Properties

Label 2352.4.a.s.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +50.0000 q^{11} -36.0000 q^{13} -54.0000 q^{15} +126.000 q^{17} +72.0000 q^{19} -14.0000 q^{23} +199.000 q^{25} -27.0000 q^{27} +158.000 q^{29} +36.0000 q^{31} -150.000 q^{33} -162.000 q^{37} +108.000 q^{39} -270.000 q^{41} +324.000 q^{43} +162.000 q^{45} +72.0000 q^{47} -378.000 q^{51} -22.0000 q^{53} +900.000 q^{55} -216.000 q^{57} -468.000 q^{59} +792.000 q^{61} -648.000 q^{65} -232.000 q^{67} +42.0000 q^{69} +734.000 q^{71} +180.000 q^{73} -597.000 q^{75} -236.000 q^{79} +81.0000 q^{81} -36.0000 q^{83} +2268.00 q^{85} -474.000 q^{87} +234.000 q^{89} -108.000 q^{93} +1296.00 q^{95} +468.000 q^{97} +450.000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 50.0000 1.37051 0.685253 0.728305i \(-0.259692\pi\)
0.685253 + 0.728305i \(0.259692\pi\)
\(12\) 0 0
\(13\) −36.0000 −0.768046 −0.384023 0.923323i \(-0.625462\pi\)
−0.384023 + 0.923323i \(0.625462\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) 0 0
\(17\) 126.000 1.79762 0.898808 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) 0 0
\(19\) 72.0000 0.869365 0.434682 0.900584i \(-0.356861\pi\)
0.434682 + 0.900584i \(0.356861\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −14.0000 −0.126922 −0.0634609 0.997984i \(-0.520214\pi\)
−0.0634609 + 0.997984i \(0.520214\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 158.000 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(30\) 0 0
\(31\) 36.0000 0.208574 0.104287 0.994547i \(-0.466744\pi\)
0.104287 + 0.994547i \(0.466744\pi\)
\(32\) 0 0
\(33\) −150.000 −0.791262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −162.000 −0.719801 −0.359900 0.932991i \(-0.617189\pi\)
−0.359900 + 0.932991i \(0.617189\pi\)
\(38\) 0 0
\(39\) 108.000 0.443432
\(40\) 0 0
\(41\) −270.000 −1.02846 −0.514231 0.857652i \(-0.671922\pi\)
−0.514231 + 0.857652i \(0.671922\pi\)
\(42\) 0 0
\(43\) 324.000 1.14906 0.574529 0.818484i \(-0.305185\pi\)
0.574529 + 0.818484i \(0.305185\pi\)
\(44\) 0 0
\(45\) 162.000 0.536656
\(46\) 0 0
\(47\) 72.0000 0.223453 0.111726 0.993739i \(-0.464362\pi\)
0.111726 + 0.993739i \(0.464362\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −378.000 −1.03785
\(52\) 0 0
\(53\) −22.0000 −0.0570176 −0.0285088 0.999594i \(-0.509076\pi\)
−0.0285088 + 0.999594i \(0.509076\pi\)
\(54\) 0 0
\(55\) 900.000 2.20647
\(56\) 0 0
\(57\) −216.000 −0.501928
\(58\) 0 0
\(59\) −468.000 −1.03268 −0.516342 0.856382i \(-0.672707\pi\)
−0.516342 + 0.856382i \(0.672707\pi\)
\(60\) 0 0
\(61\) 792.000 1.66238 0.831190 0.555988i \(-0.187660\pi\)
0.831190 + 0.555988i \(0.187660\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −648.000 −1.23653
\(66\) 0 0
\(67\) −232.000 −0.423034 −0.211517 0.977374i \(-0.567840\pi\)
−0.211517 + 0.977374i \(0.567840\pi\)
\(68\) 0 0
\(69\) 42.0000 0.0732783
\(70\) 0 0
\(71\) 734.000 1.22690 0.613449 0.789734i \(-0.289782\pi\)
0.613449 + 0.789734i \(0.289782\pi\)
\(72\) 0 0
\(73\) 180.000 0.288595 0.144297 0.989534i \(-0.453908\pi\)
0.144297 + 0.989534i \(0.453908\pi\)
\(74\) 0 0
\(75\) −597.000 −0.919142
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −236.000 −0.336102 −0.168051 0.985778i \(-0.553747\pi\)
−0.168051 + 0.985778i \(0.553747\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −36.0000 −0.0476086 −0.0238043 0.999717i \(-0.507578\pi\)
−0.0238043 + 0.999717i \(0.507578\pi\)
\(84\) 0 0
\(85\) 2268.00 2.89411
\(86\) 0 0
\(87\) −474.000 −0.584116
\(88\) 0 0
\(89\) 234.000 0.278696 0.139348 0.990243i \(-0.455499\pi\)
0.139348 + 0.990243i \(0.455499\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −108.000 −0.120420
\(94\) 0 0
\(95\) 1296.00 1.39965
\(96\) 0 0
\(97\) 468.000 0.489878 0.244939 0.969538i \(-0.421232\pi\)
0.244939 + 0.969538i \(0.421232\pi\)
\(98\) 0 0
\(99\) 450.000 0.456835
\(100\) 0 0
\(101\) −666.000 −0.656133 −0.328067 0.944655i \(-0.606397\pi\)
−0.328067 + 0.944655i \(0.606397\pi\)
\(102\) 0 0
\(103\) 252.000 0.241071 0.120535 0.992709i \(-0.461539\pi\)
0.120535 + 0.992709i \(0.461539\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −670.000 −0.605340 −0.302670 0.953095i \(-0.597878\pi\)
−0.302670 + 0.953095i \(0.597878\pi\)
\(108\) 0 0
\(109\) 162.000 0.142356 0.0711779 0.997464i \(-0.477324\pi\)
0.0711779 + 0.997464i \(0.477324\pi\)
\(110\) 0 0
\(111\) 486.000 0.415577
\(112\) 0 0
\(113\) −1390.00 −1.15717 −0.578585 0.815622i \(-0.696395\pi\)
−0.578585 + 0.815622i \(0.696395\pi\)
\(114\) 0 0
\(115\) −252.000 −0.204340
\(116\) 0 0
\(117\) −324.000 −0.256015
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) 0 0
\(123\) 810.000 0.593782
\(124\) 0 0
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) −916.000 −0.640015 −0.320007 0.947415i \(-0.603685\pi\)
−0.320007 + 0.947415i \(0.603685\pi\)
\(128\) 0 0
\(129\) −972.000 −0.663410
\(130\) 0 0
\(131\) −2268.00 −1.51264 −0.756321 0.654201i \(-0.773005\pi\)
−0.756321 + 0.654201i \(0.773005\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −486.000 −0.309839
\(136\) 0 0
\(137\) 806.000 0.502637 0.251318 0.967904i \(-0.419136\pi\)
0.251318 + 0.967904i \(0.419136\pi\)
\(138\) 0 0
\(139\) −2628.00 −1.60363 −0.801813 0.597575i \(-0.796131\pi\)
−0.801813 + 0.597575i \(0.796131\pi\)
\(140\) 0 0
\(141\) −216.000 −0.129011
\(142\) 0 0
\(143\) −1800.00 −1.05261
\(144\) 0 0
\(145\) 2844.00 1.62884
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2390.00 −1.31407 −0.657035 0.753860i \(-0.728190\pi\)
−0.657035 + 0.753860i \(0.728190\pi\)
\(150\) 0 0
\(151\) −3240.00 −1.74614 −0.873071 0.487593i \(-0.837875\pi\)
−0.873071 + 0.487593i \(0.837875\pi\)
\(152\) 0 0
\(153\) 1134.00 0.599206
\(154\) 0 0
\(155\) 648.000 0.335798
\(156\) 0 0
\(157\) 3024.00 1.53721 0.768603 0.639726i \(-0.220952\pi\)
0.768603 + 0.639726i \(0.220952\pi\)
\(158\) 0 0
\(159\) 66.0000 0.0329191
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1784.00 0.857262 0.428631 0.903480i \(-0.358996\pi\)
0.428631 + 0.903480i \(0.358996\pi\)
\(164\) 0 0
\(165\) −2700.00 −1.27391
\(166\) 0 0
\(167\) 3024.00 1.40122 0.700611 0.713543i \(-0.252911\pi\)
0.700611 + 0.713543i \(0.252911\pi\)
\(168\) 0 0
\(169\) −901.000 −0.410105
\(170\) 0 0
\(171\) 648.000 0.289788
\(172\) 0 0
\(173\) 1566.00 0.688213 0.344106 0.938931i \(-0.388182\pi\)
0.344106 + 0.938931i \(0.388182\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1404.00 0.596221
\(178\) 0 0
\(179\) −3802.00 −1.58757 −0.793784 0.608199i \(-0.791892\pi\)
−0.793784 + 0.608199i \(0.791892\pi\)
\(180\) 0 0
\(181\) −468.000 −0.192189 −0.0960944 0.995372i \(-0.530635\pi\)
−0.0960944 + 0.995372i \(0.530635\pi\)
\(182\) 0 0
\(183\) −2376.00 −0.959776
\(184\) 0 0
\(185\) −2916.00 −1.15886
\(186\) 0 0
\(187\) 6300.00 2.46365
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −482.000 −0.182598 −0.0912992 0.995824i \(-0.529102\pi\)
−0.0912992 + 0.995824i \(0.529102\pi\)
\(192\) 0 0
\(193\) −810.000 −0.302099 −0.151049 0.988526i \(-0.548265\pi\)
−0.151049 + 0.988526i \(0.548265\pi\)
\(194\) 0 0
\(195\) 1944.00 0.713911
\(196\) 0 0
\(197\) −2462.00 −0.890407 −0.445204 0.895429i \(-0.646869\pi\)
−0.445204 + 0.895429i \(0.646869\pi\)
\(198\) 0 0
\(199\) 4536.00 1.61582 0.807911 0.589305i \(-0.200598\pi\)
0.807911 + 0.589305i \(0.200598\pi\)
\(200\) 0 0
\(201\) 696.000 0.244239
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4860.00 −1.65579
\(206\) 0 0
\(207\) −126.000 −0.0423073
\(208\) 0 0
\(209\) 3600.00 1.19147
\(210\) 0 0
\(211\) −2916.00 −0.951402 −0.475701 0.879607i \(-0.657805\pi\)
−0.475701 + 0.879607i \(0.657805\pi\)
\(212\) 0 0
\(213\) −2202.00 −0.708350
\(214\) 0 0
\(215\) 5832.00 1.84995
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −540.000 −0.166620
\(220\) 0 0
\(221\) −4536.00 −1.38065
\(222\) 0 0
\(223\) 1080.00 0.324315 0.162157 0.986765i \(-0.448155\pi\)
0.162157 + 0.986765i \(0.448155\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) 1332.00 0.389462 0.194731 0.980857i \(-0.437617\pi\)
0.194731 + 0.980857i \(0.437617\pi\)
\(228\) 0 0
\(229\) 1620.00 0.467479 0.233739 0.972299i \(-0.424904\pi\)
0.233739 + 0.972299i \(0.424904\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6718.00 1.88889 0.944444 0.328673i \(-0.106601\pi\)
0.944444 + 0.328673i \(0.106601\pi\)
\(234\) 0 0
\(235\) 1296.00 0.359752
\(236\) 0 0
\(237\) 708.000 0.194049
\(238\) 0 0
\(239\) 3578.00 0.968375 0.484187 0.874964i \(-0.339115\pi\)
0.484187 + 0.874964i \(0.339115\pi\)
\(240\) 0 0
\(241\) −756.000 −0.202067 −0.101034 0.994883i \(-0.532215\pi\)
−0.101034 + 0.994883i \(0.532215\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2592.00 −0.667713
\(248\) 0 0
\(249\) 108.000 0.0274868
\(250\) 0 0
\(251\) 6516.00 1.63859 0.819295 0.573372i \(-0.194365\pi\)
0.819295 + 0.573372i \(0.194365\pi\)
\(252\) 0 0
\(253\) −700.000 −0.173947
\(254\) 0 0
\(255\) −6804.00 −1.67091
\(256\) 0 0
\(257\) −6030.00 −1.46358 −0.731792 0.681528i \(-0.761316\pi\)
−0.731792 + 0.681528i \(0.761316\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1422.00 0.337240
\(262\) 0 0
\(263\) −590.000 −0.138331 −0.0691653 0.997605i \(-0.522034\pi\)
−0.0691653 + 0.997605i \(0.522034\pi\)
\(264\) 0 0
\(265\) −396.000 −0.0917966
\(266\) 0 0
\(267\) −702.000 −0.160905
\(268\) 0 0
\(269\) −990.000 −0.224392 −0.112196 0.993686i \(-0.535788\pi\)
−0.112196 + 0.993686i \(0.535788\pi\)
\(270\) 0 0
\(271\) 3420.00 0.766606 0.383303 0.923623i \(-0.374787\pi\)
0.383303 + 0.923623i \(0.374787\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9950.00 2.18185
\(276\) 0 0
\(277\) −2734.00 −0.593033 −0.296516 0.955028i \(-0.595825\pi\)
−0.296516 + 0.955028i \(0.595825\pi\)
\(278\) 0 0
\(279\) 324.000 0.0695246
\(280\) 0 0
\(281\) 598.000 0.126953 0.0634763 0.997983i \(-0.479781\pi\)
0.0634763 + 0.997983i \(0.479781\pi\)
\(282\) 0 0
\(283\) −3600.00 −0.756176 −0.378088 0.925770i \(-0.623418\pi\)
−0.378088 + 0.925770i \(0.623418\pi\)
\(284\) 0 0
\(285\) −3888.00 −0.808089
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) −1404.00 −0.282831
\(292\) 0 0
\(293\) 7902.00 1.57556 0.787781 0.615955i \(-0.211230\pi\)
0.787781 + 0.615955i \(0.211230\pi\)
\(294\) 0 0
\(295\) −8424.00 −1.66259
\(296\) 0 0
\(297\) −1350.00 −0.263754
\(298\) 0 0
\(299\) 504.000 0.0974818
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1998.00 0.378819
\(304\) 0 0
\(305\) 14256.0 2.67638
\(306\) 0 0
\(307\) 10224.0 1.90070 0.950349 0.311185i \(-0.100726\pi\)
0.950349 + 0.311185i \(0.100726\pi\)
\(308\) 0 0
\(309\) −756.000 −0.139182
\(310\) 0 0
\(311\) −3888.00 −0.708901 −0.354451 0.935075i \(-0.615332\pi\)
−0.354451 + 0.935075i \(0.615332\pi\)
\(312\) 0 0
\(313\) 5112.00 0.923154 0.461577 0.887100i \(-0.347284\pi\)
0.461577 + 0.887100i \(0.347284\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10102.0 −1.78986 −0.894929 0.446209i \(-0.852774\pi\)
−0.894929 + 0.446209i \(0.852774\pi\)
\(318\) 0 0
\(319\) 7900.00 1.38657
\(320\) 0 0
\(321\) 2010.00 0.349493
\(322\) 0 0
\(323\) 9072.00 1.56279
\(324\) 0 0
\(325\) −7164.00 −1.22273
\(326\) 0 0
\(327\) −486.000 −0.0821892
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5508.00 −0.914644 −0.457322 0.889301i \(-0.651191\pi\)
−0.457322 + 0.889301i \(0.651191\pi\)
\(332\) 0 0
\(333\) −1458.00 −0.239934
\(334\) 0 0
\(335\) −4176.00 −0.681072
\(336\) 0 0
\(337\) −9234.00 −1.49261 −0.746303 0.665607i \(-0.768173\pi\)
−0.746303 + 0.665607i \(0.768173\pi\)
\(338\) 0 0
\(339\) 4170.00 0.668092
\(340\) 0 0
\(341\) 1800.00 0.285852
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 756.000 0.117976
\(346\) 0 0
\(347\) 6494.00 1.00466 0.502329 0.864677i \(-0.332477\pi\)
0.502329 + 0.864677i \(0.332477\pi\)
\(348\) 0 0
\(349\) 10080.0 1.54605 0.773023 0.634378i \(-0.218744\pi\)
0.773023 + 0.634378i \(0.218744\pi\)
\(350\) 0 0
\(351\) 972.000 0.147811
\(352\) 0 0
\(353\) −738.000 −0.111274 −0.0556371 0.998451i \(-0.517719\pi\)
−0.0556371 + 0.998451i \(0.517719\pi\)
\(354\) 0 0
\(355\) 13212.0 1.97527
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −194.000 −0.0285207 −0.0142603 0.999898i \(-0.504539\pi\)
−0.0142603 + 0.999898i \(0.504539\pi\)
\(360\) 0 0
\(361\) −1675.00 −0.244205
\(362\) 0 0
\(363\) −3507.00 −0.507079
\(364\) 0 0
\(365\) 3240.00 0.464628
\(366\) 0 0
\(367\) 4752.00 0.675892 0.337946 0.941165i \(-0.390268\pi\)
0.337946 + 0.941165i \(0.390268\pi\)
\(368\) 0 0
\(369\) −2430.00 −0.342820
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2306.00 −0.320108 −0.160054 0.987108i \(-0.551167\pi\)
−0.160054 + 0.987108i \(0.551167\pi\)
\(374\) 0 0
\(375\) −3996.00 −0.550273
\(376\) 0 0
\(377\) −5688.00 −0.777047
\(378\) 0 0
\(379\) 7452.00 1.00998 0.504991 0.863124i \(-0.331496\pi\)
0.504991 + 0.863124i \(0.331496\pi\)
\(380\) 0 0
\(381\) 2748.00 0.369513
\(382\) 0 0
\(383\) 1152.00 0.153693 0.0768465 0.997043i \(-0.475515\pi\)
0.0768465 + 0.997043i \(0.475515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2916.00 0.383020
\(388\) 0 0
\(389\) 1894.00 0.246863 0.123431 0.992353i \(-0.460610\pi\)
0.123431 + 0.992353i \(0.460610\pi\)
\(390\) 0 0
\(391\) −1764.00 −0.228157
\(392\) 0 0
\(393\) 6804.00 0.873324
\(394\) 0 0
\(395\) −4248.00 −0.541114
\(396\) 0 0
\(397\) 9216.00 1.16508 0.582541 0.812801i \(-0.302058\pi\)
0.582541 + 0.812801i \(0.302058\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11650.0 −1.45081 −0.725403 0.688324i \(-0.758347\pi\)
−0.725403 + 0.688324i \(0.758347\pi\)
\(402\) 0 0
\(403\) −1296.00 −0.160194
\(404\) 0 0
\(405\) 1458.00 0.178885
\(406\) 0 0
\(407\) −8100.00 −0.986492
\(408\) 0 0
\(409\) −7524.00 −0.909628 −0.454814 0.890586i \(-0.650294\pi\)
−0.454814 + 0.890586i \(0.650294\pi\)
\(410\) 0 0
\(411\) −2418.00 −0.290197
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −648.000 −0.0766484
\(416\) 0 0
\(417\) 7884.00 0.925854
\(418\) 0 0
\(419\) 3852.00 0.449123 0.224561 0.974460i \(-0.427905\pi\)
0.224561 + 0.974460i \(0.427905\pi\)
\(420\) 0 0
\(421\) 10402.0 1.20419 0.602093 0.798426i \(-0.294334\pi\)
0.602093 + 0.798426i \(0.294334\pi\)
\(422\) 0 0
\(423\) 648.000 0.0744843
\(424\) 0 0
\(425\) 25074.0 2.86181
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5400.00 0.607726
\(430\) 0 0
\(431\) 10390.0 1.16118 0.580590 0.814196i \(-0.302822\pi\)
0.580590 + 0.814196i \(0.302822\pi\)
\(432\) 0 0
\(433\) 11232.0 1.24659 0.623297 0.781985i \(-0.285793\pi\)
0.623297 + 0.781985i \(0.285793\pi\)
\(434\) 0 0
\(435\) −8532.00 −0.940409
\(436\) 0 0
\(437\) −1008.00 −0.110341
\(438\) 0 0
\(439\) −14616.0 −1.58903 −0.794514 0.607245i \(-0.792275\pi\)
−0.794514 + 0.607245i \(0.792275\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11938.0 −1.28034 −0.640171 0.768232i \(-0.721137\pi\)
−0.640171 + 0.768232i \(0.721137\pi\)
\(444\) 0 0
\(445\) 4212.00 0.448692
\(446\) 0 0
\(447\) 7170.00 0.758679
\(448\) 0 0
\(449\) 8186.00 0.860404 0.430202 0.902733i \(-0.358442\pi\)
0.430202 + 0.902733i \(0.358442\pi\)
\(450\) 0 0
\(451\) −13500.0 −1.40951
\(452\) 0 0
\(453\) 9720.00 1.00814
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2106.00 0.215568 0.107784 0.994174i \(-0.465625\pi\)
0.107784 + 0.994174i \(0.465625\pi\)
\(458\) 0 0
\(459\) −3402.00 −0.345952
\(460\) 0 0
\(461\) 9486.00 0.958367 0.479183 0.877715i \(-0.340933\pi\)
0.479183 + 0.877715i \(0.340933\pi\)
\(462\) 0 0
\(463\) 12652.0 1.26995 0.634977 0.772531i \(-0.281010\pi\)
0.634977 + 0.772531i \(0.281010\pi\)
\(464\) 0 0
\(465\) −1944.00 −0.193873
\(466\) 0 0
\(467\) 3708.00 0.367421 0.183711 0.982980i \(-0.441189\pi\)
0.183711 + 0.982980i \(0.441189\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9072.00 −0.887507
\(472\) 0 0
\(473\) 16200.0 1.57479
\(474\) 0 0
\(475\) 14328.0 1.38403
\(476\) 0 0
\(477\) −198.000 −0.0190059
\(478\) 0 0
\(479\) 8064.00 0.769214 0.384607 0.923080i \(-0.374337\pi\)
0.384607 + 0.923080i \(0.374337\pi\)
\(480\) 0 0
\(481\) 5832.00 0.552841
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8424.00 0.788689
\(486\) 0 0
\(487\) 11664.0 1.08531 0.542655 0.839955i \(-0.317419\pi\)
0.542655 + 0.839955i \(0.317419\pi\)
\(488\) 0 0
\(489\) −5352.00 −0.494940
\(490\) 0 0
\(491\) 9814.00 0.902036 0.451018 0.892515i \(-0.351061\pi\)
0.451018 + 0.892515i \(0.351061\pi\)
\(492\) 0 0
\(493\) 19908.0 1.81868
\(494\) 0 0
\(495\) 8100.00 0.735491
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 15228.0 1.36613 0.683065 0.730358i \(-0.260647\pi\)
0.683065 + 0.730358i \(0.260647\pi\)
\(500\) 0 0
\(501\) −9072.00 −0.808996
\(502\) 0 0
\(503\) 11088.0 0.982882 0.491441 0.870911i \(-0.336470\pi\)
0.491441 + 0.870911i \(0.336470\pi\)
\(504\) 0 0
\(505\) −11988.0 −1.05635
\(506\) 0 0
\(507\) 2703.00 0.236774
\(508\) 0 0
\(509\) −5814.00 −0.506289 −0.253144 0.967429i \(-0.581465\pi\)
−0.253144 + 0.967429i \(0.581465\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1944.00 −0.167309
\(514\) 0 0
\(515\) 4536.00 0.388117
\(516\) 0 0
\(517\) 3600.00 0.306243
\(518\) 0 0
\(519\) −4698.00 −0.397340
\(520\) 0 0
\(521\) −11682.0 −0.982337 −0.491169 0.871065i \(-0.663430\pi\)
−0.491169 + 0.871065i \(0.663430\pi\)
\(522\) 0 0
\(523\) 2988.00 0.249820 0.124910 0.992168i \(-0.460136\pi\)
0.124910 + 0.992168i \(0.460136\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4536.00 0.374936
\(528\) 0 0
\(529\) −11971.0 −0.983891
\(530\) 0 0
\(531\) −4212.00 −0.344228
\(532\) 0 0
\(533\) 9720.00 0.789906
\(534\) 0 0
\(535\) −12060.0 −0.974578
\(536\) 0 0
\(537\) 11406.0 0.916583
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7130.00 0.566622 0.283311 0.959028i \(-0.408567\pi\)
0.283311 + 0.959028i \(0.408567\pi\)
\(542\) 0 0
\(543\) 1404.00 0.110960
\(544\) 0 0
\(545\) 2916.00 0.229188
\(546\) 0 0
\(547\) 5488.00 0.428976 0.214488 0.976727i \(-0.431192\pi\)
0.214488 + 0.976727i \(0.431192\pi\)
\(548\) 0 0
\(549\) 7128.00 0.554127
\(550\) 0 0
\(551\) 11376.0 0.879553
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8748.00 0.669067
\(556\) 0 0
\(557\) 5746.00 0.437102 0.218551 0.975826i \(-0.429867\pi\)
0.218551 + 0.975826i \(0.429867\pi\)
\(558\) 0 0
\(559\) −11664.0 −0.882531
\(560\) 0 0
\(561\) −18900.0 −1.42239
\(562\) 0 0
\(563\) −13068.0 −0.978243 −0.489121 0.872216i \(-0.662682\pi\)
−0.489121 + 0.872216i \(0.662682\pi\)
\(564\) 0 0
\(565\) −25020.0 −1.86301
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1130.00 −0.0832549 −0.0416275 0.999133i \(-0.513254\pi\)
−0.0416275 + 0.999133i \(0.513254\pi\)
\(570\) 0 0
\(571\) −16864.0 −1.23597 −0.617983 0.786192i \(-0.712050\pi\)
−0.617983 + 0.786192i \(0.712050\pi\)
\(572\) 0 0
\(573\) 1446.00 0.105423
\(574\) 0 0
\(575\) −2786.00 −0.202060
\(576\) 0 0
\(577\) −2088.00 −0.150649 −0.0753246 0.997159i \(-0.523999\pi\)
−0.0753246 + 0.997159i \(0.523999\pi\)
\(578\) 0 0
\(579\) 2430.00 0.174417
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1100.00 −0.0781430
\(584\) 0 0
\(585\) −5832.00 −0.412177
\(586\) 0 0
\(587\) −10260.0 −0.721423 −0.360712 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(588\) 0 0
\(589\) 2592.00 0.181327
\(590\) 0 0
\(591\) 7386.00 0.514077
\(592\) 0 0
\(593\) 3582.00 0.248052 0.124026 0.992279i \(-0.460419\pi\)
0.124026 + 0.992279i \(0.460419\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −13608.0 −0.932895
\(598\) 0 0
\(599\) −7034.00 −0.479802 −0.239901 0.970797i \(-0.577115\pi\)
−0.239901 + 0.970797i \(0.577115\pi\)
\(600\) 0 0
\(601\) −18072.0 −1.22658 −0.613288 0.789859i \(-0.710154\pi\)
−0.613288 + 0.789859i \(0.710154\pi\)
\(602\) 0 0
\(603\) −2088.00 −0.141011
\(604\) 0 0
\(605\) 21042.0 1.41401
\(606\) 0 0
\(607\) −28584.0 −1.91135 −0.955674 0.294425i \(-0.904872\pi\)
−0.955674 + 0.294425i \(0.904872\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2592.00 −0.171622
\(612\) 0 0
\(613\) −10910.0 −0.718843 −0.359421 0.933175i \(-0.617026\pi\)
−0.359421 + 0.933175i \(0.617026\pi\)
\(614\) 0 0
\(615\) 14580.0 0.955971
\(616\) 0 0
\(617\) −5522.00 −0.360304 −0.180152 0.983639i \(-0.557659\pi\)
−0.180152 + 0.983639i \(0.557659\pi\)
\(618\) 0 0
\(619\) −2412.00 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(620\) 0 0
\(621\) 378.000 0.0244261
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) −10800.0 −0.687895
\(628\) 0 0
\(629\) −20412.0 −1.29393
\(630\) 0 0
\(631\) −24676.0 −1.55679 −0.778396 0.627773i \(-0.783966\pi\)
−0.778396 + 0.627773i \(0.783966\pi\)
\(632\) 0 0
\(633\) 8748.00 0.549292
\(634\) 0 0
\(635\) −16488.0 −1.03040
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6606.00 0.408966
\(640\) 0 0
\(641\) −27482.0 −1.69341 −0.846703 0.532065i \(-0.821416\pi\)
−0.846703 + 0.532065i \(0.821416\pi\)
\(642\) 0 0
\(643\) 22752.0 1.39541 0.697707 0.716383i \(-0.254204\pi\)
0.697707 + 0.716383i \(0.254204\pi\)
\(644\) 0 0
\(645\) −17496.0 −1.06807
\(646\) 0 0
\(647\) −14832.0 −0.901246 −0.450623 0.892714i \(-0.648798\pi\)
−0.450623 + 0.892714i \(0.648798\pi\)
\(648\) 0 0
\(649\) −23400.0 −1.41530
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2822.00 0.169117 0.0845585 0.996419i \(-0.473052\pi\)
0.0845585 + 0.996419i \(0.473052\pi\)
\(654\) 0 0
\(655\) −40824.0 −2.43531
\(656\) 0 0
\(657\) 1620.00 0.0961982
\(658\) 0 0
\(659\) 15826.0 0.935498 0.467749 0.883861i \(-0.345065\pi\)
0.467749 + 0.883861i \(0.345065\pi\)
\(660\) 0 0
\(661\) −23832.0 −1.40236 −0.701178 0.712986i \(-0.747342\pi\)
−0.701178 + 0.712986i \(0.747342\pi\)
\(662\) 0 0
\(663\) 13608.0 0.797121
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2212.00 −0.128409
\(668\) 0 0
\(669\) −3240.00 −0.187243
\(670\) 0 0
\(671\) 39600.0 2.27830
\(672\) 0 0
\(673\) 13770.0 0.788699 0.394350 0.918961i \(-0.370970\pi\)
0.394350 + 0.918961i \(0.370970\pi\)
\(674\) 0 0
\(675\) −5373.00 −0.306381
\(676\) 0 0
\(677\) 8334.00 0.473119 0.236560 0.971617i \(-0.423980\pi\)
0.236560 + 0.971617i \(0.423980\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3996.00 −0.224856
\(682\) 0 0
\(683\) 18598.0 1.04192 0.520961 0.853580i \(-0.325574\pi\)
0.520961 + 0.853580i \(0.325574\pi\)
\(684\) 0 0
\(685\) 14508.0 0.809229
\(686\) 0 0
\(687\) −4860.00 −0.269899
\(688\) 0 0
\(689\) 792.000 0.0437922
\(690\) 0 0
\(691\) −8964.00 −0.493497 −0.246749 0.969080i \(-0.579362\pi\)
−0.246749 + 0.969080i \(0.579362\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −47304.0 −2.58179
\(696\) 0 0
\(697\) −34020.0 −1.84878
\(698\) 0 0
\(699\) −20154.0 −1.09055
\(700\) 0 0
\(701\) 3542.00 0.190841 0.0954205 0.995437i \(-0.469580\pi\)
0.0954205 + 0.995437i \(0.469580\pi\)
\(702\) 0 0
\(703\) −11664.0 −0.625770
\(704\) 0 0
\(705\) −3888.00 −0.207703
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −486.000 −0.0257435 −0.0128717 0.999917i \(-0.504097\pi\)
−0.0128717 + 0.999917i \(0.504097\pi\)
\(710\) 0 0
\(711\) −2124.00 −0.112034
\(712\) 0 0
\(713\) −504.000 −0.0264726
\(714\) 0 0
\(715\) −32400.0 −1.69467
\(716\) 0 0
\(717\) −10734.0 −0.559091
\(718\) 0 0
\(719\) 26928.0 1.39672 0.698362 0.715744i \(-0.253912\pi\)
0.698362 + 0.715744i \(0.253912\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2268.00 0.116664
\(724\) 0 0
\(725\) 31442.0 1.61066
\(726\) 0 0
\(727\) −20628.0 −1.05234 −0.526169 0.850380i \(-0.676372\pi\)
−0.526169 + 0.850380i \(0.676372\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 40824.0 2.06557
\(732\) 0 0
\(733\) −9756.00 −0.491604 −0.245802 0.969320i \(-0.579051\pi\)
−0.245802 + 0.969320i \(0.579051\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11600.0 −0.579771
\(738\) 0 0
\(739\) −19064.0 −0.948959 −0.474479 0.880267i \(-0.657364\pi\)
−0.474479 + 0.880267i \(0.657364\pi\)
\(740\) 0 0
\(741\) 7776.00 0.385504
\(742\) 0 0
\(743\) 3766.00 0.185950 0.0929752 0.995668i \(-0.470362\pi\)
0.0929752 + 0.995668i \(0.470362\pi\)
\(744\) 0 0
\(745\) −43020.0 −2.11561
\(746\) 0 0
\(747\) −324.000 −0.0158695
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11664.0 0.566745 0.283372 0.959010i \(-0.408547\pi\)
0.283372 + 0.959010i \(0.408547\pi\)
\(752\) 0 0
\(753\) −19548.0 −0.946041
\(754\) 0 0
\(755\) −58320.0 −2.81123
\(756\) 0 0
\(757\) −34182.0 −1.64117 −0.820585 0.571524i \(-0.806352\pi\)
−0.820585 + 0.571524i \(0.806352\pi\)
\(758\) 0 0
\(759\) 2100.00 0.100428
\(760\) 0 0
\(761\) −4734.00 −0.225502 −0.112751 0.993623i \(-0.535966\pi\)
−0.112751 + 0.993623i \(0.535966\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 20412.0 0.964703
\(766\) 0 0
\(767\) 16848.0 0.793150
\(768\) 0 0
\(769\) −30240.0 −1.41805 −0.709026 0.705182i \(-0.750865\pi\)
−0.709026 + 0.705182i \(0.750865\pi\)
\(770\) 0 0
\(771\) 18090.0 0.845001
\(772\) 0 0
\(773\) 27702.0 1.28897 0.644484 0.764618i \(-0.277072\pi\)
0.644484 + 0.764618i \(0.277072\pi\)
\(774\) 0 0
\(775\) 7164.00 0.332050
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −19440.0 −0.894108
\(780\) 0 0
\(781\) 36700.0 1.68147
\(782\) 0 0
\(783\) −4266.00 −0.194705
\(784\) 0 0
\(785\) 54432.0 2.47486
\(786\) 0 0
\(787\) 22644.0 1.02563 0.512815 0.858499i \(-0.328603\pi\)
0.512815 + 0.858499i \(0.328603\pi\)
\(788\) 0 0
\(789\) 1770.00 0.0798652
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −28512.0 −1.27679
\(794\) 0 0
\(795\) 1188.00 0.0529988
\(796\) 0 0
\(797\) 30150.0 1.33998 0.669992 0.742368i \(-0.266297\pi\)
0.669992 + 0.742368i \(0.266297\pi\)
\(798\) 0 0
\(799\) 9072.00 0.401682
\(800\) 0 0
\(801\) 2106.00 0.0928987
\(802\) 0 0
\(803\) 9000.00 0.395521
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2970.00 0.129553
\(808\) 0 0
\(809\) −11318.0 −0.491866 −0.245933 0.969287i \(-0.579094\pi\)
−0.245933 + 0.969287i \(0.579094\pi\)
\(810\) 0 0
\(811\) 29628.0 1.28284 0.641418 0.767192i \(-0.278347\pi\)
0.641418 + 0.767192i \(0.278347\pi\)
\(812\) 0 0
\(813\) −10260.0 −0.442600
\(814\) 0 0
\(815\) 32112.0 1.38016
\(816\) 0 0
\(817\) 23328.0 0.998952
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17770.0 0.755393 0.377696 0.925930i \(-0.376716\pi\)
0.377696 + 0.925930i \(0.376716\pi\)
\(822\) 0 0
\(823\) −7868.00 −0.333246 −0.166623 0.986021i \(-0.553286\pi\)
−0.166623 + 0.986021i \(0.553286\pi\)
\(824\) 0 0
\(825\) −29850.0 −1.25969
\(826\) 0 0
\(827\) −35726.0 −1.50219 −0.751097 0.660192i \(-0.770475\pi\)
−0.751097 + 0.660192i \(0.770475\pi\)
\(828\) 0 0
\(829\) 27108.0 1.13571 0.567853 0.823130i \(-0.307774\pi\)
0.567853 + 0.823130i \(0.307774\pi\)
\(830\) 0 0
\(831\) 8202.00 0.342388
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 54432.0 2.25592
\(836\) 0 0
\(837\) −972.000 −0.0401401
\(838\) 0 0
\(839\) −23256.0 −0.956956 −0.478478 0.878099i \(-0.658811\pi\)
−0.478478 + 0.878099i \(0.658811\pi\)
\(840\) 0 0
\(841\) 575.000 0.0235762
\(842\) 0 0
\(843\) −1794.00 −0.0732961
\(844\) 0 0
\(845\) −16218.0 −0.660256
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 10800.0 0.436578
\(850\) 0 0
\(851\) 2268.00 0.0913584
\(852\) 0 0
\(853\) 35280.0 1.41614 0.708068 0.706144i \(-0.249567\pi\)
0.708068 + 0.706144i \(0.249567\pi\)
\(854\) 0 0
\(855\) 11664.0 0.466550
\(856\) 0 0
\(857\) −19710.0 −0.785625 −0.392813 0.919619i \(-0.628498\pi\)
−0.392813 + 0.919619i \(0.628498\pi\)
\(858\) 0 0
\(859\) −3888.00 −0.154432 −0.0772159 0.997014i \(-0.524603\pi\)
−0.0772159 + 0.997014i \(0.524603\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36634.0 1.44500 0.722500 0.691370i \(-0.242993\pi\)
0.722500 + 0.691370i \(0.242993\pi\)
\(864\) 0 0
\(865\) 28188.0 1.10800
\(866\) 0 0
\(867\) −32889.0 −1.28831
\(868\) 0 0
\(869\) −11800.0 −0.460630
\(870\) 0 0
\(871\) 8352.00 0.324910
\(872\) 0 0
\(873\) 4212.00 0.163293
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1226.00 0.0472053 0.0236027 0.999721i \(-0.492486\pi\)
0.0236027 + 0.999721i \(0.492486\pi\)
\(878\) 0 0
\(879\) −23706.0 −0.909651
\(880\) 0 0
\(881\) −38538.0 −1.47376 −0.736878 0.676026i \(-0.763701\pi\)
−0.736878 + 0.676026i \(0.763701\pi\)
\(882\) 0 0
\(883\) 37260.0 1.42004 0.710022 0.704180i \(-0.248685\pi\)
0.710022 + 0.704180i \(0.248685\pi\)
\(884\) 0 0
\(885\) 25272.0 0.959897
\(886\) 0 0
\(887\) −26640.0 −1.00844 −0.504219 0.863576i \(-0.668219\pi\)
−0.504219 + 0.863576i \(0.668219\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4050.00 0.152278
\(892\) 0 0
\(893\) 5184.00 0.194262
\(894\) 0 0
\(895\) −68436.0 −2.55594
\(896\) 0 0
\(897\) −1512.00 −0.0562812
\(898\) 0 0
\(899\) 5688.00 0.211018
\(900\) 0 0
\(901\) −2772.00 −0.102496
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8424.00 −0.309418
\(906\) 0 0
\(907\) 12636.0 0.462593 0.231296 0.972883i \(-0.425703\pi\)
0.231296 + 0.972883i \(0.425703\pi\)
\(908\) 0 0
\(909\) −5994.00 −0.218711
\(910\) 0 0
\(911\) 33638.0 1.22336 0.611678 0.791107i \(-0.290495\pi\)
0.611678 + 0.791107i \(0.290495\pi\)
\(912\) 0 0
\(913\) −1800.00 −0.0652479
\(914\) 0 0
\(915\) −42768.0 −1.54521
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −36936.0 −1.32580 −0.662898 0.748710i \(-0.730674\pi\)
−0.662898 + 0.748710i \(0.730674\pi\)
\(920\) 0 0
\(921\) −30672.0 −1.09737
\(922\) 0 0
\(923\) −26424.0 −0.942315
\(924\) 0 0
\(925\) −32238.0 −1.14592
\(926\) 0 0
\(927\) 2268.00 0.0803570
\(928\) 0 0
\(929\) −22302.0 −0.787626 −0.393813 0.919191i \(-0.628844\pi\)
−0.393813 + 0.919191i \(0.628844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 11664.0 0.409284
\(934\) 0 0
\(935\) 113400. 3.96639
\(936\) 0 0
\(937\) 13824.0 0.481975 0.240987 0.970528i \(-0.422529\pi\)
0.240987 + 0.970528i \(0.422529\pi\)
\(938\) 0 0
\(939\) −15336.0 −0.532983
\(940\) 0 0
\(941\) 13554.0 0.469551 0.234776 0.972050i \(-0.424564\pi\)
0.234776 + 0.972050i \(0.424564\pi\)
\(942\) 0 0
\(943\) 3780.00 0.130534
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −44878.0 −1.53996 −0.769978 0.638070i \(-0.779733\pi\)
−0.769978 + 0.638070i \(0.779733\pi\)
\(948\) 0 0
\(949\) −6480.00 −0.221654
\(950\) 0 0
\(951\) 30306.0 1.03337
\(952\) 0 0
\(953\) 38362.0 1.30395 0.651976 0.758239i \(-0.273940\pi\)
0.651976 + 0.758239i \(0.273940\pi\)
\(954\) 0 0
\(955\) −8676.00 −0.293978
\(956\) 0 0
\(957\) −23700.0 −0.800535
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28495.0 −0.956497
\(962\) 0 0
\(963\) −6030.00 −0.201780
\(964\) 0 0
\(965\) −14580.0 −0.486370
\(966\) 0 0
\(967\) −26444.0 −0.879402 −0.439701 0.898144i \(-0.644916\pi\)
−0.439701 + 0.898144i \(0.644916\pi\)
\(968\) 0 0
\(969\) −27216.0 −0.902274
\(970\) 0 0
\(971\) −17820.0 −0.588951 −0.294475 0.955659i \(-0.595145\pi\)
−0.294475 + 0.955659i \(0.595145\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 21492.0 0.705943
\(976\) 0 0
\(977\) 34438.0 1.12771 0.563853 0.825875i \(-0.309318\pi\)
0.563853 + 0.825875i \(0.309318\pi\)
\(978\) 0 0
\(979\) 11700.0 0.381955
\(980\) 0 0
\(981\) 1458.00 0.0474519
\(982\) 0 0
\(983\) −26064.0 −0.845689 −0.422845 0.906202i \(-0.638968\pi\)
−0.422845 + 0.906202i \(0.638968\pi\)
\(984\) 0 0
\(985\) −44316.0 −1.43353
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4536.00 −0.145841
\(990\) 0 0
\(991\) −33696.0 −1.08011 −0.540055 0.841630i \(-0.681597\pi\)
−0.540055 + 0.841630i \(0.681597\pi\)
\(992\) 0 0
\(993\) 16524.0 0.528070
\(994\) 0 0
\(995\) 81648.0 2.60142
\(996\) 0 0
\(997\) 36072.0 1.14585 0.572925 0.819608i \(-0.305809\pi\)
0.572925 + 0.819608i \(0.305809\pi\)
\(998\) 0 0
\(999\) 4374.00 0.138526
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 2352.4.a.s.1.1 1
4.3 odd 2 147.4.a.h.1.1 yes 1
7.6 odd 2 2352.4.a.t.1.1 1
12.11 even 2 441.4.a.a.1.1 1
28.3 even 6 147.4.e.d.79.1 2
28.11 odd 6 147.4.e.a.79.1 2
28.19 even 6 147.4.e.d.67.1 2
28.23 odd 6 147.4.e.a.67.1 2
28.27 even 2 147.4.a.f.1.1 1
84.11 even 6 441.4.e.o.226.1 2
84.23 even 6 441.4.e.o.361.1 2
84.47 odd 6 441.4.e.l.361.1 2
84.59 odd 6 441.4.e.l.226.1 2
84.83 odd 2 441.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.f.1.1 1 28.27 even 2
147.4.a.h.1.1 yes 1 4.3 odd 2
147.4.e.a.67.1 2 28.23 odd 6
147.4.e.a.79.1 2 28.11 odd 6
147.4.e.d.67.1 2 28.19 even 6
147.4.e.d.79.1 2 28.3 even 6
441.4.a.a.1.1 1 12.11 even 2
441.4.a.c.1.1 1 84.83 odd 2
441.4.e.l.226.1 2 84.59 odd 6
441.4.e.l.361.1 2 84.47 odd 6
441.4.e.o.226.1 2 84.11 even 6
441.4.e.o.361.1 2 84.23 even 6
2352.4.a.s.1.1 1 1.1 even 1 trivial
2352.4.a.t.1.1 1 7.6 odd 2