Properties

Label 9408.2.a.ei.1.2
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.454904 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.454904 q^{5} +1.00000 q^{9} +5.79306 q^{11} -5.88325 q^{13} +0.454904 q^{15} -2.90981 q^{17} +5.88325 q^{19} -2.90981 q^{23} -4.79306 q^{25} +1.00000 q^{27} -3.54510 q^{29} +4.33816 q^{31} +5.79306 q^{33} -7.70287 q^{37} -5.88325 q^{39} -9.58612 q^{41} -10.7931 q^{43} +0.454904 q^{45} -4.90981 q^{47} -2.90981 q^{51} -13.1312 q^{53} +2.63529 q^{55} +5.88325 q^{57} -1.79306 q^{59} +4.67632 q^{61} -2.67632 q^{65} -7.88325 q^{67} -2.90981 q^{69} -0.909808 q^{71} -5.20694 q^{73} -4.79306 q^{75} -2.75203 q^{79} +1.00000 q^{81} +9.97345 q^{83} -1.32368 q^{85} -3.54510 q^{87} +4.90981 q^{89} +4.33816 q^{93} +2.67632 q^{95} +5.79306 q^{97} +5.79306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{9} - 3q^{13} - 6q^{17} + 3q^{19} - 6q^{23} + 3q^{25} + 3q^{27} - 12q^{29} - 3q^{31} - 3q^{37} - 3q^{39} + 6q^{41} - 15q^{43} - 12q^{47} - 6q^{51} - 6q^{53} + 12q^{55} + 3q^{57} + 12q^{59} - 18q^{61} + 24q^{65} - 9q^{67} - 6q^{69} - 33q^{73} + 3q^{75} - 27q^{79} + 3q^{81} + 18q^{83} - 36q^{85} - 12q^{87} + 12q^{89} - 3q^{93} - 24q^{95} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.454904 0.203439 0.101720 0.994813i \(-0.467566\pi\)
0.101720 + 0.994813i \(0.467566\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.79306 1.74667 0.873337 0.487117i \(-0.161951\pi\)
0.873337 + 0.487117i \(0.161951\pi\)
\(12\) 0 0
\(13\) −5.88325 −1.63172 −0.815861 0.578249i \(-0.803736\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(14\) 0 0
\(15\) 0.454904 0.117456
\(16\) 0 0
\(17\) −2.90981 −0.705732 −0.352866 0.935674i \(-0.614793\pi\)
−0.352866 + 0.935674i \(0.614793\pi\)
\(18\) 0 0
\(19\) 5.88325 1.34971 0.674856 0.737950i \(-0.264206\pi\)
0.674856 + 0.737950i \(0.264206\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.90981 −0.606737 −0.303368 0.952873i \(-0.598111\pi\)
−0.303368 + 0.952873i \(0.598111\pi\)
\(24\) 0 0
\(25\) −4.79306 −0.958612
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.54510 −0.658308 −0.329154 0.944276i \(-0.606763\pi\)
−0.329154 + 0.944276i \(0.606763\pi\)
\(30\) 0 0
\(31\) 4.33816 0.779156 0.389578 0.920993i \(-0.372621\pi\)
0.389578 + 0.920993i \(0.372621\pi\)
\(32\) 0 0
\(33\) 5.79306 1.00844
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.70287 −1.26634 −0.633172 0.774011i \(-0.718247\pi\)
−0.633172 + 0.774011i \(0.718247\pi\)
\(38\) 0 0
\(39\) −5.88325 −0.942075
\(40\) 0 0
\(41\) −9.58612 −1.49710 −0.748551 0.663078i \(-0.769250\pi\)
−0.748551 + 0.663078i \(0.769250\pi\)
\(42\) 0 0
\(43\) −10.7931 −1.64593 −0.822963 0.568095i \(-0.807681\pi\)
−0.822963 + 0.568095i \(0.807681\pi\)
\(44\) 0 0
\(45\) 0.454904 0.0678131
\(46\) 0 0
\(47\) −4.90981 −0.716169 −0.358085 0.933689i \(-0.616570\pi\)
−0.358085 + 0.933689i \(0.616570\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.90981 −0.407455
\(52\) 0 0
\(53\) −13.1312 −1.80371 −0.901856 0.432037i \(-0.857795\pi\)
−0.901856 + 0.432037i \(0.857795\pi\)
\(54\) 0 0
\(55\) 2.63529 0.355342
\(56\) 0 0
\(57\) 5.88325 0.779256
\(58\) 0 0
\(59\) −1.79306 −0.233437 −0.116718 0.993165i \(-0.537238\pi\)
−0.116718 + 0.993165i \(0.537238\pi\)
\(60\) 0 0
\(61\) 4.67632 0.598741 0.299370 0.954137i \(-0.403223\pi\)
0.299370 + 0.954137i \(0.403223\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.67632 −0.331956
\(66\) 0 0
\(67\) −7.88325 −0.963093 −0.481546 0.876421i \(-0.659925\pi\)
−0.481546 + 0.876421i \(0.659925\pi\)
\(68\) 0 0
\(69\) −2.90981 −0.350300
\(70\) 0 0
\(71\) −0.909808 −0.107974 −0.0539872 0.998542i \(-0.517193\pi\)
−0.0539872 + 0.998542i \(0.517193\pi\)
\(72\) 0 0
\(73\) −5.20694 −0.609426 −0.304713 0.952444i \(-0.598561\pi\)
−0.304713 + 0.952444i \(0.598561\pi\)
\(74\) 0 0
\(75\) −4.79306 −0.553455
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.75203 −0.309628 −0.154814 0.987944i \(-0.549478\pi\)
−0.154814 + 0.987944i \(0.549478\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.97345 1.09473 0.547364 0.836895i \(-0.315631\pi\)
0.547364 + 0.836895i \(0.315631\pi\)
\(84\) 0 0
\(85\) −1.32368 −0.143574
\(86\) 0 0
\(87\) −3.54510 −0.380074
\(88\) 0 0
\(89\) 4.90981 0.520439 0.260219 0.965550i \(-0.416205\pi\)
0.260219 + 0.965550i \(0.416205\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.33816 0.449846
\(94\) 0 0
\(95\) 2.67632 0.274584
\(96\) 0 0
\(97\) 5.79306 0.588196 0.294098 0.955775i \(-0.404981\pi\)
0.294098 + 0.955775i \(0.404981\pi\)
\(98\) 0 0
\(99\) 5.79306 0.582225
\(100\) 0 0
\(101\) 16.6763 1.65936 0.829678 0.558243i \(-0.188524\pi\)
0.829678 + 0.558243i \(0.188524\pi\)
\(102\) 0 0
\(103\) 0.793062 0.0781427 0.0390714 0.999236i \(-0.487560\pi\)
0.0390714 + 0.999236i \(0.487560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.9734 −1.15752 −0.578758 0.815499i \(-0.696463\pi\)
−0.578758 + 0.815499i \(0.696463\pi\)
\(108\) 0 0
\(109\) 4.61268 0.441814 0.220907 0.975295i \(-0.429098\pi\)
0.220907 + 0.975295i \(0.429098\pi\)
\(110\) 0 0
\(111\) −7.70287 −0.731124
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −1.32368 −0.123434
\(116\) 0 0
\(117\) −5.88325 −0.543907
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 22.5596 2.05087
\(122\) 0 0
\(123\) −9.58612 −0.864352
\(124\) 0 0
\(125\) −4.45490 −0.398459
\(126\) 0 0
\(127\) −1.24797 −0.110739 −0.0553696 0.998466i \(-0.517634\pi\)
−0.0553696 + 0.998466i \(0.517634\pi\)
\(128\) 0 0
\(129\) −10.7931 −0.950276
\(130\) 0 0
\(131\) −5.79306 −0.506142 −0.253071 0.967448i \(-0.581441\pi\)
−0.253071 + 0.967448i \(0.581441\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.454904 0.0391519
\(136\) 0 0
\(137\) −3.09019 −0.264013 −0.132006 0.991249i \(-0.542142\pi\)
−0.132006 + 0.991249i \(0.542142\pi\)
\(138\) 0 0
\(139\) −5.70287 −0.483711 −0.241856 0.970312i \(-0.577756\pi\)
−0.241856 + 0.970312i \(0.577756\pi\)
\(140\) 0 0
\(141\) −4.90981 −0.413480
\(142\) 0 0
\(143\) −34.0821 −2.85008
\(144\) 0 0
\(145\) −1.61268 −0.133926
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.09019 −0.253158 −0.126579 0.991957i \(-0.540400\pi\)
−0.126579 + 0.991957i \(0.540400\pi\)
\(150\) 0 0
\(151\) −1.72548 −0.140418 −0.0702088 0.997532i \(-0.522367\pi\)
−0.0702088 + 0.997532i \(0.522367\pi\)
\(152\) 0 0
\(153\) −2.90981 −0.235244
\(154\) 0 0
\(155\) 1.97345 0.158511
\(156\) 0 0
\(157\) −5.58612 −0.445821 −0.222911 0.974839i \(-0.571556\pi\)
−0.222911 + 0.974839i \(0.571556\pi\)
\(158\) 0 0
\(159\) −13.1312 −1.04137
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.85670 0.693710 0.346855 0.937919i \(-0.387250\pi\)
0.346855 + 0.937919i \(0.387250\pi\)
\(164\) 0 0
\(165\) 2.63529 0.205157
\(166\) 0 0
\(167\) 17.4057 1.34690 0.673448 0.739234i \(-0.264812\pi\)
0.673448 + 0.739234i \(0.264812\pi\)
\(168\) 0 0
\(169\) 21.6127 1.66251
\(170\) 0 0
\(171\) 5.88325 0.449904
\(172\) 0 0
\(173\) −16.6763 −1.26788 −0.633938 0.773384i \(-0.718563\pi\)
−0.633938 + 0.773384i \(0.718563\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.79306 −0.134775
\(178\) 0 0
\(179\) −6.18038 −0.461944 −0.230972 0.972960i \(-0.574191\pi\)
−0.230972 + 0.972960i \(0.574191\pi\)
\(180\) 0 0
\(181\) −3.20694 −0.238370 −0.119185 0.992872i \(-0.538028\pi\)
−0.119185 + 0.992872i \(0.538028\pi\)
\(182\) 0 0
\(183\) 4.67632 0.345683
\(184\) 0 0
\(185\) −3.50407 −0.257624
\(186\) 0 0
\(187\) −16.8567 −1.23268
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −21.3526 −1.54502 −0.772511 0.635001i \(-0.781000\pi\)
−0.772511 + 0.635001i \(0.781000\pi\)
\(192\) 0 0
\(193\) 16.4057 1.18091 0.590456 0.807070i \(-0.298948\pi\)
0.590456 + 0.807070i \(0.298948\pi\)
\(194\) 0 0
\(195\) −2.67632 −0.191655
\(196\) 0 0
\(197\) −24.6763 −1.75811 −0.879057 0.476716i \(-0.841827\pi\)
−0.879057 + 0.476716i \(0.841827\pi\)
\(198\) 0 0
\(199\) 13.8196 0.979647 0.489823 0.871822i \(-0.337061\pi\)
0.489823 + 0.871822i \(0.337061\pi\)
\(200\) 0 0
\(201\) −7.88325 −0.556042
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.36077 −0.304569
\(206\) 0 0
\(207\) −2.90981 −0.202246
\(208\) 0 0
\(209\) 34.0821 2.35751
\(210\) 0 0
\(211\) 11.5861 0.797622 0.398811 0.917033i \(-0.369423\pi\)
0.398811 + 0.917033i \(0.369423\pi\)
\(212\) 0 0
\(213\) −0.909808 −0.0623390
\(214\) 0 0
\(215\) −4.90981 −0.334846
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.20694 −0.351852
\(220\) 0 0
\(221\) 17.1191 1.15156
\(222\) 0 0
\(223\) 20.4549 1.36976 0.684881 0.728655i \(-0.259854\pi\)
0.684881 + 0.728655i \(0.259854\pi\)
\(224\) 0 0
\(225\) −4.79306 −0.319537
\(226\) 0 0
\(227\) −17.7400 −1.17744 −0.588721 0.808336i \(-0.700368\pi\)
−0.588721 + 0.808336i \(0.700368\pi\)
\(228\) 0 0
\(229\) −24.1988 −1.59910 −0.799551 0.600598i \(-0.794929\pi\)
−0.799551 + 0.600598i \(0.794929\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.6763 1.48557 0.742787 0.669527i \(-0.233503\pi\)
0.742787 + 0.669527i \(0.233503\pi\)
\(234\) 0 0
\(235\) −2.23349 −0.145697
\(236\) 0 0
\(237\) −2.75203 −0.178764
\(238\) 0 0
\(239\) −2.90981 −0.188220 −0.0941099 0.995562i \(-0.530001\pi\)
−0.0941099 + 0.995562i \(0.530001\pi\)
\(240\) 0 0
\(241\) −21.7400 −1.40039 −0.700197 0.713950i \(-0.746904\pi\)
−0.700197 + 0.713950i \(0.746904\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −34.6127 −2.20235
\(248\) 0 0
\(249\) 9.97345 0.632041
\(250\) 0 0
\(251\) 8.02655 0.506632 0.253316 0.967384i \(-0.418479\pi\)
0.253316 + 0.967384i \(0.418479\pi\)
\(252\) 0 0
\(253\) −16.8567 −1.05977
\(254\) 0 0
\(255\) −1.32368 −0.0828923
\(256\) 0 0
\(257\) 12.9098 0.805292 0.402646 0.915356i \(-0.368091\pi\)
0.402646 + 0.915356i \(0.368091\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.54510 −0.219436
\(262\) 0 0
\(263\) −8.90981 −0.549402 −0.274701 0.961530i \(-0.588579\pi\)
−0.274701 + 0.961530i \(0.588579\pi\)
\(264\) 0 0
\(265\) −5.97345 −0.366946
\(266\) 0 0
\(267\) 4.90981 0.300475
\(268\) 0 0
\(269\) −6.04103 −0.368328 −0.184164 0.982896i \(-0.558958\pi\)
−0.184164 + 0.982896i \(0.558958\pi\)
\(270\) 0 0
\(271\) 13.3647 0.811848 0.405924 0.913907i \(-0.366950\pi\)
0.405924 + 0.913907i \(0.366950\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −27.7665 −1.67438
\(276\) 0 0
\(277\) −4.79306 −0.287987 −0.143994 0.989579i \(-0.545994\pi\)
−0.143994 + 0.989579i \(0.545994\pi\)
\(278\) 0 0
\(279\) 4.33816 0.259719
\(280\) 0 0
\(281\) 7.32368 0.436894 0.218447 0.975849i \(-0.429901\pi\)
0.218447 + 0.975849i \(0.429901\pi\)
\(282\) 0 0
\(283\) −13.2069 −0.785071 −0.392535 0.919737i \(-0.628402\pi\)
−0.392535 + 0.919737i \(0.628402\pi\)
\(284\) 0 0
\(285\) 2.67632 0.158531
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.53302 −0.501942
\(290\) 0 0
\(291\) 5.79306 0.339595
\(292\) 0 0
\(293\) 7.54510 0.440789 0.220395 0.975411i \(-0.429265\pi\)
0.220395 + 0.975411i \(0.429265\pi\)
\(294\) 0 0
\(295\) −0.815671 −0.0474902
\(296\) 0 0
\(297\) 5.79306 0.336148
\(298\) 0 0
\(299\) 17.1191 0.990025
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 16.6763 0.958029
\(304\) 0 0
\(305\) 2.12728 0.121807
\(306\) 0 0
\(307\) 23.1086 1.31888 0.659439 0.751758i \(-0.270794\pi\)
0.659439 + 0.751758i \(0.270794\pi\)
\(308\) 0 0
\(309\) 0.793062 0.0451157
\(310\) 0 0
\(311\) 7.40574 0.419941 0.209971 0.977708i \(-0.432663\pi\)
0.209971 + 0.977708i \(0.432663\pi\)
\(312\) 0 0
\(313\) −14.5861 −0.824457 −0.412228 0.911081i \(-0.635249\pi\)
−0.412228 + 0.911081i \(0.635249\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.36471 −0.0766499 −0.0383249 0.999265i \(-0.512202\pi\)
−0.0383249 + 0.999265i \(0.512202\pi\)
\(318\) 0 0
\(319\) −20.5370 −1.14985
\(320\) 0 0
\(321\) −11.9734 −0.668293
\(322\) 0 0
\(323\) −17.1191 −0.952534
\(324\) 0 0
\(325\) 28.1988 1.56419
\(326\) 0 0
\(327\) 4.61268 0.255082
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.9734 −0.603155 −0.301578 0.953442i \(-0.597513\pi\)
−0.301578 + 0.953442i \(0.597513\pi\)
\(332\) 0 0
\(333\) −7.70287 −0.422115
\(334\) 0 0
\(335\) −3.58612 −0.195931
\(336\) 0 0
\(337\) −22.1722 −1.20780 −0.603900 0.797060i \(-0.706387\pi\)
−0.603900 + 0.797060i \(0.706387\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 25.1312 1.36093
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.32368 −0.0712647
\(346\) 0 0
\(347\) −0.233492 −0.0125345 −0.00626725 0.999980i \(-0.501995\pi\)
−0.00626725 + 0.999980i \(0.501995\pi\)
\(348\) 0 0
\(349\) 7.35263 0.393577 0.196789 0.980446i \(-0.436949\pi\)
0.196789 + 0.980446i \(0.436949\pi\)
\(350\) 0 0
\(351\) −5.88325 −0.314025
\(352\) 0 0
\(353\) −19.1722 −1.02044 −0.510218 0.860045i \(-0.670435\pi\)
−0.510218 + 0.860045i \(0.670435\pi\)
\(354\) 0 0
\(355\) −0.413875 −0.0219662
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.9388 1.10511 0.552553 0.833478i \(-0.313654\pi\)
0.552553 + 0.833478i \(0.313654\pi\)
\(360\) 0 0
\(361\) 15.6127 0.821720
\(362\) 0 0
\(363\) 22.5596 1.18407
\(364\) 0 0
\(365\) −2.36866 −0.123981
\(366\) 0 0
\(367\) 25.0145 1.30574 0.652872 0.757468i \(-0.273564\pi\)
0.652872 + 0.757468i \(0.273564\pi\)
\(368\) 0 0
\(369\) −9.58612 −0.499034
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 20.3261 1.05244 0.526222 0.850347i \(-0.323608\pi\)
0.526222 + 0.850347i \(0.323608\pi\)
\(374\) 0 0
\(375\) −4.45490 −0.230050
\(376\) 0 0
\(377\) 20.8567 1.07417
\(378\) 0 0
\(379\) −20.7931 −1.06807 −0.534034 0.845463i \(-0.679325\pi\)
−0.534034 + 0.845463i \(0.679325\pi\)
\(380\) 0 0
\(381\) −1.24797 −0.0639353
\(382\) 0 0
\(383\) 25.1722 1.28624 0.643121 0.765765i \(-0.277639\pi\)
0.643121 + 0.765765i \(0.277639\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.7931 −0.548642
\(388\) 0 0
\(389\) −24.4428 −1.23930 −0.619650 0.784878i \(-0.712726\pi\)
−0.619650 + 0.784878i \(0.712726\pi\)
\(390\) 0 0
\(391\) 8.46698 0.428194
\(392\) 0 0
\(393\) −5.79306 −0.292221
\(394\) 0 0
\(395\) −1.25191 −0.0629905
\(396\) 0 0
\(397\) −8.79306 −0.441311 −0.220656 0.975352i \(-0.570820\pi\)
−0.220656 + 0.975352i \(0.570820\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 33.1191 1.65389 0.826945 0.562282i \(-0.190076\pi\)
0.826945 + 0.562282i \(0.190076\pi\)
\(402\) 0 0
\(403\) −25.5225 −1.27137
\(404\) 0 0
\(405\) 0.454904 0.0226044
\(406\) 0 0
\(407\) −44.6232 −2.21189
\(408\) 0 0
\(409\) 6.94689 0.343502 0.171751 0.985140i \(-0.445058\pi\)
0.171751 + 0.985140i \(0.445058\pi\)
\(410\) 0 0
\(411\) −3.09019 −0.152428
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.53696 0.222711
\(416\) 0 0
\(417\) −5.70287 −0.279271
\(418\) 0 0
\(419\) 29.2254 1.42775 0.713876 0.700272i \(-0.246938\pi\)
0.713876 + 0.700272i \(0.246938\pi\)
\(420\) 0 0
\(421\) −33.2359 −1.61982 −0.809909 0.586556i \(-0.800484\pi\)
−0.809909 + 0.586556i \(0.800484\pi\)
\(422\) 0 0
\(423\) −4.90981 −0.238723
\(424\) 0 0
\(425\) 13.9469 0.676524
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −34.0821 −1.64550
\(430\) 0 0
\(431\) −6.18038 −0.297699 −0.148849 0.988860i \(-0.547557\pi\)
−0.148849 + 0.988860i \(0.547557\pi\)
\(432\) 0 0
\(433\) −16.6127 −0.798354 −0.399177 0.916874i \(-0.630704\pi\)
−0.399177 + 0.916874i \(0.630704\pi\)
\(434\) 0 0
\(435\) −1.61268 −0.0773220
\(436\) 0 0
\(437\) −17.1191 −0.818920
\(438\) 0 0
\(439\) −27.6272 −1.31857 −0.659286 0.751892i \(-0.729141\pi\)
−0.659286 + 0.751892i \(0.729141\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.0266 −0.951490 −0.475745 0.879583i \(-0.657822\pi\)
−0.475745 + 0.879583i \(0.657822\pi\)
\(444\) 0 0
\(445\) 2.23349 0.105878
\(446\) 0 0
\(447\) −3.09019 −0.146161
\(448\) 0 0
\(449\) 15.0371 0.709644 0.354822 0.934934i \(-0.384541\pi\)
0.354822 + 0.934934i \(0.384541\pi\)
\(450\) 0 0
\(451\) −55.5330 −2.61495
\(452\) 0 0
\(453\) −1.72548 −0.0810701
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −42.1191 −1.97025 −0.985125 0.171838i \(-0.945029\pi\)
−0.985125 + 0.171838i \(0.945029\pi\)
\(458\) 0 0
\(459\) −2.90981 −0.135818
\(460\) 0 0
\(461\) 20.9098 0.973867 0.486933 0.873439i \(-0.338115\pi\)
0.486933 + 0.873439i \(0.338115\pi\)
\(462\) 0 0
\(463\) 38.1457 1.77278 0.886390 0.462939i \(-0.153205\pi\)
0.886390 + 0.462939i \(0.153205\pi\)
\(464\) 0 0
\(465\) 1.97345 0.0915164
\(466\) 0 0
\(467\) −11.5330 −0.533684 −0.266842 0.963740i \(-0.585980\pi\)
−0.266842 + 0.963740i \(0.585980\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −5.58612 −0.257395
\(472\) 0 0
\(473\) −62.5249 −2.87490
\(474\) 0 0
\(475\) −28.1988 −1.29385
\(476\) 0 0
\(477\) −13.1312 −0.601237
\(478\) 0 0
\(479\) −7.32368 −0.334628 −0.167314 0.985904i \(-0.553509\pi\)
−0.167314 + 0.985904i \(0.553509\pi\)
\(480\) 0 0
\(481\) 45.3179 2.06632
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.63529 0.119662
\(486\) 0 0
\(487\) 26.0516 1.18051 0.590254 0.807217i \(-0.299027\pi\)
0.590254 + 0.807217i \(0.299027\pi\)
\(488\) 0 0
\(489\) 8.85670 0.400514
\(490\) 0 0
\(491\) 7.37919 0.333018 0.166509 0.986040i \(-0.446751\pi\)
0.166509 + 0.986040i \(0.446751\pi\)
\(492\) 0 0
\(493\) 10.3155 0.464589
\(494\) 0 0
\(495\) 2.63529 0.118447
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.350238 0.0156788 0.00783940 0.999969i \(-0.497505\pi\)
0.00783940 + 0.999969i \(0.497505\pi\)
\(500\) 0 0
\(501\) 17.4057 0.777631
\(502\) 0 0
\(503\) −14.9098 −0.664795 −0.332398 0.943139i \(-0.607858\pi\)
−0.332398 + 0.943139i \(0.607858\pi\)
\(504\) 0 0
\(505\) 7.58612 0.337578
\(506\) 0 0
\(507\) 21.6127 0.959853
\(508\) 0 0
\(509\) 16.8977 0.748979 0.374489 0.927231i \(-0.377818\pi\)
0.374489 + 0.927231i \(0.377818\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.88325 0.259752
\(514\) 0 0
\(515\) 0.360767 0.0158973
\(516\) 0 0
\(517\) −28.4428 −1.25091
\(518\) 0 0
\(519\) −16.6763 −0.732009
\(520\) 0 0
\(521\) −7.58612 −0.332354 −0.166177 0.986096i \(-0.553142\pi\)
−0.166177 + 0.986096i \(0.553142\pi\)
\(522\) 0 0
\(523\) −42.5144 −1.85902 −0.929511 0.368793i \(-0.879771\pi\)
−0.929511 + 0.368793i \(0.879771\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.6232 −0.549876
\(528\) 0 0
\(529\) −14.5330 −0.631870
\(530\) 0 0
\(531\) −1.79306 −0.0778123
\(532\) 0 0
\(533\) 56.3976 2.44285
\(534\) 0 0
\(535\) −5.44677 −0.235484
\(536\) 0 0
\(537\) −6.18038 −0.266703
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.0636 −0.604643 −0.302322 0.953206i \(-0.597762\pi\)
−0.302322 + 0.953206i \(0.597762\pi\)
\(542\) 0 0
\(543\) −3.20694 −0.137623
\(544\) 0 0
\(545\) 2.09833 0.0898824
\(546\) 0 0
\(547\) 4.49593 0.192232 0.0961161 0.995370i \(-0.469358\pi\)
0.0961161 + 0.995370i \(0.469358\pi\)
\(548\) 0 0
\(549\) 4.67632 0.199580
\(550\) 0 0
\(551\) −20.8567 −0.888525
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.50407 −0.148739
\(556\) 0 0
\(557\) −31.8606 −1.34998 −0.674989 0.737827i \(-0.735852\pi\)
−0.674989 + 0.737827i \(0.735852\pi\)
\(558\) 0 0
\(559\) 63.4983 2.68569
\(560\) 0 0
\(561\) −16.8567 −0.711690
\(562\) 0 0
\(563\) 46.1376 1.94447 0.972233 0.234014i \(-0.0751862\pi\)
0.972233 + 0.234014i \(0.0751862\pi\)
\(564\) 0 0
\(565\) 3.63923 0.153104
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.6232 −0.696881 −0.348441 0.937331i \(-0.613289\pi\)
−0.348441 + 0.937331i \(0.613289\pi\)
\(570\) 0 0
\(571\) −36.5065 −1.52775 −0.763874 0.645365i \(-0.776705\pi\)
−0.763874 + 0.645365i \(0.776705\pi\)
\(572\) 0 0
\(573\) −21.3526 −0.892019
\(574\) 0 0
\(575\) 13.9469 0.581626
\(576\) 0 0
\(577\) 15.1804 0.631968 0.315984 0.948765i \(-0.397665\pi\)
0.315984 + 0.948765i \(0.397665\pi\)
\(578\) 0 0
\(579\) 16.4057 0.681799
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −76.0700 −3.15050
\(584\) 0 0
\(585\) −2.67632 −0.110652
\(586\) 0 0
\(587\) 8.20694 0.338737 0.169368 0.985553i \(-0.445827\pi\)
0.169368 + 0.985553i \(0.445827\pi\)
\(588\) 0 0
\(589\) 25.5225 1.05164
\(590\) 0 0
\(591\) −24.6763 −1.01505
\(592\) 0 0
\(593\) 5.55718 0.228206 0.114103 0.993469i \(-0.463601\pi\)
0.114103 + 0.993469i \(0.463601\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.8196 0.565599
\(598\) 0 0
\(599\) −35.5861 −1.45401 −0.727005 0.686632i \(-0.759088\pi\)
−0.727005 + 0.686632i \(0.759088\pi\)
\(600\) 0 0
\(601\) −1.18038 −0.0481489 −0.0240744 0.999710i \(-0.507664\pi\)
−0.0240744 + 0.999710i \(0.507664\pi\)
\(602\) 0 0
\(603\) −7.88325 −0.321031
\(604\) 0 0
\(605\) 10.2624 0.417228
\(606\) 0 0
\(607\) −12.3382 −0.500790 −0.250395 0.968144i \(-0.580561\pi\)
−0.250395 + 0.968144i \(0.580561\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.8856 1.16859
\(612\) 0 0
\(613\) 21.0902 0.851825 0.425912 0.904764i \(-0.359953\pi\)
0.425912 + 0.904764i \(0.359953\pi\)
\(614\) 0 0
\(615\) −4.36077 −0.175843
\(616\) 0 0
\(617\) 15.3526 0.618074 0.309037 0.951050i \(-0.399993\pi\)
0.309037 + 0.951050i \(0.399993\pi\)
\(618\) 0 0
\(619\) −18.4323 −0.740856 −0.370428 0.928861i \(-0.620789\pi\)
−0.370428 + 0.928861i \(0.620789\pi\)
\(620\) 0 0
\(621\) −2.90981 −0.116767
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 21.9388 0.877550
\(626\) 0 0
\(627\) 34.0821 1.36111
\(628\) 0 0
\(629\) 22.4139 0.893700
\(630\) 0 0
\(631\) −5.91375 −0.235423 −0.117711 0.993048i \(-0.537556\pi\)
−0.117711 + 0.993048i \(0.537556\pi\)
\(632\) 0 0
\(633\) 11.5861 0.460507
\(634\) 0 0
\(635\) −0.567705 −0.0225287
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.909808 −0.0359915
\(640\) 0 0
\(641\) 46.1641 1.82337 0.911686 0.410887i \(-0.134781\pi\)
0.911686 + 0.410887i \(0.134781\pi\)
\(642\) 0 0
\(643\) −31.0555 −1.22471 −0.612355 0.790583i \(-0.709778\pi\)
−0.612355 + 0.790583i \(0.709778\pi\)
\(644\) 0 0
\(645\) −4.90981 −0.193324
\(646\) 0 0
\(647\) −17.0371 −0.669797 −0.334898 0.942254i \(-0.608702\pi\)
−0.334898 + 0.942254i \(0.608702\pi\)
\(648\) 0 0
\(649\) −10.3873 −0.407738
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.13936 0.318518 0.159259 0.987237i \(-0.449090\pi\)
0.159259 + 0.987237i \(0.449090\pi\)
\(654\) 0 0
\(655\) −2.63529 −0.102969
\(656\) 0 0
\(657\) −5.20694 −0.203142
\(658\) 0 0
\(659\) 41.1191 1.60177 0.800887 0.598815i \(-0.204362\pi\)
0.800887 + 0.598815i \(0.204362\pi\)
\(660\) 0 0
\(661\) −13.3421 −0.518948 −0.259474 0.965750i \(-0.583549\pi\)
−0.259474 + 0.965750i \(0.583549\pi\)
\(662\) 0 0
\(663\) 17.1191 0.664852
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3155 0.399420
\(668\) 0 0
\(669\) 20.4549 0.790832
\(670\) 0 0
\(671\) 27.0902 1.04581
\(672\) 0 0
\(673\) 9.75837 0.376158 0.188079 0.982154i \(-0.439774\pi\)
0.188079 + 0.982154i \(0.439774\pi\)
\(674\) 0 0
\(675\) −4.79306 −0.184485
\(676\) 0 0
\(677\) 32.4549 1.24734 0.623672 0.781686i \(-0.285640\pi\)
0.623672 + 0.781686i \(0.285640\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −17.7400 −0.679797
\(682\) 0 0
\(683\) −36.9653 −1.41444 −0.707219 0.706994i \(-0.750051\pi\)
−0.707219 + 0.706994i \(0.750051\pi\)
\(684\) 0 0
\(685\) −1.40574 −0.0537106
\(686\) 0 0
\(687\) −24.1988 −0.923242
\(688\) 0 0
\(689\) 77.2543 2.94315
\(690\) 0 0
\(691\) 14.2519 0.542168 0.271084 0.962556i \(-0.412618\pi\)
0.271084 + 0.962556i \(0.412618\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.59426 −0.0984059
\(696\) 0 0
\(697\) 27.8938 1.05655
\(698\) 0 0
\(699\) 22.6763 0.857697
\(700\) 0 0
\(701\) 51.3937 1.94111 0.970556 0.240876i \(-0.0774347\pi\)
0.970556 + 0.240876i \(0.0774347\pi\)
\(702\) 0 0
\(703\) −45.3179 −1.70920
\(704\) 0 0
\(705\) −2.23349 −0.0841182
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −25.1722 −0.945364 −0.472682 0.881233i \(-0.656714\pi\)
−0.472682 + 0.881233i \(0.656714\pi\)
\(710\) 0 0
\(711\) −2.75203 −0.103209
\(712\) 0 0
\(713\) −12.6232 −0.472743
\(714\) 0 0
\(715\) −15.5041 −0.579819
\(716\) 0 0
\(717\) −2.90981 −0.108669
\(718\) 0 0
\(719\) −8.18038 −0.305077 −0.152538 0.988298i \(-0.548745\pi\)
−0.152538 + 0.988298i \(0.548745\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.7400 −0.808518
\(724\) 0 0
\(725\) 16.9919 0.631062
\(726\) 0 0
\(727\) −26.9614 −0.999942 −0.499971 0.866042i \(-0.666656\pi\)
−0.499971 + 0.866042i \(0.666656\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.4057 1.16158
\(732\) 0 0
\(733\) −21.4694 −0.792990 −0.396495 0.918037i \(-0.629774\pi\)
−0.396495 + 0.918037i \(0.629774\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −45.6682 −1.68221
\(738\) 0 0
\(739\) 1.34210 0.0493701 0.0246850 0.999695i \(-0.492142\pi\)
0.0246850 + 0.999695i \(0.492142\pi\)
\(740\) 0 0
\(741\) −34.6127 −1.27153
\(742\) 0 0
\(743\) 26.9919 0.990236 0.495118 0.868826i \(-0.335125\pi\)
0.495118 + 0.868826i \(0.335125\pi\)
\(744\) 0 0
\(745\) −1.40574 −0.0515024
\(746\) 0 0
\(747\) 9.97345 0.364909
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.8872 −0.470261 −0.235130 0.971964i \(-0.575552\pi\)
−0.235130 + 0.971964i \(0.575552\pi\)
\(752\) 0 0
\(753\) 8.02655 0.292504
\(754\) 0 0
\(755\) −0.784928 −0.0285664
\(756\) 0 0
\(757\) 26.3897 0.959151 0.479575 0.877501i \(-0.340791\pi\)
0.479575 + 0.877501i \(0.340791\pi\)
\(758\) 0 0
\(759\) −16.8567 −0.611859
\(760\) 0 0
\(761\) −30.4428 −1.10355 −0.551776 0.833993i \(-0.686050\pi\)
−0.551776 + 0.833993i \(0.686050\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.32368 −0.0478579
\(766\) 0 0
\(767\) 10.5490 0.380904
\(768\) 0 0
\(769\) −42.1191 −1.51886 −0.759428 0.650592i \(-0.774521\pi\)
−0.759428 + 0.650592i \(0.774521\pi\)
\(770\) 0 0
\(771\) 12.9098 0.464935
\(772\) 0 0
\(773\) 38.0289 1.36781 0.683903 0.729573i \(-0.260281\pi\)
0.683903 + 0.729573i \(0.260281\pi\)
\(774\) 0 0
\(775\) −20.7931 −0.746909
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −56.3976 −2.02065
\(780\) 0 0
\(781\) −5.27058 −0.188596
\(782\) 0 0
\(783\) −3.54510 −0.126691
\(784\) 0 0
\(785\) −2.54115 −0.0906976
\(786\) 0 0
\(787\) −31.5861 −1.12592 −0.562962 0.826483i \(-0.690338\pi\)
−0.562962 + 0.826483i \(0.690338\pi\)
\(788\) 0 0
\(789\) −8.90981 −0.317198
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −27.5120 −0.976978
\(794\) 0 0
\(795\) −5.97345 −0.211856
\(796\) 0 0
\(797\) 45.2133 1.60154 0.800768 0.598974i \(-0.204425\pi\)
0.800768 + 0.598974i \(0.204425\pi\)
\(798\) 0 0
\(799\) 14.2866 0.505424
\(800\) 0 0
\(801\) 4.90981 0.173480
\(802\) 0 0
\(803\) −30.1641 −1.06447
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.04103 −0.212654
\(808\) 0 0
\(809\) 0.991865 0.0348721 0.0174361 0.999848i \(-0.494450\pi\)
0.0174361 + 0.999848i \(0.494450\pi\)
\(810\) 0 0
\(811\) 23.6151 0.829237 0.414619 0.909995i \(-0.363915\pi\)
0.414619 + 0.909995i \(0.363915\pi\)
\(812\) 0 0
\(813\) 13.3647 0.468721
\(814\) 0 0
\(815\) 4.02895 0.141128
\(816\) 0 0
\(817\) −63.4983 −2.22153
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35.2664 1.23081 0.615403 0.788213i \(-0.288993\pi\)
0.615403 + 0.788213i \(0.288993\pi\)
\(822\) 0 0
\(823\) −10.5412 −0.367441 −0.183721 0.982978i \(-0.558814\pi\)
−0.183721 + 0.982978i \(0.558814\pi\)
\(824\) 0 0
\(825\) −27.7665 −0.966706
\(826\) 0 0
\(827\) −31.4323 −1.09301 −0.546504 0.837456i \(-0.684042\pi\)
−0.546504 + 0.837456i \(0.684042\pi\)
\(828\) 0 0
\(829\) −25.5225 −0.886433 −0.443216 0.896415i \(-0.646163\pi\)
−0.443216 + 0.896415i \(0.646163\pi\)
\(830\) 0 0
\(831\) −4.79306 −0.166269
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.91794 0.274012
\(836\) 0 0
\(837\) 4.33816 0.149949
\(838\) 0 0
\(839\) −2.78253 −0.0960637 −0.0480318 0.998846i \(-0.515295\pi\)
−0.0480318 + 0.998846i \(0.515295\pi\)
\(840\) 0 0
\(841\) −16.4323 −0.566631
\(842\) 0 0
\(843\) 7.32368 0.252241
\(844\) 0 0
\(845\) 9.83170 0.338221
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −13.2069 −0.453261
\(850\) 0 0
\(851\) 22.4139 0.768338
\(852\) 0 0
\(853\) 18.7931 0.643462 0.321731 0.946831i \(-0.395735\pi\)
0.321731 + 0.946831i \(0.395735\pi\)
\(854\) 0 0
\(855\) 2.67632 0.0915281
\(856\) 0 0
\(857\) −20.9388 −0.715254 −0.357627 0.933864i \(-0.616414\pi\)
−0.357627 + 0.933864i \(0.616414\pi\)
\(858\) 0 0
\(859\) 11.5861 0.395313 0.197657 0.980271i \(-0.436667\pi\)
0.197657 + 0.980271i \(0.436667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.5330 1.00532 0.502658 0.864485i \(-0.332356\pi\)
0.502658 + 0.864485i \(0.332356\pi\)
\(864\) 0 0
\(865\) −7.58612 −0.257936
\(866\) 0 0
\(867\) −8.53302 −0.289796
\(868\) 0 0
\(869\) −15.9427 −0.540819
\(870\) 0 0
\(871\) 46.3792 1.57150
\(872\) 0 0
\(873\) 5.79306 0.196065
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.22536 −0.176448 −0.0882239 0.996101i \(-0.528119\pi\)
−0.0882239 + 0.996101i \(0.528119\pi\)
\(878\) 0 0
\(879\) 7.54510 0.254490
\(880\) 0 0
\(881\) 13.1433 0.442809 0.221405 0.975182i \(-0.428936\pi\)
0.221405 + 0.975182i \(0.428936\pi\)
\(882\) 0 0
\(883\) −7.96531 −0.268054 −0.134027 0.990978i \(-0.542791\pi\)
−0.134027 + 0.990978i \(0.542791\pi\)
\(884\) 0 0
\(885\) −0.815671 −0.0274185
\(886\) 0 0
\(887\) −39.7955 −1.33620 −0.668100 0.744071i \(-0.732892\pi\)
−0.668100 + 0.744071i \(0.732892\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.79306 0.194075
\(892\) 0 0
\(893\) −28.8856 −0.966621
\(894\) 0 0
\(895\) −2.81148 −0.0939775
\(896\) 0 0
\(897\) 17.1191 0.571591
\(898\) 0 0
\(899\) −15.3792 −0.512925
\(900\) 0 0
\(901\) 38.2093 1.27294
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.45885 −0.0484938
\(906\) 0 0
\(907\) 11.5967 0.385061 0.192530 0.981291i \(-0.438331\pi\)
0.192530 + 0.981291i \(0.438331\pi\)
\(908\) 0 0
\(909\) 16.6763 0.553118
\(910\) 0 0
\(911\) 34.4959 1.14290 0.571451 0.820636i \(-0.306381\pi\)
0.571451 + 0.820636i \(0.306381\pi\)
\(912\) 0 0
\(913\) 57.7768 1.91213
\(914\) 0 0
\(915\) 2.12728 0.0703256
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.85431 −0.0611679 −0.0305839 0.999532i \(-0.509737\pi\)
−0.0305839 + 0.999532i \(0.509737\pi\)
\(920\) 0 0
\(921\) 23.1086 0.761455
\(922\) 0 0
\(923\) 5.35263 0.176184
\(924\) 0 0
\(925\) 36.9203 1.21393
\(926\) 0 0
\(927\) 0.793062 0.0260476
\(928\) 0 0
\(929\) 23.1191 0.758514 0.379257 0.925291i \(-0.376180\pi\)
0.379257 + 0.925291i \(0.376180\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 7.40574 0.242453
\(934\) 0 0
\(935\) −7.66818 −0.250776
\(936\) 0 0
\(937\) 43.3445 1.41600 0.708002 0.706211i \(-0.249597\pi\)
0.708002 + 0.706211i \(0.249597\pi\)
\(938\) 0 0
\(939\) −14.5861 −0.476000
\(940\) 0 0
\(941\) −29.1312 −0.949651 −0.474825 0.880080i \(-0.657489\pi\)
−0.474825 + 0.880080i \(0.657489\pi\)
\(942\) 0 0
\(943\) 27.8938 0.908347
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.9469 0.453213 0.226607 0.973986i \(-0.427237\pi\)
0.226607 + 0.973986i \(0.427237\pi\)
\(948\) 0 0
\(949\) 30.6337 0.994413
\(950\) 0 0
\(951\) −1.36471 −0.0442538
\(952\) 0 0
\(953\) −39.7134 −1.28644 −0.643222 0.765680i \(-0.722403\pi\)
−0.643222 + 0.765680i \(0.722403\pi\)
\(954\) 0 0
\(955\) −9.71340 −0.314318
\(956\) 0 0
\(957\) −20.5370 −0.663866
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −12.1804 −0.392916
\(962\) 0 0
\(963\) −11.9734 −0.385839
\(964\) 0 0
\(965\) 7.46304 0.240244
\(966\) 0 0
\(967\) −12.5717 −0.404277 −0.202139 0.979357i \(-0.564789\pi\)
−0.202139 + 0.979357i \(0.564789\pi\)
\(968\) 0 0
\(969\) −17.1191 −0.549946
\(970\) 0 0
\(971\) 14.1538 0.454218 0.227109 0.973869i \(-0.427073\pi\)
0.227109 + 0.973869i \(0.427073\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 28.1988 0.903084
\(976\) 0 0
\(977\) −46.5249 −1.48846 −0.744231 0.667922i \(-0.767184\pi\)
−0.744231 + 0.667922i \(0.767184\pi\)
\(978\) 0 0
\(979\) 28.4428 0.909037
\(980\) 0 0
\(981\) 4.61268 0.147271
\(982\) 0 0
\(983\) 50.4718 1.60980 0.804900 0.593411i \(-0.202219\pi\)
0.804900 + 0.593411i \(0.202219\pi\)
\(984\) 0 0
\(985\) −11.2254 −0.357670
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 31.4057 0.998644
\(990\) 0 0
\(991\) −3.97739 −0.126346 −0.0631730 0.998003i \(-0.520122\pi\)
−0.0631730 + 0.998003i \(0.520122\pi\)
\(992\) 0 0
\(993\) −10.9734 −0.348232
\(994\) 0 0
\(995\) 6.28660 0.199299
\(996\) 0 0
\(997\) −44.0184 −1.39408 −0.697039 0.717034i \(-0.745499\pi\)
−0.697039 + 0.717034i \(0.745499\pi\)
\(998\) 0 0
\(999\) −7.70287 −0.243708
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 9408.2.a.ei.1.2 3
4.3 odd 2 9408.2.a.eg.1.2 3
7.3 odd 6 1344.2.q.z.961.2 6
7.5 odd 6 1344.2.q.z.193.2 6
7.6 odd 2 9408.2.a.eh.1.2 3
8.3 odd 2 4704.2.a.bv.1.2 3
8.5 even 2 4704.2.a.bt.1.2 3
28.3 even 6 1344.2.q.y.961.2 6
28.19 even 6 1344.2.q.y.193.2 6
28.27 even 2 9408.2.a.ej.1.2 3
56.3 even 6 672.2.q.l.289.2 yes 6
56.5 odd 6 672.2.q.k.193.2 6
56.13 odd 2 4704.2.a.bu.1.2 3
56.19 even 6 672.2.q.l.193.2 yes 6
56.27 even 2 4704.2.a.bs.1.2 3
56.45 odd 6 672.2.q.k.289.2 yes 6
168.5 even 6 2016.2.s.u.865.2 6
168.59 odd 6 2016.2.s.v.289.2 6
168.101 even 6 2016.2.s.u.289.2 6
168.131 odd 6 2016.2.s.v.865.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.2 6 56.5 odd 6
672.2.q.k.289.2 yes 6 56.45 odd 6
672.2.q.l.193.2 yes 6 56.19 even 6
672.2.q.l.289.2 yes 6 56.3 even 6
1344.2.q.y.193.2 6 28.19 even 6
1344.2.q.y.961.2 6 28.3 even 6
1344.2.q.z.193.2 6 7.5 odd 6
1344.2.q.z.961.2 6 7.3 odd 6
2016.2.s.u.289.2 6 168.101 even 6
2016.2.s.u.865.2 6 168.5 even 6
2016.2.s.v.289.2 6 168.59 odd 6
2016.2.s.v.865.2 6 168.131 odd 6
4704.2.a.bs.1.2 3 56.27 even 2
4704.2.a.bt.1.2 3 8.5 even 2
4704.2.a.bu.1.2 3 56.13 odd 2
4704.2.a.bv.1.2 3 8.3 odd 2
9408.2.a.eg.1.2 3 4.3 odd 2
9408.2.a.eh.1.2 3 7.6 odd 2
9408.2.a.ei.1.2 3 1.1 even 1 trivial
9408.2.a.ej.1.2 3 28.27 even 2