Properties

Label 441.4.a.c.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
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 \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +8.00000 q^{4} +18.0000 q^{5} +O(q^{10})\) \(q-4.00000 q^{2} +8.00000 q^{4} +18.0000 q^{5} -72.0000 q^{10} +50.0000 q^{11} +36.0000 q^{13} -64.0000 q^{16} +126.000 q^{17} +72.0000 q^{19} +144.000 q^{20} -200.000 q^{22} -14.0000 q^{23} +199.000 q^{25} -144.000 q^{26} -158.000 q^{29} +36.0000 q^{31} +256.000 q^{32} -504.000 q^{34} -162.000 q^{37} -288.000 q^{38} -270.000 q^{41} -324.000 q^{43} +400.000 q^{44} +56.0000 q^{46} -72.0000 q^{47} -796.000 q^{50} +288.000 q^{52} +22.0000 q^{53} +900.000 q^{55} +632.000 q^{58} +468.000 q^{59} -792.000 q^{61} -144.000 q^{62} -512.000 q^{64} +648.000 q^{65} +232.000 q^{67} +1008.00 q^{68} +734.000 q^{71} -180.000 q^{73} +648.000 q^{74} +576.000 q^{76} +236.000 q^{79} -1152.00 q^{80} +1080.00 q^{82} +36.0000 q^{83} +2268.00 q^{85} +1296.00 q^{86} +234.000 q^{89} -112.000 q^{92} +288.000 q^{94} +1296.00 q^{95} -468.000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 8.00000 1.00000
\(5\) 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
\(9\) 0 0
\(10\) −72.0000 −2.27684
\(11\) 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 \(-0.374538\pi\)
0.384023 + 0.923323i \(0.374538\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 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 \(0.356861\pi\)
\(20\) 144.000 1.60997
\(21\) 0 0
\(22\) −200.000 −1.93819
\(23\) −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
\(26\) −144.000 −1.08618
\(27\) 0 0
\(28\) 0 0
\(29\) −158.000 −1.01172 −0.505860 0.862616i \(-0.668825\pi\)
−0.505860 + 0.862616i \(0.668825\pi\)
\(30\) 0 0
\(31\) 36.0000 0.208574 0.104287 0.994547i \(-0.466744\pi\)
0.104287 + 0.994547i \(0.466744\pi\)
\(32\) 256.000 1.41421
\(33\) 0 0
\(34\) −504.000 −2.54221
\(35\) 0 0
\(36\) 0 0
\(37\) −162.000 −0.719801 −0.359900 0.932991i \(-0.617189\pi\)
−0.359900 + 0.932991i \(0.617189\pi\)
\(38\) −288.000 −1.22947
\(39\) 0 0
\(40\) 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 \(-0.694815\pi\)
−0.574529 + 0.818484i \(0.694815\pi\)
\(44\) 400.000 1.37051
\(45\) 0 0
\(46\) 56.0000 0.179495
\(47\) −72.0000 −0.223453 −0.111726 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −796.000 −2.25143
\(51\) 0 0
\(52\) 288.000 0.768046
\(53\) 22.0000 0.0570176 0.0285088 0.999594i \(-0.490924\pi\)
0.0285088 + 0.999594i \(0.490924\pi\)
\(54\) 0 0
\(55\) 900.000 2.20647
\(56\) 0 0
\(57\) 0 0
\(58\) 632.000 1.43079
\(59\) 468.000 1.03268 0.516342 0.856382i \(-0.327293\pi\)
0.516342 + 0.856382i \(0.327293\pi\)
\(60\) 0 0
\(61\) −792.000 −1.66238 −0.831190 0.555988i \(-0.812340\pi\)
−0.831190 + 0.555988i \(0.812340\pi\)
\(62\) −144.000 −0.294968
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 648.000 1.23653
\(66\) 0 0
\(67\) 232.000 0.423034 0.211517 0.977374i \(-0.432160\pi\)
0.211517 + 0.977374i \(0.432160\pi\)
\(68\) 1008.00 1.79762
\(69\) 0 0
\(70\) 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 \(-0.546092\pi\)
−0.144297 + 0.989534i \(0.546092\pi\)
\(74\) 648.000 1.01795
\(75\) 0 0
\(76\) 576.000 0.869365
\(77\) 0 0
\(78\) 0 0
\(79\) 236.000 0.336102 0.168051 0.985778i \(-0.446253\pi\)
0.168051 + 0.985778i \(0.446253\pi\)
\(80\) −1152.00 −1.60997
\(81\) 0 0
\(82\) 1080.00 1.45446
\(83\) 36.0000 0.0476086 0.0238043 0.999717i \(-0.492422\pi\)
0.0238043 + 0.999717i \(0.492422\pi\)
\(84\) 0 0
\(85\) 2268.00 2.89411
\(86\) 1296.00 1.62501
\(87\) 0 0
\(88\) 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
\(92\) −112.000 −0.126922
\(93\) 0 0
\(94\) 288.000 0.316010
\(95\) 1296.00 1.39965
\(96\) 0 0
\(97\) −468.000 −0.489878 −0.244939 0.969538i \(-0.578768\pi\)
−0.244939 + 0.969538i \(0.578768\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1592.00 1.59200
\(101\) −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
\(106\) −88.0000 −0.0806351
\(107\) −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 \(0.477324\pi\)
\(110\) −3600.00 −3.12042
\(111\) 0 0
\(112\) 0 0
\(113\) 1390.00 1.15717 0.578585 0.815622i \(-0.303605\pi\)
0.578585 + 0.815622i \(0.303605\pi\)
\(114\) 0 0
\(115\) −252.000 −0.204340
\(116\) −1264.00 −1.01172
\(117\) 0 0
\(118\) −1872.00 −1.46044
\(119\) 0 0
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) 3168.00 2.35096
\(123\) 0 0
\(124\) 288.000 0.208574
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) 916.000 0.640015 0.320007 0.947415i \(-0.396315\pi\)
0.320007 + 0.947415i \(0.396315\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −2592.00 −1.74872
\(131\) 2268.00 1.51264 0.756321 0.654201i \(-0.226995\pi\)
0.756321 + 0.654201i \(0.226995\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −928.000 −0.598261
\(135\) 0 0
\(136\) 0 0
\(137\) −806.000 −0.502637 −0.251318 0.967904i \(-0.580864\pi\)
−0.251318 + 0.967904i \(0.580864\pi\)
\(138\) 0 0
\(139\) −2628.00 −1.60363 −0.801813 0.597575i \(-0.796131\pi\)
−0.801813 + 0.597575i \(0.796131\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2936.00 −1.73510
\(143\) 1800.00 1.05261
\(144\) 0 0
\(145\) −2844.00 −1.62884
\(146\) 720.000 0.408134
\(147\) 0 0
\(148\) −1296.00 −0.719801
\(149\) 2390.00 1.31407 0.657035 0.753860i \(-0.271810\pi\)
0.657035 + 0.753860i \(0.271810\pi\)
\(150\) 0 0
\(151\) 3240.00 1.74614 0.873071 0.487593i \(-0.162125\pi\)
0.873071 + 0.487593i \(0.162125\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 648.000 0.335798
\(156\) 0 0
\(157\) −3024.00 −1.53721 −0.768603 0.639726i \(-0.779048\pi\)
−0.768603 + 0.639726i \(0.779048\pi\)
\(158\) −944.000 −0.475320
\(159\) 0 0
\(160\) 4608.00 2.27684
\(161\) 0 0
\(162\) 0 0
\(163\) −1784.00 −0.857262 −0.428631 0.903480i \(-0.641004\pi\)
−0.428631 + 0.903480i \(0.641004\pi\)
\(164\) −2160.00 −1.02846
\(165\) 0 0
\(166\) −144.000 −0.0673287
\(167\) −3024.00 −1.40122 −0.700611 0.713543i \(-0.747089\pi\)
−0.700611 + 0.713543i \(0.747089\pi\)
\(168\) 0 0
\(169\) −901.000 −0.410105
\(170\) −9072.00 −4.09289
\(171\) 0 0
\(172\) −2592.00 −1.14906
\(173\) 1566.00 0.688213 0.344106 0.938931i \(-0.388182\pi\)
0.344106 + 0.938931i \(0.388182\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3200.00 −1.37051
\(177\) 0 0
\(178\) −936.000 −0.394136
\(179\) −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 \(-0.469365\pi\)
0.0960944 + 0.995372i \(0.469365\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2916.00 −1.15886
\(186\) 0 0
\(187\) 6300.00 2.46365
\(188\) −576.000 −0.223453
\(189\) 0 0
\(190\) −5184.00 −1.97940
\(191\) −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 \(0.548265\pi\)
\(194\) 1872.00 0.692793
\(195\) 0 0
\(196\) 0 0
\(197\) 2462.00 0.890407 0.445204 0.895429i \(-0.353131\pi\)
0.445204 + 0.895429i \(0.353131\pi\)
\(198\) 0 0
\(199\) 4536.00 1.61582 0.807911 0.589305i \(-0.200598\pi\)
0.807911 + 0.589305i \(0.200598\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2664.00 0.927913
\(203\) 0 0
\(204\) 0 0
\(205\) −4860.00 −1.65579
\(206\) −1008.00 −0.340926
\(207\) 0 0
\(208\) −2304.00 −0.768046
\(209\) 3600.00 1.19147
\(210\) 0 0
\(211\) 2916.00 0.951402 0.475701 0.879607i \(-0.342195\pi\)
0.475701 + 0.879607i \(0.342195\pi\)
\(212\) 176.000 0.0570176
\(213\) 0 0
\(214\) 2680.00 0.856080
\(215\) −5832.00 −1.84995
\(216\) 0 0
\(217\) 0 0
\(218\) −648.000 −0.201322
\(219\) 0 0
\(220\) 7200.00 2.20647
\(221\) 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
\(225\) 0 0
\(226\) −5560.00 −1.63649
\(227\) −1332.00 −0.389462 −0.194731 0.980857i \(-0.562383\pi\)
−0.194731 + 0.980857i \(0.562383\pi\)
\(228\) 0 0
\(229\) −1620.00 −0.467479 −0.233739 0.972299i \(-0.575096\pi\)
−0.233739 + 0.972299i \(0.575096\pi\)
\(230\) 1008.00 0.288981
\(231\) 0 0
\(232\) 0 0
\(233\) −6718.00 −1.88889 −0.944444 0.328673i \(-0.893399\pi\)
−0.944444 + 0.328673i \(0.893399\pi\)
\(234\) 0 0
\(235\) −1296.00 −0.359752
\(236\) 3744.00 1.03268
\(237\) 0 0
\(238\) 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 \(-0.467785\pi\)
0.101034 + 0.994883i \(0.467785\pi\)
\(242\) −4676.00 −1.24209
\(243\) 0 0
\(244\) −6336.00 −1.66238
\(245\) 0 0
\(246\) 0 0
\(247\) 2592.00 0.667713
\(248\) 0 0
\(249\) 0 0
\(250\) −5328.00 −1.34789
\(251\) −6516.00 −1.63859 −0.819295 0.573372i \(-0.805635\pi\)
−0.819295 + 0.573372i \(0.805635\pi\)
\(252\) 0 0
\(253\) −700.000 −0.173947
\(254\) −3664.00 −0.905117
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) −6030.00 −1.46358 −0.731792 0.681528i \(-0.761316\pi\)
−0.731792 + 0.681528i \(0.761316\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 5184.00 1.23653
\(261\) 0 0
\(262\) −9072.00 −2.13920
\(263\) −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
\(267\) 0 0
\(268\) 1856.00 0.423034
\(269\) −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 \(0.374787\pi\)
\(272\) −8064.00 −1.79762
\(273\) 0 0
\(274\) 3224.00 0.710836
\(275\) 9950.00 2.18185
\(276\) 0 0
\(277\) −2734.00 −0.593033 −0.296516 0.955028i \(-0.595825\pi\)
−0.296516 + 0.955028i \(0.595825\pi\)
\(278\) 10512.0 2.26787
\(279\) 0 0
\(280\) 0 0
\(281\) −598.000 −0.126953 −0.0634763 0.997983i \(-0.520219\pi\)
−0.0634763 + 0.997983i \(0.520219\pi\)
\(282\) 0 0
\(283\) −3600.00 −0.756176 −0.378088 0.925770i \(-0.623418\pi\)
−0.378088 + 0.925770i \(0.623418\pi\)
\(284\) 5872.00 1.22690
\(285\) 0 0
\(286\) −7200.00 −1.48862
\(287\) 0 0
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 11376.0 2.30352
\(291\) 0 0
\(292\) −1440.00 −0.288595
\(293\) 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
\(297\) 0 0
\(298\) −9560.00 −1.85838
\(299\) −504.000 −0.0974818
\(300\) 0 0
\(301\) 0 0
\(302\) −12960.0 −2.46942
\(303\) 0 0
\(304\) −4608.00 −0.869365
\(305\) −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
\(309\) 0 0
\(310\) −2592.00 −0.474889
\(311\) 3888.00 0.708901 0.354451 0.935075i \(-0.384668\pi\)
0.354451 + 0.935075i \(0.384668\pi\)
\(312\) 0 0
\(313\) −5112.00 −0.923154 −0.461577 0.887100i \(-0.652716\pi\)
−0.461577 + 0.887100i \(0.652716\pi\)
\(314\) 12096.0 2.17394
\(315\) 0 0
\(316\) 1888.00 0.336102
\(317\) 10102.0 1.78986 0.894929 0.446209i \(-0.147226\pi\)
0.894929 + 0.446209i \(0.147226\pi\)
\(318\) 0 0
\(319\) −7900.00 −1.38657
\(320\) −9216.00 −1.60997
\(321\) 0 0
\(322\) 0 0
\(323\) 9072.00 1.56279
\(324\) 0 0
\(325\) 7164.00 1.22273
\(326\) 7136.00 1.21235
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5508.00 0.914644 0.457322 0.889301i \(-0.348809\pi\)
0.457322 + 0.889301i \(0.348809\pi\)
\(332\) 288.000 0.0476086
\(333\) 0 0
\(334\) 12096.0 1.98163
\(335\) 4176.00 0.681072
\(336\) 0 0
\(337\) −9234.00 −1.49261 −0.746303 0.665607i \(-0.768173\pi\)
−0.746303 + 0.665607i \(0.768173\pi\)
\(338\) 3604.00 0.579976
\(339\) 0 0
\(340\) 18144.0 2.89411
\(341\) 1800.00 0.285852
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −6264.00 −0.973280
\(347\) 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 \(-0.781256\pi\)
−0.773023 + 0.634378i \(0.781256\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12800.0 1.93819
\(353\) −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
\(356\) 1872.00 0.278696
\(357\) 0 0
\(358\) 15208.0 2.24516
\(359\) −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
\(362\) −1872.00 −0.271796
\(363\) 0 0
\(364\) 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 \(0.390268\pi\)
\(368\) 896.000 0.126922
\(369\) 0 0
\(370\) 11664.0 1.63887
\(371\) 0 0
\(372\) 0 0
\(373\) −2306.00 −0.320108 −0.160054 0.987108i \(-0.551167\pi\)
−0.160054 + 0.987108i \(0.551167\pi\)
\(374\) −25200.0 −3.48412
\(375\) 0 0
\(376\) 0 0
\(377\) −5688.00 −0.777047
\(378\) 0 0
\(379\) −7452.00 −1.00998 −0.504991 0.863124i \(-0.668504\pi\)
−0.504991 + 0.863124i \(0.668504\pi\)
\(380\) 10368.0 1.39965
\(381\) 0 0
\(382\) 1928.00 0.258233
\(383\) −1152.00 −0.153693 −0.0768465 0.997043i \(-0.524485\pi\)
−0.0768465 + 0.997043i \(0.524485\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3240.00 0.427232
\(387\) 0 0
\(388\) −3744.00 −0.489878
\(389\) −1894.00 −0.246863 −0.123431 0.992353i \(-0.539390\pi\)
−0.123431 + 0.992353i \(0.539390\pi\)
\(390\) 0 0
\(391\) −1764.00 −0.228157
\(392\) 0 0
\(393\) 0 0
\(394\) −9848.00 −1.25923
\(395\) 4248.00 0.541114
\(396\) 0 0
\(397\) −9216.00 −1.16508 −0.582541 0.812801i \(-0.697942\pi\)
−0.582541 + 0.812801i \(0.697942\pi\)
\(398\) −18144.0 −2.28512
\(399\) 0 0
\(400\) −12736.0 −1.59200
\(401\) 11650.0 1.45081 0.725403 0.688324i \(-0.241653\pi\)
0.725403 + 0.688324i \(0.241653\pi\)
\(402\) 0 0
\(403\) 1296.00 0.160194
\(404\) −5328.00 −0.656133
\(405\) 0 0
\(406\) 0 0
\(407\) −8100.00 −0.986492
\(408\) 0 0
\(409\) 7524.00 0.909628 0.454814 0.890586i \(-0.349706\pi\)
0.454814 + 0.890586i \(0.349706\pi\)
\(410\) 19440.0 2.34164
\(411\) 0 0
\(412\) 2016.00 0.241071
\(413\) 0 0
\(414\) 0 0
\(415\) 648.000 0.0766484
\(416\) 9216.00 1.08618
\(417\) 0 0
\(418\) −14400.0 −1.68499
\(419\) −3852.00 −0.449123 −0.224561 0.974460i \(-0.572095\pi\)
−0.224561 + 0.974460i \(0.572095\pi\)
\(420\) 0 0
\(421\) 10402.0 1.20419 0.602093 0.798426i \(-0.294334\pi\)
0.602093 + 0.798426i \(0.294334\pi\)
\(422\) −11664.0 −1.34548
\(423\) 0 0
\(424\) 0 0
\(425\) 25074.0 2.86181
\(426\) 0 0
\(427\) 0 0
\(428\) −5360.00 −0.605340
\(429\) 0 0
\(430\) 23328.0 2.61622
\(431\) 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 \(-0.714207\pi\)
−0.623297 + 0.781985i \(0.714207\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1296.00 0.142356
\(437\) −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
\(442\) −18144.0 −1.95254
\(443\) −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
\(446\) −4320.00 −0.458650
\(447\) 0 0
\(448\) 0 0
\(449\) −8186.00 −0.860404 −0.430202 0.902733i \(-0.641558\pi\)
−0.430202 + 0.902733i \(0.641558\pi\)
\(450\) 0 0
\(451\) −13500.0 −1.40951
\(452\) 11120.0 1.15717
\(453\) 0 0
\(454\) 5328.00 0.550783
\(455\) 0 0
\(456\) 0 0
\(457\) 2106.00 0.215568 0.107784 0.994174i \(-0.465625\pi\)
0.107784 + 0.994174i \(0.465625\pi\)
\(458\) 6480.00 0.661115
\(459\) 0 0
\(460\) −2016.00 −0.204340
\(461\) 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 \(-0.718990\pi\)
−0.634977 + 0.772531i \(0.718990\pi\)
\(464\) 10112.0 1.01172
\(465\) 0 0
\(466\) 26872.0 2.67129
\(467\) −3708.00 −0.367421 −0.183711 0.982980i \(-0.558811\pi\)
−0.183711 + 0.982980i \(0.558811\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5184.00 0.508766
\(471\) 0 0
\(472\) 0 0
\(473\) −16200.0 −1.57479
\(474\) 0 0
\(475\) 14328.0 1.38403
\(476\) 0 0
\(477\) 0 0
\(478\) −14312.0 −1.36949
\(479\) −8064.00 −0.769214 −0.384607 0.923080i \(-0.625663\pi\)
−0.384607 + 0.923080i \(0.625663\pi\)
\(480\) 0 0
\(481\) −5832.00 −0.552841
\(482\) −3024.00 −0.285766
\(483\) 0 0
\(484\) 9352.00 0.878287
\(485\) −8424.00 −0.788689
\(486\) 0 0
\(487\) −11664.0 −1.08531 −0.542655 0.839955i \(-0.682581\pi\)
−0.542655 + 0.839955i \(0.682581\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(494\) −10368.0 −0.944288
\(495\) 0 0
\(496\) −2304.00 −0.208574
\(497\) 0 0
\(498\) 0 0
\(499\) −15228.0 −1.36613 −0.683065 0.730358i \(-0.739353\pi\)
−0.683065 + 0.730358i \(0.739353\pi\)
\(500\) 10656.0 0.953102
\(501\) 0 0
\(502\) 26064.0 2.31732
\(503\) −11088.0 −0.982882 −0.491441 0.870911i \(-0.663530\pi\)
−0.491441 + 0.870911i \(0.663530\pi\)
\(504\) 0 0
\(505\) −11988.0 −1.05635
\(506\) 2800.00 0.245998
\(507\) 0 0
\(508\) 7328.00 0.640015
\(509\) −5814.00 −0.506289 −0.253144 0.967429i \(-0.581465\pi\)
−0.253144 + 0.967429i \(0.581465\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16384.0 −1.41421
\(513\) 0 0
\(514\) 24120.0 2.06982
\(515\) 4536.00 0.388117
\(516\) 0 0
\(517\) −3600.00 −0.306243
\(518\) 0 0
\(519\) 0 0
\(520\) 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 \(0.460136\pi\)
\(524\) 18144.0 1.51264
\(525\) 0 0
\(526\) 2360.00 0.195629
\(527\) 4536.00 0.374936
\(528\) 0 0
\(529\) −11971.0 −0.983891
\(530\) −1584.00 −0.129820
\(531\) 0 0
\(532\) 0 0
\(533\) −9720.00 −0.789906
\(534\) 0 0
\(535\) −12060.0 −0.974578
\(536\) 0 0
\(537\) 0 0
\(538\) 3960.00 0.317338
\(539\) 0 0
\(540\) 0 0
\(541\) 7130.00 0.566622 0.283311 0.959028i \(-0.408567\pi\)
0.283311 + 0.959028i \(0.408567\pi\)
\(542\) −13680.0 −1.08414
\(543\) 0 0
\(544\) 32256.0 2.54221
\(545\) 2916.00 0.229188
\(546\) 0 0
\(547\) −5488.00 −0.428976 −0.214488 0.976727i \(-0.568808\pi\)
−0.214488 + 0.976727i \(0.568808\pi\)
\(548\) −6448.00 −0.502637
\(549\) 0 0
\(550\) −39800.0 −3.08560
\(551\) −11376.0 −0.879553
\(552\) 0 0
\(553\) 0 0
\(554\) 10936.0 0.838675
\(555\) 0 0
\(556\) −21024.0 −1.60363
\(557\) −5746.00 −0.437102 −0.218551 0.975826i \(-0.570133\pi\)
−0.218551 + 0.975826i \(0.570133\pi\)
\(558\) 0 0
\(559\) −11664.0 −0.882531
\(560\) 0 0
\(561\) 0 0
\(562\) 2392.00 0.179538
\(563\) 13068.0 0.978243 0.489121 0.872216i \(-0.337318\pi\)
0.489121 + 0.872216i \(0.337318\pi\)
\(564\) 0 0
\(565\) 25020.0 1.86301
\(566\) 14400.0 1.06939
\(567\) 0 0
\(568\) 0 0
\(569\) 1130.00 0.0832549 0.0416275 0.999133i \(-0.486746\pi\)
0.0416275 + 0.999133i \(0.486746\pi\)
\(570\) 0 0
\(571\) 16864.0 1.23597 0.617983 0.786192i \(-0.287950\pi\)
0.617983 + 0.786192i \(0.287950\pi\)
\(572\) 14400.0 1.05261
\(573\) 0 0
\(574\) 0 0
\(575\) −2786.00 −0.202060
\(576\) 0 0
\(577\) 2088.00 0.150649 0.0753246 0.997159i \(-0.476001\pi\)
0.0753246 + 0.997159i \(0.476001\pi\)
\(578\) −43852.0 −3.15571
\(579\) 0 0
\(580\) −22752.0 −1.62884
\(581\) 0 0
\(582\) 0 0
\(583\) 1100.00 0.0781430
\(584\) 0 0
\(585\) 0 0
\(586\) −31608.0 −2.22818
\(587\) 10260.0 0.721423 0.360712 0.932677i \(-0.382534\pi\)
0.360712 + 0.932677i \(0.382534\pi\)
\(588\) 0 0
\(589\) 2592.00 0.181327
\(590\) −33696.0 −2.35126
\(591\) 0 0
\(592\) 10368.0 0.719801
\(593\) 3582.00 0.248052 0.124026 0.992279i \(-0.460419\pi\)
0.124026 + 0.992279i \(0.460419\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19120.0 1.31407
\(597\) 0 0
\(598\) 2016.00 0.137860
\(599\) −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 \(-0.289846\pi\)
0.613288 + 0.789859i \(0.289846\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 25920.0 1.74614
\(605\) 21042.0 1.41401
\(606\) 0 0
\(607\) −28584.0 −1.91135 −0.955674 0.294425i \(-0.904872\pi\)
−0.955674 + 0.294425i \(0.904872\pi\)
\(608\) 18432.0 1.22947
\(609\) 0 0
\(610\) 57024.0 3.78497
\(611\) −2592.00 −0.171622
\(612\) 0 0
\(613\) −10910.0 −0.718843 −0.359421 0.933175i \(-0.617026\pi\)
−0.359421 + 0.933175i \(0.617026\pi\)
\(614\) −40896.0 −2.68799
\(615\) 0 0
\(616\) 0 0
\(617\) 5522.00 0.360304 0.180152 0.983639i \(-0.442341\pi\)
0.180152 + 0.983639i \(0.442341\pi\)
\(618\) 0 0
\(619\) −2412.00 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(620\) 5184.00 0.335798
\(621\) 0 0
\(622\) −15552.0 −1.00254
\(623\) 0 0
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 20448.0 1.30554
\(627\) 0 0
\(628\) −24192.0 −1.53721
\(629\) −20412.0 −1.29393
\(630\) 0 0
\(631\) 24676.0 1.55679 0.778396 0.627773i \(-0.216034\pi\)
0.778396 + 0.627773i \(0.216034\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −40408.0 −2.53124
\(635\) 16488.0 1.03040
\(636\) 0 0
\(637\) 0 0
\(638\) 31600.0 1.96090
\(639\) 0 0
\(640\) 0 0
\(641\) 27482.0 1.69341 0.846703 0.532065i \(-0.178584\pi\)
0.846703 + 0.532065i \(0.178584\pi\)
\(642\) 0 0
\(643\) 22752.0 1.39541 0.697707 0.716383i \(-0.254204\pi\)
0.697707 + 0.716383i \(0.254204\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −36288.0 −2.21011
\(647\) 14832.0 0.901246 0.450623 0.892714i \(-0.351202\pi\)
0.450623 + 0.892714i \(0.351202\pi\)
\(648\) 0 0
\(649\) 23400.0 1.41530
\(650\) −28656.0 −1.72920
\(651\) 0 0
\(652\) −14272.0 −0.857262
\(653\) −2822.00 −0.169117 −0.0845585 0.996419i \(-0.526948\pi\)
−0.0845585 + 0.996419i \(0.526948\pi\)
\(654\) 0 0
\(655\) 40824.0 2.43531
\(656\) 17280.0 1.02846
\(657\) 0 0
\(658\) 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 \(-0.252658\pi\)
0.701178 + 0.712986i \(0.252658\pi\)
\(662\) −22032.0 −1.29350
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2212.00 0.128409
\(668\) −24192.0 −1.40122
\(669\) 0 0
\(670\) −16704.0 −0.963182
\(671\) −39600.0 −2.27830
\(672\) 0 0
\(673\) 13770.0 0.788699 0.394350 0.918961i \(-0.370970\pi\)
0.394350 + 0.918961i \(0.370970\pi\)
\(674\) 36936.0 2.11086
\(675\) 0 0
\(676\) −7208.00 −0.410105
\(677\) 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
\(681\) 0 0
\(682\) −7200.00 −0.404255
\(683\) 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
\(687\) 0 0
\(688\) 20736.0 1.14906
\(689\) 792.000 0.0437922
\(690\) 0 0
\(691\) −8964.00 −0.493497 −0.246749 0.969080i \(-0.579362\pi\)
−0.246749 + 0.969080i \(0.579362\pi\)
\(692\) 12528.0 0.688213
\(693\) 0 0
\(694\) −25976.0 −1.42080
\(695\) −47304.0 −2.58179
\(696\) 0 0
\(697\) −34020.0 −1.84878
\(698\) 40320.0 2.18644
\(699\) 0 0
\(700\) 0 0
\(701\) −3542.00 −0.190841 −0.0954205 0.995437i \(-0.530420\pi\)
−0.0954205 + 0.995437i \(0.530420\pi\)
\(702\) 0 0
\(703\) −11664.0 −0.625770
\(704\) −25600.0 −1.37051
\(705\) 0 0
\(706\) 2952.00 0.157365
\(707\) 0 0
\(708\) 0 0
\(709\) −486.000 −0.0257435 −0.0128717 0.999917i \(-0.504097\pi\)
−0.0128717 + 0.999917i \(0.504097\pi\)
\(710\) −52848.0 −2.79345
\(711\) 0 0
\(712\) 0 0
\(713\) −504.000 −0.0264726
\(714\) 0 0
\(715\) 32400.0 1.69467
\(716\) −30416.0 −1.58757
\(717\) 0 0
\(718\) 776.000 0.0403343
\(719\) −26928.0 −1.39672 −0.698362 0.715744i \(-0.746088\pi\)
−0.698362 + 0.715744i \(0.746088\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6700.00 0.345358
\(723\) 0 0
\(724\) 3744.00 0.192189
\(725\) −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
\(729\) 0 0
\(730\) 12960.0 0.657084
\(731\) −40824.0 −2.06557
\(732\) 0 0
\(733\) 9756.00 0.491604 0.245802 0.969320i \(-0.420949\pi\)
0.245802 + 0.969320i \(0.420949\pi\)
\(734\) −19008.0 −0.955856
\(735\) 0 0
\(736\) −3584.00 −0.179495
\(737\) 11600.0 0.579771
\(738\) 0 0
\(739\) 19064.0 0.948959 0.474479 0.880267i \(-0.342636\pi\)
0.474479 + 0.880267i \(0.342636\pi\)
\(740\) −23328.0 −1.15886
\(741\) 0 0
\(742\) 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
\(746\) 9224.00 0.452701
\(747\) 0 0
\(748\) 50400.0 2.46365
\(749\) 0 0
\(750\) 0 0
\(751\) −11664.0 −0.566745 −0.283372 0.959010i \(-0.591453\pi\)
−0.283372 + 0.959010i \(0.591453\pi\)
\(752\) 4608.00 0.223453
\(753\) 0 0
\(754\) 22752.0 1.09891
\(755\) 58320.0 2.81123
\(756\) 0 0
\(757\) −34182.0 −1.64117 −0.820585 0.571524i \(-0.806352\pi\)
−0.820585 + 0.571524i \(0.806352\pi\)
\(758\) 29808.0 1.42833
\(759\) 0 0
\(760\) 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
\(764\) −3856.00 −0.182598
\(765\) 0 0
\(766\) 4608.00 0.217355
\(767\) 16848.0 0.793150
\(768\) 0 0
\(769\) 30240.0 1.41805 0.709026 0.705182i \(-0.249135\pi\)
0.709026 + 0.705182i \(0.249135\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6480.00 −0.302099
\(773\) 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
\(778\) 7576.00 0.349117
\(779\) −19440.0 −0.894108
\(780\) 0 0
\(781\) 36700.0 1.68147
\(782\) 7056.00 0.322662
\(783\) 0 0
\(784\) 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 \(0.328603\pi\)
\(788\) 19696.0 0.890407
\(789\) 0 0
\(790\) −16992.0 −0.765251
\(791\) 0 0
\(792\) 0 0
\(793\) −28512.0 −1.27679
\(794\) 36864.0 1.64768
\(795\) 0 0
\(796\) 36288.0 1.61582
\(797\) 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
\(800\) 50944.0 2.25143
\(801\) 0 0
\(802\) −46600.0 −2.05175
\(803\) −9000.00 −0.395521
\(804\) 0 0
\(805\) 0 0
\(806\) −5184.00 −0.226549
\(807\) 0 0
\(808\) 0 0
\(809\) 11318.0 0.491866 0.245933 0.969287i \(-0.420906\pi\)
0.245933 + 0.969287i \(0.420906\pi\)
\(810\) 0 0
\(811\) 29628.0 1.28284 0.641418 0.767192i \(-0.278347\pi\)
0.641418 + 0.767192i \(0.278347\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 32400.0 1.39511
\(815\) −32112.0 −1.38016
\(816\) 0 0
\(817\) −23328.0 −0.998952
\(818\) −30096.0 −1.28641
\(819\) 0 0
\(820\) −38880.0 −1.65579
\(821\) −17770.0 −0.755393 −0.377696 0.925930i \(-0.623284\pi\)
−0.377696 + 0.925930i \(0.623284\pi\)
\(822\) 0 0
\(823\) 7868.00 0.333246 0.166623 0.986021i \(-0.446714\pi\)
0.166623 + 0.986021i \(0.446714\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 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 \(-0.692226\pi\)
−0.567853 + 0.823130i \(0.692226\pi\)
\(830\) −2592.00 −0.108397
\(831\) 0 0
\(832\) −18432.0 −0.768046
\(833\) 0 0
\(834\) 0 0
\(835\) −54432.0 −2.25592
\(836\) 28800.0 1.19147
\(837\) 0 0
\(838\) 15408.0 0.635156
\(839\) 23256.0 0.956956 0.478478 0.878099i \(-0.341189\pi\)
0.478478 + 0.878099i \(0.341189\pi\)
\(840\) 0 0
\(841\) 575.000 0.0235762
\(842\) −41608.0 −1.70298
\(843\) 0 0
\(844\) 23328.0 0.951402
\(845\) −16218.0 −0.660256
\(846\) 0 0
\(847\) 0 0
\(848\) −1408.00 −0.0570176
\(849\) 0 0
\(850\) −100296. −4.04721
\(851\) 2268.00 0.0913584
\(852\) 0 0
\(853\) −35280.0 −1.41614 −0.708068 0.706144i \(-0.750433\pi\)
−0.708068 + 0.706144i \(0.750433\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 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 \(0.524603\pi\)
\(860\) −46656.0 −1.84995
\(861\) 0 0
\(862\) −41560.0 −1.64216
\(863\) 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
\(866\) 44928.0 1.76295
\(867\) 0 0
\(868\) 0 0
\(869\) 11800.0 0.460630
\(870\) 0 0
\(871\) 8352.00 0.324910
\(872\) 0 0
\(873\) 0 0
\(874\) 4032.00 0.156046
\(875\) 0 0
\(876\) 0 0
\(877\) 1226.00 0.0472053 0.0236027 0.999721i \(-0.492486\pi\)
0.0236027 + 0.999721i \(0.492486\pi\)
\(878\) 58464.0 2.24723
\(879\) 0 0
\(880\) −57600.0 −2.20647
\(881\) −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 \(-0.751315\pi\)
−0.710022 + 0.704180i \(0.751315\pi\)
\(884\) 36288.0 1.38065
\(885\) 0 0
\(886\) 47752.0 1.81068
\(887\) 26640.0 1.00844 0.504219 0.863576i \(-0.331781\pi\)
0.504219 + 0.863576i \(0.331781\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16848.0 −0.634546
\(891\) 0 0
\(892\) 8640.00 0.324315
\(893\) −5184.00 −0.194262
\(894\) 0 0
\(895\) −68436.0 −2.55594
\(896\) 0 0
\(897\) 0 0
\(898\) 32744.0 1.21679
\(899\) −5688.00 −0.211018
\(900\) 0 0
\(901\) 2772.00 0.102496
\(902\) 54000.0 1.99335
\(903\) 0 0
\(904\) 0 0
\(905\) 8424.00 0.309418
\(906\) 0 0
\(907\) −12636.0 −0.462593 −0.231296 0.972883i \(-0.574297\pi\)
−0.231296 + 0.972883i \(0.574297\pi\)
\(908\) −10656.0 −0.389462
\(909\) 0 0
\(910\) 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
\(914\) −8424.00 −0.304859
\(915\) 0 0
\(916\) −12960.0 −0.467479
\(917\) 0 0
\(918\) 0 0
\(919\) 36936.0 1.32580 0.662898 0.748710i \(-0.269326\pi\)
0.662898 + 0.748710i \(0.269326\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −37944.0 −1.35534
\(923\) 26424.0 0.942315
\(924\) 0 0
\(925\) −32238.0 −1.14592
\(926\) 50608.0 1.79598
\(927\) 0 0
\(928\) −40448.0 −1.43079
\(929\) −22302.0 −0.787626 −0.393813 0.919191i \(-0.628844\pi\)
−0.393813 + 0.919191i \(0.628844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −53744.0 −1.88889
\(933\) 0 0
\(934\) 14832.0 0.519612
\(935\) 113400. 3.96639
\(936\) 0 0
\(937\) −13824.0 −0.481975 −0.240987 0.970528i \(-0.577471\pi\)
−0.240987 + 0.970528i \(0.577471\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10368.0 −0.359752
\(941\) 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
\(944\) −29952.0 −1.03268
\(945\) 0 0
\(946\) 64800.0 2.22709
\(947\) −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
\(950\) −57312.0 −1.95731
\(951\) 0 0
\(952\) 0 0
\(953\) −38362.0 −1.30395 −0.651976 0.758239i \(-0.726060\pi\)
−0.651976 + 0.758239i \(0.726060\pi\)
\(954\) 0 0
\(955\) −8676.00 −0.293978
\(956\) 28624.0 0.968375
\(957\) 0 0
\(958\) 32256.0 1.08783
\(959\) 0 0
\(960\) 0 0
\(961\) −28495.0 −0.956497
\(962\) 23328.0 0.781835
\(963\) 0 0
\(964\) 6048.00 0.202067
\(965\) −14580.0 −0.486370
\(966\) 0 0
\(967\) 26444.0 0.879402 0.439701 0.898144i \(-0.355084\pi\)
0.439701 + 0.898144i \(0.355084\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 33696.0 1.11537
\(971\) 17820.0 0.588951 0.294475 0.955659i \(-0.404855\pi\)
0.294475 + 0.955659i \(0.404855\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 46656.0 1.53486
\(975\) 0 0
\(976\) 50688.0 1.66238
\(977\) −34438.0 −1.12771 −0.563853 0.825875i \(-0.690682\pi\)
−0.563853 + 0.825875i \(0.690682\pi\)
\(978\) 0 0
\(979\) 11700.0 0.381955
\(980\) 0 0
\(981\) 0 0
\(982\) −39256.0 −1.27567
\(983\) 26064.0 0.845689 0.422845 0.906202i \(-0.361032\pi\)
0.422845 + 0.906202i \(0.361032\pi\)
\(984\) 0 0
\(985\) 44316.0 1.43353
\(986\) 79632.0 2.57201
\(987\) 0 0
\(988\) 20736.0 0.667713
\(989\) 4536.00 0.145841
\(990\) 0 0
\(991\) 33696.0 1.08011 0.540055 0.841630i \(-0.318403\pi\)
0.540055 + 0.841630i \(0.318403\pi\)
\(992\) 9216.00 0.294968
\(993\) 0 0
\(994\) 0 0
\(995\) 81648.0 2.60142
\(996\) 0 0
\(997\) −36072.0 −1.14585 −0.572925 0.819608i \(-0.694191\pi\)
−0.572925 + 0.819608i \(0.694191\pi\)
\(998\) 60912.0 1.93200
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.a.c.1.1 1
3.2 odd 2 147.4.a.f.1.1 1
7.2 even 3 441.4.e.l.361.1 2
7.3 odd 6 441.4.e.o.226.1 2
7.4 even 3 441.4.e.l.226.1 2
7.5 odd 6 441.4.e.o.361.1 2
7.6 odd 2 441.4.a.a.1.1 1
12.11 even 2 2352.4.a.t.1.1 1
21.2 odd 6 147.4.e.d.67.1 2
21.5 even 6 147.4.e.a.67.1 2
21.11 odd 6 147.4.e.d.79.1 2
21.17 even 6 147.4.e.a.79.1 2
21.20 even 2 147.4.a.h.1.1 yes 1
84.83 odd 2 2352.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.f.1.1 1 3.2 odd 2
147.4.a.h.1.1 yes 1 21.20 even 2
147.4.e.a.67.1 2 21.5 even 6
147.4.e.a.79.1 2 21.17 even 6
147.4.e.d.67.1 2 21.2 odd 6
147.4.e.d.79.1 2 21.11 odd 6
441.4.a.a.1.1 1 7.6 odd 2
441.4.a.c.1.1 1 1.1 even 1 trivial
441.4.e.l.226.1 2 7.4 even 3
441.4.e.l.361.1 2 7.2 even 3
441.4.e.o.226.1 2 7.3 odd 6
441.4.e.o.361.1 2 7.5 odd 6
2352.4.a.s.1.1 1 84.83 odd 2
2352.4.a.t.1.1 1 12.11 even 2