Properties

Label 2352.4.a.cc.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{505}) \)
Defining polynomial: \(x^{2} - x - 126\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-10.7361\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +15.7361 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +15.7361 q^{5} +9.00000 q^{9} -58.6805 q^{11} -20.7361 q^{13} +47.2083 q^{15} +42.9444 q^{17} -137.153 q^{19} -29.0556 q^{23} +122.625 q^{25} +27.0000 q^{27} +8.68051 q^{29} +202.472 q^{31} -176.042 q^{33} -15.2639 q^{37} -62.2083 q^{39} +117.250 q^{41} +101.875 q^{43} +141.625 q^{45} -588.500 q^{47} +128.833 q^{51} +404.681 q^{53} -923.403 q^{55} -411.458 q^{57} -10.8748 q^{59} -894.611 q^{61} -326.305 q^{65} +703.735 q^{67} -87.1668 q^{69} -1161.94 q^{71} -1116.37 q^{73} +367.875 q^{75} -138.333 q^{79} +81.0000 q^{81} +894.319 q^{83} +675.778 q^{85} +26.0415 q^{87} -681.806 q^{89} +607.417 q^{93} -2158.25 q^{95} +246.514 q^{97} -528.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 9q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 9q^{5} + 18q^{9} - 5q^{11} - 19q^{13} + 27q^{15} - 4q^{17} - 117q^{19} - 148q^{23} + 43q^{25} + 54q^{27} - 95q^{29} + 360q^{31} - 15q^{33} - 53q^{37} - 57q^{39} - 170q^{41} - 403q^{43} + 81q^{45} - 368q^{47} - 12q^{51} + 697q^{53} - 1285q^{55} - 351q^{57} + 585q^{59} - 1160q^{61} - 338q^{65} - 233q^{67} - 444q^{69} - 616q^{71} - 817q^{73} + 129q^{75} + 802q^{79} + 162q^{81} + 283q^{83} + 992q^{85} - 285q^{87} - 1858q^{89} + 1080q^{93} - 2294q^{95} + 1729q^{97} - 45q^{99} + O(q^{100}) \)

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
\(5\) 15.7361 1.40748 0.703740 0.710458i \(-0.251512\pi\)
0.703740 + 0.710458i \(0.251512\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −58.6805 −1.60844 −0.804220 0.594332i \(-0.797417\pi\)
−0.804220 + 0.594332i \(0.797417\pi\)
\(12\) 0 0
\(13\) −20.7361 −0.442397 −0.221198 0.975229i \(-0.570997\pi\)
−0.221198 + 0.975229i \(0.570997\pi\)
\(14\) 0 0
\(15\) 47.2083 0.812609
\(16\) 0 0
\(17\) 42.9444 0.612679 0.306340 0.951922i \(-0.400896\pi\)
0.306340 + 0.951922i \(0.400896\pi\)
\(18\) 0 0
\(19\) −137.153 −1.65605 −0.828026 0.560690i \(-0.810536\pi\)
−0.828026 + 0.560690i \(0.810536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −29.0556 −0.263413 −0.131707 0.991289i \(-0.542046\pi\)
−0.131707 + 0.991289i \(0.542046\pi\)
\(24\) 0 0
\(25\) 122.625 0.980999
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 8.68051 0.0555838 0.0277919 0.999614i \(-0.491152\pi\)
0.0277919 + 0.999614i \(0.491152\pi\)
\(30\) 0 0
\(31\) 202.472 1.17307 0.586534 0.809925i \(-0.300492\pi\)
0.586534 + 0.809925i \(0.300492\pi\)
\(32\) 0 0
\(33\) −176.042 −0.928633
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −15.2639 −0.0678208 −0.0339104 0.999425i \(-0.510796\pi\)
−0.0339104 + 0.999425i \(0.510796\pi\)
\(38\) 0 0
\(39\) −62.2083 −0.255418
\(40\) 0 0
\(41\) 117.250 0.446618 0.223309 0.974748i \(-0.428314\pi\)
0.223309 + 0.974748i \(0.428314\pi\)
\(42\) 0 0
\(43\) 101.875 0.361297 0.180648 0.983548i \(-0.442180\pi\)
0.180648 + 0.983548i \(0.442180\pi\)
\(44\) 0 0
\(45\) 141.625 0.469160
\(46\) 0 0
\(47\) −588.500 −1.82641 −0.913207 0.407495i \(-0.866402\pi\)
−0.913207 + 0.407495i \(0.866402\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 128.833 0.353731
\(52\) 0 0
\(53\) 404.681 1.04881 0.524407 0.851468i \(-0.324287\pi\)
0.524407 + 0.851468i \(0.324287\pi\)
\(54\) 0 0
\(55\) −923.403 −2.26385
\(56\) 0 0
\(57\) −411.458 −0.956122
\(58\) 0 0
\(59\) −10.8748 −0.0239962 −0.0119981 0.999928i \(-0.503819\pi\)
−0.0119981 + 0.999928i \(0.503819\pi\)
\(60\) 0 0
\(61\) −894.611 −1.87776 −0.938879 0.344249i \(-0.888134\pi\)
−0.938879 + 0.344249i \(0.888134\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −326.305 −0.622665
\(66\) 0 0
\(67\) 703.735 1.28321 0.641604 0.767036i \(-0.278269\pi\)
0.641604 + 0.767036i \(0.278269\pi\)
\(68\) 0 0
\(69\) −87.1668 −0.152082
\(70\) 0 0
\(71\) −1161.94 −1.94222 −0.971108 0.238640i \(-0.923298\pi\)
−0.971108 + 0.238640i \(0.923298\pi\)
\(72\) 0 0
\(73\) −1116.37 −1.78989 −0.894943 0.446180i \(-0.852784\pi\)
−0.894943 + 0.446180i \(0.852784\pi\)
\(74\) 0 0
\(75\) 367.875 0.566380
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −138.333 −0.197008 −0.0985042 0.995137i \(-0.531406\pi\)
−0.0985042 + 0.995137i \(0.531406\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 894.319 1.18270 0.591351 0.806414i \(-0.298595\pi\)
0.591351 + 0.806414i \(0.298595\pi\)
\(84\) 0 0
\(85\) 675.778 0.862334
\(86\) 0 0
\(87\) 26.0415 0.0320913
\(88\) 0 0
\(89\) −681.806 −0.812037 −0.406018 0.913865i \(-0.633083\pi\)
−0.406018 + 0.913865i \(0.633083\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 607.417 0.677271
\(94\) 0 0
\(95\) −2158.25 −2.33086
\(96\) 0 0
\(97\) 246.514 0.258039 0.129019 0.991642i \(-0.458817\pi\)
0.129019 + 0.991642i \(0.458817\pi\)
\(98\) 0 0
\(99\) −528.125 −0.536147
\(100\) 0 0
\(101\) −1089.94 −1.07380 −0.536898 0.843647i \(-0.680404\pi\)
−0.536898 + 0.843647i \(0.680404\pi\)
\(102\) 0 0
\(103\) 30.2364 0.0289251 0.0144625 0.999895i \(-0.495396\pi\)
0.0144625 + 0.999895i \(0.495396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1887.62 −1.70545 −0.852727 0.522357i \(-0.825053\pi\)
−0.852727 + 0.522357i \(0.825053\pi\)
\(108\) 0 0
\(109\) −964.597 −0.847630 −0.423815 0.905749i \(-0.639309\pi\)
−0.423815 + 0.905749i \(0.639309\pi\)
\(110\) 0 0
\(111\) −45.7917 −0.0391564
\(112\) 0 0
\(113\) 119.806 0.0997378 0.0498689 0.998756i \(-0.484120\pi\)
0.0498689 + 0.998756i \(0.484120\pi\)
\(114\) 0 0
\(115\) −457.222 −0.370749
\(116\) 0 0
\(117\) −186.625 −0.147466
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2112.40 1.58708
\(122\) 0 0
\(123\) 351.750 0.257855
\(124\) 0 0
\(125\) −37.3745 −0.0267430
\(126\) 0 0
\(127\) −683.722 −0.477721 −0.238860 0.971054i \(-0.576774\pi\)
−0.238860 + 0.971054i \(0.576774\pi\)
\(128\) 0 0
\(129\) 305.624 0.208595
\(130\) 0 0
\(131\) −260.237 −0.173565 −0.0867825 0.996227i \(-0.527659\pi\)
−0.0867825 + 0.996227i \(0.527659\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 424.875 0.270870
\(136\) 0 0
\(137\) −731.695 −0.456298 −0.228149 0.973626i \(-0.573267\pi\)
−0.228149 + 0.973626i \(0.573267\pi\)
\(138\) 0 0
\(139\) −2287.74 −1.39599 −0.697997 0.716101i \(-0.745925\pi\)
−0.697997 + 0.716101i \(0.745925\pi\)
\(140\) 0 0
\(141\) −1765.50 −1.05448
\(142\) 0 0
\(143\) 1216.81 0.711569
\(144\) 0 0
\(145\) 136.597 0.0782331
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3439.19 1.89094 0.945469 0.325712i \(-0.105604\pi\)
0.945469 + 0.325712i \(0.105604\pi\)
\(150\) 0 0
\(151\) −2851.76 −1.53691 −0.768455 0.639904i \(-0.778974\pi\)
−0.768455 + 0.639904i \(0.778974\pi\)
\(152\) 0 0
\(153\) 386.500 0.204226
\(154\) 0 0
\(155\) 3186.12 1.65107
\(156\) 0 0
\(157\) 2440.25 1.24046 0.620232 0.784418i \(-0.287038\pi\)
0.620232 + 0.784418i \(0.287038\pi\)
\(158\) 0 0
\(159\) 1214.04 0.605533
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −241.390 −0.115995 −0.0579974 0.998317i \(-0.518472\pi\)
−0.0579974 + 0.998317i \(0.518472\pi\)
\(164\) 0 0
\(165\) −2770.21 −1.30703
\(166\) 0 0
\(167\) −326.445 −0.151264 −0.0756319 0.997136i \(-0.524097\pi\)
−0.0756319 + 0.997136i \(0.524097\pi\)
\(168\) 0 0
\(169\) −1767.01 −0.804285
\(170\) 0 0
\(171\) −1234.37 −0.552017
\(172\) 0 0
\(173\) −1595.69 −0.701262 −0.350631 0.936514i \(-0.614033\pi\)
−0.350631 + 0.936514i \(0.614033\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −32.6243 −0.0138542
\(178\) 0 0
\(179\) −2882.86 −1.20377 −0.601886 0.798582i \(-0.705584\pi\)
−0.601886 + 0.798582i \(0.705584\pi\)
\(180\) 0 0
\(181\) 1128.60 0.463470 0.231735 0.972779i \(-0.425560\pi\)
0.231735 + 0.972779i \(0.425560\pi\)
\(182\) 0 0
\(183\) −2683.83 −1.08412
\(184\) 0 0
\(185\) −240.194 −0.0954564
\(186\) 0 0
\(187\) −2520.00 −0.985458
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2205.25 −0.835426 −0.417713 0.908579i \(-0.637168\pi\)
−0.417713 + 0.908579i \(0.637168\pi\)
\(192\) 0 0
\(193\) 3795.36 1.41552 0.707761 0.706451i \(-0.249705\pi\)
0.707761 + 0.706451i \(0.249705\pi\)
\(194\) 0 0
\(195\) −978.916 −0.359496
\(196\) 0 0
\(197\) 3701.25 1.33859 0.669297 0.742995i \(-0.266595\pi\)
0.669297 + 0.742995i \(0.266595\pi\)
\(198\) 0 0
\(199\) 3600.39 1.28254 0.641268 0.767317i \(-0.278409\pi\)
0.641268 + 0.767317i \(0.278409\pi\)
\(200\) 0 0
\(201\) 2111.21 0.740861
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1845.06 0.628606
\(206\) 0 0
\(207\) −261.500 −0.0878045
\(208\) 0 0
\(209\) 8048.19 2.66366
\(210\) 0 0
\(211\) 4137.64 1.34998 0.674992 0.737825i \(-0.264147\pi\)
0.674992 + 0.737825i \(0.264147\pi\)
\(212\) 0 0
\(213\) −3485.83 −1.12134
\(214\) 0 0
\(215\) 1603.11 0.508518
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3349.12 −1.03339
\(220\) 0 0
\(221\) −890.500 −0.271047
\(222\) 0 0
\(223\) −4261.62 −1.27973 −0.639864 0.768488i \(-0.721009\pi\)
−0.639864 + 0.768488i \(0.721009\pi\)
\(224\) 0 0
\(225\) 1103.62 0.327000
\(226\) 0 0
\(227\) 1534.13 0.448562 0.224281 0.974525i \(-0.427997\pi\)
0.224281 + 0.974525i \(0.427997\pi\)
\(228\) 0 0
\(229\) −574.903 −0.165898 −0.0829491 0.996554i \(-0.526434\pi\)
−0.0829491 + 0.996554i \(0.526434\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4152.14 −1.16745 −0.583724 0.811952i \(-0.698405\pi\)
−0.583724 + 0.811952i \(0.698405\pi\)
\(234\) 0 0
\(235\) −9260.69 −2.57064
\(236\) 0 0
\(237\) −414.999 −0.113743
\(238\) 0 0
\(239\) −4548.36 −1.23100 −0.615500 0.788137i \(-0.711046\pi\)
−0.615500 + 0.788137i \(0.711046\pi\)
\(240\) 0 0
\(241\) 7008.79 1.87334 0.936672 0.350208i \(-0.113889\pi\)
0.936672 + 0.350208i \(0.113889\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2844.01 0.732632
\(248\) 0 0
\(249\) 2682.96 0.682833
\(250\) 0 0
\(251\) −682.819 −0.171710 −0.0858548 0.996308i \(-0.527362\pi\)
−0.0858548 + 0.996308i \(0.527362\pi\)
\(252\) 0 0
\(253\) 1705.00 0.423685
\(254\) 0 0
\(255\) 2027.33 0.497869
\(256\) 0 0
\(257\) 1702.55 0.413238 0.206619 0.978421i \(-0.433754\pi\)
0.206619 + 0.978421i \(0.433754\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 78.1246 0.0185279
\(262\) 0 0
\(263\) 2924.25 0.685615 0.342808 0.939406i \(-0.388622\pi\)
0.342808 + 0.939406i \(0.388622\pi\)
\(264\) 0 0
\(265\) 6368.09 1.47618
\(266\) 0 0
\(267\) −2045.42 −0.468830
\(268\) 0 0
\(269\) 1373.82 0.311388 0.155694 0.987805i \(-0.450239\pi\)
0.155694 + 0.987805i \(0.450239\pi\)
\(270\) 0 0
\(271\) 781.511 0.175179 0.0875894 0.996157i \(-0.472084\pi\)
0.0875894 + 0.996157i \(0.472084\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7195.69 −1.57788
\(276\) 0 0
\(277\) −2151.62 −0.466709 −0.233355 0.972392i \(-0.574970\pi\)
−0.233355 + 0.972392i \(0.574970\pi\)
\(278\) 0 0
\(279\) 1822.25 0.391022
\(280\) 0 0
\(281\) −2388.25 −0.507014 −0.253507 0.967334i \(-0.581584\pi\)
−0.253507 + 0.967334i \(0.581584\pi\)
\(282\) 0 0
\(283\) 3294.65 0.692038 0.346019 0.938228i \(-0.387533\pi\)
0.346019 + 0.938228i \(0.387533\pi\)
\(284\) 0 0
\(285\) −6474.75 −1.34572
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3068.78 −0.624624
\(290\) 0 0
\(291\) 739.543 0.148979
\(292\) 0 0
\(293\) −4355.37 −0.868408 −0.434204 0.900815i \(-0.642970\pi\)
−0.434204 + 0.900815i \(0.642970\pi\)
\(294\) 0 0
\(295\) −171.126 −0.0337741
\(296\) 0 0
\(297\) −1584.37 −0.309544
\(298\) 0 0
\(299\) 602.500 0.116533
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3269.83 −0.619957
\(304\) 0 0
\(305\) −14077.7 −2.64291
\(306\) 0 0
\(307\) 8189.87 1.52254 0.761271 0.648434i \(-0.224576\pi\)
0.761271 + 0.648434i \(0.224576\pi\)
\(308\) 0 0
\(309\) 90.7092 0.0166999
\(310\) 0 0
\(311\) −1165.30 −0.212470 −0.106235 0.994341i \(-0.533880\pi\)
−0.106235 + 0.994341i \(0.533880\pi\)
\(312\) 0 0
\(313\) −5784.52 −1.04460 −0.522301 0.852761i \(-0.674926\pi\)
−0.522301 + 0.852761i \(0.674926\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4567.37 −0.809240 −0.404620 0.914485i \(-0.632596\pi\)
−0.404620 + 0.914485i \(0.632596\pi\)
\(318\) 0 0
\(319\) −509.377 −0.0894032
\(320\) 0 0
\(321\) −5662.87 −0.984644
\(322\) 0 0
\(323\) −5889.94 −1.01463
\(324\) 0 0
\(325\) −2542.76 −0.433991
\(326\) 0 0
\(327\) −2893.79 −0.489379
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3540.62 −0.587946 −0.293973 0.955814i \(-0.594978\pi\)
−0.293973 + 0.955814i \(0.594978\pi\)
\(332\) 0 0
\(333\) −137.375 −0.0226069
\(334\) 0 0
\(335\) 11074.1 1.80609
\(336\) 0 0
\(337\) −9183.50 −1.48444 −0.742221 0.670155i \(-0.766227\pi\)
−0.742221 + 0.670155i \(0.766227\pi\)
\(338\) 0 0
\(339\) 359.417 0.0575837
\(340\) 0 0
\(341\) −11881.2 −1.88681
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1371.67 −0.214052
\(346\) 0 0
\(347\) 5955.00 0.921271 0.460636 0.887589i \(-0.347622\pi\)
0.460636 + 0.887589i \(0.347622\pi\)
\(348\) 0 0
\(349\) 2816.11 0.431929 0.215964 0.976401i \(-0.430711\pi\)
0.215964 + 0.976401i \(0.430711\pi\)
\(350\) 0 0
\(351\) −559.875 −0.0851393
\(352\) 0 0
\(353\) −381.306 −0.0574926 −0.0287463 0.999587i \(-0.509151\pi\)
−0.0287463 + 0.999587i \(0.509151\pi\)
\(354\) 0 0
\(355\) −18284.5 −2.73363
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2994.75 0.440270 0.220135 0.975469i \(-0.429350\pi\)
0.220135 + 0.975469i \(0.429350\pi\)
\(360\) 0 0
\(361\) 11951.9 1.74251
\(362\) 0 0
\(363\) 6337.21 0.916301
\(364\) 0 0
\(365\) −17567.4 −2.51923
\(366\) 0 0
\(367\) 4650.03 0.661388 0.330694 0.943738i \(-0.392717\pi\)
0.330694 + 0.943738i \(0.392717\pi\)
\(368\) 0 0
\(369\) 1055.25 0.148873
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4828.34 0.670247 0.335123 0.942174i \(-0.391222\pi\)
0.335123 + 0.942174i \(0.391222\pi\)
\(374\) 0 0
\(375\) −112.123 −0.0154401
\(376\) 0 0
\(377\) −180.000 −0.0245901
\(378\) 0 0
\(379\) −13230.5 −1.79315 −0.896577 0.442887i \(-0.853954\pi\)
−0.896577 + 0.442887i \(0.853954\pi\)
\(380\) 0 0
\(381\) −2051.17 −0.275812
\(382\) 0 0
\(383\) 4409.75 0.588323 0.294161 0.955756i \(-0.404960\pi\)
0.294161 + 0.955756i \(0.404960\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 916.873 0.120432
\(388\) 0 0
\(389\) 575.383 0.0749950 0.0374975 0.999297i \(-0.488061\pi\)
0.0374975 + 0.999297i \(0.488061\pi\)
\(390\) 0 0
\(391\) −1247.78 −0.161388
\(392\) 0 0
\(393\) −780.711 −0.100208
\(394\) 0 0
\(395\) −2176.82 −0.277285
\(396\) 0 0
\(397\) 7241.62 0.915482 0.457741 0.889086i \(-0.348659\pi\)
0.457741 + 0.889086i \(0.348659\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2247.00 0.279825 0.139912 0.990164i \(-0.455318\pi\)
0.139912 + 0.990164i \(0.455318\pi\)
\(402\) 0 0
\(403\) −4198.48 −0.518961
\(404\) 0 0
\(405\) 1274.62 0.156387
\(406\) 0 0
\(407\) 895.693 0.109086
\(408\) 0 0
\(409\) −1650.64 −0.199557 −0.0997784 0.995010i \(-0.531813\pi\)
−0.0997784 + 0.995010i \(0.531813\pi\)
\(410\) 0 0
\(411\) −2195.08 −0.263444
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 14073.1 1.66463
\(416\) 0 0
\(417\) −6863.21 −0.805978
\(418\) 0 0
\(419\) 3178.64 0.370613 0.185307 0.982681i \(-0.440672\pi\)
0.185307 + 0.982681i \(0.440672\pi\)
\(420\) 0 0
\(421\) 8781.12 1.01655 0.508273 0.861196i \(-0.330284\pi\)
0.508273 + 0.861196i \(0.330284\pi\)
\(422\) 0 0
\(423\) −5296.50 −0.608805
\(424\) 0 0
\(425\) 5266.05 0.601038
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 3650.42 0.410825
\(430\) 0 0
\(431\) −9086.69 −1.01552 −0.507762 0.861498i \(-0.669527\pi\)
−0.507762 + 0.861498i \(0.669527\pi\)
\(432\) 0 0
\(433\) 3150.88 0.349703 0.174852 0.984595i \(-0.444055\pi\)
0.174852 + 0.984595i \(0.444055\pi\)
\(434\) 0 0
\(435\) 409.792 0.0451679
\(436\) 0 0
\(437\) 3985.05 0.436226
\(438\) 0 0
\(439\) −10331.4 −1.12321 −0.561605 0.827405i \(-0.689816\pi\)
−0.561605 + 0.827405i \(0.689816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8272.07 0.887173 0.443587 0.896231i \(-0.353706\pi\)
0.443587 + 0.896231i \(0.353706\pi\)
\(444\) 0 0
\(445\) −10729.0 −1.14293
\(446\) 0 0
\(447\) 10317.6 1.09173
\(448\) 0 0
\(449\) 10111.1 1.06275 0.531374 0.847137i \(-0.321676\pi\)
0.531374 + 0.847137i \(0.321676\pi\)
\(450\) 0 0
\(451\) −6880.28 −0.718359
\(452\) 0 0
\(453\) −8555.29 −0.887335
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17840.7 −1.82616 −0.913080 0.407781i \(-0.866303\pi\)
−0.913080 + 0.407781i \(0.866303\pi\)
\(458\) 0 0
\(459\) 1159.50 0.117910
\(460\) 0 0
\(461\) −15787.7 −1.59503 −0.797515 0.603299i \(-0.793852\pi\)
−0.797515 + 0.603299i \(0.793852\pi\)
\(462\) 0 0
\(463\) 5960.87 0.598326 0.299163 0.954202i \(-0.403293\pi\)
0.299163 + 0.954202i \(0.403293\pi\)
\(464\) 0 0
\(465\) 9558.37 0.953245
\(466\) 0 0
\(467\) 11058.3 1.09575 0.547877 0.836559i \(-0.315436\pi\)
0.547877 + 0.836559i \(0.315436\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 7320.74 0.716183
\(472\) 0 0
\(473\) −5978.06 −0.581124
\(474\) 0 0
\(475\) −16818.3 −1.62459
\(476\) 0 0
\(477\) 3642.12 0.349605
\(478\) 0 0
\(479\) 11863.8 1.13167 0.565836 0.824518i \(-0.308554\pi\)
0.565836 + 0.824518i \(0.308554\pi\)
\(480\) 0 0
\(481\) 316.514 0.0300037
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3879.18 0.363184
\(486\) 0 0
\(487\) 5924.97 0.551306 0.275653 0.961257i \(-0.411106\pi\)
0.275653 + 0.961257i \(0.411106\pi\)
\(488\) 0 0
\(489\) −724.171 −0.0669696
\(490\) 0 0
\(491\) 9203.12 0.845888 0.422944 0.906156i \(-0.360997\pi\)
0.422944 + 0.906156i \(0.360997\pi\)
\(492\) 0 0
\(493\) 372.779 0.0340551
\(494\) 0 0
\(495\) −8310.62 −0.754616
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9807.60 0.879857 0.439928 0.898033i \(-0.355004\pi\)
0.439928 + 0.898033i \(0.355004\pi\)
\(500\) 0 0
\(501\) −979.334 −0.0873322
\(502\) 0 0
\(503\) −10462.6 −0.927446 −0.463723 0.885980i \(-0.653487\pi\)
−0.463723 + 0.885980i \(0.653487\pi\)
\(504\) 0 0
\(505\) −17151.5 −1.51135
\(506\) 0 0
\(507\) −5301.04 −0.464354
\(508\) 0 0
\(509\) 5341.83 0.465171 0.232586 0.972576i \(-0.425281\pi\)
0.232586 + 0.972576i \(0.425281\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3703.12 −0.318707
\(514\) 0 0
\(515\) 475.803 0.0407115
\(516\) 0 0
\(517\) 34533.5 2.93768
\(518\) 0 0
\(519\) −4787.08 −0.404874
\(520\) 0 0
\(521\) 11553.3 0.971511 0.485755 0.874095i \(-0.338545\pi\)
0.485755 + 0.874095i \(0.338545\pi\)
\(522\) 0 0
\(523\) −5251.51 −0.439068 −0.219534 0.975605i \(-0.570454\pi\)
−0.219534 + 0.975605i \(0.570454\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8695.05 0.718714
\(528\) 0 0
\(529\) −11322.8 −0.930613
\(530\) 0 0
\(531\) −97.8729 −0.00799872
\(532\) 0 0
\(533\) −2431.30 −0.197583
\(534\) 0 0
\(535\) −29703.9 −2.40039
\(536\) 0 0
\(537\) −8648.58 −0.694998
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −20617.6 −1.63848 −0.819241 0.573449i \(-0.805605\pi\)
−0.819241 + 0.573449i \(0.805605\pi\)
\(542\) 0 0
\(543\) 3385.79 0.267584
\(544\) 0 0
\(545\) −15179.0 −1.19302
\(546\) 0 0
\(547\) −10669.4 −0.833985 −0.416993 0.908910i \(-0.636916\pi\)
−0.416993 + 0.908910i \(0.636916\pi\)
\(548\) 0 0
\(549\) −8051.50 −0.625919
\(550\) 0 0
\(551\) −1190.56 −0.0920497
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −720.583 −0.0551118
\(556\) 0 0
\(557\) −5377.87 −0.409098 −0.204549 0.978856i \(-0.565573\pi\)
−0.204549 + 0.978856i \(0.565573\pi\)
\(558\) 0 0
\(559\) −2112.49 −0.159837
\(560\) 0 0
\(561\) −7560.00 −0.568954
\(562\) 0 0
\(563\) 3518.87 0.263415 0.131708 0.991289i \(-0.457954\pi\)
0.131708 + 0.991289i \(0.457954\pi\)
\(564\) 0 0
\(565\) 1885.28 0.140379
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2550.30 0.187898 0.0939492 0.995577i \(-0.470051\pi\)
0.0939492 + 0.995577i \(0.470051\pi\)
\(570\) 0 0
\(571\) −13655.3 −1.00080 −0.500401 0.865794i \(-0.666814\pi\)
−0.500401 + 0.865794i \(0.666814\pi\)
\(572\) 0 0
\(573\) −6615.75 −0.482333
\(574\) 0 0
\(575\) −3562.94 −0.258408
\(576\) 0 0
\(577\) −16695.9 −1.20461 −0.602304 0.798267i \(-0.705750\pi\)
−0.602304 + 0.798267i \(0.705750\pi\)
\(578\) 0 0
\(579\) 11386.1 0.817253
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −23746.9 −1.68695
\(584\) 0 0
\(585\) −2936.75 −0.207555
\(586\) 0 0
\(587\) 16341.4 1.14903 0.574517 0.818493i \(-0.305190\pi\)
0.574517 + 0.818493i \(0.305190\pi\)
\(588\) 0 0
\(589\) −27769.6 −1.94266
\(590\) 0 0
\(591\) 11103.7 0.772838
\(592\) 0 0
\(593\) 19925.2 1.37981 0.689906 0.723899i \(-0.257652\pi\)
0.689906 + 0.723899i \(0.257652\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10801.2 0.740472
\(598\) 0 0
\(599\) −14781.7 −1.00829 −0.504143 0.863620i \(-0.668191\pi\)
−0.504143 + 0.863620i \(0.668191\pi\)
\(600\) 0 0
\(601\) −20487.7 −1.39054 −0.695268 0.718751i \(-0.744714\pi\)
−0.695268 + 0.718751i \(0.744714\pi\)
\(602\) 0 0
\(603\) 6333.62 0.427736
\(604\) 0 0
\(605\) 33241.0 2.23378
\(606\) 0 0
\(607\) −7524.04 −0.503116 −0.251558 0.967842i \(-0.580943\pi\)
−0.251558 + 0.967842i \(0.580943\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12203.2 0.808000
\(612\) 0 0
\(613\) 3414.64 0.224985 0.112493 0.993653i \(-0.464117\pi\)
0.112493 + 0.993653i \(0.464117\pi\)
\(614\) 0 0
\(615\) 5535.17 0.362926
\(616\) 0 0
\(617\) 22219.4 1.44979 0.724895 0.688859i \(-0.241888\pi\)
0.724895 + 0.688859i \(0.241888\pi\)
\(618\) 0 0
\(619\) −21863.5 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(620\) 0 0
\(621\) −784.501 −0.0506939
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −15916.2 −1.01864
\(626\) 0 0
\(627\) 24144.6 1.53787
\(628\) 0 0
\(629\) −655.499 −0.0415524
\(630\) 0 0
\(631\) 16080.8 1.01452 0.507262 0.861792i \(-0.330658\pi\)
0.507262 + 0.861792i \(0.330658\pi\)
\(632\) 0 0
\(633\) 12412.9 0.779414
\(634\) 0 0
\(635\) −10759.1 −0.672382
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −10457.5 −0.647405
\(640\) 0 0
\(641\) −18463.8 −1.13772 −0.568858 0.822436i \(-0.692615\pi\)
−0.568858 + 0.822436i \(0.692615\pi\)
\(642\) 0 0
\(643\) 3110.87 0.190794 0.0953971 0.995439i \(-0.469588\pi\)
0.0953971 + 0.995439i \(0.469588\pi\)
\(644\) 0 0
\(645\) 4809.34 0.293593
\(646\) 0 0
\(647\) 24729.0 1.50263 0.751313 0.659946i \(-0.229421\pi\)
0.751313 + 0.659946i \(0.229421\pi\)
\(648\) 0 0
\(649\) 638.137 0.0385964
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7052.63 0.422651 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(654\) 0 0
\(655\) −4095.12 −0.244289
\(656\) 0 0
\(657\) −10047.4 −0.596629
\(658\) 0 0
\(659\) −7895.55 −0.466718 −0.233359 0.972391i \(-0.574972\pi\)
−0.233359 + 0.972391i \(0.574972\pi\)
\(660\) 0 0
\(661\) −13101.1 −0.770914 −0.385457 0.922726i \(-0.625956\pi\)
−0.385457 + 0.922726i \(0.625956\pi\)
\(662\) 0 0
\(663\) −2671.50 −0.156489
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −252.217 −0.0146415
\(668\) 0 0
\(669\) −12784.9 −0.738852
\(670\) 0 0
\(671\) 52496.2 3.02026
\(672\) 0 0
\(673\) −9128.25 −0.522836 −0.261418 0.965226i \(-0.584190\pi\)
−0.261418 + 0.965226i \(0.584190\pi\)
\(674\) 0 0
\(675\) 3310.87 0.188793
\(676\) 0 0
\(677\) 16029.9 0.910011 0.455006 0.890489i \(-0.349637\pi\)
0.455006 + 0.890489i \(0.349637\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4602.38 0.258977
\(682\) 0 0
\(683\) 22176.4 1.24239 0.621197 0.783655i \(-0.286647\pi\)
0.621197 + 0.783655i \(0.286647\pi\)
\(684\) 0 0
\(685\) −11514.0 −0.642231
\(686\) 0 0
\(687\) −1724.71 −0.0957814
\(688\) 0 0
\(689\) −8391.50 −0.463992
\(690\) 0 0
\(691\) 1184.14 0.0651909 0.0325955 0.999469i \(-0.489623\pi\)
0.0325955 + 0.999469i \(0.489623\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −36000.0 −1.96483
\(696\) 0 0
\(697\) 5035.23 0.273634
\(698\) 0 0
\(699\) −12456.4 −0.674027
\(700\) 0 0
\(701\) 31064.6 1.67374 0.836870 0.547401i \(-0.184383\pi\)
0.836870 + 0.547401i \(0.184383\pi\)
\(702\) 0 0
\(703\) 2093.49 0.112315
\(704\) 0 0
\(705\) −27782.1 −1.48416
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 575.891 0.0305050 0.0152525 0.999884i \(-0.495145\pi\)
0.0152525 + 0.999884i \(0.495145\pi\)
\(710\) 0 0
\(711\) −1245.00 −0.0656695
\(712\) 0 0
\(713\) −5882.95 −0.309002
\(714\) 0 0
\(715\) 19147.8 1.00152
\(716\) 0 0
\(717\) −13645.1 −0.710718
\(718\) 0 0
\(719\) −4835.54 −0.250814 −0.125407 0.992105i \(-0.540024\pi\)
−0.125407 + 0.992105i \(0.540024\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 21026.4 1.08158
\(724\) 0 0
\(725\) 1064.45 0.0545277
\(726\) 0 0
\(727\) 17999.8 0.918259 0.459129 0.888369i \(-0.348161\pi\)
0.459129 + 0.888369i \(0.348161\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 4374.95 0.221359
\(732\) 0 0
\(733\) 5113.40 0.257664 0.128832 0.991666i \(-0.458877\pi\)
0.128832 + 0.991666i \(0.458877\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −41295.6 −2.06396
\(738\) 0 0
\(739\) −13056.1 −0.649902 −0.324951 0.945731i \(-0.605348\pi\)
−0.324951 + 0.945731i \(0.605348\pi\)
\(740\) 0 0
\(741\) 8532.04 0.422986
\(742\) 0 0
\(743\) 28402.9 1.40242 0.701212 0.712953i \(-0.252643\pi\)
0.701212 + 0.712953i \(0.252643\pi\)
\(744\) 0 0
\(745\) 54119.5 2.66146
\(746\) 0 0
\(747\) 8048.87 0.394234
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 33270.3 1.61658 0.808288 0.588787i \(-0.200394\pi\)
0.808288 + 0.588787i \(0.200394\pi\)
\(752\) 0 0
\(753\) −2048.46 −0.0991366
\(754\) 0 0
\(755\) −44875.7 −2.16317
\(756\) 0 0
\(757\) 18331.2 0.880129 0.440064 0.897966i \(-0.354956\pi\)
0.440064 + 0.897966i \(0.354956\pi\)
\(758\) 0 0
\(759\) 5114.99 0.244614
\(760\) 0 0
\(761\) 4997.01 0.238031 0.119015 0.992892i \(-0.462026\pi\)
0.119015 + 0.992892i \(0.462026\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 6082.00 0.287445
\(766\) 0 0
\(767\) 225.500 0.0106158
\(768\) 0 0
\(769\) −9834.47 −0.461171 −0.230585 0.973052i \(-0.574064\pi\)
−0.230585 + 0.973052i \(0.574064\pi\)
\(770\) 0 0
\(771\) 5107.66 0.238583
\(772\) 0 0
\(773\) −13983.0 −0.650625 −0.325312 0.945607i \(-0.605469\pi\)
−0.325312 + 0.945607i \(0.605469\pi\)
\(774\) 0 0
\(775\) 24828.1 1.15078
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16081.1 −0.739623
\(780\) 0 0
\(781\) 68183.5 3.12394
\(782\) 0 0
\(783\) 234.374 0.0106971
\(784\) 0 0
\(785\) 38400.0 1.74593
\(786\) 0 0
\(787\) 31278.6 1.41672 0.708362 0.705849i \(-0.249434\pi\)
0.708362 + 0.705849i \(0.249434\pi\)
\(788\) 0 0
\(789\) 8772.74 0.395840
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18550.7 0.830714
\(794\) 0 0
\(795\) 19104.3 0.852276
\(796\) 0 0
\(797\) 30546.9 1.35762 0.678811 0.734313i \(-0.262495\pi\)
0.678811 + 0.734313i \(0.262495\pi\)
\(798\) 0 0
\(799\) −25272.8 −1.11901
\(800\) 0 0
\(801\) −6136.25 −0.270679
\(802\) 0 0
\(803\) 65509.4 2.87893
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4121.46 0.179780
\(808\) 0 0
\(809\) −29572.7 −1.28519 −0.642596 0.766205i \(-0.722142\pi\)
−0.642596 + 0.766205i \(0.722142\pi\)
\(810\) 0 0
\(811\) −45069.0 −1.95140 −0.975701 0.219107i \(-0.929686\pi\)
−0.975701 + 0.219107i \(0.929686\pi\)
\(812\) 0 0
\(813\) 2344.53 0.101139
\(814\) 0 0
\(815\) −3798.54 −0.163260
\(816\) 0 0
\(817\) −13972.4 −0.598326
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9400.93 0.399628 0.199814 0.979834i \(-0.435966\pi\)
0.199814 + 0.979834i \(0.435966\pi\)
\(822\) 0 0
\(823\) 28008.9 1.18630 0.593152 0.805091i \(-0.297883\pi\)
0.593152 + 0.805091i \(0.297883\pi\)
\(824\) 0 0
\(825\) −21587.1 −0.910989
\(826\) 0 0
\(827\) 28650.8 1.20470 0.602349 0.798232i \(-0.294231\pi\)
0.602349 + 0.798232i \(0.294231\pi\)
\(828\) 0 0
\(829\) −4249.64 −0.178041 −0.0890205 0.996030i \(-0.528374\pi\)
−0.0890205 + 0.996030i \(0.528374\pi\)
\(830\) 0 0
\(831\) −6454.87 −0.269455
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −5136.97 −0.212901
\(836\) 0 0
\(837\) 5466.75 0.225757
\(838\) 0 0
\(839\) −9265.50 −0.381264 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(840\) 0 0
\(841\) −24313.6 −0.996910
\(842\) 0 0
\(843\) −7164.75 −0.292725
\(844\) 0 0
\(845\) −27805.9 −1.13201
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9883.96 0.399548
\(850\) 0 0
\(851\) 443.502 0.0178649
\(852\) 0 0
\(853\) 36321.1 1.45793 0.728964 0.684552i \(-0.240002\pi\)
0.728964 + 0.684552i \(0.240002\pi\)
\(854\) 0 0
\(855\) −19424.2 −0.776953
\(856\) 0 0
\(857\) −23676.2 −0.943716 −0.471858 0.881675i \(-0.656417\pi\)
−0.471858 + 0.881675i \(0.656417\pi\)
\(858\) 0 0
\(859\) −27412.3 −1.08882 −0.544410 0.838819i \(-0.683247\pi\)
−0.544410 + 0.838819i \(0.683247\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22725.5 0.896390 0.448195 0.893936i \(-0.352067\pi\)
0.448195 + 0.893936i \(0.352067\pi\)
\(864\) 0 0
\(865\) −25110.0 −0.987013
\(866\) 0 0
\(867\) −9206.33 −0.360627
\(868\) 0 0
\(869\) 8117.45 0.316876
\(870\) 0 0
\(871\) −14592.7 −0.567688
\(872\) 0 0
\(873\) 2218.63 0.0860129
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11981.9 0.461344 0.230672 0.973032i \(-0.425908\pi\)
0.230672 + 0.973032i \(0.425908\pi\)
\(878\) 0 0
\(879\) −13066.1 −0.501376
\(880\) 0 0
\(881\) −22891.8 −0.875419 −0.437709 0.899117i \(-0.644210\pi\)
−0.437709 + 0.899117i \(0.644210\pi\)
\(882\) 0 0
\(883\) −36489.6 −1.39068 −0.695341 0.718680i \(-0.744747\pi\)
−0.695341 + 0.718680i \(0.744747\pi\)
\(884\) 0 0
\(885\) −513.379 −0.0194995
\(886\) 0 0
\(887\) −24105.8 −0.912506 −0.456253 0.889850i \(-0.650809\pi\)
−0.456253 + 0.889850i \(0.650809\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4753.12 −0.178716
\(892\) 0 0
\(893\) 80714.3 3.02464
\(894\) 0 0
\(895\) −45365.0 −1.69428
\(896\) 0 0
\(897\) 1807.50 0.0672805
\(898\) 0 0
\(899\) 1757.56 0.0652036
\(900\) 0 0
\(901\) 17378.8 0.642587
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17759.7 0.652324
\(906\) 0 0
\(907\) −22363.3 −0.818699 −0.409350 0.912378i \(-0.634244\pi\)
−0.409350 + 0.912378i \(0.634244\pi\)
\(908\) 0 0
\(909\) −9809.49 −0.357932
\(910\) 0 0
\(911\) 21426.3 0.779238 0.389619 0.920976i \(-0.372607\pi\)
0.389619 + 0.920976i \(0.372607\pi\)
\(912\) 0 0
\(913\) −52479.1 −1.90230
\(914\) 0 0
\(915\) −42233.1 −1.52588
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 30526.1 1.09572 0.547858 0.836571i \(-0.315443\pi\)
0.547858 + 0.836571i \(0.315443\pi\)
\(920\) 0 0
\(921\) 24569.6 0.879040
\(922\) 0 0
\(923\) 24094.2 0.859231
\(924\) 0 0
\(925\) −1871.73 −0.0665322
\(926\) 0 0
\(927\) 272.128 0.00964169
\(928\) 0 0
\(929\) 4638.90 0.163829 0.0819146 0.996639i \(-0.473897\pi\)
0.0819146 + 0.996639i \(0.473897\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3495.90 −0.122670
\(934\) 0 0
\(935\) −39655.0 −1.38701
\(936\) 0 0
\(937\) −7713.38 −0.268928 −0.134464 0.990919i \(-0.542931\pi\)
−0.134464 + 0.990919i \(0.542931\pi\)
\(938\) 0 0
\(939\) −17353.6 −0.603102
\(940\) 0 0
\(941\) −7490.63 −0.259498 −0.129749 0.991547i \(-0.541417\pi\)
−0.129749 + 0.991547i \(0.541417\pi\)
\(942\) 0 0
\(943\) −3406.76 −0.117645
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25139.1 −0.862632 −0.431316 0.902201i \(-0.641951\pi\)
−0.431316 + 0.902201i \(0.641951\pi\)
\(948\) 0 0
\(949\) 23149.3 0.791840
\(950\) 0 0
\(951\) −13702.1 −0.467215
\(952\) 0 0
\(953\) −2505.29 −0.0851568 −0.0425784 0.999093i \(-0.513557\pi\)
−0.0425784 + 0.999093i \(0.513557\pi\)
\(954\) 0 0
\(955\) −34702.0 −1.17584
\(956\) 0 0
\(957\) −1528.13 −0.0516170
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 11204.0 0.376087
\(962\) 0 0
\(963\) −16988.6 −0.568485
\(964\) 0 0
\(965\) 59724.2 1.99232
\(966\) 0 0
\(967\) 9331.61 0.310325 0.155162 0.987889i \(-0.450410\pi\)
0.155162 + 0.987889i \(0.450410\pi\)
\(968\) 0 0
\(969\) −17669.8 −0.585796
\(970\) 0 0
\(971\) 41985.9 1.38763 0.693817 0.720152i \(-0.255928\pi\)
0.693817 + 0.720152i \(0.255928\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7628.29 −0.250565
\(976\) 0 0
\(977\) −7950.95 −0.260362 −0.130181 0.991490i \(-0.541556\pi\)
−0.130181 + 0.991490i \(0.541556\pi\)
\(978\) 0 0
\(979\) 40008.7 1.30611
\(980\) 0 0
\(981\) −8681.37 −0.282543
\(982\) 0 0
\(983\) −10539.5 −0.341971 −0.170986 0.985274i \(-0.554695\pi\)
−0.170986 + 0.985274i \(0.554695\pi\)
\(984\) 0 0
\(985\) 58243.2 1.88404
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2960.03 −0.0951704
\(990\) 0 0
\(991\) 4431.86 0.142061 0.0710306 0.997474i \(-0.477371\pi\)
0.0710306 + 0.997474i \(0.477371\pi\)
\(992\) 0 0
\(993\) −10621.9 −0.339451
\(994\) 0 0
\(995\) 56656.0 1.80514
\(996\) 0 0
\(997\) −39415.6 −1.25206 −0.626031 0.779799i \(-0.715321\pi\)
−0.626031 + 0.779799i \(0.715321\pi\)
\(998\) 0 0
\(999\) −412.125 −0.0130521
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.cc.1.2 2
4.3 odd 2 1176.4.a.r.1.2 2
7.2 even 3 336.4.q.g.193.1 4
7.4 even 3 336.4.q.g.289.1 4
7.6 odd 2 2352.4.a.bo.1.1 2
28.11 odd 6 168.4.q.d.121.1 yes 4
28.23 odd 6 168.4.q.d.25.1 4
28.27 even 2 1176.4.a.u.1.1 2
84.11 even 6 504.4.s.f.289.2 4
84.23 even 6 504.4.s.f.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.d.25.1 4 28.23 odd 6
168.4.q.d.121.1 yes 4 28.11 odd 6
336.4.q.g.193.1 4 7.2 even 3
336.4.q.g.289.1 4 7.4 even 3
504.4.s.f.289.2 4 84.11 even 6
504.4.s.f.361.2 4 84.23 even 6
1176.4.a.r.1.2 2 4.3 odd 2
1176.4.a.u.1.1 2 28.27 even 2
2352.4.a.bo.1.1 2 7.6 odd 2
2352.4.a.cc.1.2 2 1.1 even 1 trivial