i −0.967247 0.253837i \(-0.918307\pi\)
0.967247 0.253837i \(-0.0816925\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −3.46410 −0.341328 −0.170664 0.985329i \(-0.554591\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) − 17.3205i − 1.64399i
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 15.0000i 1.38675i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 21.0000 1.84895
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 15.0000i 1.30066i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) − 17.3205i − 1.46911i −0.678551 0.734553i \(-0.737392\pi\)
0.678551 0.734553i \(-0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.92820 −0.571429
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) − 8.66025i − 0.704761i −0.935857 0.352381i \(-0.885372\pi\)
0.935857 0.352381i \(-0.114628\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 25.0000i − 1.99522i −0.0691164 0.997609i \(-0.522018\pi\)
0.0691164 0.997609i \(-0.477982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.19615 0.406994 0.203497 0.979076i \(-0.434769\pi\)
0.203497 + 0.979076i \(0.434769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 25.9808i 1.98680i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) −22.5167 −1.66448
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 25.0000i 1.79954i 0.436365 + 0.899770i \(0.356266\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) − 25.9808i − 1.84173i −0.389885 0.920864i \(-0.627485\pi\)
0.389885 0.920864i \(-0.372515\pi\)
\(200\) 0 0
\(201\) 27.0000 1.90443
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 25.9808i 1.78859i 0.447478 + 0.894295i \(0.352322\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 15.0000i − 1.01827i
\(218\) 0 0
\(219\) − 17.3205i − 1.17041i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −29.4449 −1.97177 −0.985887 0.167412i \(-0.946459\pi\)
−0.985887 + 0.167412i \(0.946459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 29.0000 1.91637 0.958187 0.286143i \(-0.0923732\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 30.0000i 1.94871i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −43.3013 −2.75519
\(248\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) − 17.3205i − 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) − 17.3205i − 1.05215i −0.850439 0.526073i \(-0.823664\pi\)
0.850439 0.526073i \(-0.176336\pi\)
\(272\) 0 0
\(273\) 15.0000i 0.907841i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 5.00000i − 0.300421i −0.988654 0.150210i \(-0.952005\pi\)
0.988654 0.150210i \(-0.0479951\pi\)
\(278\) 0 0
\(279\) − 25.9808i − 1.55543i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −22.5167 −1.33848 −0.669238 0.743048i \(-0.733379\pi\)
−0.669238 + 0.743048i \(0.733379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) − 8.66025i − 0.507673i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 21.0000 1.21042
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.73205 0.0988534 0.0494267 0.998778i \(-0.484261\pi\)
0.0494267 + 0.998778i \(0.484261\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) − 35.0000i − 1.97832i −0.146852 0.989158i \(-0.546914\pi\)
0.146852 0.989158i \(-0.453086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 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
\(327\) −32.9090 −1.81987
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.3205i 0.952021i 0.879440 + 0.476011i \(0.157918\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) − 30.0000i − 1.64399i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 5.00000i − 0.272367i −0.990684 0.136184i \(-0.956516\pi\)
0.990684 0.136184i \(-0.0434837\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 25.9808i 1.38675i
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −56.0000 −2.94737
\(362\) 0 0
\(363\) −19.0526 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.5885 0.813711 0.406855 0.913493i \(-0.366625\pi\)
0.406855 + 0.913493i \(0.366625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0000i 1.29445i 0.762299 + 0.647225i \(0.224071\pi\)
−0.762299 + 0.647225i \(0.775929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 25.9808i − 1.33454i −0.744815 0.667271i \(-0.767462\pi\)
0.744815 0.667271i \(-0.232538\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 36.3731 1.84895
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.0000i 1.75660i 0.478110 + 0.878300i \(0.341322\pi\)
−0.478110 + 0.878300i \(0.658678\pi\)
\(398\) 0 0
\(399\) 25.9808i 1.30066i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 43.3013 2.15699
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −31.0000 −1.53285 −0.766426 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 30.0000i − 1.46911i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −22.5167 −1.08966
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 35.0000i − 1.68199i −0.541041 0.840996i \(-0.681970\pi\)
0.541041 0.840996i \(-0.318030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 8.66025i 0.413331i 0.978412 + 0.206666i \(0.0662612\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 15.0000i − 0.704761i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 38.1051 1.77090 0.885448 0.464739i \(-0.153852\pi\)
0.885448 + 0.464739i \(0.153852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 27.0000 1.24674
\(470\) 0 0
\(471\) − 43.3013i − 1.99522i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 50.0000 2.27980
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 36.3731 1.64822 0.824110 0.566429i \(-0.191675\pi\)
0.824110 + 0.566429i \(0.191675\pi\)
\(488\) 0 0
\(489\) 9.00000 0.406994
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 43.3013i 1.93843i 0.246214 + 0.969216i \(0.420813\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 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
\(507\) −20.7846 −0.923077
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) − 17.3205i − 0.766214i
\(512\) 0 0
\(513\) 45.0000i 1.98680i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −29.4449 −1.28753 −0.643767 0.765222i \(-0.722629\pi\)
−0.643767 + 0.765222i \(0.722629\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 0 0
\(543\) 12.1244 0.520306
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24.2487 1.03680 0.518400 0.855138i \(-0.326528\pi\)
0.518400 + 0.855138i \(0.326528\pi\)
\(548\) 0 0
\(549\) −39.0000 −1.66448
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 30.0000i 1.27573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 60.6218i 2.56403i
\(560\) 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
\(567\) 15.5885 0.654654
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) − 8.66025i − 0.362420i −0.983444 0.181210i \(-0.941999\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 35.0000i 1.45707i 0.685009 + 0.728535i \(0.259798\pi\)
−0.685009 + 0.728535i \(0.740202\pi\)
\(578\) 0 0
\(579\) 43.3013i 1.79954i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 75.0000 3.09032
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 45.0000i − 1.84173i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 46.7654 1.90443
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 45.0333 1.82785 0.913923 0.405887i \(-0.133038\pi\)
0.913923 + 0.405887i \(0.133038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 8.66025i 0.348085i 0.984738 + 0.174042i \(0.0556830\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 25.9808i 1.03428i 0.855901 + 0.517139i \(0.173003\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 45.0000i 1.78859i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 20.0000i − 0.792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 31.1769 1.22950 0.614749 0.788723i \(-0.289257\pi\)
0.614749 + 0.788723i \(0.289257\pi\)
\(644\) 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
\(651\) − 25.9808i − 1.01827i
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 30.0000i − 1.17041i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −51.0000 −1.97177
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 50.0000i − 1.92736i −0.267063 0.963679i \(-0.586053\pi\)
0.267063 0.963679i \(-0.413947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) − 8.66025i − 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 50.2295 1.91637
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 51.9615i − 1.97671i −0.152167 0.988355i \(-0.548625\pi\)
0.152167 0.988355i \(-0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 86.6025 3.26628
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 51.9615i 1.94871i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) −29.4449 −1.09507
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −53.6936 −1.99138 −0.995692 0.0927199i \(-0.970444\pi\)
−0.995692 + 0.0927199i \(0.970444\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 51.9615i 1.91144i 0.294285 + 0.955718i \(0.404919\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −75.0000 −2.75519
\(742\) 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
\(751\) − 17.3205i − 0.632034i −0.948753 0.316017i \(-0.897654\pi\)
0.948753 0.316017i \(-0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 55.0000i 1.99901i 0.0314762 + 0.999505i \(0.489979\pi\)
−0.0314762 + 0.999505i \(0.510021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −32.9090 −1.19138
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 30.0000i − 1.07624i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 50.2295 1.79049 0.895244 0.445577i \(-0.147001\pi\)
0.895244 + 0.445577i \(0.147001\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 65.0000i − 2.30822i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 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
\(811\) − 43.3013i − 1.52051i −0.649623 0.760257i \(-0.725073\pi\)
0.649623 0.760257i \(-0.274927\pi\)
\(812\) 0 0
\(813\) − 30.0000i − 1.05215i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 105.000i 3.67348i
\(818\) 0 0
\(819\) 25.9808i 0.907841i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −57.1577 −1.99239 −0.996196 0.0871445i \(-0.972226\pi\)
−0.996196 + 0.0871445i \(0.972226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) − 8.66025i − 0.300421i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 45.0000i − 1.55543i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −19.0526 −0.654654
\(848\) 0 0
\(849\) −39.0000 −1.33848
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 35.0000i − 1.19838i −0.800608 0.599189i \(-0.795490\pi\)
0.800608 0.599189i \(-0.204510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) − 17.3205i − 0.590968i −0.955348 0.295484i \(-0.904519\pi\)
0.955348 0.295484i \(-0.0954809\pi\)
\(860\) 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
\(867\) −29.4449 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 77.9423i 2.64097i
\(872\) 0 0
\(873\) − 15.0000i − 0.507673i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 25.0000i − 0.844190i −0.906552 0.422095i \(-0.861295\pi\)
0.906552 0.422095i \(-0.138705\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −22.5167 −0.757746 −0.378873 0.925449i \(-0.623688\pi\)
−0.378873 + 0.925449i \(0.623688\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 36.3731 1.21042
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.0333 −1.49531 −0.747653 0.664089i \(-0.768820\pi\)
−0.747653 + 0.664089i \(0.768820\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(917\) 0 0
\(918\) 0 0
\(919\) − 60.6218i − 1.99973i −0.0164935 0.999864i \(-0.505250\pi\)
0.0164935 0.999864i \(-0.494750\pi\)
\(920\) 0 0
\(921\) 3.00000 0.0988534
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10.3923 −0.341328
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) − 34.6410i − 1.13531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000i 1.14340i 0.820463 + 0.571700i \(0.193716\pi\)
−0.820463 + 0.571700i \(0.806284\pi\)
\(938\) 0 0
\(939\) − 60.6218i − 1.97832i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 50.0000 1.62307
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −44.0000 −1.41935
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 58.8897 1.89377 0.946883 0.321578i \(-0.104213\pi\)
0.946883 + 0.321578i \(0.104213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) − 30.0000i − 0.961756i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −57.0000 −1.81987
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 60.6218i 1.92571i 0.270011 + 0.962857i \(0.412973\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) 30.0000i 0.952021i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 0 0
\(999\) − 51.9615i − 1.64399i
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 1200.2.o.h.1199.4 4
3.2 odd 2 CM 1200.2.o.h.1199.4 4
4.3 odd 2 inner 1200.2.o.h.1199.2 4
5.2 odd 4 1200.2.h.f.1151.1 yes 2
5.3 odd 4 1200.2.h.d.1151.2 yes 2
5.4 even 2 inner 1200.2.o.h.1199.1 4
12.11 even 2 inner 1200.2.o.h.1199.2 4
15.2 even 4 1200.2.h.f.1151.1 yes 2
15.8 even 4 1200.2.h.d.1151.2 yes 2
15.14 odd 2 inner 1200.2.o.h.1199.1 4
20.3 even 4 1200.2.h.d.1151.1 2
20.7 even 4 1200.2.h.f.1151.2 yes 2
20.19 odd 2 inner 1200.2.o.h.1199.3 4
60.23 odd 4 1200.2.h.d.1151.1 2
60.47 odd 4 1200.2.h.f.1151.2 yes 2
60.59 even 2 inner 1200.2.o.h.1199.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1200.2.h.d.1151.1 2 20.3 even 4
1200.2.h.d.1151.1 2 60.23 odd 4
1200.2.h.d.1151.2 yes 2 5.3 odd 4
1200.2.h.d.1151.2 yes 2 15.8 even 4
1200.2.h.f.1151.1 yes 2 5.2 odd 4
1200.2.h.f.1151.1 yes 2 15.2 even 4
1200.2.h.f.1151.2 yes 2 20.7 even 4
1200.2.h.f.1151.2 yes 2 60.47 odd 4
1200.2.o.h.1199.1 4 5.4 even 2 inner
1200.2.o.h.1199.1 4 15.14 odd 2 inner
1200.2.o.h.1199.2 4 4.3 odd 2 inner
1200.2.o.h.1199.2 4 12.11 even 2 inner
1200.2.o.h.1199.3 4 20.19 odd 2 inner
1200.2.o.h.1199.3 4 60.59 even 2 inner
1200.2.o.h.1199.4 4 1.1 even 1 trivial
1200.2.o.h.1199.4 4 3.2 odd 2 CM