Properties

Label 2960.2.a.u.1.2
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.91729\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.406728 q^{3} -1.00000 q^{5} -2.91729 q^{7} -2.83457 q^{9} +O(q^{10})\) \(q-0.406728 q^{3} -1.00000 q^{5} -2.91729 q^{7} -2.83457 q^{9} -6.51056 q^{11} -0.813457 q^{13} +0.406728 q^{15} -2.51056 q^{17} -0.406728 q^{19} +1.18654 q^{21} -5.02112 q^{23} +1.00000 q^{25} +2.37309 q^{27} -5.32401 q^{29} +8.75186 q^{31} +2.64803 q^{33} +2.91729 q^{35} -1.00000 q^{37} +0.330856 q^{39} +6.34513 q^{41} +7.32401 q^{43} +2.83457 q^{45} +5.42784 q^{47} +1.51056 q^{49} +1.02112 q^{51} -2.34513 q^{53} +6.51056 q^{55} +0.165428 q^{57} +1.42784 q^{59} -1.32401 q^{61} +8.26926 q^{63} +0.813457 q^{65} +5.42784 q^{67} +2.04223 q^{69} -14.6480 q^{71} -11.0211 q^{73} -0.406728 q^{75} +18.9932 q^{77} +1.75870 q^{79} +7.53851 q^{81} +7.05476 q^{83} +2.51056 q^{85} +2.16543 q^{87} +6.00000 q^{89} +2.37309 q^{91} -3.55963 q^{93} +0.406728 q^{95} +2.34513 q^{97} +18.4546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45} - 2 q^{47} - 4 q^{49} - 14 q^{51} + 21 q^{53} + 11 q^{55} + 20 q^{57} - 14 q^{59} + 7 q^{61} + 37 q^{63} - 2 q^{67} - 28 q^{69} - 22 q^{71} - 16 q^{73} + 7 q^{77} + 26 q^{79} + 47 q^{81} - 2 q^{83} - q^{85} + 26 q^{87} + 18 q^{89} + 12 q^{91} - 18 q^{93} - 21 q^{97} - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.406728 −0.234825 −0.117412 0.993083i \(-0.537460\pi\)
−0.117412 + 0.993083i \(0.537460\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.91729 −1.10263 −0.551315 0.834297i \(-0.685874\pi\)
−0.551315 + 0.834297i \(0.685874\pi\)
\(8\) 0 0
\(9\) −2.83457 −0.944857
\(10\) 0 0
\(11\) −6.51056 −1.96301 −0.981503 0.191444i \(-0.938683\pi\)
−0.981503 + 0.191444i \(0.938683\pi\)
\(12\) 0 0
\(13\) −0.813457 −0.225612 −0.112806 0.993617i \(-0.535984\pi\)
−0.112806 + 0.993617i \(0.535984\pi\)
\(14\) 0 0
\(15\) 0.406728 0.105017
\(16\) 0 0
\(17\) −2.51056 −0.608900 −0.304450 0.952528i \(-0.598473\pi\)
−0.304450 + 0.952528i \(0.598473\pi\)
\(18\) 0 0
\(19\) −0.406728 −0.0933099 −0.0466550 0.998911i \(-0.514856\pi\)
−0.0466550 + 0.998911i \(0.514856\pi\)
\(20\) 0 0
\(21\) 1.18654 0.258925
\(22\) 0 0
\(23\) −5.02112 −1.04697 −0.523487 0.852033i \(-0.675369\pi\)
−0.523487 + 0.852033i \(0.675369\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.37309 0.456701
\(28\) 0 0
\(29\) −5.32401 −0.988645 −0.494322 0.869279i \(-0.664584\pi\)
−0.494322 + 0.869279i \(0.664584\pi\)
\(30\) 0 0
\(31\) 8.75186 1.57188 0.785940 0.618303i \(-0.212179\pi\)
0.785940 + 0.618303i \(0.212179\pi\)
\(32\) 0 0
\(33\) 2.64803 0.460963
\(34\) 0 0
\(35\) 2.91729 0.493111
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0.330856 0.0529794
\(40\) 0 0
\(41\) 6.34513 0.990943 0.495471 0.868624i \(-0.334995\pi\)
0.495471 + 0.868624i \(0.334995\pi\)
\(42\) 0 0
\(43\) 7.32401 1.11690 0.558451 0.829538i \(-0.311396\pi\)
0.558451 + 0.829538i \(0.311396\pi\)
\(44\) 0 0
\(45\) 2.83457 0.422553
\(46\) 0 0
\(47\) 5.42784 0.791732 0.395866 0.918308i \(-0.370444\pi\)
0.395866 + 0.918308i \(0.370444\pi\)
\(48\) 0 0
\(49\) 1.51056 0.215794
\(50\) 0 0
\(51\) 1.02112 0.142985
\(52\) 0 0
\(53\) −2.34513 −0.322128 −0.161064 0.986944i \(-0.551493\pi\)
−0.161064 + 0.986944i \(0.551493\pi\)
\(54\) 0 0
\(55\) 6.51056 0.877883
\(56\) 0 0
\(57\) 0.165428 0.0219115
\(58\) 0 0
\(59\) 1.42784 0.185889 0.0929447 0.995671i \(-0.470372\pi\)
0.0929447 + 0.995671i \(0.470372\pi\)
\(60\) 0 0
\(61\) −1.32401 −0.169523 −0.0847613 0.996401i \(-0.527013\pi\)
−0.0847613 + 0.996401i \(0.527013\pi\)
\(62\) 0 0
\(63\) 8.26926 1.04183
\(64\) 0 0
\(65\) 0.813457 0.100897
\(66\) 0 0
\(67\) 5.42784 0.663117 0.331558 0.943435i \(-0.392426\pi\)
0.331558 + 0.943435i \(0.392426\pi\)
\(68\) 0 0
\(69\) 2.04223 0.245856
\(70\) 0 0
\(71\) −14.6480 −1.73840 −0.869201 0.494460i \(-0.835366\pi\)
−0.869201 + 0.494460i \(0.835366\pi\)
\(72\) 0 0
\(73\) −11.0211 −1.28992 −0.644962 0.764215i \(-0.723127\pi\)
−0.644962 + 0.764215i \(0.723127\pi\)
\(74\) 0 0
\(75\) −0.406728 −0.0469650
\(76\) 0 0
\(77\) 18.9932 2.16447
\(78\) 0 0
\(79\) 1.75870 0.197869 0.0989346 0.995094i \(-0.468457\pi\)
0.0989346 + 0.995094i \(0.468457\pi\)
\(80\) 0 0
\(81\) 7.53851 0.837613
\(82\) 0 0
\(83\) 7.05476 0.774360 0.387180 0.922004i \(-0.373449\pi\)
0.387180 + 0.922004i \(0.373449\pi\)
\(84\) 0 0
\(85\) 2.51056 0.272308
\(86\) 0 0
\(87\) 2.16543 0.232158
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.37309 0.248767
\(92\) 0 0
\(93\) −3.55963 −0.369116
\(94\) 0 0
\(95\) 0.406728 0.0417295
\(96\) 0 0
\(97\) 2.34513 0.238112 0.119056 0.992888i \(-0.462013\pi\)
0.119056 + 0.992888i \(0.462013\pi\)
\(98\) 0 0
\(99\) 18.4546 1.85476
\(100\) 0 0
\(101\) −8.20766 −0.816693 −0.408346 0.912827i \(-0.633894\pi\)
−0.408346 + 0.912827i \(0.633894\pi\)
\(102\) 0 0
\(103\) 8.81346 0.868416 0.434208 0.900813i \(-0.357028\pi\)
0.434208 + 0.900813i \(0.357028\pi\)
\(104\) 0 0
\(105\) −1.18654 −0.115795
\(106\) 0 0
\(107\) −15.2624 −1.47547 −0.737737 0.675089i \(-0.764105\pi\)
−0.737737 + 0.675089i \(0.764105\pi\)
\(108\) 0 0
\(109\) 4.30290 0.412143 0.206072 0.978537i \(-0.433932\pi\)
0.206072 + 0.978537i \(0.433932\pi\)
\(110\) 0 0
\(111\) 0.406728 0.0386050
\(112\) 0 0
\(113\) −9.15859 −0.861567 −0.430784 0.902455i \(-0.641763\pi\)
−0.430784 + 0.902455i \(0.641763\pi\)
\(114\) 0 0
\(115\) 5.02112 0.468221
\(116\) 0 0
\(117\) 2.30580 0.213171
\(118\) 0 0
\(119\) 7.32401 0.671391
\(120\) 0 0
\(121\) 31.3874 2.85340
\(122\) 0 0
\(123\) −2.58074 −0.232698
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.61439 0.764403 0.382202 0.924079i \(-0.375166\pi\)
0.382202 + 0.924079i \(0.375166\pi\)
\(128\) 0 0
\(129\) −2.97888 −0.262276
\(130\) 0 0
\(131\) −13.4278 −1.17320 −0.586598 0.809878i \(-0.699533\pi\)
−0.586598 + 0.809878i \(0.699533\pi\)
\(132\) 0 0
\(133\) 1.18654 0.102886
\(134\) 0 0
\(135\) −2.37309 −0.204243
\(136\) 0 0
\(137\) −14.6903 −1.25507 −0.627537 0.778587i \(-0.715937\pi\)
−0.627537 + 0.778587i \(0.715937\pi\)
\(138\) 0 0
\(139\) −17.3662 −1.47299 −0.736493 0.676445i \(-0.763519\pi\)
−0.736493 + 0.676445i \(0.763519\pi\)
\(140\) 0 0
\(141\) −2.20766 −0.185918
\(142\) 0 0
\(143\) 5.29606 0.442879
\(144\) 0 0
\(145\) 5.32401 0.442135
\(146\) 0 0
\(147\) −0.614387 −0.0506738
\(148\) 0 0
\(149\) −0.207658 −0.0170120 −0.00850601 0.999964i \(-0.502708\pi\)
−0.00850601 + 0.999964i \(0.502708\pi\)
\(150\) 0 0
\(151\) −0.813457 −0.0661982 −0.0330991 0.999452i \(-0.510538\pi\)
−0.0330991 + 0.999452i \(0.510538\pi\)
\(152\) 0 0
\(153\) 7.11636 0.575323
\(154\) 0 0
\(155\) −8.75186 −0.702966
\(156\) 0 0
\(157\) 5.32401 0.424903 0.212451 0.977172i \(-0.431855\pi\)
0.212451 + 0.977172i \(0.431855\pi\)
\(158\) 0 0
\(159\) 0.953831 0.0756437
\(160\) 0 0
\(161\) 14.6480 1.15443
\(162\) 0 0
\(163\) −15.2008 −1.19062 −0.595310 0.803496i \(-0.702971\pi\)
−0.595310 + 0.803496i \(0.702971\pi\)
\(164\) 0 0
\(165\) −2.64803 −0.206149
\(166\) 0 0
\(167\) 12.4826 0.965933 0.482966 0.875639i \(-0.339559\pi\)
0.482966 + 0.875639i \(0.339559\pi\)
\(168\) 0 0
\(169\) −12.3383 −0.949099
\(170\) 0 0
\(171\) 1.15290 0.0881645
\(172\) 0 0
\(173\) 23.0354 1.75135 0.875674 0.482903i \(-0.160417\pi\)
0.875674 + 0.482903i \(0.160417\pi\)
\(174\) 0 0
\(175\) −2.91729 −0.220526
\(176\) 0 0
\(177\) −0.580745 −0.0436514
\(178\) 0 0
\(179\) −24.2835 −1.81504 −0.907518 0.420013i \(-0.862026\pi\)
−0.907518 + 0.420013i \(0.862026\pi\)
\(180\) 0 0
\(181\) 2.16543 0.160955 0.0804775 0.996756i \(-0.474355\pi\)
0.0804775 + 0.996756i \(0.474355\pi\)
\(182\) 0 0
\(183\) 0.538514 0.0398081
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 16.3451 1.19527
\(188\) 0 0
\(189\) −6.92297 −0.503572
\(190\) 0 0
\(191\) 19.3326 1.39886 0.699429 0.714702i \(-0.253438\pi\)
0.699429 + 0.714702i \(0.253438\pi\)
\(192\) 0 0
\(193\) −20.0422 −1.44267 −0.721336 0.692586i \(-0.756471\pi\)
−0.721336 + 0.692586i \(0.756471\pi\)
\(194\) 0 0
\(195\) −0.330856 −0.0236931
\(196\) 0 0
\(197\) 15.6269 1.11337 0.556686 0.830723i \(-0.312073\pi\)
0.556686 + 0.830723i \(0.312073\pi\)
\(198\) 0 0
\(199\) −20.2835 −1.43786 −0.718931 0.695082i \(-0.755368\pi\)
−0.718931 + 0.695082i \(0.755368\pi\)
\(200\) 0 0
\(201\) −2.20766 −0.155716
\(202\) 0 0
\(203\) 15.5317 1.09011
\(204\) 0 0
\(205\) −6.34513 −0.443163
\(206\) 0 0
\(207\) 14.2327 0.989242
\(208\) 0 0
\(209\) 2.64803 0.183168
\(210\) 0 0
\(211\) −18.7182 −1.28862 −0.644308 0.764766i \(-0.722854\pi\)
−0.644308 + 0.764766i \(0.722854\pi\)
\(212\) 0 0
\(213\) 5.95777 0.408220
\(214\) 0 0
\(215\) −7.32401 −0.499494
\(216\) 0 0
\(217\) −25.5317 −1.73320
\(218\) 0 0
\(219\) 4.48260 0.302906
\(220\) 0 0
\(221\) 2.04223 0.137375
\(222\) 0 0
\(223\) 15.7307 1.05341 0.526704 0.850049i \(-0.323428\pi\)
0.526704 + 0.850049i \(0.323428\pi\)
\(224\) 0 0
\(225\) −2.83457 −0.188971
\(226\) 0 0
\(227\) −12.8836 −0.855117 −0.427559 0.903988i \(-0.640626\pi\)
−0.427559 + 0.903988i \(0.640626\pi\)
\(228\) 0 0
\(229\) 5.25383 0.347183 0.173591 0.984818i \(-0.444463\pi\)
0.173591 + 0.984818i \(0.444463\pi\)
\(230\) 0 0
\(231\) −7.72506 −0.508271
\(232\) 0 0
\(233\) 28.3172 1.85512 0.927560 0.373675i \(-0.121902\pi\)
0.927560 + 0.373675i \(0.121902\pi\)
\(234\) 0 0
\(235\) −5.42784 −0.354073
\(236\) 0 0
\(237\) −0.715313 −0.0464646
\(238\) 0 0
\(239\) −14.3788 −0.930085 −0.465043 0.885288i \(-0.653961\pi\)
−0.465043 + 0.885288i \(0.653961\pi\)
\(240\) 0 0
\(241\) 18.2749 1.17719 0.588596 0.808427i \(-0.299681\pi\)
0.588596 + 0.808427i \(0.299681\pi\)
\(242\) 0 0
\(243\) −10.1854 −0.653393
\(244\) 0 0
\(245\) −1.51056 −0.0965060
\(246\) 0 0
\(247\) 0.330856 0.0210519
\(248\) 0 0
\(249\) −2.86937 −0.181839
\(250\) 0 0
\(251\) 1.22019 0.0770174 0.0385087 0.999258i \(-0.487739\pi\)
0.0385087 + 0.999258i \(0.487739\pi\)
\(252\) 0 0
\(253\) 32.6903 2.05522
\(254\) 0 0
\(255\) −1.02112 −0.0639447
\(256\) 0 0
\(257\) 27.1306 1.69236 0.846181 0.532895i \(-0.178896\pi\)
0.846181 + 0.532895i \(0.178896\pi\)
\(258\) 0 0
\(259\) 2.91729 0.181271
\(260\) 0 0
\(261\) 15.0913 0.934128
\(262\) 0 0
\(263\) 20.2133 1.24641 0.623204 0.782059i \(-0.285831\pi\)
0.623204 + 0.782059i \(0.285831\pi\)
\(264\) 0 0
\(265\) 2.34513 0.144060
\(266\) 0 0
\(267\) −2.44037 −0.149348
\(268\) 0 0
\(269\) −0.207658 −0.0126611 −0.00633057 0.999980i \(-0.502015\pi\)
−0.00633057 + 0.999980i \(0.502015\pi\)
\(270\) 0 0
\(271\) 30.6480 1.86174 0.930868 0.365357i \(-0.119053\pi\)
0.930868 + 0.365357i \(0.119053\pi\)
\(272\) 0 0
\(273\) −0.965202 −0.0584167
\(274\) 0 0
\(275\) −6.51056 −0.392601
\(276\) 0 0
\(277\) −16.8135 −1.01022 −0.505111 0.863054i \(-0.668549\pi\)
−0.505111 + 0.863054i \(0.668549\pi\)
\(278\) 0 0
\(279\) −24.8078 −1.48520
\(280\) 0 0
\(281\) −14.2749 −0.851572 −0.425786 0.904824i \(-0.640002\pi\)
−0.425786 + 0.904824i \(0.640002\pi\)
\(282\) 0 0
\(283\) 13.2288 0.786369 0.393184 0.919460i \(-0.371373\pi\)
0.393184 + 0.919460i \(0.371373\pi\)
\(284\) 0 0
\(285\) −0.165428 −0.00979911
\(286\) 0 0
\(287\) −18.5106 −1.09264
\(288\) 0 0
\(289\) −10.6971 −0.629241
\(290\) 0 0
\(291\) −0.953831 −0.0559146
\(292\) 0 0
\(293\) 2.34513 0.137004 0.0685020 0.997651i \(-0.478178\pi\)
0.0685020 + 0.997651i \(0.478178\pi\)
\(294\) 0 0
\(295\) −1.42784 −0.0831323
\(296\) 0 0
\(297\) −15.4501 −0.896507
\(298\) 0 0
\(299\) 4.08446 0.236210
\(300\) 0 0
\(301\) −21.3662 −1.23153
\(302\) 0 0
\(303\) 3.33829 0.191780
\(304\) 0 0
\(305\) 1.32401 0.0758128
\(306\) 0 0
\(307\) 23.2624 1.32766 0.663828 0.747885i \(-0.268931\pi\)
0.663828 + 0.747885i \(0.268931\pi\)
\(308\) 0 0
\(309\) −3.58468 −0.203926
\(310\) 0 0
\(311\) 31.3999 1.78052 0.890262 0.455449i \(-0.150521\pi\)
0.890262 + 0.455449i \(0.150521\pi\)
\(312\) 0 0
\(313\) −8.04223 −0.454574 −0.227287 0.973828i \(-0.572986\pi\)
−0.227287 + 0.973828i \(0.572986\pi\)
\(314\) 0 0
\(315\) −8.26926 −0.465920
\(316\) 0 0
\(317\) −12.3029 −0.691000 −0.345500 0.938419i \(-0.612291\pi\)
−0.345500 + 0.938419i \(0.612291\pi\)
\(318\) 0 0
\(319\) 34.6623 1.94072
\(320\) 0 0
\(321\) 6.20766 0.346478
\(322\) 0 0
\(323\) 1.02112 0.0568164
\(324\) 0 0
\(325\) −0.813457 −0.0451225
\(326\) 0 0
\(327\) −1.75011 −0.0967814
\(328\) 0 0
\(329\) −15.8346 −0.872988
\(330\) 0 0
\(331\) 6.77981 0.372652 0.186326 0.982488i \(-0.440342\pi\)
0.186326 + 0.982488i \(0.440342\pi\)
\(332\) 0 0
\(333\) 2.83457 0.155334
\(334\) 0 0
\(335\) −5.42784 −0.296555
\(336\) 0 0
\(337\) 10.6903 0.582336 0.291168 0.956672i \(-0.405956\pi\)
0.291168 + 0.956672i \(0.405956\pi\)
\(338\) 0 0
\(339\) 3.72506 0.202317
\(340\) 0 0
\(341\) −56.9795 −3.08561
\(342\) 0 0
\(343\) 16.0143 0.864689
\(344\) 0 0
\(345\) −2.04223 −0.109950
\(346\) 0 0
\(347\) −5.62691 −0.302069 −0.151034 0.988529i \(-0.548260\pi\)
−0.151034 + 0.988529i \(0.548260\pi\)
\(348\) 0 0
\(349\) 15.1306 0.809924 0.404962 0.914334i \(-0.367285\pi\)
0.404962 + 0.914334i \(0.367285\pi\)
\(350\) 0 0
\(351\) −1.93040 −0.103037
\(352\) 0 0
\(353\) 35.7816 1.90446 0.952230 0.305381i \(-0.0987839\pi\)
0.952230 + 0.305381i \(0.0987839\pi\)
\(354\) 0 0
\(355\) 14.6480 0.777437
\(356\) 0 0
\(357\) −2.97888 −0.157659
\(358\) 0 0
\(359\) −4.48260 −0.236583 −0.118291 0.992979i \(-0.537742\pi\)
−0.118291 + 0.992979i \(0.537742\pi\)
\(360\) 0 0
\(361\) −18.8346 −0.991293
\(362\) 0 0
\(363\) −12.7661 −0.670048
\(364\) 0 0
\(365\) 11.0211 0.576872
\(366\) 0 0
\(367\) 21.5653 1.12570 0.562850 0.826559i \(-0.309705\pi\)
0.562850 + 0.826559i \(0.309705\pi\)
\(368\) 0 0
\(369\) −17.9857 −0.936300
\(370\) 0 0
\(371\) 6.84141 0.355188
\(372\) 0 0
\(373\) 16.9230 0.876238 0.438119 0.898917i \(-0.355645\pi\)
0.438119 + 0.898917i \(0.355645\pi\)
\(374\) 0 0
\(375\) 0.406728 0.0210034
\(376\) 0 0
\(377\) 4.33086 0.223050
\(378\) 0 0
\(379\) −31.2710 −1.60628 −0.803142 0.595788i \(-0.796840\pi\)
−0.803142 + 0.595788i \(0.796840\pi\)
\(380\) 0 0
\(381\) −3.50372 −0.179501
\(382\) 0 0
\(383\) −1.62691 −0.0831314 −0.0415657 0.999136i \(-0.513235\pi\)
−0.0415657 + 0.999136i \(0.513235\pi\)
\(384\) 0 0
\(385\) −18.9932 −0.967981
\(386\) 0 0
\(387\) −20.7604 −1.05531
\(388\) 0 0
\(389\) −31.6412 −1.60427 −0.802136 0.597141i \(-0.796303\pi\)
−0.802136 + 0.597141i \(0.796303\pi\)
\(390\) 0 0
\(391\) 12.6058 0.637503
\(392\) 0 0
\(393\) 5.46149 0.275496
\(394\) 0 0
\(395\) −1.75870 −0.0884898
\(396\) 0 0
\(397\) −39.2961 −1.97221 −0.986106 0.166116i \(-0.946878\pi\)
−0.986106 + 0.166116i \(0.946878\pi\)
\(398\) 0 0
\(399\) −0.482601 −0.0241603
\(400\) 0 0
\(401\) −9.66914 −0.482854 −0.241427 0.970419i \(-0.577615\pi\)
−0.241427 + 0.970419i \(0.577615\pi\)
\(402\) 0 0
\(403\) −7.11926 −0.354636
\(404\) 0 0
\(405\) −7.53851 −0.374592
\(406\) 0 0
\(407\) 6.51056 0.322716
\(408\) 0 0
\(409\) −26.2749 −1.29921 −0.649606 0.760271i \(-0.725066\pi\)
−0.649606 + 0.760271i \(0.725066\pi\)
\(410\) 0 0
\(411\) 5.97495 0.294722
\(412\) 0 0
\(413\) −4.16543 −0.204967
\(414\) 0 0
\(415\) −7.05476 −0.346304
\(416\) 0 0
\(417\) 7.06335 0.345894
\(418\) 0 0
\(419\) 30.1940 1.47507 0.737536 0.675308i \(-0.235989\pi\)
0.737536 + 0.675308i \(0.235989\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −15.3856 −0.748074
\(424\) 0 0
\(425\) −2.51056 −0.121780
\(426\) 0 0
\(427\) 3.86253 0.186921
\(428\) 0 0
\(429\) −2.15406 −0.103999
\(430\) 0 0
\(431\) −25.5653 −1.23144 −0.615719 0.787966i \(-0.711134\pi\)
−0.615719 + 0.787966i \(0.711134\pi\)
\(432\) 0 0
\(433\) 32.0422 1.53985 0.769926 0.638134i \(-0.220293\pi\)
0.769926 + 0.638134i \(0.220293\pi\)
\(434\) 0 0
\(435\) −2.16543 −0.103824
\(436\) 0 0
\(437\) 2.04223 0.0976931
\(438\) 0 0
\(439\) −3.93840 −0.187970 −0.0939848 0.995574i \(-0.529961\pi\)
−0.0939848 + 0.995574i \(0.529961\pi\)
\(440\) 0 0
\(441\) −4.28178 −0.203894
\(442\) 0 0
\(443\) 0.0758724 0.00360481 0.00180240 0.999998i \(-0.499426\pi\)
0.00180240 + 0.999998i \(0.499426\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 0.0844605 0.00399485
\(448\) 0 0
\(449\) 19.2961 0.910637 0.455319 0.890329i \(-0.349525\pi\)
0.455319 + 0.890329i \(0.349525\pi\)
\(450\) 0 0
\(451\) −41.3103 −1.94523
\(452\) 0 0
\(453\) 0.330856 0.0155450
\(454\) 0 0
\(455\) −2.37309 −0.111252
\(456\) 0 0
\(457\) 3.00684 0.140654 0.0703271 0.997524i \(-0.477596\pi\)
0.0703271 + 0.997524i \(0.477596\pi\)
\(458\) 0 0
\(459\) −5.95777 −0.278085
\(460\) 0 0
\(461\) −0.633755 −0.0295169 −0.0147585 0.999891i \(-0.504698\pi\)
−0.0147585 + 0.999891i \(0.504698\pi\)
\(462\) 0 0
\(463\) −18.0422 −0.838494 −0.419247 0.907872i \(-0.637706\pi\)
−0.419247 + 0.907872i \(0.637706\pi\)
\(464\) 0 0
\(465\) 3.55963 0.165074
\(466\) 0 0
\(467\) 23.4758 1.08633 0.543164 0.839626i \(-0.317226\pi\)
0.543164 + 0.839626i \(0.317226\pi\)
\(468\) 0 0
\(469\) −15.8346 −0.731173
\(470\) 0 0
\(471\) −2.16543 −0.0997777
\(472\) 0 0
\(473\) −47.6834 −2.19249
\(474\) 0 0
\(475\) −0.406728 −0.0186620
\(476\) 0 0
\(477\) 6.64744 0.304365
\(478\) 0 0
\(479\) 8.19907 0.374625 0.187313 0.982300i \(-0.440022\pi\)
0.187313 + 0.982300i \(0.440022\pi\)
\(480\) 0 0
\(481\) 0.813457 0.0370904
\(482\) 0 0
\(483\) −5.95777 −0.271088
\(484\) 0 0
\(485\) −2.34513 −0.106487
\(486\) 0 0
\(487\) −0.953831 −0.0432222 −0.0216111 0.999766i \(-0.506880\pi\)
−0.0216111 + 0.999766i \(0.506880\pi\)
\(488\) 0 0
\(489\) 6.18260 0.279587
\(490\) 0 0
\(491\) −19.3383 −0.872725 −0.436362 0.899771i \(-0.643733\pi\)
−0.436362 + 0.899771i \(0.643733\pi\)
\(492\) 0 0
\(493\) 13.3662 0.601985
\(494\) 0 0
\(495\) −18.4546 −0.829475
\(496\) 0 0
\(497\) 42.7325 1.91681
\(498\) 0 0
\(499\) 5.63550 0.252280 0.126140 0.992012i \(-0.459741\pi\)
0.126140 + 0.992012i \(0.459741\pi\)
\(500\) 0 0
\(501\) −5.07703 −0.226825
\(502\) 0 0
\(503\) 33.6269 1.49935 0.749675 0.661806i \(-0.230210\pi\)
0.749675 + 0.661806i \(0.230210\pi\)
\(504\) 0 0
\(505\) 8.20766 0.365236
\(506\) 0 0
\(507\) 5.01833 0.222872
\(508\) 0 0
\(509\) −18.6903 −0.828431 −0.414216 0.910179i \(-0.635944\pi\)
−0.414216 + 0.910179i \(0.635944\pi\)
\(510\) 0 0
\(511\) 32.1517 1.42231
\(512\) 0 0
\(513\) −0.965202 −0.0426147
\(514\) 0 0
\(515\) −8.81346 −0.388367
\(516\) 0 0
\(517\) −35.3383 −1.55418
\(518\) 0 0
\(519\) −9.36915 −0.411260
\(520\) 0 0
\(521\) 25.1586 1.10222 0.551109 0.834433i \(-0.314205\pi\)
0.551109 + 0.834433i \(0.314205\pi\)
\(522\) 0 0
\(523\) 7.25383 0.317188 0.158594 0.987344i \(-0.449304\pi\)
0.158594 + 0.987344i \(0.449304\pi\)
\(524\) 0 0
\(525\) 1.18654 0.0517850
\(526\) 0 0
\(527\) −21.9720 −0.957117
\(528\) 0 0
\(529\) 2.21160 0.0961564
\(530\) 0 0
\(531\) −4.04733 −0.175639
\(532\) 0 0
\(533\) −5.16149 −0.223569
\(534\) 0 0
\(535\) 15.2624 0.659852
\(536\) 0 0
\(537\) 9.87680 0.426215
\(538\) 0 0
\(539\) −9.83457 −0.423605
\(540\) 0 0
\(541\) 16.0422 0.689709 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(542\) 0 0
\(543\) −0.880741 −0.0377962
\(544\) 0 0
\(545\) −4.30290 −0.184316
\(546\) 0 0
\(547\) −16.0143 −0.684721 −0.342360 0.939569i \(-0.611226\pi\)
−0.342360 + 0.939569i \(0.611226\pi\)
\(548\) 0 0
\(549\) 3.75301 0.160175
\(550\) 0 0
\(551\) 2.16543 0.0922503
\(552\) 0 0
\(553\) −5.13063 −0.218177
\(554\) 0 0
\(555\) −0.406728 −0.0172647
\(556\) 0 0
\(557\) −10.8557 −0.459970 −0.229985 0.973194i \(-0.573868\pi\)
−0.229985 + 0.973194i \(0.573868\pi\)
\(558\) 0 0
\(559\) −5.95777 −0.251987
\(560\) 0 0
\(561\) −6.64803 −0.280680
\(562\) 0 0
\(563\) −31.0776 −1.30977 −0.654883 0.755731i \(-0.727282\pi\)
−0.654883 + 0.755731i \(0.727282\pi\)
\(564\) 0 0
\(565\) 9.15859 0.385305
\(566\) 0 0
\(567\) −21.9920 −0.923577
\(568\) 0 0
\(569\) 30.0845 1.26121 0.630603 0.776105i \(-0.282808\pi\)
0.630603 + 0.776105i \(0.282808\pi\)
\(570\) 0 0
\(571\) −5.55673 −0.232542 −0.116271 0.993218i \(-0.537094\pi\)
−0.116271 + 0.993218i \(0.537094\pi\)
\(572\) 0 0
\(573\) −7.86312 −0.328487
\(574\) 0 0
\(575\) −5.02112 −0.209395
\(576\) 0 0
\(577\) −22.2499 −0.926275 −0.463137 0.886286i \(-0.653276\pi\)
−0.463137 + 0.886286i \(0.653276\pi\)
\(578\) 0 0
\(579\) 8.15174 0.338775
\(580\) 0 0
\(581\) −20.5807 −0.853833
\(582\) 0 0
\(583\) 15.2681 0.632340
\(584\) 0 0
\(585\) −2.30580 −0.0953332
\(586\) 0 0
\(587\) −45.3103 −1.87016 −0.935079 0.354440i \(-0.884672\pi\)
−0.935079 + 0.354440i \(0.884672\pi\)
\(588\) 0 0
\(589\) −3.55963 −0.146672
\(590\) 0 0
\(591\) −6.35591 −0.261447
\(592\) 0 0
\(593\) −0.232712 −0.00955635 −0.00477817 0.999989i \(-0.501521\pi\)
−0.00477817 + 0.999989i \(0.501521\pi\)
\(594\) 0 0
\(595\) −7.32401 −0.300255
\(596\) 0 0
\(597\) 8.24989 0.337645
\(598\) 0 0
\(599\) −43.4056 −1.77350 −0.886752 0.462246i \(-0.847044\pi\)
−0.886752 + 0.462246i \(0.847044\pi\)
\(600\) 0 0
\(601\) 35.5317 1.44937 0.724684 0.689082i \(-0.241986\pi\)
0.724684 + 0.689082i \(0.241986\pi\)
\(602\) 0 0
\(603\) −15.3856 −0.626551
\(604\) 0 0
\(605\) −31.3874 −1.27608
\(606\) 0 0
\(607\) −26.7325 −1.08504 −0.542519 0.840043i \(-0.682529\pi\)
−0.542519 + 0.840043i \(0.682529\pi\)
\(608\) 0 0
\(609\) −6.31717 −0.255985
\(610\) 0 0
\(611\) −4.41532 −0.178625
\(612\) 0 0
\(613\) 48.0565 1.94098 0.970492 0.241134i \(-0.0775192\pi\)
0.970492 + 0.241134i \(0.0775192\pi\)
\(614\) 0 0
\(615\) 2.58074 0.104066
\(616\) 0 0
\(617\) 12.9789 0.522510 0.261255 0.965270i \(-0.415864\pi\)
0.261255 + 0.965270i \(0.415864\pi\)
\(618\) 0 0
\(619\) −25.3662 −1.01956 −0.509778 0.860306i \(-0.670272\pi\)
−0.509778 + 0.860306i \(0.670272\pi\)
\(620\) 0 0
\(621\) −11.9155 −0.478154
\(622\) 0 0
\(623\) −17.5037 −0.701272
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.07703 −0.0430124
\(628\) 0 0
\(629\) 2.51056 0.100102
\(630\) 0 0
\(631\) 31.6075 1.25828 0.629138 0.777293i \(-0.283408\pi\)
0.629138 + 0.777293i \(0.283408\pi\)
\(632\) 0 0
\(633\) 7.61323 0.302599
\(634\) 0 0
\(635\) −8.61439 −0.341852
\(636\) 0 0
\(637\) −1.22877 −0.0486858
\(638\) 0 0
\(639\) 41.5209 1.64254
\(640\) 0 0
\(641\) −23.8625 −0.942513 −0.471257 0.881996i \(-0.656199\pi\)
−0.471257 + 0.881996i \(0.656199\pi\)
\(642\) 0 0
\(643\) −32.8277 −1.29460 −0.647300 0.762236i \(-0.724102\pi\)
−0.647300 + 0.762236i \(0.724102\pi\)
\(644\) 0 0
\(645\) 2.97888 0.117293
\(646\) 0 0
\(647\) 13.9578 0.548737 0.274368 0.961625i \(-0.411531\pi\)
0.274368 + 0.961625i \(0.411531\pi\)
\(648\) 0 0
\(649\) −9.29606 −0.364902
\(650\) 0 0
\(651\) 10.3845 0.406999
\(652\) 0 0
\(653\) 28.5921 1.11890 0.559448 0.828865i \(-0.311013\pi\)
0.559448 + 0.828865i \(0.311013\pi\)
\(654\) 0 0
\(655\) 13.4278 0.524669
\(656\) 0 0
\(657\) 31.2401 1.21879
\(658\) 0 0
\(659\) 5.62691 0.219193 0.109597 0.993976i \(-0.465044\pi\)
0.109597 + 0.993976i \(0.465044\pi\)
\(660\) 0 0
\(661\) 13.3240 0.518244 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(662\) 0 0
\(663\) −0.830633 −0.0322591
\(664\) 0 0
\(665\) −1.18654 −0.0460122
\(666\) 0 0
\(667\) 26.7325 1.03509
\(668\) 0 0
\(669\) −6.39814 −0.247366
\(670\) 0 0
\(671\) 8.62007 0.332774
\(672\) 0 0
\(673\) 29.6691 1.14366 0.571831 0.820372i \(-0.306233\pi\)
0.571831 + 0.820372i \(0.306233\pi\)
\(674\) 0 0
\(675\) 2.37309 0.0913401
\(676\) 0 0
\(677\) −1.25383 −0.0481885 −0.0240943 0.999710i \(-0.507670\pi\)
−0.0240943 + 0.999710i \(0.507670\pi\)
\(678\) 0 0
\(679\) −6.84141 −0.262549
\(680\) 0 0
\(681\) 5.24014 0.200803
\(682\) 0 0
\(683\) 4.76045 0.182153 0.0910767 0.995844i \(-0.470969\pi\)
0.0910767 + 0.995844i \(0.470969\pi\)
\(684\) 0 0
\(685\) 14.6903 0.561286
\(686\) 0 0
\(687\) −2.13688 −0.0815271
\(688\) 0 0
\(689\) 1.90766 0.0726761
\(690\) 0 0
\(691\) 20.2219 0.769279 0.384639 0.923067i \(-0.374326\pi\)
0.384639 + 0.923067i \(0.374326\pi\)
\(692\) 0 0
\(693\) −53.8375 −2.04512
\(694\) 0 0
\(695\) 17.3662 0.658739
\(696\) 0 0
\(697\) −15.9298 −0.603385
\(698\) 0 0
\(699\) −11.5174 −0.435628
\(700\) 0 0
\(701\) −22.0845 −0.834119 −0.417059 0.908879i \(-0.636939\pi\)
−0.417059 + 0.908879i \(0.636939\pi\)
\(702\) 0 0
\(703\) 0.406728 0.0153401
\(704\) 0 0
\(705\) 2.20766 0.0831452
\(706\) 0 0
\(707\) 23.9441 0.900510
\(708\) 0 0
\(709\) 5.59896 0.210273 0.105137 0.994458i \(-0.466472\pi\)
0.105137 + 0.994458i \(0.466472\pi\)
\(710\) 0 0
\(711\) −4.98516 −0.186958
\(712\) 0 0
\(713\) −43.9441 −1.64572
\(714\) 0 0
\(715\) −5.29606 −0.198061
\(716\) 0 0
\(717\) 5.84826 0.218407
\(718\) 0 0
\(719\) −40.2921 −1.50264 −0.751321 0.659937i \(-0.770583\pi\)
−0.751321 + 0.659937i \(0.770583\pi\)
\(720\) 0 0
\(721\) −25.7114 −0.957542
\(722\) 0 0
\(723\) −7.43294 −0.276434
\(724\) 0 0
\(725\) −5.32401 −0.197729
\(726\) 0 0
\(727\) −11.2538 −0.417381 −0.208691 0.977982i \(-0.566920\pi\)
−0.208691 + 0.977982i \(0.566920\pi\)
\(728\) 0 0
\(729\) −18.4729 −0.684180
\(730\) 0 0
\(731\) −18.3874 −0.680081
\(732\) 0 0
\(733\) −42.2892 −1.56199 −0.780994 0.624539i \(-0.785287\pi\)
−0.780994 + 0.624539i \(0.785287\pi\)
\(734\) 0 0
\(735\) 0.614387 0.0226620
\(736\) 0 0
\(737\) −35.3383 −1.30170
\(738\) 0 0
\(739\) 29.3103 1.07820 0.539099 0.842242i \(-0.318765\pi\)
0.539099 + 0.842242i \(0.318765\pi\)
\(740\) 0 0
\(741\) −0.134569 −0.00494350
\(742\) 0 0
\(743\) 4.39245 0.161144 0.0805718 0.996749i \(-0.474325\pi\)
0.0805718 + 0.996749i \(0.474325\pi\)
\(744\) 0 0
\(745\) 0.207658 0.00760801
\(746\) 0 0
\(747\) −19.9972 −0.731660
\(748\) 0 0
\(749\) 44.5248 1.62690
\(750\) 0 0
\(751\) −2.58074 −0.0941727 −0.0470864 0.998891i \(-0.514994\pi\)
−0.0470864 + 0.998891i \(0.514994\pi\)
\(752\) 0 0
\(753\) −0.496284 −0.0180856
\(754\) 0 0
\(755\) 0.813457 0.0296047
\(756\) 0 0
\(757\) 27.3805 0.995162 0.497581 0.867418i \(-0.334222\pi\)
0.497581 + 0.867418i \(0.334222\pi\)
\(758\) 0 0
\(759\) −13.2961 −0.482616
\(760\) 0 0
\(761\) −42.9509 −1.55697 −0.778485 0.627663i \(-0.784011\pi\)
−0.778485 + 0.627663i \(0.784011\pi\)
\(762\) 0 0
\(763\) −12.5528 −0.454441
\(764\) 0 0
\(765\) −7.11636 −0.257292
\(766\) 0 0
\(767\) −1.16149 −0.0419389
\(768\) 0 0
\(769\) −13.0633 −0.471076 −0.235538 0.971865i \(-0.575685\pi\)
−0.235538 + 0.971865i \(0.575685\pi\)
\(770\) 0 0
\(771\) −11.0348 −0.397409
\(772\) 0 0
\(773\) −15.3662 −0.552685 −0.276343 0.961059i \(-0.589122\pi\)
−0.276343 + 0.961059i \(0.589122\pi\)
\(774\) 0 0
\(775\) 8.75186 0.314376
\(776\) 0 0
\(777\) −1.18654 −0.0425670
\(778\) 0 0
\(779\) −2.58074 −0.0924648
\(780\) 0 0
\(781\) 95.3668 3.41249
\(782\) 0 0
\(783\) −12.6343 −0.451515
\(784\) 0 0
\(785\) −5.32401 −0.190022
\(786\) 0 0
\(787\) −10.9316 −0.389668 −0.194834 0.980836i \(-0.562417\pi\)
−0.194834 + 0.980836i \(0.562417\pi\)
\(788\) 0 0
\(789\) −8.22134 −0.292688
\(790\) 0 0
\(791\) 26.7182 0.949990
\(792\) 0 0
\(793\) 1.07703 0.0382464
\(794\) 0 0
\(795\) −0.953831 −0.0338289
\(796\) 0 0
\(797\) −24.2921 −0.860471 −0.430235 0.902717i \(-0.641569\pi\)
−0.430235 + 0.902717i \(0.641569\pi\)
\(798\) 0 0
\(799\) −13.6269 −0.482086
\(800\) 0 0
\(801\) −17.0074 −0.600928
\(802\) 0 0
\(803\) 71.7536 2.53213
\(804\) 0 0
\(805\) −14.6480 −0.516275
\(806\) 0 0
\(807\) 0.0844605 0.00297315
\(808\) 0 0
\(809\) −40.0422 −1.40781 −0.703905 0.710294i \(-0.748562\pi\)
−0.703905 + 0.710294i \(0.748562\pi\)
\(810\) 0 0
\(811\) 40.6343 1.42686 0.713432 0.700724i \(-0.247140\pi\)
0.713432 + 0.700724i \(0.247140\pi\)
\(812\) 0 0
\(813\) −12.4654 −0.437182
\(814\) 0 0
\(815\) 15.2008 0.532461
\(816\) 0 0
\(817\) −2.97888 −0.104218
\(818\) 0 0
\(819\) −6.72668 −0.235049
\(820\) 0 0
\(821\) −41.4787 −1.44762 −0.723808 0.690002i \(-0.757610\pi\)
−0.723808 + 0.690002i \(0.757610\pi\)
\(822\) 0 0
\(823\) 5.27610 0.183913 0.0919566 0.995763i \(-0.470688\pi\)
0.0919566 + 0.995763i \(0.470688\pi\)
\(824\) 0 0
\(825\) 2.64803 0.0921925
\(826\) 0 0
\(827\) −1.76438 −0.0613537 −0.0306768 0.999529i \(-0.509766\pi\)
−0.0306768 + 0.999529i \(0.509766\pi\)
\(828\) 0 0
\(829\) 16.3029 0.566223 0.283112 0.959087i \(-0.408633\pi\)
0.283112 + 0.959087i \(0.408633\pi\)
\(830\) 0 0
\(831\) 6.83851 0.237225
\(832\) 0 0
\(833\) −3.79234 −0.131397
\(834\) 0 0
\(835\) −12.4826 −0.431978
\(836\) 0 0
\(837\) 20.7689 0.717879
\(838\) 0 0
\(839\) −14.3731 −0.496214 −0.248107 0.968733i \(-0.579808\pi\)
−0.248107 + 0.968733i \(0.579808\pi\)
\(840\) 0 0
\(841\) −0.654870 −0.0225817
\(842\) 0 0
\(843\) 5.80602 0.199970
\(844\) 0 0
\(845\) 12.3383 0.424450
\(846\) 0 0
\(847\) −91.5659 −3.14624
\(848\) 0 0
\(849\) −5.38052 −0.184659
\(850\) 0 0
\(851\) 5.02112 0.172122
\(852\) 0 0
\(853\) 43.2961 1.48243 0.741214 0.671268i \(-0.234250\pi\)
0.741214 + 0.671268i \(0.234250\pi\)
\(854\) 0 0
\(855\) −1.15290 −0.0394284
\(856\) 0 0
\(857\) −49.0776 −1.67646 −0.838230 0.545317i \(-0.816409\pi\)
−0.838230 + 0.545317i \(0.816409\pi\)
\(858\) 0 0
\(859\) −15.5374 −0.530128 −0.265064 0.964231i \(-0.585393\pi\)
−0.265064 + 0.964231i \(0.585393\pi\)
\(860\) 0 0
\(861\) 7.52877 0.256580
\(862\) 0 0
\(863\) 36.0901 1.22852 0.614261 0.789103i \(-0.289454\pi\)
0.614261 + 0.789103i \(0.289454\pi\)
\(864\) 0 0
\(865\) −23.0354 −0.783227
\(866\) 0 0
\(867\) 4.35081 0.147761
\(868\) 0 0
\(869\) −11.4501 −0.388419
\(870\) 0 0
\(871\) −4.41532 −0.149607
\(872\) 0 0
\(873\) −6.64744 −0.224982
\(874\) 0 0
\(875\) 2.91729 0.0986223
\(876\) 0 0
\(877\) 29.3240 0.990202 0.495101 0.868836i \(-0.335131\pi\)
0.495101 + 0.868836i \(0.335131\pi\)
\(878\) 0 0
\(879\) −0.953831 −0.0321719
\(880\) 0 0
\(881\) 23.8066 0.802065 0.401033 0.916064i \(-0.368651\pi\)
0.401033 + 0.916064i \(0.368651\pi\)
\(882\) 0 0
\(883\) 3.94699 0.132827 0.0664134 0.997792i \(-0.478844\pi\)
0.0664134 + 0.997792i \(0.478844\pi\)
\(884\) 0 0
\(885\) 0.580745 0.0195215
\(886\) 0 0
\(887\) −1.96346 −0.0659264 −0.0329632 0.999457i \(-0.510494\pi\)
−0.0329632 + 0.999457i \(0.510494\pi\)
\(888\) 0 0
\(889\) −25.1306 −0.842854
\(890\) 0 0
\(891\) −49.0799 −1.64424
\(892\) 0 0
\(893\) −2.20766 −0.0738765
\(894\) 0 0
\(895\) 24.2835 0.811709
\(896\) 0 0
\(897\) −1.66127 −0.0554681
\(898\) 0 0
\(899\) −46.5950 −1.55403
\(900\) 0 0
\(901\) 5.88758 0.196144
\(902\) 0 0
\(903\) 8.69026 0.289194
\(904\) 0 0
\(905\) −2.16543 −0.0719813
\(906\) 0 0
\(907\) −5.89049 −0.195590 −0.0977952 0.995207i \(-0.531179\pi\)
−0.0977952 + 0.995207i \(0.531179\pi\)
\(908\) 0 0
\(909\) 23.2652 0.771658
\(910\) 0 0
\(911\) 27.9527 0.926113 0.463057 0.886329i \(-0.346753\pi\)
0.463057 + 0.886329i \(0.346753\pi\)
\(912\) 0 0
\(913\) −45.9304 −1.52007
\(914\) 0 0
\(915\) −0.538514 −0.0178027
\(916\) 0 0
\(917\) 39.1729 1.29360
\(918\) 0 0
\(919\) −17.7587 −0.585805 −0.292903 0.956142i \(-0.594621\pi\)
−0.292903 + 0.956142i \(0.594621\pi\)
\(920\) 0 0
\(921\) −9.46149 −0.311767
\(922\) 0 0
\(923\) 11.9155 0.392205
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) −24.9824 −0.820529
\(928\) 0 0
\(929\) 34.2892 1.12499 0.562496 0.826800i \(-0.309841\pi\)
0.562496 + 0.826800i \(0.309841\pi\)
\(930\) 0 0
\(931\) −0.614387 −0.0201357
\(932\) 0 0
\(933\) −12.7712 −0.418111
\(934\) 0 0
\(935\) −16.3451 −0.534543
\(936\) 0 0
\(937\) −14.0845 −0.460119 −0.230060 0.973177i \(-0.573892\pi\)
−0.230060 + 0.973177i \(0.573892\pi\)
\(938\) 0 0
\(939\) 3.27100 0.106745
\(940\) 0 0
\(941\) 14.2499 0.464533 0.232267 0.972652i \(-0.425386\pi\)
0.232267 + 0.972652i \(0.425386\pi\)
\(942\) 0 0
\(943\) −31.8596 −1.03749
\(944\) 0 0
\(945\) 6.92297 0.225204
\(946\) 0 0
\(947\) 29.8203 0.969029 0.484515 0.874783i \(-0.338996\pi\)
0.484515 + 0.874783i \(0.338996\pi\)
\(948\) 0 0
\(949\) 8.96520 0.291023
\(950\) 0 0
\(951\) 5.00394 0.162264
\(952\) 0 0
\(953\) 8.64803 0.280137 0.140069 0.990142i \(-0.455268\pi\)
0.140069 + 0.990142i \(0.455268\pi\)
\(954\) 0 0
\(955\) −19.3326 −0.625588
\(956\) 0 0
\(957\) −14.0981 −0.455728
\(958\) 0 0
\(959\) 42.8557 1.38388
\(960\) 0 0
\(961\) 45.5950 1.47081
\(962\) 0 0
\(963\) 43.2624 1.39411
\(964\) 0 0
\(965\) 20.0422 0.645182
\(966\) 0 0
\(967\) 45.0325 1.44815 0.724074 0.689723i \(-0.242268\pi\)
0.724074 + 0.689723i \(0.242268\pi\)
\(968\) 0 0
\(969\) −0.415317 −0.0133419
\(970\) 0 0
\(971\) −26.3029 −0.844100 −0.422050 0.906573i \(-0.638689\pi\)
−0.422050 + 0.906573i \(0.638689\pi\)
\(972\) 0 0
\(973\) 50.6623 1.62416
\(974\) 0 0
\(975\) 0.330856 0.0105959
\(976\) 0 0
\(977\) 25.6549 0.820772 0.410386 0.911912i \(-0.365394\pi\)
0.410386 + 0.911912i \(0.365394\pi\)
\(978\) 0 0
\(979\) −39.0633 −1.24847
\(980\) 0 0
\(981\) −12.1969 −0.389416
\(982\) 0 0
\(983\) 55.3611 1.76575 0.882873 0.469611i \(-0.155606\pi\)
0.882873 + 0.469611i \(0.155606\pi\)
\(984\) 0 0
\(985\) −15.6269 −0.497915
\(986\) 0 0
\(987\) 6.44037 0.204999
\(988\) 0 0
\(989\) −36.7747 −1.16937
\(990\) 0 0
\(991\) 3.24814 0.103181 0.0515903 0.998668i \(-0.483571\pi\)
0.0515903 + 0.998668i \(0.483571\pi\)
\(992\) 0 0
\(993\) −2.75754 −0.0875080
\(994\) 0 0
\(995\) 20.2835 0.643031
\(996\) 0 0
\(997\) −12.6480 −0.400567 −0.200284 0.979738i \(-0.564186\pi\)
−0.200284 + 0.979738i \(0.564186\pi\)
\(998\) 0 0
\(999\) −2.37309 −0.0750811
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 2960.2.a.u.1.2 3
4.3 odd 2 370.2.a.g.1.2 3
12.11 even 2 3330.2.a.bg.1.3 3
20.3 even 4 1850.2.b.o.149.2 6
20.7 even 4 1850.2.b.o.149.5 6
20.19 odd 2 1850.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.2 3 4.3 odd 2
1850.2.a.z.1.2 3 20.19 odd 2
1850.2.b.o.149.2 6 20.3 even 4
1850.2.b.o.149.5 6 20.7 even 4
2960.2.a.u.1.2 3 1.1 even 1 trivial
3330.2.a.bg.1.3 3 12.11 even 2