Properties

Label 1200.2.o.g.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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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.g.1199.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} -5.19615 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} -5.19615 q^{7} +3.00000 q^{9} +7.00000i q^{13} +5.19615i q^{19} -9.00000 q^{21} +5.19615 q^{27} +1.73205i q^{31} +10.0000i q^{37} +12.1244i q^{39} -1.73205 q^{43} +20.0000 q^{49} +9.00000i q^{57} -1.00000 q^{61} -15.5885 q^{63} -12.1244 q^{67} +10.0000i q^{73} -17.3205i q^{79} +9.00000 q^{81} -36.3731i q^{91} +3.00000i q^{93} -19.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 36q^{21} + 80q^{49} - 4q^{61} + 36q^{81} + O(q^{100}) \)

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
\(7\) −5.19615 −1.96396 −0.981981 0.188982i \(-0.939481\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 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
\(13\) 7.00000i 1.94145i 0.240192 + 0.970725i \(0.422790\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i 0.802955 + 0.596040i \(0.203260\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) −9.00000 −1.96396
\(22\) 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
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 12.1244i 1.94145i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.73205 −0.264135 −0.132068 0.991241i \(-0.542162\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 20.0000 2.85714
\(50\) 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
\(57\) 9.00000i 1.19208i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −15.5885 −1.96396
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.1244 −1.48123 −0.740613 0.671932i \(-0.765465\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 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 \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 17.3205i − 1.94871i −0.225018 0.974355i \(-0.572244\pi\)
0.225018 0.974355i \(-0.427756\pi\)
\(80\) 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
\(91\) − 36.3731i − 3.81293i
\(92\) 0 0
\(93\) 3.00000i 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 19.0000i − 1.92916i −0.263795 0.964579i \(-0.584974\pi\)
0.263795 0.964579i \(-0.415026\pi\)
\(98\) 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
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 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
\(117\) 21.0000i 1.94145i
\(118\) 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
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 27.0000i − 2.34120i
\(134\) 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 \(0.262608\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 34.6410 2.85714
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 15.5885i 1.26857i 0.773099 + 0.634285i \(0.218706\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 11.0000i − 0.877896i −0.898513 0.438948i \(-0.855351\pi\)
0.898513 0.438948i \(-0.144649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0526 1.49231 0.746156 0.665771i \(-0.231897\pi\)
0.746156 + 0.665771i \(0.231897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −36.0000 −2.76923
\(170\) 0 0
\(171\) 15.5885i 1.19208i
\(172\) 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
\(181\) 19.0000 1.41226 0.706129 0.708083i \(-0.250440\pi\)
0.706129 + 0.708083i \(0.250440\pi\)
\(182\) 0 0
\(183\) −1.73205 −0.128037
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −27.0000 −1.96396
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 23.0000i 1.65558i 0.561041 + 0.827788i \(0.310401\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) − 22.5167i − 1.59616i −0.602549 0.798082i \(-0.705848\pi\)
0.602549 0.798082i \(-0.294152\pi\)
\(200\) 0 0
\(201\) −21.0000 −1.48123
\(202\) 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
\(211\) 1.73205i 0.119239i 0.998221 + 0.0596196i \(0.0189888\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.00000i − 0.610960i
\(218\) 0 0
\(219\) 17.3205i 1.17041i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0526 1.27585 0.637927 0.770097i \(-0.279792\pi\)
0.637927 + 0.770097i \(0.279792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 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
\(241\) 31.0000 1.99689 0.998443 0.0557856i \(-0.0177663\pi\)
0.998443 + 0.0557856i \(0.0177663\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −36.3731 −2.31436
\(248\) 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
\(259\) − 51.9615i − 3.22873i
\(260\) 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 \(0.176336\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) − 63.0000i − 3.81293i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 31.0000i − 1.86261i −0.364241 0.931305i \(-0.618672\pi\)
0.364241 0.931305i \(-0.381328\pi\)
\(278\) 0 0
\(279\) 5.19615i 0.311086i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 32.9090 1.95623 0.978117 0.208053i \(-0.0667128\pi\)
0.978117 + 0.208053i \(0.0667128\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) − 32.9090i − 1.92916i
\(292\) 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
\(301\) 9.00000 0.518751
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.4449 1.68051 0.840254 0.542194i \(-0.182406\pi\)
0.840254 + 0.542194i \(0.182406\pi\)
\(308\) 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
\(313\) − 13.0000i − 0.734803i −0.930062 0.367402i \(-0.880247\pi\)
0.930062 0.367402i \(-0.119753\pi\)
\(314\) 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
\(327\) 29.4449 1.62830
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 17.3205i − 0.952021i −0.879440 0.476011i \(-0.842082\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 30.0000i 1.64399i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −67.5500 −3.64736
\(344\) 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
\(351\) 36.3731i 1.94145i
\(352\) 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
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) −19.0526 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 22.5167 1.17536 0.587680 0.809093i \(-0.300041\pi\)
0.587680 + 0.809093i \(0.300041\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 13.0000i − 0.673114i −0.941663 0.336557i \(-0.890737\pi\)
0.941663 0.336557i \(-0.109263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.1244i 0.622786i 0.950281 + 0.311393i \(0.100796\pi\)
−0.950281 + 0.311393i \(0.899204\pi\)
\(380\) 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
\(387\) −5.19615 −0.264135
\(388\) 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
\(397\) 1.00000i 0.0501886i 0.999685 + 0.0250943i \(0.00798860\pi\)
−0.999685 + 0.0250943i \(0.992011\pi\)
\(398\) 0 0
\(399\) − 46.7654i − 2.34120i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −12.1244 −0.603957
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 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
\(427\) 5.19615 0.251459
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 37.0000i − 1.77811i −0.457804 0.889053i \(-0.651364\pi\)
0.457804 0.889053i \(-0.348636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 39.8372i 1.90132i 0.310228 + 0.950662i \(0.399595\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 60.0000 2.85714
\(442\) 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
\(453\) 27.0000i 1.26857i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 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
\(469\) 63.0000 2.90907
\(470\) 0 0
\(471\) − 19.0526i − 0.877896i
\(472\) 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
\(481\) −70.0000 −3.19173
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −39.8372 −1.80519 −0.902597 0.430486i \(-0.858342\pi\)
−0.902597 + 0.430486i \(0.858342\pi\)
\(488\) 0 0
\(489\) 33.0000 1.49231
\(490\) 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
\(499\) 12.1244i 0.542761i 0.962472 + 0.271380i \(0.0874801\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 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
\(507\) −62.3538 −2.76923
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) − 51.9615i − 2.29864i
\(512\) 0 0
\(513\) 27.0000i 1.19208i
\(514\) 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
\(523\) −15.5885 −0.681636 −0.340818 0.940129i \(-0.610704\pi\)
−0.340818 + 0.940129i \(0.610704\pi\)
\(524\) 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
\(541\) −29.0000 −1.24681 −0.623404 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 0 0
\(543\) 32.9090 1.41226
\(544\) 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
\(549\) −3.00000 −0.128037
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 90.0000i 3.82719i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) − 12.1244i − 0.512806i
\(560\) 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
\(567\) −46.7654 −1.96396
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 36.3731i 1.52217i 0.648655 + 0.761083i \(0.275332\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 11.0000i − 0.457936i −0.973434 0.228968i \(-0.926465\pi\)
0.973434 0.228968i \(-0.0735351\pi\)
\(578\) 0 0
\(579\) 39.8372i 1.65558i
\(580\) 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
\(589\) −9.00000 −0.370839
\(590\) 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
\(597\) − 39.0000i − 1.59616i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −49.0000 −1.99875 −0.999376 0.0353259i \(-0.988753\pi\)
−0.999376 + 0.0353259i \(0.988753\pi\)
\(602\) 0 0
\(603\) −36.3731 −1.48123
\(604\) 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 \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 46.7654i 1.87966i 0.341644 + 0.939829i \(0.389016\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 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
\(631\) 50.2295i 1.99960i 0.0199047 + 0.999802i \(0.493664\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 3.00000i 0.119239i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 140.000i 5.54700i
\(638\) 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
\(651\) − 15.5885i − 0.610960i
\(652\) 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
\(669\) 33.0000 1.27585
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 50.0000i 1.92736i 0.267063 + 0.963679i \(0.413947\pi\)
−0.267063 + 0.963679i \(0.586053\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 98.7269i 3.78879i
\(680\) 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
\(687\) −12.1244 −0.462573
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 51.9615i 1.97671i 0.152167 + 0.988355i \(0.451375\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 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
\(703\) −51.9615 −1.95977
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 53.0000 1.99046 0.995228 0.0975728i \(-0.0311079\pi\)
0.995228 + 0.0975728i \(0.0311079\pi\)
\(710\) 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
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 53.6936 1.99689
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 22.5167 0.835097 0.417548 0.908655i \(-0.362889\pi\)
0.417548 + 0.908655i \(0.362889\pi\)
\(728\) 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 \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 51.9615i − 1.91144i −0.294285 0.955718i \(-0.595081\pi\)
0.294285 0.955718i \(-0.404919\pi\)
\(740\) 0 0
\(741\) −63.0000 −2.31436
\(742\) 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 \(0.102346\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.0000i 1.05402i 0.849858 + 0.527011i \(0.176688\pi\)
−0.849858 + 0.527011i \(0.823312\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −88.3346 −3.19793
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 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
\(777\) − 90.0000i − 3.22873i
\(778\) 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
\(787\) −46.7654 −1.66701 −0.833503 0.552515i \(-0.813668\pi\)
−0.833503 + 0.552515i \(0.813668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 7.00000i − 0.248577i
\(794\) 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
\(811\) − 53.6936i − 1.88544i −0.333590 0.942718i \(-0.608260\pi\)
0.333590 0.942718i \(-0.391740\pi\)
\(812\) 0 0
\(813\) 30.0000i 1.05215i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 9.00000i − 0.314870i
\(818\) 0 0
\(819\) − 109.119i − 3.81293i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 32.9090 1.14713 0.573567 0.819159i \(-0.305559\pi\)
0.573567 + 0.819159i \(0.305559\pi\)
\(824\) 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
\(831\) − 53.6936i − 1.86261i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.00000i 0.311086i
\(838\) 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
\(847\) 57.1577 1.96396
\(848\) 0 0
\(849\) 57.0000 1.95623
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 23.0000i 0.787505i 0.919216 + 0.393753i \(0.128823\pi\)
−0.919216 + 0.393753i \(0.871177\pi\)
\(854\) 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 \(0.0954809\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 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
\(871\) − 84.8705i − 2.87573i
\(872\) 0 0
\(873\) − 57.0000i − 1.92916i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 59.0000i − 1.99229i −0.0877308 0.996144i \(-0.527962\pi\)
0.0877308 0.996144i \(-0.472038\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −36.3731 −1.22405 −0.612026 0.790838i \(-0.709645\pi\)
−0.612026 + 0.790838i \(0.709645\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 54.0000 1.81110
\(890\) 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
\(903\) 15.5885 0.518751
\(904\) 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
\(919\) − 29.4449i − 0.971296i −0.874154 0.485648i \(-0.838584\pi\)
0.874154 0.485648i \(-0.161416\pi\)
\(920\) 0 0
\(921\) 51.0000 1.68051
\(922\) 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
\(931\) 103.923i 3.40594i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 61.0000i 1.99278i 0.0848755 + 0.996392i \(0.472951\pi\)
−0.0848755 + 0.996392i \(0.527049\pi\)
\(938\) 0 0
\(939\) − 22.5167i − 0.734803i
\(940\) 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
\(949\) −70.0000 −2.27230
\(950\) 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
\(961\) 28.0000 0.903226
\(962\) 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
\(973\) − 90.0000i − 2.88527i
\(974\) 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
\(981\) 51.0000 1.62830
\(982\) 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
\(991\) 15.5885i 0.495184i 0.968864 + 0.247592i \(0.0796392\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 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 \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) 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.g.1199.4 4
3.2 odd 2 CM 1200.2.o.g.1199.4 4
4.3 odd 2 inner 1200.2.o.g.1199.2 4
5.2 odd 4 1200.2.h.g.1151.1 yes 2
5.3 odd 4 1200.2.h.c.1151.2 yes 2
5.4 even 2 inner 1200.2.o.g.1199.1 4
12.11 even 2 inner 1200.2.o.g.1199.2 4
15.2 even 4 1200.2.h.g.1151.1 yes 2
15.8 even 4 1200.2.h.c.1151.2 yes 2
15.14 odd 2 inner 1200.2.o.g.1199.1 4
20.3 even 4 1200.2.h.c.1151.1 2
20.7 even 4 1200.2.h.g.1151.2 yes 2
20.19 odd 2 inner 1200.2.o.g.1199.3 4
60.23 odd 4 1200.2.h.c.1151.1 2
60.47 odd 4 1200.2.h.g.1151.2 yes 2
60.59 even 2 inner 1200.2.o.g.1199.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1200.2.h.c.1151.1 2 20.3 even 4
1200.2.h.c.1151.1 2 60.23 odd 4
1200.2.h.c.1151.2 yes 2 5.3 odd 4
1200.2.h.c.1151.2 yes 2 15.8 even 4
1200.2.h.g.1151.1 yes 2 5.2 odd 4
1200.2.h.g.1151.1 yes 2 15.2 even 4
1200.2.h.g.1151.2 yes 2 20.7 even 4
1200.2.h.g.1151.2 yes 2 60.47 odd 4
1200.2.o.g.1199.1 4 5.4 even 2 inner
1200.2.o.g.1199.1 4 15.14 odd 2 inner
1200.2.o.g.1199.2 4 4.3 odd 2 inner
1200.2.o.g.1199.2 4 12.11 even 2 inner
1200.2.o.g.1199.3 4 20.19 odd 2 inner
1200.2.o.g.1199.3 4 60.59 even 2 inner
1200.2.o.g.1199.4 4 1.1 even 1 trivial
1200.2.o.g.1199.4 4 3.2 odd 2 CM