Properties

Label 6724.2.a.c.1.3
Level $6724$
Weight $2$
Character 6724.1
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6914103191\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.31526\) of defining polynomial
Character \(\chi\) \(=\) 6724.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +O(q^{10})\) \(q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +1.67570 q^{11} -6.63052 q^{13} -0.111283 q^{15} -5.16120 q^{17} +4.72562 q^{19} +0.299072 q^{21} -8.82071 q^{23} -3.63052 q^{25} +0.569704 q^{27} +1.80981 q^{29} -1.65951 q^{31} -0.159349 q^{33} -3.68044 q^{35} -1.99096 q^{37} +0.630522 q^{39} +1.46932 q^{43} -3.50016 q^{45} +8.53543 q^{47} +2.89112 q^{49} +0.490799 q^{51} +9.35139 q^{53} +1.96098 q^{55} -0.449377 q^{57} +8.82071 q^{59} +12.6305 q^{61} +9.40660 q^{63} -7.75935 q^{65} -9.67570 q^{67} +0.838796 q^{69} +0.776081 q^{71} +8.33145 q^{73} +0.345241 q^{75} -5.27009 q^{77} -0.915804 q^{79} +8.91870 q^{81} -10.0998 q^{83} -6.03988 q^{85} -0.172102 q^{87} +6.44033 q^{89} +20.8531 q^{91} +0.157809 q^{93} +5.53014 q^{95} +8.32241 q^{97} -5.01193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} - 4q^{11} + 10q^{15} + 4q^{17} - 6q^{19} - 12q^{23} + 12q^{25} + 10q^{27} + 4q^{29} - 8q^{31} - 20q^{33} + 26q^{35} + 16q^{37} - 24q^{39} + 4q^{43} + 4q^{45} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} + 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 4q^{65} - 28q^{67} + 28q^{69} + 2q^{71} + 8q^{73} - 30q^{75} + 8q^{77} + 18q^{79} + 28q^{81} - 12q^{83} - 32q^{85} + 44q^{87} - 4q^{89} + 36q^{91} + 28q^{93} - 14q^{95} - 16q^{97} - 58q^{99} + O(q^{100}) \)

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
\(3\) −0.0950939 −0.0549025 −0.0274513 0.999623i \(-0.508739\pi\)
−0.0274513 + 0.999623i \(0.508739\pi\)
\(4\) 0 0
\(5\) 1.17025 0.523350 0.261675 0.965156i \(-0.415725\pi\)
0.261675 + 0.965156i \(0.415725\pi\)
\(6\) 0 0
\(7\) −3.14501 −1.18870 −0.594352 0.804205i \(-0.702591\pi\)
−0.594352 + 0.804205i \(0.702591\pi\)
\(8\) 0 0
\(9\) −2.99096 −0.996986
\(10\) 0 0
\(11\) 1.67570 0.505241 0.252621 0.967565i \(-0.418708\pi\)
0.252621 + 0.967565i \(0.418708\pi\)
\(12\) 0 0
\(13\) −6.63052 −1.83898 −0.919488 0.393118i \(-0.871396\pi\)
−0.919488 + 0.393118i \(0.871396\pi\)
\(14\) 0 0
\(15\) −0.111283 −0.0287332
\(16\) 0 0
\(17\) −5.16120 −1.25178 −0.625888 0.779913i \(-0.715263\pi\)
−0.625888 + 0.779913i \(0.715263\pi\)
\(18\) 0 0
\(19\) 4.72562 1.08413 0.542065 0.840336i \(-0.317643\pi\)
0.542065 + 0.840336i \(0.317643\pi\)
\(20\) 0 0
\(21\) 0.299072 0.0652628
\(22\) 0 0
\(23\) −8.82071 −1.83925 −0.919623 0.392803i \(-0.871505\pi\)
−0.919623 + 0.392803i \(0.871505\pi\)
\(24\) 0 0
\(25\) −3.63052 −0.726104
\(26\) 0 0
\(27\) 0.569704 0.109640
\(28\) 0 0
\(29\) 1.80981 0.336074 0.168037 0.985781i \(-0.446257\pi\)
0.168037 + 0.985781i \(0.446257\pi\)
\(30\) 0 0
\(31\) −1.65951 −0.298056 −0.149028 0.988833i \(-0.547614\pi\)
−0.149028 + 0.988833i \(0.547614\pi\)
\(32\) 0 0
\(33\) −0.159349 −0.0277390
\(34\) 0 0
\(35\) −3.68044 −0.622108
\(36\) 0 0
\(37\) −1.99096 −0.327311 −0.163656 0.986518i \(-0.552329\pi\)
−0.163656 + 0.986518i \(0.552329\pi\)
\(38\) 0 0
\(39\) 0.630522 0.100964
\(40\) 0 0
\(41\) 0 0
\(42\) 0 0
\(43\) 1.46932 0.224069 0.112034 0.993704i \(-0.464263\pi\)
0.112034 + 0.993704i \(0.464263\pi\)
\(44\) 0 0
\(45\) −3.50016 −0.521773
\(46\) 0 0
\(47\) 8.53543 1.24502 0.622510 0.782612i \(-0.286113\pi\)
0.622510 + 0.782612i \(0.286113\pi\)
\(48\) 0 0
\(49\) 2.89112 0.413017
\(50\) 0 0
\(51\) 0.490799 0.0687256
\(52\) 0 0
\(53\) 9.35139 1.28451 0.642256 0.766490i \(-0.277999\pi\)
0.642256 + 0.766490i \(0.277999\pi\)
\(54\) 0 0
\(55\) 1.96098 0.264418
\(56\) 0 0
\(57\) −0.449377 −0.0595215
\(58\) 0 0
\(59\) 8.82071 1.14836 0.574179 0.818730i \(-0.305321\pi\)
0.574179 + 0.818730i \(0.305321\pi\)
\(60\) 0 0
\(61\) 12.6305 1.61717 0.808586 0.588378i \(-0.200233\pi\)
0.808586 + 0.588378i \(0.200233\pi\)
\(62\) 0 0
\(63\) 9.40660 1.18512
\(64\) 0 0
\(65\) −7.75935 −0.962429
\(66\) 0 0
\(67\) −9.67570 −1.18207 −0.591037 0.806644i \(-0.701281\pi\)
−0.591037 + 0.806644i \(0.701281\pi\)
\(68\) 0 0
\(69\) 0.838796 0.100979
\(70\) 0 0
\(71\) 0.776081 0.0921039 0.0460519 0.998939i \(-0.485336\pi\)
0.0460519 + 0.998939i \(0.485336\pi\)
\(72\) 0 0
\(73\) 8.33145 0.975123 0.487561 0.873089i \(-0.337887\pi\)
0.487561 + 0.873089i \(0.337887\pi\)
\(74\) 0 0
\(75\) 0.345241 0.0398650
\(76\) 0 0
\(77\) −5.27009 −0.600582
\(78\) 0 0
\(79\) −0.915804 −0.103036 −0.0515180 0.998672i \(-0.516406\pi\)
−0.0515180 + 0.998672i \(0.516406\pi\)
\(80\) 0 0
\(81\) 8.91870 0.990966
\(82\) 0 0
\(83\) −10.0998 −1.10860 −0.554301 0.832316i \(-0.687014\pi\)
−0.554301 + 0.832316i \(0.687014\pi\)
\(84\) 0 0
\(85\) −6.03988 −0.655117
\(86\) 0 0
\(87\) −0.172102 −0.0184513
\(88\) 0 0
\(89\) 6.44033 0.682674 0.341337 0.939941i \(-0.389120\pi\)
0.341337 + 0.939941i \(0.389120\pi\)
\(90\) 0 0
\(91\) 20.8531 2.18600
\(92\) 0 0
\(93\) 0.157809 0.0163640
\(94\) 0 0
\(95\) 5.53014 0.567380
\(96\) 0 0
\(97\) 8.32241 0.845012 0.422506 0.906360i \(-0.361150\pi\)
0.422506 + 0.906360i \(0.361150\pi\)
\(98\) 0 0
\(99\) −5.01193 −0.503718
\(100\) 0 0
\(101\) −9.79173 −0.974313 −0.487157 0.873315i \(-0.661966\pi\)
−0.487157 + 0.873315i \(0.661966\pi\)
\(102\) 0 0
\(103\) −0.648608 −0.0639093 −0.0319546 0.999489i \(-0.510173\pi\)
−0.0319546 + 0.999489i \(0.510173\pi\)
\(104\) 0 0
\(105\) 0.349988 0.0341553
\(106\) 0 0
\(107\) 16.8026 1.62437 0.812186 0.583399i \(-0.198278\pi\)
0.812186 + 0.583399i \(0.198278\pi\)
\(108\) 0 0
\(109\) −3.84969 −0.368734 −0.184367 0.982857i \(-0.559023\pi\)
−0.184367 + 0.982857i \(0.559023\pi\)
\(110\) 0 0
\(111\) 0.189328 0.0179702
\(112\) 0 0
\(113\) −7.65046 −0.719695 −0.359848 0.933011i \(-0.617171\pi\)
−0.359848 + 0.933011i \(0.617171\pi\)
\(114\) 0 0
\(115\) −10.3224 −0.962569
\(116\) 0 0
\(117\) 19.8316 1.83343
\(118\) 0 0
\(119\) 16.2321 1.48799
\(120\) 0 0
\(121\) −8.19204 −0.744731
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.0998 −0.903357
\(126\) 0 0
\(127\) 14.9206 1.32398 0.661992 0.749511i \(-0.269711\pi\)
0.661992 + 0.749511i \(0.269711\pi\)
\(128\) 0 0
\(129\) −0.139723 −0.0123019
\(130\) 0 0
\(131\) 4.19019 0.366098 0.183049 0.983104i \(-0.441403\pi\)
0.183049 + 0.983104i \(0.441403\pi\)
\(132\) 0 0
\(133\) −14.8621 −1.28871
\(134\) 0 0
\(135\) 0.666694 0.0573799
\(136\) 0 0
\(137\) −4.09984 −0.350273 −0.175137 0.984544i \(-0.556037\pi\)
−0.175137 + 0.984544i \(0.556037\pi\)
\(138\) 0 0
\(139\) −11.9819 −1.01629 −0.508146 0.861271i \(-0.669669\pi\)
−0.508146 + 0.861271i \(0.669669\pi\)
\(140\) 0 0
\(141\) −0.811668 −0.0683547
\(142\) 0 0
\(143\) −11.1107 −0.929127
\(144\) 0 0
\(145\) 2.11793 0.175884
\(146\) 0 0
\(147\) −0.274928 −0.0226756
\(148\) 0 0
\(149\) 1.80981 0.148266 0.0741328 0.997248i \(-0.476381\pi\)
0.0741328 + 0.997248i \(0.476381\pi\)
\(150\) 0 0
\(151\) 21.3561 1.73794 0.868969 0.494867i \(-0.164783\pi\)
0.868969 + 0.494867i \(0.164783\pi\)
\(152\) 0 0
\(153\) 15.4369 1.24800
\(154\) 0 0
\(155\) −1.94203 −0.155988
\(156\) 0 0
\(157\) −13.1107 −1.04635 −0.523175 0.852225i \(-0.675253\pi\)
−0.523175 + 0.852225i \(0.675253\pi\)
\(158\) 0 0
\(159\) −0.889261 −0.0705230
\(160\) 0 0
\(161\) 27.7413 2.18632
\(162\) 0 0
\(163\) 10.7808 0.844420 0.422210 0.906498i \(-0.361255\pi\)
0.422210 + 0.906498i \(0.361255\pi\)
\(164\) 0 0
\(165\) −0.186477 −0.0145172
\(166\) 0 0
\(167\) −13.6757 −1.05826 −0.529129 0.848542i \(-0.677481\pi\)
−0.529129 + 0.848542i \(0.677481\pi\)
\(168\) 0 0
\(169\) 30.9638 2.38183
\(170\) 0 0
\(171\) −14.1341 −1.08086
\(172\) 0 0
\(173\) 10.5801 0.804387 0.402193 0.915555i \(-0.368248\pi\)
0.402193 + 0.915555i \(0.368248\pi\)
\(174\) 0 0
\(175\) 11.4180 0.863123
\(176\) 0 0
\(177\) −0.838796 −0.0630478
\(178\) 0 0
\(179\) 1.23536 0.0923352 0.0461676 0.998934i \(-0.485299\pi\)
0.0461676 + 0.998934i \(0.485299\pi\)
\(180\) 0 0
\(181\) 3.90965 0.290602 0.145301 0.989387i \(-0.453585\pi\)
0.145301 + 0.989387i \(0.453585\pi\)
\(182\) 0 0
\(183\) −1.20109 −0.0887868
\(184\) 0 0
\(185\) −2.32991 −0.171298
\(186\) 0 0
\(187\) −8.64861 −0.632449
\(188\) 0 0
\(189\) −1.79173 −0.130329
\(190\) 0 0
\(191\) 7.31712 0.529448 0.264724 0.964324i \(-0.414719\pi\)
0.264724 + 0.964324i \(0.414719\pi\)
\(192\) 0 0
\(193\) 13.3009 0.957422 0.478711 0.877973i \(-0.341104\pi\)
0.478711 + 0.877973i \(0.341104\pi\)
\(194\) 0 0
\(195\) 0.737867 0.0528397
\(196\) 0 0
\(197\) −15.5692 −1.10926 −0.554628 0.832098i \(-0.687140\pi\)
−0.554628 + 0.832098i \(0.687140\pi\)
\(198\) 0 0
\(199\) 27.8972 1.97758 0.988789 0.149319i \(-0.0477080\pi\)
0.988789 + 0.149319i \(0.0477080\pi\)
\(200\) 0 0
\(201\) 0.920100 0.0648989
\(202\) 0 0
\(203\) −5.69189 −0.399492
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 26.3824 1.83370
\(208\) 0 0
\(209\) 7.91870 0.547748
\(210\) 0 0
\(211\) −18.1484 −1.24939 −0.624694 0.780870i \(-0.714776\pi\)
−0.624694 + 0.780870i \(0.714776\pi\)
\(212\) 0 0
\(213\) −0.0738006 −0.00505673
\(214\) 0 0
\(215\) 1.71947 0.117267
\(216\) 0 0
\(217\) 5.21917 0.354300
\(218\) 0 0
\(219\) −0.792270 −0.0535367
\(220\) 0 0
\(221\) 34.2215 2.30199
\(222\) 0 0
\(223\) −6.55826 −0.439174 −0.219587 0.975593i \(-0.570471\pi\)
−0.219587 + 0.975593i \(0.570471\pi\)
\(224\) 0 0
\(225\) 10.8587 0.723916
\(226\) 0 0
\(227\) −23.4884 −1.55898 −0.779489 0.626416i \(-0.784521\pi\)
−0.779489 + 0.626416i \(0.784521\pi\)
\(228\) 0 0
\(229\) −12.1503 −0.802915 −0.401457 0.915878i \(-0.631496\pi\)
−0.401457 + 0.915878i \(0.631496\pi\)
\(230\) 0 0
\(231\) 0.501153 0.0329735
\(232\) 0 0
\(233\) 21.8422 1.43093 0.715465 0.698649i \(-0.246215\pi\)
0.715465 + 0.698649i \(0.246215\pi\)
\(234\) 0 0
\(235\) 9.98856 0.651582
\(236\) 0 0
\(237\) 0.0870874 0.00565694
\(238\) 0 0
\(239\) −21.5968 −1.39698 −0.698490 0.715620i \(-0.746144\pi\)
−0.698490 + 0.715620i \(0.746144\pi\)
\(240\) 0 0
\(241\) 12.6305 0.813603 0.406802 0.913516i \(-0.366644\pi\)
0.406802 + 0.913516i \(0.366644\pi\)
\(242\) 0 0
\(243\) −2.55723 −0.164046
\(244\) 0 0
\(245\) 3.38332 0.216152
\(246\) 0 0
\(247\) −31.3333 −1.99369
\(248\) 0 0
\(249\) 0.960434 0.0608650
\(250\) 0 0
\(251\) 20.7702 1.31101 0.655503 0.755192i \(-0.272457\pi\)
0.655503 + 0.755192i \(0.272457\pi\)
\(252\) 0 0
\(253\) −14.7808 −0.929263
\(254\) 0 0
\(255\) 0.574356 0.0359676
\(256\) 0 0
\(257\) −14.8207 −0.924491 −0.462245 0.886752i \(-0.652956\pi\)
−0.462245 + 0.886752i \(0.652956\pi\)
\(258\) 0 0
\(259\) 6.26159 0.389076
\(260\) 0 0
\(261\) −5.41307 −0.335061
\(262\) 0 0
\(263\) −18.4175 −1.13567 −0.567836 0.823142i \(-0.692219\pi\)
−0.567836 + 0.823142i \(0.692219\pi\)
\(264\) 0 0
\(265\) 10.9434 0.672250
\(266\) 0 0
\(267\) −0.612437 −0.0374805
\(268\) 0 0
\(269\) −25.8916 −1.57864 −0.789318 0.613984i \(-0.789566\pi\)
−0.789318 + 0.613984i \(0.789566\pi\)
\(270\) 0 0
\(271\) −17.5017 −1.06315 −0.531576 0.847010i \(-0.678400\pi\)
−0.531576 + 0.847010i \(0.678400\pi\)
\(272\) 0 0
\(273\) −1.98300 −0.120017
\(274\) 0 0
\(275\) −6.08365 −0.366858
\(276\) 0 0
\(277\) 8.77179 0.527045 0.263523 0.964653i \(-0.415116\pi\)
0.263523 + 0.964653i \(0.415116\pi\)
\(278\) 0 0
\(279\) 4.96351 0.297158
\(280\) 0 0
\(281\) −16.2215 −0.967692 −0.483846 0.875153i \(-0.660761\pi\)
−0.483846 + 0.875153i \(0.660761\pi\)
\(282\) 0 0
\(283\) −11.9819 −0.712251 −0.356125 0.934438i \(-0.615902\pi\)
−0.356125 + 0.934438i \(0.615902\pi\)
\(284\) 0 0
\(285\) −0.525883 −0.0311506
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.63803 0.566943
\(290\) 0 0
\(291\) −0.791411 −0.0463933
\(292\) 0 0
\(293\) 11.2516 0.657323 0.328661 0.944448i \(-0.393403\pi\)
0.328661 + 0.944448i \(0.393403\pi\)
\(294\) 0 0
\(295\) 10.3224 0.600994
\(296\) 0 0
\(297\) 0.954650 0.0553944
\(298\) 0 0
\(299\) 58.4859 3.38233
\(300\) 0 0
\(301\) −4.62103 −0.266352
\(302\) 0 0
\(303\) 0.931134 0.0534922
\(304\) 0 0
\(305\) 14.7808 0.846348
\(306\) 0 0
\(307\) 8.61244 0.491538 0.245769 0.969328i \(-0.420960\pi\)
0.245769 + 0.969328i \(0.420960\pi\)
\(308\) 0 0
\(309\) 0.0616787 0.00350878
\(310\) 0 0
\(311\) −31.7575 −1.80080 −0.900400 0.435063i \(-0.856726\pi\)
−0.900400 + 0.435063i \(0.856726\pi\)
\(312\) 0 0
\(313\) −2.21777 −0.125356 −0.0626778 0.998034i \(-0.519964\pi\)
−0.0626778 + 0.998034i \(0.519964\pi\)
\(314\) 0 0
\(315\) 11.0080 0.620233
\(316\) 0 0
\(317\) 18.3947 1.03315 0.516574 0.856243i \(-0.327207\pi\)
0.516574 + 0.856243i \(0.327207\pi\)
\(318\) 0 0
\(319\) 3.03269 0.169798
\(320\) 0 0
\(321\) −1.59783 −0.0891820
\(322\) 0 0
\(323\) −24.3899 −1.35709
\(324\) 0 0
\(325\) 24.0723 1.33529
\(326\) 0 0
\(327\) 0.366083 0.0202444
\(328\) 0 0
\(329\) −26.8440 −1.47996
\(330\) 0 0
\(331\) −10.1160 −0.556027 −0.278014 0.960577i \(-0.589676\pi\)
−0.278014 + 0.960577i \(0.589676\pi\)
\(332\) 0 0
\(333\) 5.95487 0.326325
\(334\) 0 0
\(335\) −11.3230 −0.618639
\(336\) 0 0
\(337\) 7.81135 0.425511 0.212756 0.977105i \(-0.431756\pi\)
0.212756 + 0.977105i \(0.431756\pi\)
\(338\) 0 0
\(339\) 0.727513 0.0395131
\(340\) 0 0
\(341\) −2.78083 −0.150590
\(342\) 0 0
\(343\) 12.9225 0.697749
\(344\) 0 0
\(345\) 0.981598 0.0528475
\(346\) 0 0
\(347\) 15.6281 0.838959 0.419480 0.907765i \(-0.362213\pi\)
0.419480 + 0.907765i \(0.362213\pi\)
\(348\) 0 0
\(349\) 7.49265 0.401073 0.200536 0.979686i \(-0.435732\pi\)
0.200536 + 0.979686i \(0.435732\pi\)
\(350\) 0 0
\(351\) −3.77743 −0.201624
\(352\) 0 0
\(353\) 22.4313 1.19390 0.596949 0.802279i \(-0.296380\pi\)
0.596949 + 0.802279i \(0.296380\pi\)
\(354\) 0 0
\(355\) 0.908206 0.0482026
\(356\) 0 0
\(357\) −1.54357 −0.0816944
\(358\) 0 0
\(359\) −15.6196 −0.824372 −0.412186 0.911100i \(-0.635235\pi\)
−0.412186 + 0.911100i \(0.635235\pi\)
\(360\) 0 0
\(361\) 3.33145 0.175340
\(362\) 0 0
\(363\) 0.779014 0.0408876
\(364\) 0 0
\(365\) 9.74985 0.510331
\(366\) 0 0
\(367\) −3.82790 −0.199815 −0.0999073 0.994997i \(-0.531855\pi\)
−0.0999073 + 0.994997i \(0.531855\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.4103 −1.52690
\(372\) 0 0
\(373\) −18.1997 −0.942344 −0.471172 0.882041i \(-0.656169\pi\)
−0.471172 + 0.882041i \(0.656169\pi\)
\(374\) 0 0
\(375\) 0.960434 0.0495966
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −14.7627 −0.758311 −0.379156 0.925333i \(-0.623785\pi\)
−0.379156 + 0.925333i \(0.623785\pi\)
\(380\) 0 0
\(381\) −1.41885 −0.0726901
\(382\) 0 0
\(383\) −4.75406 −0.242921 −0.121460 0.992596i \(-0.538758\pi\)
−0.121460 + 0.992596i \(0.538758\pi\)
\(384\) 0 0
\(385\) −6.16730 −0.314315
\(386\) 0 0
\(387\) −4.39467 −0.223394
\(388\) 0 0
\(389\) 14.3804 0.729114 0.364557 0.931181i \(-0.381220\pi\)
0.364557 + 0.931181i \(0.381220\pi\)
\(390\) 0 0
\(391\) 45.5255 2.30232
\(392\) 0 0
\(393\) −0.398461 −0.0200997
\(394\) 0 0
\(395\) −1.07172 −0.0539239
\(396\) 0 0
\(397\) 10.9311 0.548618 0.274309 0.961642i \(-0.411551\pi\)
0.274309 + 0.961642i \(0.411551\pi\)
\(398\) 0 0
\(399\) 1.41330 0.0707534
\(400\) 0 0
\(401\) 30.4132 1.51876 0.759382 0.650646i \(-0.225502\pi\)
0.759382 + 0.650646i \(0.225502\pi\)
\(402\) 0 0
\(403\) 11.0034 0.548118
\(404\) 0 0
\(405\) 10.4371 0.518622
\(406\) 0 0
\(407\) −3.33624 −0.165371
\(408\) 0 0
\(409\) 13.3100 0.658136 0.329068 0.944306i \(-0.393266\pi\)
0.329068 + 0.944306i \(0.393266\pi\)
\(410\) 0 0
\(411\) 0.389870 0.0192309
\(412\) 0 0
\(413\) −27.7413 −1.36506
\(414\) 0 0
\(415\) −11.8193 −0.580187
\(416\) 0 0
\(417\) 1.13941 0.0557970
\(418\) 0 0
\(419\) 15.7917 0.771476 0.385738 0.922608i \(-0.373947\pi\)
0.385738 + 0.922608i \(0.373947\pi\)
\(420\) 0 0
\(421\) 34.4317 1.67810 0.839050 0.544054i \(-0.183111\pi\)
0.839050 + 0.544054i \(0.183111\pi\)
\(422\) 0 0
\(423\) −25.5291 −1.24127
\(424\) 0 0
\(425\) 18.7379 0.908920
\(426\) 0 0
\(427\) −39.7232 −1.92234
\(428\) 0 0
\(429\) 1.05656 0.0510114
\(430\) 0 0
\(431\) 5.69189 0.274168 0.137084 0.990559i \(-0.456227\pi\)
0.137084 + 0.990559i \(0.456227\pi\)
\(432\) 0 0
\(433\) −13.0034 −0.624903 −0.312452 0.949934i \(-0.601150\pi\)
−0.312452 + 0.949934i \(0.601150\pi\)
\(434\) 0 0
\(435\) −0.201402 −0.00965649
\(436\) 0 0
\(437\) −41.6833 −1.99398
\(438\) 0 0
\(439\) −9.59593 −0.457989 −0.228994 0.973428i \(-0.573544\pi\)
−0.228994 + 0.973428i \(0.573544\pi\)
\(440\) 0 0
\(441\) −8.64720 −0.411772
\(442\) 0 0
\(443\) −2.47072 −0.117388 −0.0586938 0.998276i \(-0.518694\pi\)
−0.0586938 + 0.998276i \(0.518694\pi\)
\(444\) 0 0
\(445\) 7.53678 0.357278
\(446\) 0 0
\(447\) −0.172102 −0.00814015
\(448\) 0 0
\(449\) 35.2324 1.66272 0.831359 0.555735i \(-0.187563\pi\)
0.831359 + 0.555735i \(0.187563\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.03084 −0.0954172
\(454\) 0 0
\(455\) 24.4033 1.14404
\(456\) 0 0
\(457\) 37.0241 1.73191 0.865957 0.500118i \(-0.166710\pi\)
0.865957 + 0.500118i \(0.166710\pi\)
\(458\) 0 0
\(459\) −2.94036 −0.137244
\(460\) 0 0
\(461\) −27.6324 −1.28697 −0.643484 0.765460i \(-0.722512\pi\)
−0.643484 + 0.765460i \(0.722512\pi\)
\(462\) 0 0
\(463\) −23.1877 −1.07763 −0.538813 0.842425i \(-0.681127\pi\)
−0.538813 + 0.842425i \(0.681127\pi\)
\(464\) 0 0
\(465\) 0.184675 0.00856412
\(466\) 0 0
\(467\) −27.5235 −1.27364 −0.636818 0.771014i \(-0.719750\pi\)
−0.636818 + 0.771014i \(0.719750\pi\)
\(468\) 0 0
\(469\) 30.4302 1.40514
\(470\) 0 0
\(471\) 1.24675 0.0574473
\(472\) 0 0
\(473\) 2.46213 0.113209
\(474\) 0 0
\(475\) −17.1565 −0.787192
\(476\) 0 0
\(477\) −27.9696 −1.28064
\(478\) 0 0
\(479\) −11.1088 −0.507576 −0.253788 0.967260i \(-0.581677\pi\)
−0.253788 + 0.967260i \(0.581677\pi\)
\(480\) 0 0
\(481\) 13.2011 0.601918
\(482\) 0 0
\(483\) −2.63803 −0.120034
\(484\) 0 0
\(485\) 9.73927 0.442238
\(486\) 0 0
\(487\) −31.3933 −1.42256 −0.711282 0.702906i \(-0.751885\pi\)
−0.711282 + 0.702906i \(0.751885\pi\)
\(488\) 0 0
\(489\) −1.02519 −0.0463608
\(490\) 0 0
\(491\) 39.9590 1.80333 0.901663 0.432440i \(-0.142347\pi\)
0.901663 + 0.432440i \(0.142347\pi\)
\(492\) 0 0
\(493\) −9.34081 −0.420689
\(494\) 0 0
\(495\) −5.86520 −0.263621
\(496\) 0 0
\(497\) −2.44079 −0.109484
\(498\) 0 0
\(499\) −16.0951 −0.720515 −0.360258 0.932853i \(-0.617311\pi\)
−0.360258 + 0.932853i \(0.617311\pi\)
\(500\) 0 0
\(501\) 1.30048 0.0581010
\(502\) 0 0
\(503\) −5.97717 −0.266509 −0.133254 0.991082i \(-0.542543\pi\)
−0.133254 + 0.991082i \(0.542543\pi\)
\(504\) 0 0
\(505\) −11.4587 −0.509907
\(506\) 0 0
\(507\) −2.94447 −0.130769
\(508\) 0 0
\(509\) 12.5726 0.557269 0.278634 0.960397i \(-0.410118\pi\)
0.278634 + 0.960397i \(0.410118\pi\)
\(510\) 0 0
\(511\) −26.2025 −1.15913
\(512\) 0 0
\(513\) 2.69220 0.118864
\(514\) 0 0
\(515\) −0.759032 −0.0334469
\(516\) 0 0
\(517\) 14.3028 0.629036
\(518\) 0 0
\(519\) −1.00610 −0.0441629
\(520\) 0 0
\(521\) 31.6233 1.38544 0.692722 0.721205i \(-0.256411\pi\)
0.692722 + 0.721205i \(0.256411\pi\)
\(522\) 0 0
\(523\) −21.5405 −0.941900 −0.470950 0.882160i \(-0.656089\pi\)
−0.470950 + 0.882160i \(0.656089\pi\)
\(524\) 0 0
\(525\) −1.08579 −0.0473876
\(526\) 0 0
\(527\) 8.56505 0.373099
\(528\) 0 0
\(529\) 54.8049 2.38282
\(530\) 0 0
\(531\) −26.3824 −1.14490
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 19.6632 0.850115
\(536\) 0 0
\(537\) −0.117475 −0.00506944
\(538\) 0 0
\(539\) 4.84463 0.208673
\(540\) 0 0
\(541\) 31.4927 1.35397 0.676987 0.735995i \(-0.263285\pi\)
0.676987 + 0.735995i \(0.263285\pi\)
\(542\) 0 0
\(543\) −0.371784 −0.0159548
\(544\) 0 0
\(545\) −4.50509 −0.192977
\(546\) 0 0
\(547\) 17.6861 0.756201 0.378100 0.925765i \(-0.376577\pi\)
0.378100 + 0.925765i \(0.376577\pi\)
\(548\) 0 0
\(549\) −37.7774 −1.61230
\(550\) 0 0
\(551\) 8.55248 0.364348
\(552\) 0 0
\(553\) 2.88022 0.122479
\(554\) 0 0
\(555\) 0.221560 0.00940472
\(556\) 0 0
\(557\) 19.3296 0.819021 0.409511 0.912305i \(-0.365699\pi\)
0.409511 + 0.912305i \(0.365699\pi\)
\(558\) 0 0
\(559\) −9.74235 −0.412057
\(560\) 0 0
\(561\) 0.822430 0.0347230
\(562\) 0 0
\(563\) −35.7299 −1.50583 −0.752917 0.658115i \(-0.771354\pi\)
−0.752917 + 0.658115i \(0.771354\pi\)
\(564\) 0 0
\(565\) −8.95293 −0.376653
\(566\) 0 0
\(567\) −28.0494 −1.17797
\(568\) 0 0
\(569\) 4.65187 0.195016 0.0975082 0.995235i \(-0.468913\pi\)
0.0975082 + 0.995235i \(0.468913\pi\)
\(570\) 0 0
\(571\) 20.6791 0.865393 0.432697 0.901540i \(-0.357562\pi\)
0.432697 + 0.901540i \(0.357562\pi\)
\(572\) 0 0
\(573\) −0.695813 −0.0290680
\(574\) 0 0
\(575\) 32.0238 1.33548
\(576\) 0 0
\(577\) 24.7808 1.03164 0.515820 0.856697i \(-0.327487\pi\)
0.515820 + 0.856697i \(0.327487\pi\)
\(578\) 0 0
\(579\) −1.26484 −0.0525649
\(580\) 0 0
\(581\) 31.7641 1.31780
\(582\) 0 0
\(583\) 15.6701 0.648989
\(584\) 0 0
\(585\) 23.2079 0.959528
\(586\) 0 0
\(587\) 13.4027 0.553187 0.276594 0.960987i \(-0.410794\pi\)
0.276594 + 0.960987i \(0.410794\pi\)
\(588\) 0 0
\(589\) −7.84219 −0.323132
\(590\) 0 0
\(591\) 1.48053 0.0609010
\(592\) 0 0
\(593\) 1.83911 0.0755233 0.0377616 0.999287i \(-0.487977\pi\)
0.0377616 + 0.999287i \(0.487977\pi\)
\(594\) 0 0
\(595\) 18.9955 0.778740
\(596\) 0 0
\(597\) −2.65285 −0.108574
\(598\) 0 0
\(599\) −35.9457 −1.46870 −0.734352 0.678769i \(-0.762514\pi\)
−0.734352 + 0.678769i \(0.762514\pi\)
\(600\) 0 0
\(601\) 29.8858 1.21907 0.609533 0.792760i \(-0.291357\pi\)
0.609533 + 0.792760i \(0.291357\pi\)
\(602\) 0 0
\(603\) 28.9396 1.17851
\(604\) 0 0
\(605\) −9.58671 −0.389755
\(606\) 0 0
\(607\) 4.25015 0.172508 0.0862541 0.996273i \(-0.472510\pi\)
0.0862541 + 0.996273i \(0.472510\pi\)
\(608\) 0 0
\(609\) 0.541264 0.0219331
\(610\) 0 0
\(611\) −56.5944 −2.28956
\(612\) 0 0
\(613\) −9.35184 −0.377717 −0.188859 0.982004i \(-0.560479\pi\)
−0.188859 + 0.982004i \(0.560479\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.6019 0.507332 0.253666 0.967292i \(-0.418364\pi\)
0.253666 + 0.967292i \(0.418364\pi\)
\(618\) 0 0
\(619\) −2.54018 −0.102098 −0.0510491 0.998696i \(-0.516257\pi\)
−0.0510491 + 0.998696i \(0.516257\pi\)
\(620\) 0 0
\(621\) −5.02519 −0.201654
\(622\) 0 0
\(623\) −20.2549 −0.811497
\(624\) 0 0
\(625\) 6.33331 0.253332
\(626\) 0 0
\(627\) −0.753020 −0.0300727
\(628\) 0 0
\(629\) 10.2757 0.409720
\(630\) 0 0
\(631\) 48.0057 1.91108 0.955538 0.294867i \(-0.0952752\pi\)
0.955538 + 0.294867i \(0.0952752\pi\)
\(632\) 0 0
\(633\) 1.72580 0.0685945
\(634\) 0 0
\(635\) 17.4607 0.692908
\(636\) 0 0
\(637\) −19.1696 −0.759528
\(638\) 0 0
\(639\) −2.32123 −0.0918262
\(640\) 0 0
\(641\) −1.02148 −0.0403461 −0.0201730 0.999797i \(-0.506422\pi\)
−0.0201730 + 0.999797i \(0.506422\pi\)
\(642\) 0 0
\(643\) 8.40606 0.331503 0.165751 0.986168i \(-0.446995\pi\)
0.165751 + 0.986168i \(0.446995\pi\)
\(644\) 0 0
\(645\) −0.163511 −0.00643823
\(646\) 0 0
\(647\) 5.86778 0.230686 0.115343 0.993326i \(-0.463203\pi\)
0.115343 + 0.993326i \(0.463203\pi\)
\(648\) 0 0
\(649\) 14.7808 0.580198
\(650\) 0 0
\(651\) −0.496312 −0.0194520
\(652\) 0 0
\(653\) −37.5749 −1.47042 −0.735209 0.677841i \(-0.762916\pi\)
−0.735209 + 0.677841i \(0.762916\pi\)
\(654\) 0 0
\(655\) 4.90355 0.191598
\(656\) 0 0
\(657\) −24.9190 −0.972183
\(658\) 0 0
\(659\) 16.8360 0.655839 0.327919 0.944706i \(-0.393653\pi\)
0.327919 + 0.944706i \(0.393653\pi\)
\(660\) 0 0
\(661\) 4.84983 0.188637 0.0943183 0.995542i \(-0.469933\pi\)
0.0943183 + 0.995542i \(0.469933\pi\)
\(662\) 0 0
\(663\) −3.25426 −0.126385
\(664\) 0 0
\(665\) −17.3924 −0.674447
\(666\) 0 0
\(667\) −15.9638 −0.618122
\(668\) 0 0
\(669\) 0.623651 0.0241117
\(670\) 0 0
\(671\) 21.1649 0.817062
\(672\) 0 0
\(673\) −6.24065 −0.240559 −0.120280 0.992740i \(-0.538379\pi\)
−0.120280 + 0.992740i \(0.538379\pi\)
\(674\) 0 0
\(675\) −2.06832 −0.0796098
\(676\) 0 0
\(677\) −29.3566 −1.12827 −0.564134 0.825683i \(-0.690790\pi\)
−0.564134 + 0.825683i \(0.690790\pi\)
\(678\) 0 0
\(679\) −26.1741 −1.00447
\(680\) 0 0
\(681\) 2.23360 0.0855918
\(682\) 0 0
\(683\) 25.6861 0.982849 0.491425 0.870920i \(-0.336476\pi\)
0.491425 + 0.870920i \(0.336476\pi\)
\(684\) 0 0
\(685\) −4.79783 −0.183316
\(686\) 0 0
\(687\) 1.15542 0.0440820
\(688\) 0 0
\(689\) −62.0046 −2.36219
\(690\) 0 0
\(691\) −8.77128 −0.333675 −0.166838 0.985984i \(-0.553356\pi\)
−0.166838 + 0.985984i \(0.553356\pi\)
\(692\) 0 0
\(693\) 15.7626 0.598772
\(694\) 0 0
\(695\) −14.0218 −0.531877
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −2.07706 −0.0785616
\(700\) 0 0
\(701\) −15.1075 −0.570602 −0.285301 0.958438i \(-0.592094\pi\)
−0.285301 + 0.958438i \(0.592094\pi\)
\(702\) 0 0
\(703\) −9.40850 −0.354848
\(704\) 0 0
\(705\) −0.949851 −0.0357735
\(706\) 0 0
\(707\) 30.7951 1.15817
\(708\) 0 0
\(709\) 12.0132 0.451165 0.225583 0.974224i \(-0.427571\pi\)
0.225583 + 0.974224i \(0.427571\pi\)
\(710\) 0 0
\(711\) 2.73913 0.102725
\(712\) 0 0
\(713\) 14.6380 0.548198
\(714\) 0 0
\(715\) −13.0023 −0.486259
\(716\) 0 0
\(717\) 2.05372 0.0766977
\(718\) 0 0
\(719\) 2.27778 0.0849468 0.0424734 0.999098i \(-0.486476\pi\)
0.0424734 + 0.999098i \(0.486476\pi\)
\(720\) 0 0
\(721\) 2.03988 0.0759692
\(722\) 0 0
\(723\) −1.20109 −0.0446689
\(724\) 0 0
\(725\) −6.57056 −0.244025
\(726\) 0 0
\(727\) −7.71952 −0.286301 −0.143151 0.989701i \(-0.545723\pi\)
−0.143151 + 0.989701i \(0.545723\pi\)
\(728\) 0 0
\(729\) −26.5129 −0.981960
\(730\) 0 0
\(731\) −7.58345 −0.280484
\(732\) 0 0
\(733\) 33.2753 1.22905 0.614526 0.788896i \(-0.289347\pi\)
0.614526 + 0.788896i \(0.289347\pi\)
\(734\) 0 0
\(735\) −0.321733 −0.0118673
\(736\) 0 0
\(737\) −16.2135 −0.597233
\(738\) 0 0
\(739\) 48.0057 1.76592 0.882959 0.469450i \(-0.155548\pi\)
0.882959 + 0.469450i \(0.155548\pi\)
\(740\) 0 0
\(741\) 2.97961 0.109459
\(742\) 0 0
\(743\) 43.1345 1.58245 0.791226 0.611524i \(-0.209443\pi\)
0.791226 + 0.611524i \(0.209443\pi\)
\(744\) 0 0
\(745\) 2.11793 0.0775948
\(746\) 0 0
\(747\) 30.2082 1.10526
\(748\) 0 0
\(749\) −52.8445 −1.93090
\(750\) 0 0
\(751\) 6.46883 0.236051 0.118025 0.993011i \(-0.462344\pi\)
0.118025 + 0.993011i \(0.462344\pi\)
\(752\) 0 0
\(753\) −1.97512 −0.0719775
\(754\) 0 0
\(755\) 24.9920 0.909550
\(756\) 0 0
\(757\) 39.5531 1.43758 0.718790 0.695227i \(-0.244696\pi\)
0.718790 + 0.695227i \(0.244696\pi\)
\(758\) 0 0
\(759\) 1.40557 0.0510189
\(760\) 0 0
\(761\) 28.5130 1.03360 0.516799 0.856107i \(-0.327124\pi\)
0.516799 + 0.856107i \(0.327124\pi\)
\(762\) 0 0
\(763\) 12.1073 0.438315
\(764\) 0 0
\(765\) 18.0650 0.653142
\(766\) 0 0
\(767\) −58.4859 −2.11180
\(768\) 0 0
\(769\) 49.7184 1.79289 0.896445 0.443154i \(-0.146141\pi\)
0.896445 + 0.443154i \(0.146141\pi\)
\(770\) 0 0
\(771\) 1.40936 0.0507569
\(772\) 0 0
\(773\) 24.2539 0.872351 0.436175 0.899862i \(-0.356333\pi\)
0.436175 + 0.899862i \(0.356333\pi\)
\(774\) 0 0
\(775\) 6.02488 0.216420
\(776\) 0 0
\(777\) −0.595439 −0.0213613
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.30048 0.0465347
\(782\) 0 0
\(783\) 1.03106 0.0368470
\(784\) 0 0
\(785\) −15.3428 −0.547608
\(786\) 0 0
\(787\) 0.907346 0.0323434 0.0161717 0.999869i \(-0.494852\pi\)
0.0161717 + 0.999869i \(0.494852\pi\)
\(788\) 0 0
\(789\) 1.75139 0.0623512
\(790\) 0 0
\(791\) 24.0608 0.855504
\(792\) 0 0
\(793\) −83.7470 −2.97394
\(794\) 0 0
\(795\) −1.04065 −0.0369082
\(796\) 0 0
\(797\) −32.4934 −1.15098 −0.575488 0.817810i \(-0.695188\pi\)
−0.575488 + 0.817810i \(0.695188\pi\)
\(798\) 0 0
\(799\) −44.0531 −1.55849
\(800\) 0 0
\(801\) −19.2628 −0.680616
\(802\) 0 0
\(803\) 13.9610 0.492672
\(804\) 0 0
\(805\) 32.4641 1.14421
\(806\) 0 0
\(807\) 2.46213 0.0866711
\(808\) 0 0
\(809\) 2.07804 0.0730602 0.0365301 0.999333i \(-0.488370\pi\)
0.0365301 + 0.999333i \(0.488370\pi\)
\(810\) 0 0
\(811\) −13.7518 −0.482893 −0.241446 0.970414i \(-0.577622\pi\)
−0.241446 + 0.970414i \(0.577622\pi\)
\(812\) 0 0
\(813\) 1.66431 0.0583697
\(814\) 0 0
\(815\) 12.6162 0.441927
\(816\) 0 0
\(817\) 6.94344 0.242920
\(818\) 0 0
\(819\) −62.3707 −2.17941
\(820\) 0 0
\(821\) 6.76979 0.236267 0.118134 0.992998i \(-0.462309\pi\)
0.118134 + 0.992998i \(0.462309\pi\)
\(822\) 0 0
\(823\) −22.3575 −0.779335 −0.389667 0.920956i \(-0.627410\pi\)
−0.389667 + 0.920956i \(0.627410\pi\)
\(824\) 0 0
\(825\) 0.578518 0.0201414
\(826\) 0 0
\(827\) −26.1160 −0.908143 −0.454072 0.890965i \(-0.650029\pi\)
−0.454072 + 0.890965i \(0.650029\pi\)
\(828\) 0 0
\(829\) −19.7540 −0.686085 −0.343043 0.939320i \(-0.611458\pi\)
−0.343043 + 0.939320i \(0.611458\pi\)
\(830\) 0 0
\(831\) −0.834144 −0.0289361
\(832\) 0 0
\(833\) −14.9216 −0.517004
\(834\) 0 0
\(835\) −16.0039 −0.553839
\(836\) 0 0
\(837\) −0.945427 −0.0326787
\(838\) 0 0
\(839\) −20.8188 −0.718745 −0.359373 0.933194i \(-0.617009\pi\)
−0.359373 + 0.933194i \(0.617009\pi\)
\(840\) 0 0
\(841\) −25.7246 −0.887054
\(842\) 0 0
\(843\) 1.54256 0.0531287
\(844\) 0 0
\(845\) 36.2353 1.24653
\(846\) 0 0
\(847\) 25.7641 0.885265
\(848\) 0 0
\(849\) 1.13941 0.0391044
\(850\) 0 0
\(851\) 17.5617 0.602006
\(852\) 0 0
\(853\) 48.6666 1.66631 0.833157 0.553037i \(-0.186531\pi\)
0.833157 + 0.553037i \(0.186531\pi\)
\(854\) 0 0
\(855\) −16.5404 −0.565670
\(856\) 0 0
\(857\) −12.8515 −0.439001 −0.219500 0.975612i \(-0.570443\pi\)
−0.219500 + 0.975612i \(0.570443\pi\)
\(858\) 0 0
\(859\) −1.60904 −0.0548998 −0.0274499 0.999623i \(-0.508739\pi\)
−0.0274499 + 0.999623i \(0.508739\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −46.0494 −1.56754 −0.783770 0.621052i \(-0.786706\pi\)
−0.783770 + 0.621052i \(0.786706\pi\)
\(864\) 0 0
\(865\) 12.3813 0.420976
\(866\) 0 0
\(867\) −0.916518 −0.0311266
\(868\) 0 0
\(869\) −1.53461 −0.0520581
\(870\) 0 0
\(871\) 64.1549 2.17381
\(872\) 0 0
\(873\) −24.8920 −0.842465
\(874\) 0 0
\(875\) 31.7641 1.07382
\(876\) 0 0
\(877\) 3.63392 0.122709 0.0613543 0.998116i \(-0.480458\pi\)
0.0613543 + 0.998116i \(0.480458\pi\)
\(878\) 0 0
\(879\) −1.06995 −0.0360887
\(880\) 0 0
\(881\) 28.2426 0.951519 0.475759 0.879575i \(-0.342173\pi\)
0.475759 + 0.879575i \(0.342173\pi\)
\(882\) 0 0
\(883\) 4.23962 0.142674 0.0713372 0.997452i \(-0.477273\pi\)
0.0713372 + 0.997452i \(0.477273\pi\)
\(884\) 0 0
\(885\) −0.981598 −0.0329961
\(886\) 0 0
\(887\) 46.4003 1.55797 0.778984 0.627043i \(-0.215735\pi\)
0.778984 + 0.627043i \(0.215735\pi\)
\(888\) 0 0
\(889\) −46.9254 −1.57383
\(890\) 0 0
\(891\) 14.9450 0.500677
\(892\) 0 0
\(893\) 40.3352 1.34976
\(894\) 0 0
\(895\) 1.44568 0.0483237
\(896\) 0 0
\(897\) −5.56166 −0.185698
\(898\) 0 0
\(899\) −3.00339 −0.100169
\(900\) 0 0
\(901\) −48.2644 −1.60792
\(902\) 0 0
\(903\) 0.439432 0.0146234
\(904\) 0 0
\(905\) 4.57526 0.152087
\(906\) 0 0
\(907\) −27.0358 −0.897708 −0.448854 0.893605i \(-0.648168\pi\)
−0.448854 + 0.893605i \(0.648168\pi\)
\(908\) 0 0
\(909\) 29.2866 0.971376
\(910\) 0 0
\(911\) −42.3500 −1.40312 −0.701559 0.712612i \(-0.747512\pi\)
−0.701559 + 0.712612i \(0.747512\pi\)
\(912\) 0 0
\(913\) −16.9243 −0.560111
\(914\) 0 0
\(915\) −1.40557 −0.0464666
\(916\) 0 0
\(917\) −13.1782 −0.435183
\(918\) 0 0
\(919\) −8.17400 −0.269635 −0.134818 0.990870i \(-0.543045\pi\)
−0.134818 + 0.990870i \(0.543045\pi\)
\(920\) 0 0
\(921\) −0.818991 −0.0269867
\(922\) 0 0
\(923\) −5.14582 −0.169377
\(924\) 0 0
\(925\) 7.22821 0.237662
\(926\) 0 0
\(927\) 1.93996 0.0637166
\(928\) 0 0
\(929\) −42.2596 −1.38649 −0.693247 0.720700i \(-0.743820\pi\)
−0.693247 + 0.720700i \(0.743820\pi\)
\(930\) 0 0
\(931\) 13.6623 0.447764
\(932\) 0 0
\(933\) 3.01994 0.0988684
\(934\) 0 0
\(935\) −10.1210 −0.330992
\(936\) 0 0
\(937\) −1.16600 −0.0380917 −0.0190458 0.999819i \(-0.506063\pi\)
−0.0190458 + 0.999819i \(0.506063\pi\)
\(938\) 0 0
\(939\) 0.210896 0.00688234
\(940\) 0 0
\(941\) 4.93864 0.160995 0.0804975 0.996755i \(-0.474349\pi\)
0.0804975 + 0.996755i \(0.474349\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.09676 −0.0682077
\(946\) 0 0
\(947\) 25.3210 0.822822 0.411411 0.911450i \(-0.365036\pi\)
0.411411 + 0.911450i \(0.365036\pi\)
\(948\) 0 0
\(949\) −55.2419 −1.79323
\(950\) 0 0
\(951\) −1.74922 −0.0567224
\(952\) 0 0
\(953\) −47.6143 −1.54238 −0.771189 0.636606i \(-0.780338\pi\)
−0.771189 + 0.636606i \(0.780338\pi\)
\(954\) 0 0
\(955\) 8.56283 0.277087
\(956\) 0 0
\(957\) −0.288391 −0.00932235
\(958\) 0 0
\(959\) 12.8941 0.416371
\(960\) 0 0
\(961\) −28.2460 −0.911163
\(962\) 0 0
\(963\) −50.2559 −1.61947
\(964\) 0 0
\(965\) 15.5654 0.501067
\(966\) 0 0
\(967\) 25.9763 0.835342 0.417671 0.908598i \(-0.362847\pi\)
0.417671 + 0.908598i \(0.362847\pi\)
\(968\) 0 0
\(969\) 2.31933 0.0745076
\(970\) 0 0
\(971\) 55.8687 1.79291 0.896457 0.443132i \(-0.146133\pi\)
0.896457 + 0.443132i \(0.146133\pi\)
\(972\) 0 0
\(973\) 37.6833 1.20807
\(974\) 0 0
\(975\) −2.28913 −0.0733107
\(976\) 0 0
\(977\) 42.5241 1.36047 0.680233 0.732996i \(-0.261879\pi\)
0.680233 + 0.732996i \(0.261879\pi\)
\(978\) 0 0
\(979\) 10.7920 0.344915
\(980\) 0 0
\(981\) 11.5143 0.367622
\(982\) 0 0
\(983\) 50.8977 1.62338 0.811692 0.584086i \(-0.198547\pi\)
0.811692 + 0.584086i \(0.198547\pi\)
\(984\) 0 0
\(985\) −18.2198 −0.580530
\(986\) 0 0
\(987\) 2.55271 0.0812535
\(988\) 0 0
\(989\) −12.9604 −0.412118
\(990\) 0 0
\(991\) 27.7147 0.880387 0.440194 0.897903i \(-0.354910\pi\)
0.440194 + 0.897903i \(0.354910\pi\)
\(992\) 0 0
\(993\) 0.961973 0.0305273
\(994\) 0 0
\(995\) 32.6466 1.03497
\(996\) 0 0
\(997\) 22.3480 0.707768 0.353884 0.935289i \(-0.384861\pi\)
0.353884 + 0.935289i \(0.384861\pi\)
\(998\) 0 0
\(999\) −1.13426 −0.0358863
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 6724.2.a.c.1.3 4
41.40 even 2 164.2.a.a.1.2 4
123.122 odd 2 1476.2.a.g.1.3 4
164.163 odd 2 656.2.a.i.1.3 4
205.122 odd 4 4100.2.d.c.1149.4 8
205.163 odd 4 4100.2.d.c.1149.5 8
205.204 even 2 4100.2.a.c.1.3 4
287.286 odd 2 8036.2.a.i.1.3 4
328.163 odd 2 2624.2.a.y.1.2 4
328.245 even 2 2624.2.a.v.1.3 4
492.491 even 2 5904.2.a.bp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.2 4 41.40 even 2
656.2.a.i.1.3 4 164.163 odd 2
1476.2.a.g.1.3 4 123.122 odd 2
2624.2.a.v.1.3 4 328.245 even 2
2624.2.a.y.1.2 4 328.163 odd 2
4100.2.a.c.1.3 4 205.204 even 2
4100.2.d.c.1149.4 8 205.122 odd 4
4100.2.d.c.1149.5 8 205.163 odd 4
5904.2.a.bp.1.3 4 492.491 even 2
6724.2.a.c.1.3 4 1.1 even 1 trivial
8036.2.a.i.1.3 4 287.286 odd 2