Properties

Label 7728.2.a.u.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +3.00000 q^{13} +3.00000 q^{15} -1.00000 q^{21} +1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.00000 q^{31} -4.00000 q^{33} -3.00000 q^{35} -5.00000 q^{37} +3.00000 q^{39} +5.00000 q^{41} +7.00000 q^{43} +3.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +2.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +9.00000 q^{65} +12.0000 q^{67} +1.00000 q^{69} -10.0000 q^{71} +4.00000 q^{75} +4.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +1.00000 q^{87} +10.0000 q^{89} -3.00000 q^{91} +2.00000 q^{93} +19.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 9.00000 1.11631
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.0000 1.92916 0.964579 0.263795i \(-0.0849741\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −24.0000 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 0 0
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) −21.0000 −1.49619 −0.748094 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) 0 0
\(199\) 23.0000 1.63043 0.815213 0.579161i \(-0.196620\pi\)
0.815213 + 0.579161i \(0.196620\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −10.0000 −0.685189
\(214\) 0 0
\(215\) 21.0000 1.43219
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 5.00000 0.331862 0.165931 0.986137i \(-0.446937\pi\)
0.165931 + 0.986137i \(0.446937\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 9.00000 0.579741 0.289870 0.957066i \(-0.406388\pi\)
0.289870 + 0.957066i \(0.406388\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 0 0
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.00000 −0.295141
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 19.0000 1.11380
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 12.0000 0.665640
\(326\) 0 0
\(327\) −1.00000 −0.0553001
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 36.0000 1.96689
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 7.00000 0.380188
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) −17.0000 −0.904819 −0.452409 0.891810i \(-0.649435\pi\)
−0.452409 + 0.891810i \(0.649435\pi\)
\(354\) 0 0
\(355\) −30.0000 −1.59223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) 0 0
\(369\) 5.00000 0.260290
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 7.00000 0.355830
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −17.0000 −0.838548
\(412\) 0 0
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) −27.0000 −1.29754 −0.648769 0.760986i \(-0.724716\pi\)
−0.648769 + 0.760986i \(0.724716\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 29.0000 1.37783 0.688916 0.724841i \(-0.258087\pi\)
0.688916 + 0.724841i \(0.258087\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) 17.0000 0.798730
\(454\) 0 0
\(455\) −9.00000 −0.421927
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 0 0
\(473\) −28.0000 −1.28744
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 57.0000 2.58824
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 10.0000 0.448561
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 42.0000 1.86898
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.00000 0.132196
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) −9.00000 −0.388379
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −15.0000 −0.636715
\(556\) 0 0
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 0 0
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 21.0000 0.883477
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) −1.00000 −0.0415586
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −48.0000 −1.98796
\(584\) 0 0
\(585\) 9.00000 0.372104
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) 0 0
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.0000 0.941327
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 15.0000 0.609837
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 21.0000 0.848182 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(614\) 0 0
\(615\) 15.0000 0.604858
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −21.0000 −0.833360
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 1.00000 0.0394976 0.0197488 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 21.0000 0.826874
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) 13.0000 0.508729 0.254365 0.967108i \(-0.418134\pi\)
0.254365 + 0.967108i \(0.418134\pi\)
\(654\) 0 0
\(655\) 54.0000 2.10995
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −19.0000 −0.729153
\(680\) 0 0
\(681\) 5.00000 0.191600
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −51.0000 −1.94861
\(686\) 0 0
\(687\) 16.0000 0.610438
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −21.0000 −0.798878 −0.399439 0.916760i \(-0.630795\pi\)
−0.399439 + 0.916760i \(0.630795\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) −21.0000 −0.796575
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 9.00000 0.338960
\(706\) 0 0
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 43.0000 1.60363 0.801815 0.597573i \(-0.203868\pi\)
0.801815 + 0.597573i \(0.203868\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) 9.00000 0.334714
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 36.0000 1.31894
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −19.0000 −0.692398
\(754\) 0 0
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 1.00000 0.0362024
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 35.0000 1.25886 0.629431 0.777056i \(-0.283288\pi\)
0.629431 + 0.777056i \(0.283288\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 5.00000 0.179374
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −72.0000 −2.56979
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −7.00000 −0.248891
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 0 0
\(795\) 36.0000 1.27679
\(796\) 0 0
\(797\) 47.0000 1.66483 0.832413 0.554156i \(-0.186959\pi\)
0.832413 + 0.554156i \(0.186959\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) 0 0
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) −23.0000 −0.807639 −0.403820 0.914839i \(-0.632318\pi\)
−0.403820 + 0.914839i \(0.632318\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) −16.0000 −0.557048
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 1.00000 0.0344418
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −5.00000 −0.171398
\(852\) 0 0
\(853\) 41.0000 1.40381 0.701907 0.712269i \(-0.252332\pi\)
0.701907 + 0.712269i \(0.252332\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 0 0
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 0 0
\(861\) −5.00000 −0.170400
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 0 0
\(873\) 19.0000 0.643053
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 0 0
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) −6.00000 −0.201916 −0.100958 0.994891i \(-0.532191\pi\)
−0.100958 + 0.994891i \(0.532191\pi\)
\(884\) 0 0
\(885\) 6.00000 0.201688
\(886\) 0 0
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −27.0000 −0.902510
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −7.00000 −0.232945
\(904\) 0 0
\(905\) 54.0000 1.79502
\(906\) 0 0
\(907\) 49.0000 1.62702 0.813509 0.581552i \(-0.197554\pi\)
0.813509 + 0.581552i \(0.197554\pi\)
\(908\) 0 0
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 55.0000 1.82223 0.911116 0.412151i \(-0.135222\pi\)
0.911116 + 0.412151i \(0.135222\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) −18.0000 −0.595062
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −60.0000 −1.97922 −0.989609 0.143787i \(-0.954072\pi\)
−0.989609 + 0.143787i \(0.954072\pi\)
\(920\) 0 0
\(921\) −25.0000 −0.823778
\(922\) 0 0
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) 1.00000 0.0328443
\(928\) 0 0
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −53.0000 −1.73143 −0.865717 0.500533i \(-0.833137\pi\)
−0.865717 + 0.500533i \(0.833137\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −21.0000 −0.684580 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(942\) 0 0
\(943\) 5.00000 0.162822
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) −3.00000 −0.0974869 −0.0487435 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −33.0000 −1.07010
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 60.0000 1.94155
\(956\) 0 0
\(957\) −4.00000 −0.129302
\(958\) 0 0
\(959\) 17.0000 0.548959
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −3.00000 −0.0965734
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 7.00000 0.224410
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) 0 0
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) 0 0
\(983\) −2.00000 −0.0637901 −0.0318950 0.999491i \(-0.510154\pi\)
−0.0318950 + 0.999491i \(0.510154\pi\)
\(984\) 0 0
\(985\) −63.0000 −2.00735
\(986\) 0 0
\(987\) −3.00000 −0.0954911
\(988\) 0 0
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 69.0000 2.18745
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 0 0
\(999\) −5.00000 −0.158193
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 7728.2.a.u.1.1 1
4.3 odd 2 966.2.a.d.1.1 1
12.11 even 2 2898.2.a.k.1.1 1
28.27 even 2 6762.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.d.1.1 1 4.3 odd 2
2898.2.a.k.1.1 1 12.11 even 2
6762.2.a.l.1.1 1 28.27 even 2
7728.2.a.u.1.1 1 1.1 even 1 trivial