Properties

Label 2304.3.e.o.1025.3
Level $2304$
Weight $3$
Character 2304.1025
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.540942598144.16
Defining polynomial: \(x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.3
Root \(-2.69642i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1025
Dual form 2304.3.e.o.1025.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.07498i q^{5} +7.21110 q^{7} +O(q^{10})\) \(q-1.07498i q^{5} +7.21110 q^{7} +16.3517i q^{11} -21.6045 q^{13} -18.9819i q^{17} -17.0438 q^{19} -1.11567i q^{23} +23.8444 q^{25} -29.4784i q^{29} -5.63331 q^{31} -7.75182i q^{35} +17.0438 q^{37} -27.4671i q^{41} +52.3306 q^{43} -64.5352i q^{47} +3.00000 q^{49} +35.9283i q^{53} +17.5778 q^{55} -56.8069i q^{59} +69.3743 q^{61} +23.2245i q^{65} +69.3743 q^{67} -98.4764i q^{71} -37.6888 q^{73} +117.914i q^{77} +127.322 q^{79} +7.75182i q^{83} -20.4052 q^{85} +76.1475i q^{89} -155.792 q^{91} +18.3218i q^{95} +4.84441 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} + 128q^{31} + 24q^{49} + 256q^{55} + 160q^{73} + 384q^{79} - 192q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.07498i − 0.214997i −0.994205 0.107498i \(-0.965716\pi\)
0.994205 0.107498i \(-0.0342840\pi\)
\(6\) 0 0
\(7\) 7.21110 1.03016 0.515079 0.857143i \(-0.327763\pi\)
0.515079 + 0.857143i \(0.327763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 16.3517i 1.48652i 0.669004 + 0.743258i \(0.266721\pi\)
−0.669004 + 0.743258i \(0.733279\pi\)
\(12\) 0 0
\(13\) −21.6045 −1.66189 −0.830943 0.556357i \(-0.812199\pi\)
−0.830943 + 0.556357i \(0.812199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 18.9819i − 1.11658i −0.829646 0.558290i \(-0.811458\pi\)
0.829646 0.558290i \(-0.188542\pi\)
\(18\) 0 0
\(19\) −17.0438 −0.897040 −0.448520 0.893773i \(-0.648049\pi\)
−0.448520 + 0.893773i \(0.648049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.11567i − 0.0485074i −0.999706 0.0242537i \(-0.992279\pi\)
0.999706 0.0242537i \(-0.00772094\pi\)
\(24\) 0 0
\(25\) 23.8444 0.953776
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 29.4784i − 1.01650i −0.861210 0.508249i \(-0.830293\pi\)
0.861210 0.508249i \(-0.169707\pi\)
\(30\) 0 0
\(31\) −5.63331 −0.181720 −0.0908598 0.995864i \(-0.528962\pi\)
−0.0908598 + 0.995864i \(0.528962\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 7.75182i − 0.221480i
\(36\) 0 0
\(37\) 17.0438 0.460642 0.230321 0.973115i \(-0.426022\pi\)
0.230321 + 0.973115i \(0.426022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 27.4671i − 0.669930i −0.942231 0.334965i \(-0.891275\pi\)
0.942231 0.334965i \(-0.108725\pi\)
\(42\) 0 0
\(43\) 52.3306 1.21699 0.608495 0.793558i \(-0.291774\pi\)
0.608495 + 0.793558i \(0.291774\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 64.5352i − 1.37309i −0.727087 0.686545i \(-0.759126\pi\)
0.727087 0.686545i \(-0.240874\pi\)
\(48\) 0 0
\(49\) 3.00000 0.0612245
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 35.9283i 0.677893i 0.940806 + 0.338946i \(0.110071\pi\)
−0.940806 + 0.338946i \(0.889929\pi\)
\(54\) 0 0
\(55\) 17.5778 0.319596
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 56.8069i − 0.962828i −0.876493 0.481414i \(-0.840123\pi\)
0.876493 0.481414i \(-0.159877\pi\)
\(60\) 0 0
\(61\) 69.3743 1.13728 0.568642 0.822585i \(-0.307469\pi\)
0.568642 + 0.822585i \(0.307469\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23.2245i 0.357300i
\(66\) 0 0
\(67\) 69.3743 1.03544 0.517719 0.855551i \(-0.326781\pi\)
0.517719 + 0.855551i \(0.326781\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 98.4764i − 1.38699i −0.720461 0.693496i \(-0.756070\pi\)
0.720461 0.693496i \(-0.243930\pi\)
\(72\) 0 0
\(73\) −37.6888 −0.516285 −0.258143 0.966107i \(-0.583110\pi\)
−0.258143 + 0.966107i \(0.583110\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 117.914i 1.53135i
\(78\) 0 0
\(79\) 127.322 1.61167 0.805836 0.592138i \(-0.201716\pi\)
0.805836 + 0.592138i \(0.201716\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.75182i 0.0933954i 0.998909 + 0.0466977i \(0.0148697\pi\)
−0.998909 + 0.0466977i \(0.985130\pi\)
\(84\) 0 0
\(85\) −20.4052 −0.240061
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 76.1475i 0.855590i 0.903876 + 0.427795i \(0.140709\pi\)
−0.903876 + 0.427795i \(0.859291\pi\)
\(90\) 0 0
\(91\) −155.792 −1.71200
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.3218i 0.192861i
\(96\) 0 0
\(97\) 4.84441 0.0499424 0.0249712 0.999688i \(-0.492051\pi\)
0.0249712 + 0.999688i \(0.492051\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 105.635i − 1.04589i −0.852366 0.522946i \(-0.824833\pi\)
0.852366 0.522946i \(-0.175167\pi\)
\(102\) 0 0
\(103\) 104.789 1.01737 0.508684 0.860953i \(-0.330132\pi\)
0.508684 + 0.860953i \(0.330132\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 65.4067i 0.611278i 0.952147 + 0.305639i \(0.0988701\pi\)
−0.952147 + 0.305639i \(0.901130\pi\)
\(108\) 0 0
\(109\) 3.36144 0.0308389 0.0154195 0.999881i \(-0.495092\pi\)
0.0154195 + 0.999881i \(0.495092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 84.1927i 0.745068i 0.928019 + 0.372534i \(0.121511\pi\)
−0.928019 + 0.372534i \(0.878489\pi\)
\(114\) 0 0
\(115\) −1.19933 −0.0104289
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 136.880i − 1.15025i
\(120\) 0 0
\(121\) −146.378 −1.20973
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 52.5069i − 0.420056i
\(126\) 0 0
\(127\) 45.5223 0.358443 0.179222 0.983809i \(-0.442642\pi\)
0.179222 + 0.983809i \(0.442642\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 129.117i 0.985629i 0.870134 + 0.492814i \(0.164032\pi\)
−0.870134 + 0.492814i \(0.835968\pi\)
\(132\) 0 0
\(133\) −122.904 −0.924092
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.9367i − 0.0798296i −0.999203 0.0399148i \(-0.987291\pi\)
0.999203 0.0399148i \(-0.0127087\pi\)
\(138\) 0 0
\(139\) −1.19933 −0.00862825 −0.00431412 0.999991i \(-0.501373\pi\)
−0.00431412 + 0.999991i \(0.501373\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 353.270i − 2.47042i
\(144\) 0 0
\(145\) −31.6888 −0.218544
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 29.9323i − 0.200888i −0.994943 0.100444i \(-0.967974\pi\)
0.994943 0.100444i \(-0.0320263\pi\)
\(150\) 0 0
\(151\) 61.5223 0.407432 0.203716 0.979030i \(-0.434698\pi\)
0.203716 + 0.979030i \(0.434698\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.05571i 0.0390691i
\(156\) 0 0
\(157\) −137.549 −0.876110 −0.438055 0.898948i \(-0.644333\pi\)
−0.438055 + 0.898948i \(0.644333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 8.04521i − 0.0499702i
\(162\) 0 0
\(163\) −191.079 −1.17227 −0.586133 0.810215i \(-0.699350\pi\)
−0.586133 + 0.810215i \(0.699350\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 233.125i − 1.39596i −0.716118 0.697980i \(-0.754083\pi\)
0.716118 0.697980i \(-0.245917\pi\)
\(168\) 0 0
\(169\) 297.755 1.76187
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 323.809i − 1.87173i −0.352363 0.935864i \(-0.614622\pi\)
0.352363 0.935864i \(-0.385378\pi\)
\(174\) 0 0
\(175\) 171.944 0.982540
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 185.924i − 1.03868i −0.854567 0.519342i \(-0.826177\pi\)
0.854567 0.519342i \(-0.173823\pi\)
\(180\) 0 0
\(181\) 126.266 0.697600 0.348800 0.937197i \(-0.386589\pi\)
0.348800 + 0.937197i \(0.386589\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 18.3218i − 0.0990365i
\(186\) 0 0
\(187\) 310.385 1.65982
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 233.125i − 1.22055i −0.792189 0.610275i \(-0.791059\pi\)
0.792189 0.610275i \(-0.208941\pi\)
\(192\) 0 0
\(193\) −117.378 −0.608174 −0.304087 0.952644i \(-0.598351\pi\)
−0.304087 + 0.952644i \(0.598351\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 296.313i 1.50413i 0.659091 + 0.752064i \(0.270941\pi\)
−0.659091 + 0.752064i \(0.729059\pi\)
\(198\) 0 0
\(199\) 233.011 1.17091 0.585455 0.810705i \(-0.300916\pi\)
0.585455 + 0.810705i \(0.300916\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 212.572i − 1.04715i
\(204\) 0 0
\(205\) −29.5267 −0.144033
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 278.694i − 1.33346i
\(210\) 0 0
\(211\) 35.2868 0.167236 0.0836181 0.996498i \(-0.473352\pi\)
0.0836181 + 0.996498i \(0.473352\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 56.2545i − 0.261649i
\(216\) 0 0
\(217\) −40.6224 −0.187200
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 410.094i 1.85563i
\(222\) 0 0
\(223\) 51.8335 0.232437 0.116219 0.993224i \(-0.462923\pi\)
0.116219 + 0.993224i \(0.462923\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 334.786i − 1.47483i −0.675442 0.737413i \(-0.736047\pi\)
0.675442 0.737413i \(-0.263953\pi\)
\(228\) 0 0
\(229\) 219.407 0.958108 0.479054 0.877786i \(-0.340980\pi\)
0.479054 + 0.877786i \(0.340980\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 203.867i − 0.874965i −0.899227 0.437482i \(-0.855870\pi\)
0.899227 0.437482i \(-0.144130\pi\)
\(234\) 0 0
\(235\) −69.3743 −0.295210
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 54.0232i − 0.226038i −0.993593 0.113019i \(-0.963948\pi\)
0.993593 0.113019i \(-0.0360522\pi\)
\(240\) 0 0
\(241\) 327.600 1.35933 0.679667 0.733520i \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.22495i − 0.0131631i
\(246\) 0 0
\(247\) 368.222 1.49078
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 362.281i 1.44335i 0.692231 + 0.721676i \(0.256628\pi\)
−0.692231 + 0.721676i \(0.743372\pi\)
\(252\) 0 0
\(253\) 18.2431 0.0721070
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.2132i 0.0825416i 0.999148 + 0.0412708i \(0.0131406\pi\)
−0.999148 + 0.0412708i \(0.986859\pi\)
\(258\) 0 0
\(259\) 122.904 0.474534
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 205.878i − 0.782806i −0.920219 0.391403i \(-0.871990\pi\)
0.920219 0.391403i \(-0.128010\pi\)
\(264\) 0 0
\(265\) 38.6224 0.145745
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.77109i 0.0103014i 0.999987 + 0.00515072i \(0.00163953\pi\)
−0.999987 + 0.00515072i \(0.998360\pi\)
\(270\) 0 0
\(271\) 125.744 0.464001 0.232001 0.972716i \(-0.425473\pi\)
0.232001 + 0.972716i \(0.425473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 389.896i 1.41780i
\(276\) 0 0
\(277\) 162.752 0.587552 0.293776 0.955874i \(-0.405088\pi\)
0.293776 + 0.955874i \(0.405088\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 301.227i 1.07198i 0.844223 + 0.535992i \(0.180062\pi\)
−0.844223 + 0.535992i \(0.819938\pi\)
\(282\) 0 0
\(283\) −31.6888 −0.111975 −0.0559874 0.998431i \(-0.517831\pi\)
−0.0559874 + 0.998431i \(0.517831\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 198.068i − 0.690134i
\(288\) 0 0
\(289\) −71.3112 −0.246751
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 304.913i − 1.04066i −0.853966 0.520329i \(-0.825809\pi\)
0.853966 0.520329i \(-0.174191\pi\)
\(294\) 0 0
\(295\) −61.0665 −0.207005
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.1035i 0.0806137i
\(300\) 0 0
\(301\) 377.361 1.25369
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 74.5763i − 0.244512i
\(306\) 0 0
\(307\) 105.860 0.344822 0.172411 0.985025i \(-0.444844\pi\)
0.172411 + 0.985025i \(0.444844\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 555.157i − 1.78507i −0.450978 0.892535i \(-0.648925\pi\)
0.450978 0.892535i \(-0.351075\pi\)
\(312\) 0 0
\(313\) −158.000 −0.504792 −0.252396 0.967624i \(-0.581219\pi\)
−0.252396 + 0.967624i \(0.581219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 97.0351i − 0.306105i −0.988218 0.153052i \(-0.951090\pi\)
0.988218 0.153052i \(-0.0489103\pi\)
\(318\) 0 0
\(319\) 482.022 1.51104
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 323.522i 1.00162i
\(324\) 0 0
\(325\) −515.147 −1.58507
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 465.370i − 1.41450i
\(330\) 0 0
\(331\) −619.572 −1.87182 −0.935909 0.352242i \(-0.885419\pi\)
−0.935909 + 0.352242i \(0.885419\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 74.5763i − 0.222616i
\(336\) 0 0
\(337\) −170.755 −0.506692 −0.253346 0.967376i \(-0.581531\pi\)
−0.253346 + 0.967376i \(0.581531\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 92.1141i − 0.270129i
\(342\) 0 0
\(343\) −331.711 −0.967087
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 300.386i − 0.865666i −0.901474 0.432833i \(-0.857514\pi\)
0.901474 0.432833i \(-0.142486\pi\)
\(348\) 0 0
\(349\) −53.5299 −0.153381 −0.0766904 0.997055i \(-0.524435\pi\)
−0.0766904 + 0.997055i \(0.524435\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 600.475i 1.70106i 0.525925 + 0.850531i \(0.323719\pi\)
−0.525925 + 0.850531i \(0.676281\pi\)
\(354\) 0 0
\(355\) −105.860 −0.298199
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 148.508i 0.413671i 0.978376 + 0.206836i \(0.0663165\pi\)
−0.978376 + 0.206836i \(0.933683\pi\)
\(360\) 0 0
\(361\) −70.5106 −0.195320
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40.5149i 0.111000i
\(366\) 0 0
\(367\) −316.389 −0.862094 −0.431047 0.902329i \(-0.641856\pi\)
−0.431047 + 0.902329i \(0.641856\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 259.083i 0.698336i
\(372\) 0 0
\(373\) −626.768 −1.68034 −0.840171 0.542321i \(-0.817545\pi\)
−0.840171 + 0.542321i \(0.817545\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 636.867i 1.68930i
\(378\) 0 0
\(379\) 331.027 0.873423 0.436711 0.899602i \(-0.356143\pi\)
0.436711 + 0.899602i \(0.356143\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 407.765i − 1.06466i −0.846537 0.532330i \(-0.821317\pi\)
0.846537 0.532330i \(-0.178683\pi\)
\(384\) 0 0
\(385\) 126.755 0.329234
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 53.5819i 0.137743i 0.997626 + 0.0688714i \(0.0219398\pi\)
−0.997626 + 0.0688714i \(0.978060\pi\)
\(390\) 0 0
\(391\) −21.1775 −0.0541624
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 136.869i − 0.346504i
\(396\) 0 0
\(397\) −289.744 −0.729833 −0.364917 0.931040i \(-0.618902\pi\)
−0.364917 + 0.931040i \(0.618902\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 203.458i 0.507376i 0.967286 + 0.253688i \(0.0816436\pi\)
−0.967286 + 0.253688i \(0.918356\pi\)
\(402\) 0 0
\(403\) 121.705 0.301997
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 278.694i 0.684752i
\(408\) 0 0
\(409\) −412.844 −1.00940 −0.504700 0.863295i \(-0.668397\pi\)
−0.504700 + 0.863295i \(0.668397\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 409.640i − 0.991865i
\(414\) 0 0
\(415\) 8.33308 0.0200797
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 336.482i − 0.803059i −0.915846 0.401529i \(-0.868479\pi\)
0.915846 0.401529i \(-0.131521\pi\)
\(420\) 0 0
\(421\) 217.481 0.516582 0.258291 0.966067i \(-0.416841\pi\)
0.258291 + 0.966067i \(0.416841\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 452.611i − 1.06497i
\(426\) 0 0
\(427\) 500.265 1.17158
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 750.994i 1.74245i 0.490888 + 0.871223i \(0.336672\pi\)
−0.490888 + 0.871223i \(0.663328\pi\)
\(432\) 0 0
\(433\) −176.133 −0.406773 −0.203387 0.979098i \(-0.565195\pi\)
−0.203387 + 0.979098i \(0.565195\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19.0152i 0.0435130i
\(438\) 0 0
\(439\) 417.788 0.951681 0.475841 0.879531i \(-0.342144\pi\)
0.475841 + 0.879531i \(0.342144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 61.0471i − 0.137804i −0.997623 0.0689020i \(-0.978050\pi\)
0.997623 0.0689020i \(-0.0219496\pi\)
\(444\) 0 0
\(445\) 81.8573 0.183949
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 182.905i − 0.407360i −0.979038 0.203680i \(-0.934710\pi\)
0.979038 0.203680i \(-0.0652902\pi\)
\(450\) 0 0
\(451\) 449.134 0.995863
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 167.474i 0.368075i
\(456\) 0 0
\(457\) 443.600 0.970678 0.485339 0.874326i \(-0.338696\pi\)
0.485339 + 0.874326i \(0.338696\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 728.241i 1.57970i 0.613301 + 0.789850i \(0.289841\pi\)
−0.613301 + 0.789850i \(0.710159\pi\)
\(462\) 0 0
\(463\) 107.722 0.232662 0.116331 0.993211i \(-0.462887\pi\)
0.116331 + 0.993211i \(0.462887\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 820.248i − 1.75642i −0.478276 0.878210i \(-0.658738\pi\)
0.478276 0.878210i \(-0.341262\pi\)
\(468\) 0 0
\(469\) 500.265 1.06666
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 855.693i 1.80908i
\(474\) 0 0
\(475\) −406.398 −0.855575
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 120.790i 0.252171i 0.992019 + 0.126085i \(0.0402413\pi\)
−0.992019 + 0.126085i \(0.959759\pi\)
\(480\) 0 0
\(481\) −368.222 −0.765534
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 5.20766i − 0.0107374i
\(486\) 0 0
\(487\) −128.234 −0.263314 −0.131657 0.991295i \(-0.542030\pi\)
−0.131657 + 0.991295i \(0.542030\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 382.144i 0.778298i 0.921175 + 0.389149i \(0.127231\pi\)
−0.921175 + 0.389149i \(0.872769\pi\)
\(492\) 0 0
\(493\) −559.555 −1.13500
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 710.123i − 1.42882i
\(498\) 0 0
\(499\) −554.995 −1.11221 −0.556107 0.831111i \(-0.687705\pi\)
−0.556107 + 0.831111i \(0.687705\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 78.3943i − 0.155854i −0.996959 0.0779268i \(-0.975170\pi\)
0.996959 0.0779268i \(-0.0248300\pi\)
\(504\) 0 0
\(505\) −113.556 −0.224863
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 514.941i − 1.01167i −0.862630 0.505836i \(-0.831184\pi\)
0.862630 0.505836i \(-0.168816\pi\)
\(510\) 0 0
\(511\) −271.778 −0.531855
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 112.646i − 0.218731i
\(516\) 0 0
\(517\) 1055.26 2.04112
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 489.695i − 0.939914i −0.882689 0.469957i \(-0.844270\pi\)
0.882689 0.469957i \(-0.155730\pi\)
\(522\) 0 0
\(523\) 774.165 1.48024 0.740119 0.672476i \(-0.234769\pi\)
0.740119 + 0.672476i \(0.234769\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 106.931i 0.202905i
\(528\) 0 0
\(529\) 527.755 0.997647
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 593.415i 1.11335i
\(534\) 0 0
\(535\) 70.3112 0.131423
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 49.0551i 0.0910112i
\(540\) 0 0
\(541\) 590.045 1.09066 0.545328 0.838223i \(-0.316405\pi\)
0.545328 + 0.838223i \(0.316405\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 3.61350i − 0.00663027i
\(546\) 0 0
\(547\) −707.189 −1.29285 −0.646425 0.762977i \(-0.723737\pi\)
−0.646425 + 0.762977i \(0.723737\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 502.423i 0.911838i
\(552\) 0 0
\(553\) 918.133 1.66028
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 75.0816i 0.134796i 0.997726 + 0.0673982i \(0.0214698\pi\)
−0.997726 + 0.0673982i \(0.978530\pi\)
\(558\) 0 0
\(559\) −1130.58 −2.02250
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 114.462i − 0.203307i −0.994820 0.101653i \(-0.967587\pi\)
0.994820 0.101653i \(-0.0324133\pi\)
\(564\) 0 0
\(565\) 90.5058 0.160187
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 45.8199i − 0.0805270i −0.999189 0.0402635i \(-0.987180\pi\)
0.999189 0.0402635i \(-0.0128197\pi\)
\(570\) 0 0
\(571\) −830.093 −1.45375 −0.726877 0.686768i \(-0.759029\pi\)
−0.726877 + 0.686768i \(0.759029\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 26.6025i − 0.0462652i
\(576\) 0 0
\(577\) −69.3776 −0.120239 −0.0601193 0.998191i \(-0.519148\pi\)
−0.0601193 + 0.998191i \(0.519148\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 55.8991i 0.0962120i
\(582\) 0 0
\(583\) −587.489 −1.00770
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 428.655i 0.730248i 0.930959 + 0.365124i \(0.118973\pi\)
−0.930959 + 0.365124i \(0.881027\pi\)
\(588\) 0 0
\(589\) 96.0127 0.163010
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 77.0276i − 0.129895i −0.997889 0.0649474i \(-0.979312\pi\)
0.997889 0.0649474i \(-0.0206880\pi\)
\(594\) 0 0
\(595\) −147.144 −0.247301
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 963.566i 1.60862i 0.594207 + 0.804312i \(0.297466\pi\)
−0.594207 + 0.804312i \(0.702534\pi\)
\(600\) 0 0
\(601\) −840.133 −1.39789 −0.698946 0.715175i \(-0.746347\pi\)
−0.698946 + 0.715175i \(0.746347\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 157.354i 0.260089i
\(606\) 0 0
\(607\) −156.611 −0.258008 −0.129004 0.991644i \(-0.541178\pi\)
−0.129004 + 0.991644i \(0.541178\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1394.25i 2.28192i
\(612\) 0 0
\(613\) −426.094 −0.695096 −0.347548 0.937662i \(-0.612986\pi\)
−0.347548 + 0.937662i \(0.612986\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 963.520i − 1.56162i −0.624769 0.780810i \(-0.714807\pi\)
0.624769 0.780810i \(-0.285193\pi\)
\(618\) 0 0
\(619\) 175.235 0.283093 0.141547 0.989932i \(-0.454792\pi\)
0.141547 + 0.989932i \(0.454792\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 549.107i 0.881392i
\(624\) 0 0
\(625\) 539.666 0.863466
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 323.522i − 0.514344i
\(630\) 0 0
\(631\) −1140.72 −1.80780 −0.903900 0.427744i \(-0.859308\pi\)
−0.903900 + 0.427744i \(0.859308\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 48.9357i − 0.0770641i
\(636\) 0 0
\(637\) −64.8136 −0.101748
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 351.730i 0.548721i 0.961627 + 0.274360i \(0.0884661\pi\)
−0.961627 + 0.274360i \(0.911534\pi\)
\(642\) 0 0
\(643\) 86.4181 0.134398 0.0671991 0.997740i \(-0.478594\pi\)
0.0671991 + 0.997740i \(0.478594\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 430.723i 0.665723i 0.942976 + 0.332861i \(0.108014\pi\)
−0.942976 + 0.332861i \(0.891986\pi\)
\(648\) 0 0
\(649\) 928.888 1.43126
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 886.670i − 1.35784i −0.734212 0.678920i \(-0.762448\pi\)
0.734212 0.678920i \(-0.237552\pi\)
\(654\) 0 0
\(655\) 138.799 0.211907
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 352.833i 0.535407i 0.963501 + 0.267704i \(0.0862648\pi\)
−0.963501 + 0.267704i \(0.913735\pi\)
\(660\) 0 0
\(661\) −1206.26 −1.82489 −0.912447 0.409195i \(-0.865810\pi\)
−0.912447 + 0.409195i \(0.865810\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 132.120i 0.198677i
\(666\) 0 0
\(667\) −32.8882 −0.0493076
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1134.39i 1.69059i
\(672\) 0 0
\(673\) −956.133 −1.42070 −0.710351 0.703847i \(-0.751464\pi\)
−0.710351 + 0.703847i \(0.751464\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 420.557i 0.621207i 0.950540 + 0.310604i \(0.100531\pi\)
−0.950540 + 0.310604i \(0.899469\pi\)
\(678\) 0 0
\(679\) 34.9335 0.0514485
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 539.725i − 0.790227i −0.918632 0.395113i \(-0.870705\pi\)
0.918632 0.395113i \(-0.129295\pi\)
\(684\) 0 0
\(685\) −11.7567 −0.0171631
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 776.214i − 1.12658i
\(690\) 0 0
\(691\) 432.090 0.625312 0.312656 0.949866i \(-0.398781\pi\)
0.312656 + 0.949866i \(0.398781\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.28926i 0.00185504i
\(696\) 0 0
\(697\) −521.378 −0.748031
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 880.339i 1.25583i 0.778280 + 0.627917i \(0.216092\pi\)
−0.778280 + 0.627917i \(0.783908\pi\)
\(702\) 0 0
\(703\) −290.489 −0.413214
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 761.745i − 1.07743i
\(708\) 0 0
\(709\) −799.367 −1.12746 −0.563729 0.825960i \(-0.690634\pi\)
−0.563729 + 0.825960i \(0.690634\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.28491i 0.00881474i
\(714\) 0 0
\(715\) −379.760 −0.531133
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 258.786i 0.359924i 0.983674 + 0.179962i \(0.0575975\pi\)
−0.983674 + 0.179962i \(0.942402\pi\)
\(720\) 0 0
\(721\) 755.643 1.04805
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 702.896i − 0.969511i
\(726\) 0 0
\(727\) −663.211 −0.912257 −0.456129 0.889914i \(-0.650764\pi\)
−0.456129 + 0.889914i \(0.650764\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 993.332i − 1.35887i
\(732\) 0 0
\(733\) −908.826 −1.23987 −0.619936 0.784653i \(-0.712841\pi\)
−0.619936 + 0.784653i \(0.712841\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1134.39i 1.53920i
\(738\) 0 0
\(739\) 1112.39 1.50526 0.752631 0.658443i \(-0.228785\pi\)
0.752631 + 0.658443i \(0.228785\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 541.298i 0.728530i 0.931295 + 0.364265i \(0.118680\pi\)
−0.931295 + 0.364265i \(0.881320\pi\)
\(744\) 0 0
\(745\) −32.1767 −0.0431902
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 471.655i 0.629713i
\(750\) 0 0
\(751\) 763.699 1.01691 0.508455 0.861089i \(-0.330217\pi\)
0.508455 + 0.861089i \(0.330217\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 66.1354i − 0.0875966i
\(756\) 0 0
\(757\) 419.134 0.553678 0.276839 0.960916i \(-0.410713\pi\)
0.276839 + 0.960916i \(0.410713\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1330.24i 1.74802i 0.485912 + 0.874008i \(0.338488\pi\)
−0.485912 + 0.874008i \(0.661512\pi\)
\(762\) 0 0
\(763\) 24.2397 0.0317690
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1227.29i 1.60011i
\(768\) 0 0
\(769\) 671.511 0.873226 0.436613 0.899649i \(-0.356178\pi\)
0.436613 + 0.899649i \(0.356178\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1421.13i − 1.83846i −0.393724 0.919229i \(-0.628813\pi\)
0.393724 0.919229i \(-0.371187\pi\)
\(774\) 0 0
\(775\) −134.323 −0.173320
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 468.143i 0.600954i
\(780\) 0 0
\(781\) 1610.25 2.06179
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 147.863i 0.188361i
\(786\) 0 0
\(787\) 649.808 0.825677 0.412839 0.910804i \(-0.364537\pi\)
0.412839 + 0.910804i \(0.364537\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 607.122i 0.767538i
\(792\) 0 0
\(793\) −1498.80 −1.89004
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 422.946i − 0.530672i −0.964156 0.265336i \(-0.914517\pi\)
0.964156 0.265336i \(-0.0854830\pi\)
\(798\) 0 0
\(799\) −1225.00 −1.53317
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 616.276i − 0.767467i
\(804\) 0 0
\(805\) −8.64847 −0.0107434
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1353.87i − 1.67351i −0.547574 0.836757i \(-0.684448\pi\)
0.547574 0.836757i \(-0.315552\pi\)
\(810\) 0 0
\(811\) −842.340 −1.03864 −0.519322 0.854579i \(-0.673815\pi\)
−0.519322 + 0.854579i \(0.673815\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 205.407i 0.252033i
\(816\) 0 0
\(817\) −891.909 −1.09169
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 616.897i − 0.751397i −0.926742 0.375698i \(-0.877403\pi\)
0.926742 0.375698i \(-0.122597\pi\)
\(822\) 0 0
\(823\) 980.500 1.19137 0.595686 0.803217i \(-0.296880\pi\)
0.595686 + 0.803217i \(0.296880\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 70.4951i 0.0852419i 0.999091 + 0.0426210i \(0.0135708\pi\)
−0.999091 + 0.0426210i \(0.986429\pi\)
\(828\) 0 0
\(829\) −515.147 −0.621408 −0.310704 0.950507i \(-0.600565\pi\)
−0.310704 + 0.950507i \(0.600565\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 56.9456i − 0.0683621i
\(834\) 0 0
\(835\) −250.606 −0.300127
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 47.1556i − 0.0562045i −0.999605 0.0281022i \(-0.991054\pi\)
0.999605 0.0281022i \(-0.00894640\pi\)
\(840\) 0 0
\(841\) −27.9773 −0.0332667
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 320.082i − 0.378795i
\(846\) 0 0
\(847\) −1055.54 −1.24622
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 19.0152i − 0.0223445i
\(852\) 0 0
\(853\) 763.118 0.894628 0.447314 0.894377i \(-0.352381\pi\)
0.447314 + 0.894377i \(0.352381\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1641.05i 1.91488i 0.288631 + 0.957440i \(0.406800\pi\)
−0.288631 + 0.957440i \(0.593200\pi\)
\(858\) 0 0
\(859\) 1366.84 1.59120 0.795602 0.605819i \(-0.207155\pi\)
0.795602 + 0.605819i \(0.207155\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 836.082i 0.968809i 0.874844 + 0.484405i \(0.160964\pi\)
−0.874844 + 0.484405i \(0.839036\pi\)
\(864\) 0 0
\(865\) −348.089 −0.402415
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2081.93i 2.39578i
\(870\) 0 0
\(871\) −1498.80 −1.72078
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 378.633i − 0.432723i
\(876\) 0 0
\(877\) −158.191 −0.180377 −0.0901887 0.995925i \(-0.528747\pi\)
−0.0901887 + 0.995925i \(0.528747\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 237.430i 0.269500i 0.990880 + 0.134750i \(0.0430232\pi\)
−0.990880 + 0.134750i \(0.956977\pi\)
\(882\) 0 0
\(883\) −592.933 −0.671499 −0.335749 0.941951i \(-0.608989\pi\)
−0.335749 + 0.941951i \(0.608989\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1420.72i 1.60171i 0.598858 + 0.800855i \(0.295621\pi\)
−0.598858 + 0.800855i \(0.704379\pi\)
\(888\) 0 0
\(889\) 328.266 0.369253
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1099.92i 1.23172i
\(894\) 0 0
\(895\) −199.866 −0.223313
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 166.061i 0.184717i
\(900\) 0 0
\(901\) 681.987 0.756922
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 135.734i − 0.149982i
\(906\) 0 0
\(907\) 816.648 0.900383 0.450192 0.892932i \(-0.351356\pi\)
0.450192 + 0.892932i \(0.351356\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 469.895i − 0.515801i −0.966171 0.257901i \(-0.916969\pi\)
0.966171 0.257901i \(-0.0830307\pi\)
\(912\) 0 0
\(913\) −126.755 −0.138834
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 931.079i 1.01535i
\(918\) 0 0
\(919\) −1648.21 −1.79348 −0.896741 0.442555i \(-0.854072\pi\)
−0.896741 + 0.442555i \(0.854072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2127.53i 2.30502i
\(924\) 0 0
\(925\) 406.398 0.439349
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 421.311i 0.453510i 0.973952 + 0.226755i \(0.0728116\pi\)
−0.973952 + 0.226755i \(0.927188\pi\)
\(930\) 0 0
\(931\) −51.1313 −0.0549208
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 333.659i − 0.356855i
\(936\) 0 0
\(937\) 1434.27 1.53070 0.765350 0.643614i \(-0.222566\pi\)
0.765350 + 0.643614i \(0.222566\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 732.876i − 0.778827i −0.921063 0.389413i \(-0.872678\pi\)
0.921063 0.389413i \(-0.127322\pi\)
\(942\) 0 0
\(943\) −30.6443 −0.0324966
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1385.53i 1.46308i 0.681800 + 0.731538i \(0.261197\pi\)
−0.681800 + 0.731538i \(0.738803\pi\)
\(948\) 0 0
\(949\) 814.249 0.858007
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 957.328i 1.00454i 0.864711 + 0.502270i \(0.167502\pi\)
−0.864711 + 0.502270i \(0.832498\pi\)
\(954\) 0 0
\(955\) −250.606 −0.262414
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 78.8654i − 0.0822371i
\(960\) 0 0
\(961\) −929.266 −0.966978
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 126.179i 0.130755i
\(966\) 0 0
\(967\) 1225.90 1.26773 0.633867 0.773442i \(-0.281467\pi\)
0.633867 + 0.773442i \(0.281467\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 752.906i − 0.775393i −0.921787 0.387696i \(-0.873271\pi\)
0.921787 0.387696i \(-0.126729\pi\)
\(972\) 0 0
\(973\) −8.64847 −0.00888845
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 642.932i − 0.658068i −0.944318 0.329034i \(-0.893277\pi\)
0.944318 0.329034i \(-0.106723\pi\)
\(978\) 0 0
\(979\) −1245.14 −1.27185
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 161.251i 0.164040i 0.996631 + 0.0820200i \(0.0261371\pi\)
−0.996631 + 0.0820200i \(0.973863\pi\)
\(984\) 0 0
\(985\) 318.532 0.323382
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 58.3836i − 0.0590330i
\(990\) 0 0
\(991\) −811.500 −0.818870 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 250.483i − 0.251742i
\(996\) 0 0
\(997\) −390.554 −0.391729 −0.195864 0.980631i \(-0.562751\pi\)
−0.195864 + 0.980631i \(0.562751\pi\)
\(998\) 0 0
\(999\) 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 2304.3.e.o.1025.3 8
3.2 odd 2 inner 2304.3.e.o.1025.5 8
4.3 odd 2 2304.3.e.n.1025.3 8
8.3 odd 2 2304.3.e.n.1025.6 8
8.5 even 2 inner 2304.3.e.o.1025.6 8
12.11 even 2 2304.3.e.n.1025.5 8
16.3 odd 4 72.3.h.a.53.8 yes 8
16.5 even 4 288.3.h.a.17.6 8
16.11 odd 4 72.3.h.a.53.2 yes 8
16.13 even 4 288.3.h.a.17.3 8
24.5 odd 2 inner 2304.3.e.o.1025.4 8
24.11 even 2 2304.3.e.n.1025.4 8
48.5 odd 4 288.3.h.a.17.4 8
48.11 even 4 72.3.h.a.53.7 yes 8
48.29 odd 4 288.3.h.a.17.5 8
48.35 even 4 72.3.h.a.53.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.h.a.53.1 8 48.35 even 4
72.3.h.a.53.2 yes 8 16.11 odd 4
72.3.h.a.53.7 yes 8 48.11 even 4
72.3.h.a.53.8 yes 8 16.3 odd 4
288.3.h.a.17.3 8 16.13 even 4
288.3.h.a.17.4 8 48.5 odd 4
288.3.h.a.17.5 8 48.29 odd 4
288.3.h.a.17.6 8 16.5 even 4
2304.3.e.n.1025.3 8 4.3 odd 2
2304.3.e.n.1025.4 8 24.11 even 2
2304.3.e.n.1025.5 8 12.11 even 2
2304.3.e.n.1025.6 8 8.3 odd 2
2304.3.e.o.1025.3 8 1.1 even 1 trivial
2304.3.e.o.1025.4 8 24.5 odd 2 inner
2304.3.e.o.1025.5 8 3.2 odd 2 inner
2304.3.e.o.1025.6 8 8.5 even 2 inner