Properties

Label 8752.2.a.s.1.6
Level $8752$
Weight $2$
Character 8752.1
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8752 = 2^{4} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8752.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.8850718490\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.735255\) of defining polynomial
Character \(\chi\) \(=\) 8752.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +O(q^{10})\) \(q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +0.883096 q^{11} -4.48154 q^{13} -0.523917 q^{15} -3.18294 q^{17} -4.12466 q^{19} -1.77017 q^{21} -3.73970 q^{23} -4.07304 q^{25} +3.10387 q^{27} +6.33388 q^{29} +8.47939 q^{31} -0.480552 q^{33} +3.13193 q^{35} +11.0375 q^{37} +2.43871 q^{39} +1.41138 q^{41} +9.31097 q^{43} -2.60326 q^{45} +5.82191 q^{47} +3.58188 q^{49} +1.73205 q^{51} -7.06826 q^{53} +0.850233 q^{55} +2.24450 q^{57} -12.9274 q^{59} -3.04777 q^{61} -8.79567 q^{63} -4.31477 q^{65} +9.15218 q^{67} +2.03502 q^{69} +4.57601 q^{71} -14.4453 q^{73} +2.21642 q^{75} +2.87269 q^{77} -7.20620 q^{79} +6.42262 q^{81} -7.39056 q^{83} -3.06450 q^{85} -3.44669 q^{87} +0.838265 q^{89} -14.5784 q^{91} -4.61421 q^{93} -3.97117 q^{95} +0.475381 q^{97} -2.38779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} + O(q^{10}) \) \( 18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} - 2q^{11} - 25q^{13} - 9q^{15} - 30q^{17} - 4q^{19} - 16q^{21} + 26q^{23} + 31q^{25} + 37q^{27} - 18q^{29} + 5q^{31} - 10q^{33} + 9q^{35} - 18q^{37} - 7q^{39} - 17q^{41} - 8q^{43} - 44q^{45} + 52q^{47} + 29q^{49} - 19q^{51} - 60q^{53} - 11q^{55} + 4q^{57} + 8q^{59} - 26q^{61} + q^{63} - 6q^{65} - 12q^{67} - 38q^{69} + q^{71} - 2q^{73} + 17q^{75} - 73q^{77} - 18q^{79} + 18q^{81} + 43q^{83} + 51q^{85} - 3q^{87} - 28q^{89} + q^{91} - 60q^{93} + 18q^{95} - 34q^{97} - 15q^{99} + O(q^{100}) \)

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
\(3\) −0.544167 −0.314175 −0.157088 0.987585i \(-0.550210\pi\)
−0.157088 + 0.987585i \(0.550210\pi\)
\(4\) 0 0
\(5\) 0.962787 0.430571 0.215286 0.976551i \(-0.430932\pi\)
0.215286 + 0.976551i \(0.430932\pi\)
\(6\) 0 0
\(7\) 3.25298 1.22951 0.614755 0.788718i \(-0.289255\pi\)
0.614755 + 0.788718i \(0.289255\pi\)
\(8\) 0 0
\(9\) −2.70388 −0.901294
\(10\) 0 0
\(11\) 0.883096 0.266263 0.133132 0.991098i \(-0.457497\pi\)
0.133132 + 0.991098i \(0.457497\pi\)
\(12\) 0 0
\(13\) −4.48154 −1.24296 −0.621478 0.783431i \(-0.713468\pi\)
−0.621478 + 0.783431i \(0.713468\pi\)
\(14\) 0 0
\(15\) −0.523917 −0.135275
\(16\) 0 0
\(17\) −3.18294 −0.771977 −0.385988 0.922504i \(-0.626140\pi\)
−0.385988 + 0.922504i \(0.626140\pi\)
\(18\) 0 0
\(19\) −4.12466 −0.946261 −0.473131 0.880992i \(-0.656876\pi\)
−0.473131 + 0.880992i \(0.656876\pi\)
\(20\) 0 0
\(21\) −1.77017 −0.386282
\(22\) 0 0
\(23\) −3.73970 −0.779781 −0.389890 0.920861i \(-0.627487\pi\)
−0.389890 + 0.920861i \(0.627487\pi\)
\(24\) 0 0
\(25\) −4.07304 −0.814608
\(26\) 0 0
\(27\) 3.10387 0.597339
\(28\) 0 0
\(29\) 6.33388 1.17617 0.588086 0.808798i \(-0.299881\pi\)
0.588086 + 0.808798i \(0.299881\pi\)
\(30\) 0 0
\(31\) 8.47939 1.52294 0.761472 0.648198i \(-0.224477\pi\)
0.761472 + 0.648198i \(0.224477\pi\)
\(32\) 0 0
\(33\) −0.480552 −0.0836533
\(34\) 0 0
\(35\) 3.13193 0.529392
\(36\) 0 0
\(37\) 11.0375 1.81455 0.907276 0.420535i \(-0.138158\pi\)
0.907276 + 0.420535i \(0.138158\pi\)
\(38\) 0 0
\(39\) 2.43871 0.390506
\(40\) 0 0
\(41\) 1.41138 0.220421 0.110210 0.993908i \(-0.464848\pi\)
0.110210 + 0.993908i \(0.464848\pi\)
\(42\) 0 0
\(43\) 9.31097 1.41991 0.709955 0.704247i \(-0.248715\pi\)
0.709955 + 0.704247i \(0.248715\pi\)
\(44\) 0 0
\(45\) −2.60326 −0.388071
\(46\) 0 0
\(47\) 5.82191 0.849213 0.424606 0.905378i \(-0.360413\pi\)
0.424606 + 0.905378i \(0.360413\pi\)
\(48\) 0 0
\(49\) 3.58188 0.511697
\(50\) 0 0
\(51\) 1.73205 0.242536
\(52\) 0 0
\(53\) −7.06826 −0.970900 −0.485450 0.874264i \(-0.661344\pi\)
−0.485450 + 0.874264i \(0.661344\pi\)
\(54\) 0 0
\(55\) 0.850233 0.114645
\(56\) 0 0
\(57\) 2.24450 0.297292
\(58\) 0 0
\(59\) −12.9274 −1.68301 −0.841503 0.540252i \(-0.818329\pi\)
−0.841503 + 0.540252i \(0.818329\pi\)
\(60\) 0 0
\(61\) −3.04777 −0.390227 −0.195114 0.980781i \(-0.562508\pi\)
−0.195114 + 0.980781i \(0.562508\pi\)
\(62\) 0 0
\(63\) −8.79567 −1.10815
\(64\) 0 0
\(65\) −4.31477 −0.535182
\(66\) 0 0
\(67\) 9.15218 1.11812 0.559059 0.829128i \(-0.311163\pi\)
0.559059 + 0.829128i \(0.311163\pi\)
\(68\) 0 0
\(69\) 2.03502 0.244988
\(70\) 0 0
\(71\) 4.57601 0.543072 0.271536 0.962428i \(-0.412468\pi\)
0.271536 + 0.962428i \(0.412468\pi\)
\(72\) 0 0
\(73\) −14.4453 −1.69070 −0.845349 0.534215i \(-0.820607\pi\)
−0.845349 + 0.534215i \(0.820607\pi\)
\(74\) 0 0
\(75\) 2.21642 0.255930
\(76\) 0 0
\(77\) 2.87269 0.327374
\(78\) 0 0
\(79\) −7.20620 −0.810761 −0.405381 0.914148i \(-0.632861\pi\)
−0.405381 + 0.914148i \(0.632861\pi\)
\(80\) 0 0
\(81\) 6.42262 0.713625
\(82\) 0 0
\(83\) −7.39056 −0.811219 −0.405610 0.914046i \(-0.632941\pi\)
−0.405610 + 0.914046i \(0.632941\pi\)
\(84\) 0 0
\(85\) −3.06450 −0.332391
\(86\) 0 0
\(87\) −3.44669 −0.369524
\(88\) 0 0
\(89\) 0.838265 0.0888559 0.0444280 0.999013i \(-0.485853\pi\)
0.0444280 + 0.999013i \(0.485853\pi\)
\(90\) 0 0
\(91\) −14.5784 −1.52823
\(92\) 0 0
\(93\) −4.61421 −0.478471
\(94\) 0 0
\(95\) −3.97117 −0.407433
\(96\) 0 0
\(97\) 0.475381 0.0482676 0.0241338 0.999709i \(-0.492317\pi\)
0.0241338 + 0.999709i \(0.492317\pi\)
\(98\) 0 0
\(99\) −2.38779 −0.239982
\(100\) 0 0
\(101\) −3.07939 −0.306411 −0.153205 0.988194i \(-0.548960\pi\)
−0.153205 + 0.988194i \(0.548960\pi\)
\(102\) 0 0
\(103\) 3.71378 0.365929 0.182965 0.983119i \(-0.441431\pi\)
0.182965 + 0.983119i \(0.441431\pi\)
\(104\) 0 0
\(105\) −1.70429 −0.166322
\(106\) 0 0
\(107\) −16.1036 −1.55680 −0.778398 0.627771i \(-0.783968\pi\)
−0.778398 + 0.627771i \(0.783968\pi\)
\(108\) 0 0
\(109\) 2.61098 0.250087 0.125043 0.992151i \(-0.460093\pi\)
0.125043 + 0.992151i \(0.460093\pi\)
\(110\) 0 0
\(111\) −6.00624 −0.570087
\(112\) 0 0
\(113\) −12.2204 −1.14960 −0.574798 0.818295i \(-0.694919\pi\)
−0.574798 + 0.818295i \(0.694919\pi\)
\(114\) 0 0
\(115\) −3.60053 −0.335751
\(116\) 0 0
\(117\) 12.1176 1.12027
\(118\) 0 0
\(119\) −10.3540 −0.949154
\(120\) 0 0
\(121\) −10.2201 −0.929104
\(122\) 0 0
\(123\) −0.768028 −0.0692508
\(124\) 0 0
\(125\) −8.73541 −0.781318
\(126\) 0 0
\(127\) −5.90845 −0.524290 −0.262145 0.965028i \(-0.584430\pi\)
−0.262145 + 0.965028i \(0.584430\pi\)
\(128\) 0 0
\(129\) −5.06673 −0.446100
\(130\) 0 0
\(131\) −4.91027 −0.429012 −0.214506 0.976723i \(-0.568814\pi\)
−0.214506 + 0.976723i \(0.568814\pi\)
\(132\) 0 0
\(133\) −13.4174 −1.16344
\(134\) 0 0
\(135\) 2.98836 0.257197
\(136\) 0 0
\(137\) −8.29905 −0.709036 −0.354518 0.935049i \(-0.615355\pi\)
−0.354518 + 0.935049i \(0.615355\pi\)
\(138\) 0 0
\(139\) −21.0776 −1.78778 −0.893889 0.448288i \(-0.852034\pi\)
−0.893889 + 0.448288i \(0.852034\pi\)
\(140\) 0 0
\(141\) −3.16809 −0.266802
\(142\) 0 0
\(143\) −3.95763 −0.330954
\(144\) 0 0
\(145\) 6.09818 0.506426
\(146\) 0 0
\(147\) −1.94914 −0.160762
\(148\) 0 0
\(149\) −20.9236 −1.71413 −0.857063 0.515212i \(-0.827713\pi\)
−0.857063 + 0.515212i \(0.827713\pi\)
\(150\) 0 0
\(151\) 12.2666 0.998241 0.499121 0.866533i \(-0.333656\pi\)
0.499121 + 0.866533i \(0.333656\pi\)
\(152\) 0 0
\(153\) 8.60630 0.695778
\(154\) 0 0
\(155\) 8.16385 0.655736
\(156\) 0 0
\(157\) 9.63793 0.769191 0.384595 0.923085i \(-0.374341\pi\)
0.384595 + 0.923085i \(0.374341\pi\)
\(158\) 0 0
\(159\) 3.84632 0.305033
\(160\) 0 0
\(161\) −12.1652 −0.958749
\(162\) 0 0
\(163\) −11.8440 −0.927693 −0.463847 0.885915i \(-0.653531\pi\)
−0.463847 + 0.885915i \(0.653531\pi\)
\(164\) 0 0
\(165\) −0.462669 −0.0360187
\(166\) 0 0
\(167\) 18.8572 1.45922 0.729609 0.683865i \(-0.239702\pi\)
0.729609 + 0.683865i \(0.239702\pi\)
\(168\) 0 0
\(169\) 7.08424 0.544941
\(170\) 0 0
\(171\) 11.1526 0.852859
\(172\) 0 0
\(173\) −15.1822 −1.15428 −0.577140 0.816645i \(-0.695831\pi\)
−0.577140 + 0.816645i \(0.695831\pi\)
\(174\) 0 0
\(175\) −13.2495 −1.00157
\(176\) 0 0
\(177\) 7.03468 0.528759
\(178\) 0 0
\(179\) 2.57897 0.192761 0.0963806 0.995345i \(-0.469273\pi\)
0.0963806 + 0.995345i \(0.469273\pi\)
\(180\) 0 0
\(181\) −9.50776 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(182\) 0 0
\(183\) 1.65850 0.122600
\(184\) 0 0
\(185\) 10.6268 0.781295
\(186\) 0 0
\(187\) −2.81084 −0.205549
\(188\) 0 0
\(189\) 10.0968 0.734435
\(190\) 0 0
\(191\) −6.20885 −0.449257 −0.224628 0.974444i \(-0.572117\pi\)
−0.224628 + 0.974444i \(0.572117\pi\)
\(192\) 0 0
\(193\) −19.6307 −1.41305 −0.706526 0.707687i \(-0.749739\pi\)
−0.706526 + 0.707687i \(0.749739\pi\)
\(194\) 0 0
\(195\) 2.34796 0.168141
\(196\) 0 0
\(197\) 26.3638 1.87834 0.939171 0.343449i \(-0.111595\pi\)
0.939171 + 0.343449i \(0.111595\pi\)
\(198\) 0 0
\(199\) 5.17965 0.367176 0.183588 0.983003i \(-0.441229\pi\)
0.183588 + 0.983003i \(0.441229\pi\)
\(200\) 0 0
\(201\) −4.98032 −0.351285
\(202\) 0 0
\(203\) 20.6040 1.44612
\(204\) 0 0
\(205\) 1.35886 0.0949069
\(206\) 0 0
\(207\) 10.1117 0.702812
\(208\) 0 0
\(209\) −3.64247 −0.251955
\(210\) 0 0
\(211\) −11.3624 −0.782217 −0.391109 0.920344i \(-0.627908\pi\)
−0.391109 + 0.920344i \(0.627908\pi\)
\(212\) 0 0
\(213\) −2.49011 −0.170620
\(214\) 0 0
\(215\) 8.96448 0.611373
\(216\) 0 0
\(217\) 27.5833 1.87248
\(218\) 0 0
\(219\) 7.86068 0.531175
\(220\) 0 0
\(221\) 14.2645 0.959534
\(222\) 0 0
\(223\) −18.9951 −1.27201 −0.636003 0.771686i \(-0.719413\pi\)
−0.636003 + 0.771686i \(0.719413\pi\)
\(224\) 0 0
\(225\) 11.0130 0.734201
\(226\) 0 0
\(227\) −11.7708 −0.781254 −0.390627 0.920549i \(-0.627742\pi\)
−0.390627 + 0.920549i \(0.627742\pi\)
\(228\) 0 0
\(229\) 6.31464 0.417283 0.208641 0.977992i \(-0.433096\pi\)
0.208641 + 0.977992i \(0.433096\pi\)
\(230\) 0 0
\(231\) −1.56323 −0.102853
\(232\) 0 0
\(233\) 20.7839 1.36160 0.680799 0.732470i \(-0.261633\pi\)
0.680799 + 0.732470i \(0.261633\pi\)
\(234\) 0 0
\(235\) 5.60526 0.365647
\(236\) 0 0
\(237\) 3.92138 0.254721
\(238\) 0 0
\(239\) 23.8405 1.54212 0.771058 0.636765i \(-0.219728\pi\)
0.771058 + 0.636765i \(0.219728\pi\)
\(240\) 0 0
\(241\) 1.62031 0.104374 0.0521868 0.998637i \(-0.483381\pi\)
0.0521868 + 0.998637i \(0.483381\pi\)
\(242\) 0 0
\(243\) −12.8066 −0.821543
\(244\) 0 0
\(245\) 3.44859 0.220322
\(246\) 0 0
\(247\) 18.4848 1.17616
\(248\) 0 0
\(249\) 4.02170 0.254865
\(250\) 0 0
\(251\) 10.7109 0.676064 0.338032 0.941135i \(-0.390239\pi\)
0.338032 + 0.941135i \(0.390239\pi\)
\(252\) 0 0
\(253\) −3.30251 −0.207627
\(254\) 0 0
\(255\) 1.66760 0.104429
\(256\) 0 0
\(257\) −28.7954 −1.79621 −0.898105 0.439781i \(-0.855056\pi\)
−0.898105 + 0.439781i \(0.855056\pi\)
\(258\) 0 0
\(259\) 35.9047 2.23101
\(260\) 0 0
\(261\) −17.1261 −1.06008
\(262\) 0 0
\(263\) 27.1652 1.67508 0.837539 0.546378i \(-0.183994\pi\)
0.837539 + 0.546378i \(0.183994\pi\)
\(264\) 0 0
\(265\) −6.80523 −0.418042
\(266\) 0 0
\(267\) −0.456156 −0.0279163
\(268\) 0 0
\(269\) −19.2182 −1.17176 −0.585878 0.810399i \(-0.699250\pi\)
−0.585878 + 0.810399i \(0.699250\pi\)
\(270\) 0 0
\(271\) 9.73786 0.591533 0.295766 0.955260i \(-0.404425\pi\)
0.295766 + 0.955260i \(0.404425\pi\)
\(272\) 0 0
\(273\) 7.93308 0.480132
\(274\) 0 0
\(275\) −3.59689 −0.216900
\(276\) 0 0
\(277\) −26.8931 −1.61585 −0.807925 0.589285i \(-0.799409\pi\)
−0.807925 + 0.589285i \(0.799409\pi\)
\(278\) 0 0
\(279\) −22.9273 −1.37262
\(280\) 0 0
\(281\) −4.21282 −0.251316 −0.125658 0.992074i \(-0.540104\pi\)
−0.125658 + 0.992074i \(0.540104\pi\)
\(282\) 0 0
\(283\) −31.3587 −1.86408 −0.932039 0.362357i \(-0.881972\pi\)
−0.932039 + 0.362357i \(0.881972\pi\)
\(284\) 0 0
\(285\) 2.16098 0.128005
\(286\) 0 0
\(287\) 4.59120 0.271010
\(288\) 0 0
\(289\) −6.86888 −0.404052
\(290\) 0 0
\(291\) −0.258687 −0.0151645
\(292\) 0 0
\(293\) −15.8890 −0.928244 −0.464122 0.885771i \(-0.653630\pi\)
−0.464122 + 0.885771i \(0.653630\pi\)
\(294\) 0 0
\(295\) −12.4463 −0.724654
\(296\) 0 0
\(297\) 2.74101 0.159050
\(298\) 0 0
\(299\) 16.7596 0.969234
\(300\) 0 0
\(301\) 30.2884 1.74579
\(302\) 0 0
\(303\) 1.67570 0.0962667
\(304\) 0 0
\(305\) −2.93436 −0.168021
\(306\) 0 0
\(307\) 13.7030 0.782070 0.391035 0.920376i \(-0.372117\pi\)
0.391035 + 0.920376i \(0.372117\pi\)
\(308\) 0 0
\(309\) −2.02092 −0.114966
\(310\) 0 0
\(311\) −6.49422 −0.368253 −0.184127 0.982903i \(-0.558946\pi\)
−0.184127 + 0.982903i \(0.558946\pi\)
\(312\) 0 0
\(313\) 8.26403 0.467111 0.233555 0.972344i \(-0.424964\pi\)
0.233555 + 0.972344i \(0.424964\pi\)
\(314\) 0 0
\(315\) −8.46836 −0.477138
\(316\) 0 0
\(317\) −28.1441 −1.58073 −0.790364 0.612638i \(-0.790108\pi\)
−0.790364 + 0.612638i \(0.790108\pi\)
\(318\) 0 0
\(319\) 5.59343 0.313172
\(320\) 0 0
\(321\) 8.76307 0.489107
\(322\) 0 0
\(323\) 13.1285 0.730492
\(324\) 0 0
\(325\) 18.2535 1.01252
\(326\) 0 0
\(327\) −1.42081 −0.0785710
\(328\) 0 0
\(329\) 18.9386 1.04412
\(330\) 0 0
\(331\) 1.78319 0.0980130 0.0490065 0.998798i \(-0.484394\pi\)
0.0490065 + 0.998798i \(0.484394\pi\)
\(332\) 0 0
\(333\) −29.8441 −1.63545
\(334\) 0 0
\(335\) 8.81160 0.481429
\(336\) 0 0
\(337\) 17.5528 0.956160 0.478080 0.878316i \(-0.341333\pi\)
0.478080 + 0.878316i \(0.341333\pi\)
\(338\) 0 0
\(339\) 6.64993 0.361175
\(340\) 0 0
\(341\) 7.48811 0.405504
\(342\) 0 0
\(343\) −11.1191 −0.600374
\(344\) 0 0
\(345\) 1.95929 0.105485
\(346\) 0 0
\(347\) −14.8308 −0.796160 −0.398080 0.917351i \(-0.630323\pi\)
−0.398080 + 0.917351i \(0.630323\pi\)
\(348\) 0 0
\(349\) −34.3697 −1.83977 −0.919884 0.392190i \(-0.871718\pi\)
−0.919884 + 0.392190i \(0.871718\pi\)
\(350\) 0 0
\(351\) −13.9101 −0.742467
\(352\) 0 0
\(353\) −16.1758 −0.860949 −0.430474 0.902603i \(-0.641654\pi\)
−0.430474 + 0.902603i \(0.641654\pi\)
\(354\) 0 0
\(355\) 4.40572 0.233831
\(356\) 0 0
\(357\) 5.63433 0.298201
\(358\) 0 0
\(359\) −17.7167 −0.935053 −0.467526 0.883979i \(-0.654855\pi\)
−0.467526 + 0.883979i \(0.654855\pi\)
\(360\) 0 0
\(361\) −1.98721 −0.104590
\(362\) 0 0
\(363\) 5.56147 0.291901
\(364\) 0 0
\(365\) −13.9078 −0.727966
\(366\) 0 0
\(367\) −7.29148 −0.380612 −0.190306 0.981725i \(-0.560948\pi\)
−0.190306 + 0.981725i \(0.560948\pi\)
\(368\) 0 0
\(369\) −3.81621 −0.198664
\(370\) 0 0
\(371\) −22.9929 −1.19373
\(372\) 0 0
\(373\) 19.4135 1.00520 0.502598 0.864520i \(-0.332378\pi\)
0.502598 + 0.864520i \(0.332378\pi\)
\(374\) 0 0
\(375\) 4.75352 0.245471
\(376\) 0 0
\(377\) −28.3856 −1.46193
\(378\) 0 0
\(379\) −21.5543 −1.10717 −0.553586 0.832792i \(-0.686741\pi\)
−0.553586 + 0.832792i \(0.686741\pi\)
\(380\) 0 0
\(381\) 3.21519 0.164719
\(382\) 0 0
\(383\) 8.69462 0.444274 0.222137 0.975015i \(-0.428697\pi\)
0.222137 + 0.975015i \(0.428697\pi\)
\(384\) 0 0
\(385\) 2.76579 0.140958
\(386\) 0 0
\(387\) −25.1758 −1.27976
\(388\) 0 0
\(389\) 20.0986 1.01904 0.509520 0.860459i \(-0.329823\pi\)
0.509520 + 0.860459i \(0.329823\pi\)
\(390\) 0 0
\(391\) 11.9032 0.601973
\(392\) 0 0
\(393\) 2.67201 0.134785
\(394\) 0 0
\(395\) −6.93804 −0.349091
\(396\) 0 0
\(397\) −22.3318 −1.12080 −0.560402 0.828221i \(-0.689353\pi\)
−0.560402 + 0.828221i \(0.689353\pi\)
\(398\) 0 0
\(399\) 7.30132 0.365523
\(400\) 0 0
\(401\) −1.95708 −0.0977321 −0.0488661 0.998805i \(-0.515561\pi\)
−0.0488661 + 0.998805i \(0.515561\pi\)
\(402\) 0 0
\(403\) −38.0008 −1.89295
\(404\) 0 0
\(405\) 6.18362 0.307266
\(406\) 0 0
\(407\) 9.74716 0.483149
\(408\) 0 0
\(409\) 14.4716 0.715576 0.357788 0.933803i \(-0.383531\pi\)
0.357788 + 0.933803i \(0.383531\pi\)
\(410\) 0 0
\(411\) 4.51607 0.222761
\(412\) 0 0
\(413\) −42.0526 −2.06927
\(414\) 0 0
\(415\) −7.11554 −0.349288
\(416\) 0 0
\(417\) 11.4697 0.561676
\(418\) 0 0
\(419\) 4.92102 0.240408 0.120204 0.992749i \(-0.461645\pi\)
0.120204 + 0.992749i \(0.461645\pi\)
\(420\) 0 0
\(421\) 16.7810 0.817857 0.408928 0.912566i \(-0.365903\pi\)
0.408928 + 0.912566i \(0.365903\pi\)
\(422\) 0 0
\(423\) −15.7418 −0.765390
\(424\) 0 0
\(425\) 12.9643 0.628859
\(426\) 0 0
\(427\) −9.91435 −0.479789
\(428\) 0 0
\(429\) 2.15361 0.103977
\(430\) 0 0
\(431\) −10.5214 −0.506800 −0.253400 0.967362i \(-0.581549\pi\)
−0.253400 + 0.967362i \(0.581549\pi\)
\(432\) 0 0
\(433\) −11.8135 −0.567721 −0.283860 0.958866i \(-0.591615\pi\)
−0.283860 + 0.958866i \(0.591615\pi\)
\(434\) 0 0
\(435\) −3.31843 −0.159107
\(436\) 0 0
\(437\) 15.4250 0.737876
\(438\) 0 0
\(439\) −8.18885 −0.390833 −0.195416 0.980720i \(-0.562606\pi\)
−0.195416 + 0.980720i \(0.562606\pi\)
\(440\) 0 0
\(441\) −9.68497 −0.461189
\(442\) 0 0
\(443\) 15.7119 0.746496 0.373248 0.927732i \(-0.378244\pi\)
0.373248 + 0.927732i \(0.378244\pi\)
\(444\) 0 0
\(445\) 0.807071 0.0382588
\(446\) 0 0
\(447\) 11.3859 0.538536
\(448\) 0 0
\(449\) 9.89465 0.466957 0.233479 0.972362i \(-0.424989\pi\)
0.233479 + 0.972362i \(0.424989\pi\)
\(450\) 0 0
\(451\) 1.24639 0.0586900
\(452\) 0 0
\(453\) −6.67508 −0.313623
\(454\) 0 0
\(455\) −14.0359 −0.658012
\(456\) 0 0
\(457\) 19.4319 0.908985 0.454492 0.890751i \(-0.349821\pi\)
0.454492 + 0.890751i \(0.349821\pi\)
\(458\) 0 0
\(459\) −9.87943 −0.461132
\(460\) 0 0
\(461\) 28.4760 1.32626 0.663130 0.748504i \(-0.269228\pi\)
0.663130 + 0.748504i \(0.269228\pi\)
\(462\) 0 0
\(463\) 10.8095 0.502361 0.251181 0.967940i \(-0.419181\pi\)
0.251181 + 0.967940i \(0.419181\pi\)
\(464\) 0 0
\(465\) −4.44250 −0.206016
\(466\) 0 0
\(467\) −26.3662 −1.22008 −0.610041 0.792369i \(-0.708847\pi\)
−0.610041 + 0.792369i \(0.708847\pi\)
\(468\) 0 0
\(469\) 29.7719 1.37474
\(470\) 0 0
\(471\) −5.24465 −0.241661
\(472\) 0 0
\(473\) 8.22248 0.378070
\(474\) 0 0
\(475\) 16.7999 0.770832
\(476\) 0 0
\(477\) 19.1117 0.875067
\(478\) 0 0
\(479\) 0.194030 0.00886547 0.00443274 0.999990i \(-0.498589\pi\)
0.00443274 + 0.999990i \(0.498589\pi\)
\(480\) 0 0
\(481\) −49.4650 −2.25541
\(482\) 0 0
\(483\) 6.61988 0.301215
\(484\) 0 0
\(485\) 0.457691 0.0207827
\(486\) 0 0
\(487\) 2.65682 0.120392 0.0601959 0.998187i \(-0.480827\pi\)
0.0601959 + 0.998187i \(0.480827\pi\)
\(488\) 0 0
\(489\) 6.44512 0.291458
\(490\) 0 0
\(491\) −11.0398 −0.498219 −0.249109 0.968475i \(-0.580138\pi\)
−0.249109 + 0.968475i \(0.580138\pi\)
\(492\) 0 0
\(493\) −20.1604 −0.907978
\(494\) 0 0
\(495\) −2.29893 −0.103329
\(496\) 0 0
\(497\) 14.8857 0.667713
\(498\) 0 0
\(499\) −30.9966 −1.38760 −0.693799 0.720168i \(-0.744065\pi\)
−0.693799 + 0.720168i \(0.744065\pi\)
\(500\) 0 0
\(501\) −10.2615 −0.458450
\(502\) 0 0
\(503\) 37.0982 1.65413 0.827063 0.562109i \(-0.190010\pi\)
0.827063 + 0.562109i \(0.190010\pi\)
\(504\) 0 0
\(505\) −2.96480 −0.131932
\(506\) 0 0
\(507\) −3.85501 −0.171207
\(508\) 0 0
\(509\) −8.10275 −0.359148 −0.179574 0.983744i \(-0.557472\pi\)
−0.179574 + 0.983744i \(0.557472\pi\)
\(510\) 0 0
\(511\) −46.9904 −2.07873
\(512\) 0 0
\(513\) −12.8024 −0.565239
\(514\) 0 0
\(515\) 3.57558 0.157559
\(516\) 0 0
\(517\) 5.14130 0.226114
\(518\) 0 0
\(519\) 8.26165 0.362646
\(520\) 0 0
\(521\) 9.96721 0.436671 0.218336 0.975874i \(-0.429937\pi\)
0.218336 + 0.975874i \(0.429937\pi\)
\(522\) 0 0
\(523\) 7.12073 0.311368 0.155684 0.987807i \(-0.450242\pi\)
0.155684 + 0.987807i \(0.450242\pi\)
\(524\) 0 0
\(525\) 7.20996 0.314668
\(526\) 0 0
\(527\) −26.9894 −1.17568
\(528\) 0 0
\(529\) −9.01467 −0.391942
\(530\) 0 0
\(531\) 34.9542 1.51688
\(532\) 0 0
\(533\) −6.32517 −0.273974
\(534\) 0 0
\(535\) −15.5044 −0.670312
\(536\) 0 0
\(537\) −1.40339 −0.0605608
\(538\) 0 0
\(539\) 3.16314 0.136246
\(540\) 0 0
\(541\) 39.1006 1.68107 0.840533 0.541760i \(-0.182242\pi\)
0.840533 + 0.541760i \(0.182242\pi\)
\(542\) 0 0
\(543\) 5.17381 0.222030
\(544\) 0 0
\(545\) 2.51382 0.107680
\(546\) 0 0
\(547\) 1.00000 0.0427569
\(548\) 0 0
\(549\) 8.24082 0.351710
\(550\) 0 0
\(551\) −26.1251 −1.11297
\(552\) 0 0
\(553\) −23.4416 −0.996840
\(554\) 0 0
\(555\) −5.78273 −0.245463
\(556\) 0 0
\(557\) 39.1227 1.65768 0.828841 0.559485i \(-0.189001\pi\)
0.828841 + 0.559485i \(0.189001\pi\)
\(558\) 0 0
\(559\) −41.7275 −1.76489
\(560\) 0 0
\(561\) 1.52957 0.0645784
\(562\) 0 0
\(563\) −32.3381 −1.36289 −0.681445 0.731869i \(-0.738648\pi\)
−0.681445 + 0.731869i \(0.738648\pi\)
\(564\) 0 0
\(565\) −11.7656 −0.494983
\(566\) 0 0
\(567\) 20.8927 0.877409
\(568\) 0 0
\(569\) 0.316254 0.0132581 0.00662903 0.999978i \(-0.497890\pi\)
0.00662903 + 0.999978i \(0.497890\pi\)
\(570\) 0 0
\(571\) −4.67642 −0.195702 −0.0978511 0.995201i \(-0.531197\pi\)
−0.0978511 + 0.995201i \(0.531197\pi\)
\(572\) 0 0
\(573\) 3.37866 0.141145
\(574\) 0 0
\(575\) 15.2319 0.635216
\(576\) 0 0
\(577\) 38.4107 1.59906 0.799529 0.600627i \(-0.205082\pi\)
0.799529 + 0.600627i \(0.205082\pi\)
\(578\) 0 0
\(579\) 10.6824 0.443946
\(580\) 0 0
\(581\) −24.0413 −0.997403
\(582\) 0 0
\(583\) −6.24195 −0.258515
\(584\) 0 0
\(585\) 11.6666 0.482356
\(586\) 0 0
\(587\) −6.56995 −0.271171 −0.135586 0.990766i \(-0.543292\pi\)
−0.135586 + 0.990766i \(0.543292\pi\)
\(588\) 0 0
\(589\) −34.9746 −1.44110
\(590\) 0 0
\(591\) −14.3463 −0.590129
\(592\) 0 0
\(593\) −0.570902 −0.0234441 −0.0117221 0.999931i \(-0.503731\pi\)
−0.0117221 + 0.999931i \(0.503731\pi\)
\(594\) 0 0
\(595\) −9.96874 −0.408679
\(596\) 0 0
\(597\) −2.81860 −0.115357
\(598\) 0 0
\(599\) −8.95352 −0.365831 −0.182915 0.983129i \(-0.558553\pi\)
−0.182915 + 0.983129i \(0.558553\pi\)
\(600\) 0 0
\(601\) −12.7284 −0.519202 −0.259601 0.965716i \(-0.583591\pi\)
−0.259601 + 0.965716i \(0.583591\pi\)
\(602\) 0 0
\(603\) −24.7464 −1.00775
\(604\) 0 0
\(605\) −9.83982 −0.400046
\(606\) 0 0
\(607\) −26.7859 −1.08721 −0.543603 0.839342i \(-0.682940\pi\)
−0.543603 + 0.839342i \(0.682940\pi\)
\(608\) 0 0
\(609\) −11.2120 −0.454334
\(610\) 0 0
\(611\) −26.0911 −1.05553
\(612\) 0 0
\(613\) 31.3273 1.26530 0.632649 0.774439i \(-0.281968\pi\)
0.632649 + 0.774439i \(0.281968\pi\)
\(614\) 0 0
\(615\) −0.739448 −0.0298174
\(616\) 0 0
\(617\) −31.6103 −1.27258 −0.636292 0.771448i \(-0.719533\pi\)
−0.636292 + 0.771448i \(0.719533\pi\)
\(618\) 0 0
\(619\) −7.99712 −0.321431 −0.160716 0.987001i \(-0.551380\pi\)
−0.160716 + 0.987001i \(0.551380\pi\)
\(620\) 0 0
\(621\) −11.6075 −0.465794
\(622\) 0 0
\(623\) 2.72686 0.109249
\(624\) 0 0
\(625\) 11.9549 0.478195
\(626\) 0 0
\(627\) 1.98211 0.0791579
\(628\) 0 0
\(629\) −35.1317 −1.40079
\(630\) 0 0
\(631\) 44.2164 1.76023 0.880113 0.474764i \(-0.157466\pi\)
0.880113 + 0.474764i \(0.157466\pi\)
\(632\) 0 0
\(633\) 6.18303 0.245753
\(634\) 0 0
\(635\) −5.68858 −0.225744
\(636\) 0 0
\(637\) −16.0523 −0.636017
\(638\) 0 0
\(639\) −12.3730 −0.489468
\(640\) 0 0
\(641\) −16.8320 −0.664824 −0.332412 0.943134i \(-0.607862\pi\)
−0.332412 + 0.943134i \(0.607862\pi\)
\(642\) 0 0
\(643\) 12.4597 0.491362 0.245681 0.969351i \(-0.420988\pi\)
0.245681 + 0.969351i \(0.420988\pi\)
\(644\) 0 0
\(645\) −4.87818 −0.192078
\(646\) 0 0
\(647\) 48.1533 1.89310 0.946551 0.322555i \(-0.104542\pi\)
0.946551 + 0.322555i \(0.104542\pi\)
\(648\) 0 0
\(649\) −11.4161 −0.448123
\(650\) 0 0
\(651\) −15.0099 −0.588285
\(652\) 0 0
\(653\) −12.7505 −0.498967 −0.249484 0.968379i \(-0.580261\pi\)
−0.249484 + 0.968379i \(0.580261\pi\)
\(654\) 0 0
\(655\) −4.72754 −0.184720
\(656\) 0 0
\(657\) 39.0585 1.52382
\(658\) 0 0
\(659\) 29.8939 1.16450 0.582250 0.813010i \(-0.302172\pi\)
0.582250 + 0.813010i \(0.302172\pi\)
\(660\) 0 0
\(661\) 22.0815 0.858872 0.429436 0.903097i \(-0.358712\pi\)
0.429436 + 0.903097i \(0.358712\pi\)
\(662\) 0 0
\(663\) −7.76227 −0.301462
\(664\) 0 0
\(665\) −12.9181 −0.500943
\(666\) 0 0
\(667\) −23.6868 −0.917157
\(668\) 0 0
\(669\) 10.3365 0.399633
\(670\) 0 0
\(671\) −2.69148 −0.103903
\(672\) 0 0
\(673\) 24.6133 0.948772 0.474386 0.880317i \(-0.342670\pi\)
0.474386 + 0.880317i \(0.342670\pi\)
\(674\) 0 0
\(675\) −12.6422 −0.486598
\(676\) 0 0
\(677\) 15.1930 0.583913 0.291956 0.956432i \(-0.405694\pi\)
0.291956 + 0.956432i \(0.405694\pi\)
\(678\) 0 0
\(679\) 1.54640 0.0593456
\(680\) 0 0
\(681\) 6.40527 0.245451
\(682\) 0 0
\(683\) 23.5519 0.901189 0.450594 0.892729i \(-0.351212\pi\)
0.450594 + 0.892729i \(0.351212\pi\)
\(684\) 0 0
\(685\) −7.99022 −0.305291
\(686\) 0 0
\(687\) −3.43622 −0.131100
\(688\) 0 0
\(689\) 31.6767 1.20679
\(690\) 0 0
\(691\) 37.0300 1.40869 0.704343 0.709860i \(-0.251242\pi\)
0.704343 + 0.709860i \(0.251242\pi\)
\(692\) 0 0
\(693\) −7.76742 −0.295060
\(694\) 0 0
\(695\) −20.2932 −0.769766
\(696\) 0 0
\(697\) −4.49235 −0.170160
\(698\) 0 0
\(699\) −11.3099 −0.427781
\(700\) 0 0
\(701\) 35.5829 1.34395 0.671973 0.740576i \(-0.265447\pi\)
0.671973 + 0.740576i \(0.265447\pi\)
\(702\) 0 0
\(703\) −45.5259 −1.71704
\(704\) 0 0
\(705\) −3.05020 −0.114877
\(706\) 0 0
\(707\) −10.0172 −0.376736
\(708\) 0 0
\(709\) 40.5125 1.52148 0.760739 0.649058i \(-0.224837\pi\)
0.760739 + 0.649058i \(0.224837\pi\)
\(710\) 0 0
\(711\) 19.4847 0.730734
\(712\) 0 0
\(713\) −31.7103 −1.18756
\(714\) 0 0
\(715\) −3.81036 −0.142499
\(716\) 0 0
\(717\) −12.9732 −0.484494
\(718\) 0 0
\(719\) −6.36743 −0.237465 −0.118733 0.992926i \(-0.537883\pi\)
−0.118733 + 0.992926i \(0.537883\pi\)
\(720\) 0 0
\(721\) 12.0808 0.449914
\(722\) 0 0
\(723\) −0.881722 −0.0327916
\(724\) 0 0
\(725\) −25.7982 −0.958120
\(726\) 0 0
\(727\) 35.1164 1.30240 0.651198 0.758908i \(-0.274267\pi\)
0.651198 + 0.758908i \(0.274267\pi\)
\(728\) 0 0
\(729\) −12.2989 −0.455516
\(730\) 0 0
\(731\) −29.6363 −1.09614
\(732\) 0 0
\(733\) 26.1891 0.967315 0.483658 0.875257i \(-0.339308\pi\)
0.483658 + 0.875257i \(0.339308\pi\)
\(734\) 0 0
\(735\) −1.87661 −0.0692197
\(736\) 0 0
\(737\) 8.08225 0.297714
\(738\) 0 0
\(739\) −15.8483 −0.582988 −0.291494 0.956573i \(-0.594152\pi\)
−0.291494 + 0.956573i \(0.594152\pi\)
\(740\) 0 0
\(741\) −10.0588 −0.369521
\(742\) 0 0
\(743\) −13.1980 −0.484189 −0.242095 0.970253i \(-0.577834\pi\)
−0.242095 + 0.970253i \(0.577834\pi\)
\(744\) 0 0
\(745\) −20.1449 −0.738054
\(746\) 0 0
\(747\) 19.9832 0.731147
\(748\) 0 0
\(749\) −52.3848 −1.91410
\(750\) 0 0
\(751\) 17.3729 0.633945 0.316973 0.948435i \(-0.397334\pi\)
0.316973 + 0.948435i \(0.397334\pi\)
\(752\) 0 0
\(753\) −5.82851 −0.212403
\(754\) 0 0
\(755\) 11.8101 0.429814
\(756\) 0 0
\(757\) −38.7514 −1.40844 −0.704222 0.709980i \(-0.748704\pi\)
−0.704222 + 0.709980i \(0.748704\pi\)
\(758\) 0 0
\(759\) 1.79712 0.0652313
\(760\) 0 0
\(761\) −29.0280 −1.05226 −0.526132 0.850403i \(-0.676358\pi\)
−0.526132 + 0.850403i \(0.676358\pi\)
\(762\) 0 0
\(763\) 8.49347 0.307484
\(764\) 0 0
\(765\) 8.28603 0.299582
\(766\) 0 0
\(767\) 57.9348 2.09190
\(768\) 0 0
\(769\) −8.49451 −0.306320 −0.153160 0.988201i \(-0.548945\pi\)
−0.153160 + 0.988201i \(0.548945\pi\)
\(770\) 0 0
\(771\) 15.6695 0.564325
\(772\) 0 0
\(773\) −30.2676 −1.08865 −0.544325 0.838875i \(-0.683214\pi\)
−0.544325 + 0.838875i \(0.683214\pi\)
\(774\) 0 0
\(775\) −34.5369 −1.24060
\(776\) 0 0
\(777\) −19.5382 −0.700929
\(778\) 0 0
\(779\) −5.82147 −0.208576
\(780\) 0 0
\(781\) 4.04105 0.144600
\(782\) 0 0
\(783\) 19.6595 0.702574
\(784\) 0 0
\(785\) 9.27927 0.331191
\(786\) 0 0
\(787\) 25.5339 0.910185 0.455093 0.890444i \(-0.349606\pi\)
0.455093 + 0.890444i \(0.349606\pi\)
\(788\) 0 0
\(789\) −14.7824 −0.526268
\(790\) 0 0
\(791\) −39.7526 −1.41344
\(792\) 0 0
\(793\) 13.6587 0.485036
\(794\) 0 0
\(795\) 3.70318 0.131338
\(796\) 0 0
\(797\) −48.5429 −1.71948 −0.859738 0.510735i \(-0.829373\pi\)
−0.859738 + 0.510735i \(0.829373\pi\)
\(798\) 0 0
\(799\) −18.5308 −0.655573
\(800\) 0 0
\(801\) −2.26657 −0.0800853
\(802\) 0 0
\(803\) −12.7566 −0.450171
\(804\) 0 0
\(805\) −11.7125 −0.412810
\(806\) 0 0
\(807\) 10.4579 0.368137
\(808\) 0 0
\(809\) 30.1082 1.05855 0.529274 0.848451i \(-0.322464\pi\)
0.529274 + 0.848451i \(0.322464\pi\)
\(810\) 0 0
\(811\) 26.6166 0.934634 0.467317 0.884090i \(-0.345221\pi\)
0.467317 + 0.884090i \(0.345221\pi\)
\(812\) 0 0
\(813\) −5.29902 −0.185845
\(814\) 0 0
\(815\) −11.4032 −0.399438
\(816\) 0 0
\(817\) −38.4046 −1.34361
\(818\) 0 0
\(819\) 39.4182 1.37738
\(820\) 0 0
\(821\) −52.7820 −1.84210 −0.921052 0.389440i \(-0.872668\pi\)
−0.921052 + 0.389440i \(0.872668\pi\)
\(822\) 0 0
\(823\) 20.7992 0.725014 0.362507 0.931981i \(-0.381921\pi\)
0.362507 + 0.931981i \(0.381921\pi\)
\(824\) 0 0
\(825\) 1.95731 0.0681447
\(826\) 0 0
\(827\) −12.6722 −0.440654 −0.220327 0.975426i \(-0.570713\pi\)
−0.220327 + 0.975426i \(0.570713\pi\)
\(828\) 0 0
\(829\) 24.4262 0.848357 0.424179 0.905579i \(-0.360563\pi\)
0.424179 + 0.905579i \(0.360563\pi\)
\(830\) 0 0
\(831\) 14.6343 0.507660
\(832\) 0 0
\(833\) −11.4009 −0.395018
\(834\) 0 0
\(835\) 18.1555 0.628297
\(836\) 0 0
\(837\) 26.3189 0.909714
\(838\) 0 0
\(839\) −41.6208 −1.43691 −0.718455 0.695574i \(-0.755150\pi\)
−0.718455 + 0.695574i \(0.755150\pi\)
\(840\) 0 0
\(841\) 11.1181 0.383383
\(842\) 0 0
\(843\) 2.29248 0.0789572
\(844\) 0 0
\(845\) 6.82061 0.234636
\(846\) 0 0
\(847\) −33.2459 −1.14234
\(848\) 0 0
\(849\) 17.0644 0.585647
\(850\) 0 0
\(851\) −41.2769 −1.41495
\(852\) 0 0
\(853\) 11.9436 0.408942 0.204471 0.978873i \(-0.434453\pi\)
0.204471 + 0.978873i \(0.434453\pi\)
\(854\) 0 0
\(855\) 10.7376 0.367217
\(856\) 0 0
\(857\) −17.0803 −0.583452 −0.291726 0.956502i \(-0.594229\pi\)
−0.291726 + 0.956502i \(0.594229\pi\)
\(858\) 0 0
\(859\) 45.1499 1.54049 0.770247 0.637745i \(-0.220133\pi\)
0.770247 + 0.637745i \(0.220133\pi\)
\(860\) 0 0
\(861\) −2.49838 −0.0851446
\(862\) 0 0
\(863\) 42.9192 1.46099 0.730494 0.682919i \(-0.239290\pi\)
0.730494 + 0.682919i \(0.239290\pi\)
\(864\) 0 0
\(865\) −14.6172 −0.497000
\(866\) 0 0
\(867\) 3.73782 0.126943
\(868\) 0 0
\(869\) −6.36377 −0.215876
\(870\) 0 0
\(871\) −41.0159 −1.38977
\(872\) 0 0
\(873\) −1.28537 −0.0435033
\(874\) 0 0
\(875\) −28.4161 −0.960640
\(876\) 0 0
\(877\) −22.1529 −0.748050 −0.374025 0.927419i \(-0.622023\pi\)
−0.374025 + 0.927419i \(0.622023\pi\)
\(878\) 0 0
\(879\) 8.64627 0.291631
\(880\) 0 0
\(881\) −8.84872 −0.298121 −0.149060 0.988828i \(-0.547625\pi\)
−0.149060 + 0.988828i \(0.547625\pi\)
\(882\) 0 0
\(883\) −14.6632 −0.493455 −0.246728 0.969085i \(-0.579355\pi\)
−0.246728 + 0.969085i \(0.579355\pi\)
\(884\) 0 0
\(885\) 6.77290 0.227668
\(886\) 0 0
\(887\) −17.3973 −0.584143 −0.292072 0.956396i \(-0.594345\pi\)
−0.292072 + 0.956396i \(0.594345\pi\)
\(888\) 0 0
\(889\) −19.2201 −0.644621
\(890\) 0 0
\(891\) 5.67179 0.190012
\(892\) 0 0
\(893\) −24.0134 −0.803577
\(894\) 0 0
\(895\) 2.48300 0.0829975
\(896\) 0 0
\(897\) −9.12004 −0.304509
\(898\) 0 0
\(899\) 53.7075 1.79124
\(900\) 0 0
\(901\) 22.4979 0.749512
\(902\) 0 0
\(903\) −16.4820 −0.548485
\(904\) 0 0
\(905\) −9.15395 −0.304288
\(906\) 0 0
\(907\) 33.7438 1.12045 0.560223 0.828342i \(-0.310715\pi\)
0.560223 + 0.828342i \(0.310715\pi\)
\(908\) 0 0
\(909\) 8.32631 0.276166
\(910\) 0 0
\(911\) −12.1001 −0.400895 −0.200447 0.979704i \(-0.564240\pi\)
−0.200447 + 0.979704i \(0.564240\pi\)
\(912\) 0 0
\(913\) −6.52657 −0.215998
\(914\) 0 0
\(915\) 1.59678 0.0527880
\(916\) 0 0
\(917\) −15.9730 −0.527475
\(918\) 0 0
\(919\) 31.4919 1.03882 0.519412 0.854524i \(-0.326151\pi\)
0.519412 + 0.854524i \(0.326151\pi\)
\(920\) 0 0
\(921\) −7.45671 −0.245707
\(922\) 0 0
\(923\) −20.5076 −0.675015
\(924\) 0 0
\(925\) −44.9562 −1.47815
\(926\) 0 0
\(927\) −10.0416 −0.329810
\(928\) 0 0
\(929\) 28.6545 0.940123 0.470061 0.882634i \(-0.344232\pi\)
0.470061 + 0.882634i \(0.344232\pi\)
\(930\) 0 0
\(931\) −14.7740 −0.484199
\(932\) 0 0
\(933\) 3.53394 0.115696
\(934\) 0 0
\(935\) −2.70624 −0.0885036
\(936\) 0 0
\(937\) 6.14354 0.200701 0.100350 0.994952i \(-0.468004\pi\)
0.100350 + 0.994952i \(0.468004\pi\)
\(938\) 0 0
\(939\) −4.49702 −0.146755
\(940\) 0 0
\(941\) −25.9528 −0.846038 −0.423019 0.906121i \(-0.639030\pi\)
−0.423019 + 0.906121i \(0.639030\pi\)
\(942\) 0 0
\(943\) −5.27814 −0.171880
\(944\) 0 0
\(945\) 9.72108 0.316227
\(946\) 0 0
\(947\) −45.1337 −1.46665 −0.733324 0.679880i \(-0.762032\pi\)
−0.733324 + 0.679880i \(0.762032\pi\)
\(948\) 0 0
\(949\) 64.7374 2.10146
\(950\) 0 0
\(951\) 15.3151 0.496625
\(952\) 0 0
\(953\) −6.09060 −0.197294 −0.0986469 0.995122i \(-0.531451\pi\)
−0.0986469 + 0.995122i \(0.531451\pi\)
\(954\) 0 0
\(955\) −5.97781 −0.193437
\(956\) 0 0
\(957\) −3.04376 −0.0983908
\(958\) 0 0
\(959\) −26.9966 −0.871767
\(960\) 0 0
\(961\) 40.9000 1.31936
\(962\) 0 0
\(963\) 43.5423 1.40313
\(964\) 0 0
\(965\) −18.9002 −0.608420
\(966\) 0 0
\(967\) 32.8474 1.05630 0.528151 0.849150i \(-0.322885\pi\)
0.528151 + 0.849150i \(0.322885\pi\)
\(968\) 0 0
\(969\) −7.14412 −0.229502
\(970\) 0 0
\(971\) −45.3642 −1.45581 −0.727903 0.685680i \(-0.759505\pi\)
−0.727903 + 0.685680i \(0.759505\pi\)
\(972\) 0 0
\(973\) −68.5650 −2.19809
\(974\) 0 0
\(975\) −9.93297 −0.318110
\(976\) 0 0
\(977\) 9.96515 0.318813 0.159407 0.987213i \(-0.449042\pi\)
0.159407 + 0.987213i \(0.449042\pi\)
\(978\) 0 0
\(979\) 0.740268 0.0236591
\(980\) 0 0
\(981\) −7.05979 −0.225402
\(982\) 0 0
\(983\) 8.37150 0.267009 0.133505 0.991048i \(-0.457377\pi\)
0.133505 + 0.991048i \(0.457377\pi\)
\(984\) 0 0
\(985\) 25.3827 0.808761
\(986\) 0 0
\(987\) −10.3057 −0.328035
\(988\) 0 0
\(989\) −34.8202 −1.10722
\(990\) 0 0
\(991\) −28.8111 −0.915215 −0.457608 0.889154i \(-0.651294\pi\)
−0.457608 + 0.889154i \(0.651294\pi\)
\(992\) 0 0
\(993\) −0.970354 −0.0307933
\(994\) 0 0
\(995\) 4.98690 0.158095
\(996\) 0 0
\(997\) 12.3708 0.391788 0.195894 0.980625i \(-0.437239\pi\)
0.195894 + 0.980625i \(0.437239\pi\)
\(998\) 0 0
\(999\) 34.2589 1.08390
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 8752.2.a.s.1.6 18
4.3 odd 2 547.2.a.b.1.11 18
12.11 even 2 4923.2.a.l.1.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.11 18 4.3 odd 2
4923.2.a.l.1.8 18 12.11 even 2
8752.2.a.s.1.6 18 1.1 even 1 trivial