Properties

Label 2898.2.a.bc.1.2
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.70156 q^{10} +5.70156 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.40312 q^{19} +3.70156 q^{20} +1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} -1.00000 q^{28} -0.298438 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.70156 q^{35} +3.70156 q^{37} -5.40312 q^{38} +3.70156 q^{40} -7.70156 q^{41} +5.70156 q^{43} +1.00000 q^{46} -3.70156 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} -9.40312 q^{53} -1.00000 q^{56} -0.298438 q^{58} +0.596876 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +21.1047 q^{65} +4.00000 q^{67} -3.70156 q^{70} -7.40312 q^{71} -5.40312 q^{73} +3.70156 q^{74} -5.40312 q^{76} +3.70156 q^{80} -7.70156 q^{82} +2.59688 q^{83} +5.70156 q^{86} +15.4031 q^{89} -5.70156 q^{91} +1.00000 q^{92} -3.70156 q^{94} -20.0000 q^{95} -1.10469 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + q^{5} - 2q^{7} + 2q^{8} + q^{10} + 5q^{13} - 2q^{14} + 2q^{16} + 2q^{19} + q^{20} + 2q^{23} + 11q^{25} + 5q^{26} - 2q^{28} - 7q^{29} + 12q^{31} + 2q^{32} - q^{35} + q^{37} + 2q^{38} + q^{40} - 9q^{41} + 5q^{43} + 2q^{46} - q^{47} + 2q^{49} + 11q^{50} + 5q^{52} - 6q^{53} - 2q^{56} - 7q^{58} + 14q^{59} + 20q^{61} + 12q^{62} + 2q^{64} + 23q^{65} + 8q^{67} - q^{70} - 2q^{71} + 2q^{73} + q^{74} + 2q^{76} + q^{80} - 9q^{82} + 18q^{83} + 5q^{86} + 18q^{89} - 5q^{91} + 2q^{92} - q^{94} - 40q^{95} + 17q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.70156 1.65539 0.827694 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.70156 1.17054
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.70156 1.58133 0.790664 0.612250i \(-0.209735\pi\)
0.790664 + 0.612250i \(0.209735\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −5.40312 −1.23956 −0.619781 0.784775i \(-0.712779\pi\)
−0.619781 + 0.784775i \(0.712779\pi\)
\(20\) 3.70156 0.827694
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 8.70156 1.74031
\(26\) 5.70156 1.11817
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −0.298438 −0.0554185 −0.0277093 0.999616i \(-0.508821\pi\)
−0.0277093 + 0.999616i \(0.508821\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −3.70156 −0.625678
\(36\) 0 0
\(37\) 3.70156 0.608533 0.304267 0.952587i \(-0.401589\pi\)
0.304267 + 0.952587i \(0.401589\pi\)
\(38\) −5.40312 −0.876502
\(39\) 0 0
\(40\) 3.70156 0.585268
\(41\) −7.70156 −1.20278 −0.601391 0.798955i \(-0.705387\pi\)
−0.601391 + 0.798955i \(0.705387\pi\)
\(42\) 0 0
\(43\) 5.70156 0.869480 0.434740 0.900556i \(-0.356840\pi\)
0.434740 + 0.900556i \(0.356840\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.70156 −0.539928 −0.269964 0.962870i \(-0.587012\pi\)
−0.269964 + 0.962870i \(0.587012\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.70156 1.23059
\(51\) 0 0
\(52\) 5.70156 0.790664
\(53\) −9.40312 −1.29162 −0.645809 0.763499i \(-0.723480\pi\)
−0.645809 + 0.763499i \(0.723480\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −0.298438 −0.0391868
\(59\) 0.596876 0.0777066 0.0388533 0.999245i \(-0.487629\pi\)
0.0388533 + 0.999245i \(0.487629\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 21.1047 2.61771
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.70156 −0.442421
\(71\) −7.40312 −0.878589 −0.439295 0.898343i \(-0.644772\pi\)
−0.439295 + 0.898343i \(0.644772\pi\)
\(72\) 0 0
\(73\) −5.40312 −0.632388 −0.316194 0.948695i \(-0.602405\pi\)
−0.316194 + 0.948695i \(0.602405\pi\)
\(74\) 3.70156 0.430298
\(75\) 0 0
\(76\) −5.40312 −0.619781
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 3.70156 0.413847
\(81\) 0 0
\(82\) −7.70156 −0.850495
\(83\) 2.59688 0.285044 0.142522 0.989792i \(-0.454479\pi\)
0.142522 + 0.989792i \(0.454479\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.70156 0.614815
\(87\) 0 0
\(88\) 0 0
\(89\) 15.4031 1.63273 0.816364 0.577538i \(-0.195986\pi\)
0.816364 + 0.577538i \(0.195986\pi\)
\(90\) 0 0
\(91\) −5.70156 −0.597686
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −3.70156 −0.381787
\(95\) −20.0000 −2.05196
\(96\) 0 0
\(97\) −1.10469 −0.112164 −0.0560820 0.998426i \(-0.517861\pi\)
−0.0560820 + 0.998426i \(0.517861\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 8.70156 0.870156
\(101\) −4.59688 −0.457406 −0.228703 0.973496i \(-0.573449\pi\)
−0.228703 + 0.973496i \(0.573449\pi\)
\(102\) 0 0
\(103\) −17.1047 −1.68537 −0.842687 0.538403i \(-0.819028\pi\)
−0.842687 + 0.538403i \(0.819028\pi\)
\(104\) 5.70156 0.559084
\(105\) 0 0
\(106\) −9.40312 −0.913312
\(107\) −14.8062 −1.43137 −0.715687 0.698421i \(-0.753886\pi\)
−0.715687 + 0.698421i \(0.753886\pi\)
\(108\) 0 0
\(109\) 15.7016 1.50394 0.751968 0.659199i \(-0.229105\pi\)
0.751968 + 0.659199i \(0.229105\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 4.29844 0.404363 0.202182 0.979348i \(-0.435197\pi\)
0.202182 + 0.979348i \(0.435197\pi\)
\(114\) 0 0
\(115\) 3.70156 0.345172
\(116\) −0.298438 −0.0277093
\(117\) 0 0
\(118\) 0.596876 0.0549469
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 13.7016 1.22550
\(126\) 0 0
\(127\) 17.1047 1.51780 0.758898 0.651210i \(-0.225738\pi\)
0.758898 + 0.651210i \(0.225738\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 21.1047 1.85100
\(131\) −15.4031 −1.34578 −0.672889 0.739744i \(-0.734947\pi\)
−0.672889 + 0.739744i \(0.734947\pi\)
\(132\) 0 0
\(133\) 5.40312 0.468510
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −0.895314 −0.0764918 −0.0382459 0.999268i \(-0.512177\pi\)
−0.0382459 + 0.999268i \(0.512177\pi\)
\(138\) 0 0
\(139\) −21.1047 −1.79008 −0.895038 0.445990i \(-0.852852\pi\)
−0.895038 + 0.445990i \(0.852852\pi\)
\(140\) −3.70156 −0.312839
\(141\) 0 0
\(142\) −7.40312 −0.621256
\(143\) 0 0
\(144\) 0 0
\(145\) −1.10469 −0.0917392
\(146\) −5.40312 −0.447166
\(147\) 0 0
\(148\) 3.70156 0.304267
\(149\) 17.4031 1.42572 0.712860 0.701307i \(-0.247400\pi\)
0.712860 + 0.701307i \(0.247400\pi\)
\(150\) 0 0
\(151\) 9.10469 0.740929 0.370464 0.928847i \(-0.379199\pi\)
0.370464 + 0.928847i \(0.379199\pi\)
\(152\) −5.40312 −0.438251
\(153\) 0 0
\(154\) 0 0
\(155\) 22.2094 1.78390
\(156\) 0 0
\(157\) −13.4031 −1.06969 −0.534843 0.844952i \(-0.679629\pi\)
−0.534843 + 0.844952i \(0.679629\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 3.70156 0.292634
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −7.70156 −0.601391
\(165\) 0 0
\(166\) 2.59688 0.201557
\(167\) 21.4031 1.65622 0.828112 0.560563i \(-0.189415\pi\)
0.828112 + 0.560563i \(0.189415\pi\)
\(168\) 0 0
\(169\) 19.5078 1.50060
\(170\) 0 0
\(171\) 0 0
\(172\) 5.70156 0.434740
\(173\) 3.40312 0.258735 0.129367 0.991597i \(-0.458705\pi\)
0.129367 + 0.991597i \(0.458705\pi\)
\(174\) 0 0
\(175\) −8.70156 −0.657776
\(176\) 0 0
\(177\) 0 0
\(178\) 15.4031 1.15451
\(179\) −5.70156 −0.426155 −0.213077 0.977035i \(-0.568349\pi\)
−0.213077 + 0.977035i \(0.568349\pi\)
\(180\) 0 0
\(181\) 16.8062 1.24920 0.624599 0.780945i \(-0.285262\pi\)
0.624599 + 0.780945i \(0.285262\pi\)
\(182\) −5.70156 −0.422628
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 13.7016 1.00736
\(186\) 0 0
\(187\) 0 0
\(188\) −3.70156 −0.269964
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) −22.8062 −1.65020 −0.825101 0.564985i \(-0.808882\pi\)
−0.825101 + 0.564985i \(0.808882\pi\)
\(192\) 0 0
\(193\) −7.10469 −0.511407 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(194\) −1.10469 −0.0793119
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.50781 −0.463662 −0.231831 0.972756i \(-0.574472\pi\)
−0.231831 + 0.972756i \(0.574472\pi\)
\(198\) 0 0
\(199\) 1.10469 0.0783091 0.0391546 0.999233i \(-0.487534\pi\)
0.0391546 + 0.999233i \(0.487534\pi\)
\(200\) 8.70156 0.615293
\(201\) 0 0
\(202\) −4.59688 −0.323435
\(203\) 0.298438 0.0209462
\(204\) 0 0
\(205\) −28.5078 −1.99107
\(206\) −17.1047 −1.19174
\(207\) 0 0
\(208\) 5.70156 0.395332
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −9.40312 −0.645809
\(213\) 0 0
\(214\) −14.8062 −1.01213
\(215\) 21.1047 1.43933
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 15.7016 1.06344
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −20.8062 −1.39329 −0.696645 0.717416i \(-0.745325\pi\)
−0.696645 + 0.717416i \(0.745325\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 4.29844 0.285928
\(227\) 26.5078 1.75939 0.879693 0.475543i \(-0.157748\pi\)
0.879693 + 0.475543i \(0.157748\pi\)
\(228\) 0 0
\(229\) 28.2094 1.86413 0.932064 0.362294i \(-0.118006\pi\)
0.932064 + 0.362294i \(0.118006\pi\)
\(230\) 3.70156 0.244074
\(231\) 0 0
\(232\) −0.298438 −0.0195934
\(233\) −28.8062 −1.88716 −0.943580 0.331145i \(-0.892565\pi\)
−0.943580 + 0.331145i \(0.892565\pi\)
\(234\) 0 0
\(235\) −13.7016 −0.893791
\(236\) 0.596876 0.0388533
\(237\) 0 0
\(238\) 0 0
\(239\) 18.8062 1.21648 0.608238 0.793755i \(-0.291877\pi\)
0.608238 + 0.793755i \(0.291877\pi\)
\(240\) 0 0
\(241\) 5.10469 0.328822 0.164411 0.986392i \(-0.447428\pi\)
0.164411 + 0.986392i \(0.447428\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 3.70156 0.236484
\(246\) 0 0
\(247\) −30.8062 −1.96015
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 13.7016 0.866563
\(251\) −13.9109 −0.878050 −0.439025 0.898475i \(-0.644676\pi\)
−0.439025 + 0.898475i \(0.644676\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 17.1047 1.07324
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −23.6125 −1.47291 −0.736454 0.676488i \(-0.763501\pi\)
−0.736454 + 0.676488i \(0.763501\pi\)
\(258\) 0 0
\(259\) −3.70156 −0.230004
\(260\) 21.1047 1.30886
\(261\) 0 0
\(262\) −15.4031 −0.951608
\(263\) 16.5078 1.01792 0.508958 0.860792i \(-0.330031\pi\)
0.508958 + 0.860792i \(0.330031\pi\)
\(264\) 0 0
\(265\) −34.8062 −2.13813
\(266\) 5.40312 0.331287
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 3.40312 0.207492 0.103746 0.994604i \(-0.466917\pi\)
0.103746 + 0.994604i \(0.466917\pi\)
\(270\) 0 0
\(271\) 17.4031 1.05716 0.528582 0.848882i \(-0.322724\pi\)
0.528582 + 0.848882i \(0.322724\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.895314 −0.0540879
\(275\) 0 0
\(276\) 0 0
\(277\) −13.4031 −0.805316 −0.402658 0.915351i \(-0.631914\pi\)
−0.402658 + 0.915351i \(0.631914\pi\)
\(278\) −21.1047 −1.26577
\(279\) 0 0
\(280\) −3.70156 −0.221211
\(281\) −4.29844 −0.256423 −0.128212 0.991747i \(-0.540924\pi\)
−0.128212 + 0.991747i \(0.540924\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −7.40312 −0.439295
\(285\) 0 0
\(286\) 0 0
\(287\) 7.70156 0.454609
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) −1.10469 −0.0648694
\(291\) 0 0
\(292\) −5.40312 −0.316194
\(293\) 3.19375 0.186581 0.0932905 0.995639i \(-0.470261\pi\)
0.0932905 + 0.995639i \(0.470261\pi\)
\(294\) 0 0
\(295\) 2.20937 0.128635
\(296\) 3.70156 0.215149
\(297\) 0 0
\(298\) 17.4031 1.00814
\(299\) 5.70156 0.329730
\(300\) 0 0
\(301\) −5.70156 −0.328633
\(302\) 9.10469 0.523916
\(303\) 0 0
\(304\) −5.40312 −0.309890
\(305\) 37.0156 2.11951
\(306\) 0 0
\(307\) 5.10469 0.291340 0.145670 0.989333i \(-0.453466\pi\)
0.145670 + 0.989333i \(0.453466\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 22.2094 1.26141
\(311\) 16.2094 0.919149 0.459575 0.888139i \(-0.348002\pi\)
0.459575 + 0.888139i \(0.348002\pi\)
\(312\) 0 0
\(313\) −4.59688 −0.259831 −0.129915 0.991525i \(-0.541471\pi\)
−0.129915 + 0.991525i \(0.541471\pi\)
\(314\) −13.4031 −0.756382
\(315\) 0 0
\(316\) 0 0
\(317\) −7.70156 −0.432563 −0.216281 0.976331i \(-0.569393\pi\)
−0.216281 + 0.976331i \(0.569393\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.70156 0.206924
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) 49.6125 2.75201
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −7.70156 −0.425248
\(329\) 3.70156 0.204074
\(330\) 0 0
\(331\) −17.6125 −0.968070 −0.484035 0.875049i \(-0.660829\pi\)
−0.484035 + 0.875049i \(0.660829\pi\)
\(332\) 2.59688 0.142522
\(333\) 0 0
\(334\) 21.4031 1.17113
\(335\) 14.8062 0.808952
\(336\) 0 0
\(337\) 17.4031 0.948009 0.474004 0.880523i \(-0.342808\pi\)
0.474004 + 0.880523i \(0.342808\pi\)
\(338\) 19.5078 1.06109
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 5.70156 0.307408
\(345\) 0 0
\(346\) 3.40312 0.182953
\(347\) −20.5078 −1.10092 −0.550458 0.834863i \(-0.685547\pi\)
−0.550458 + 0.834863i \(0.685547\pi\)
\(348\) 0 0
\(349\) 34.2094 1.83119 0.915593 0.402107i \(-0.131722\pi\)
0.915593 + 0.402107i \(0.131722\pi\)
\(350\) −8.70156 −0.465118
\(351\) 0 0
\(352\) 0 0
\(353\) 33.3141 1.77313 0.886564 0.462606i \(-0.153085\pi\)
0.886564 + 0.462606i \(0.153085\pi\)
\(354\) 0 0
\(355\) −27.4031 −1.45441
\(356\) 15.4031 0.816364
\(357\) 0 0
\(358\) −5.70156 −0.301337
\(359\) −9.70156 −0.512029 −0.256014 0.966673i \(-0.582409\pi\)
−0.256014 + 0.966673i \(0.582409\pi\)
\(360\) 0 0
\(361\) 10.1938 0.536513
\(362\) 16.8062 0.883317
\(363\) 0 0
\(364\) −5.70156 −0.298843
\(365\) −20.0000 −1.04685
\(366\) 0 0
\(367\) 15.9109 0.830544 0.415272 0.909697i \(-0.363686\pi\)
0.415272 + 0.909697i \(0.363686\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 13.7016 0.712310
\(371\) 9.40312 0.488186
\(372\) 0 0
\(373\) −4.80625 −0.248858 −0.124429 0.992229i \(-0.539710\pi\)
−0.124429 + 0.992229i \(0.539710\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.70156 −0.190893
\(377\) −1.70156 −0.0876349
\(378\) 0 0
\(379\) −17.7016 −0.909268 −0.454634 0.890678i \(-0.650230\pi\)
−0.454634 + 0.890678i \(0.650230\pi\)
\(380\) −20.0000 −1.02598
\(381\) 0 0
\(382\) −22.8062 −1.16687
\(383\) −6.80625 −0.347783 −0.173892 0.984765i \(-0.555634\pi\)
−0.173892 + 0.984765i \(0.555634\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.10469 −0.361619
\(387\) 0 0
\(388\) −1.10469 −0.0560820
\(389\) −36.8062 −1.86615 −0.933075 0.359681i \(-0.882886\pi\)
−0.933075 + 0.359681i \(0.882886\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −6.50781 −0.327859
\(395\) 0 0
\(396\) 0 0
\(397\) 6.20937 0.311639 0.155820 0.987786i \(-0.450198\pi\)
0.155820 + 0.987786i \(0.450198\pi\)
\(398\) 1.10469 0.0553729
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) −28.8062 −1.43852 −0.719258 0.694743i \(-0.755518\pi\)
−0.719258 + 0.694743i \(0.755518\pi\)
\(402\) 0 0
\(403\) 34.2094 1.70409
\(404\) −4.59688 −0.228703
\(405\) 0 0
\(406\) 0.298438 0.0148112
\(407\) 0 0
\(408\) 0 0
\(409\) −32.2094 −1.59265 −0.796325 0.604868i \(-0.793226\pi\)
−0.796325 + 0.604868i \(0.793226\pi\)
\(410\) −28.5078 −1.40790
\(411\) 0 0
\(412\) −17.1047 −0.842687
\(413\) −0.596876 −0.0293703
\(414\) 0 0
\(415\) 9.61250 0.471859
\(416\) 5.70156 0.279542
\(417\) 0 0
\(418\) 0 0
\(419\) 2.59688 0.126866 0.0634328 0.997986i \(-0.479795\pi\)
0.0634328 + 0.997986i \(0.479795\pi\)
\(420\) 0 0
\(421\) 15.7016 0.765247 0.382624 0.923904i \(-0.375021\pi\)
0.382624 + 0.923904i \(0.375021\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) −9.40312 −0.456656
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −14.8062 −0.715687
\(429\) 0 0
\(430\) 21.1047 1.01776
\(431\) −5.10469 −0.245884 −0.122942 0.992414i \(-0.539233\pi\)
−0.122942 + 0.992414i \(0.539233\pi\)
\(432\) 0 0
\(433\) 17.1047 0.821999 0.410999 0.911636i \(-0.365180\pi\)
0.410999 + 0.911636i \(0.365180\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 15.7016 0.751968
\(437\) −5.40312 −0.258466
\(438\) 0 0
\(439\) 3.19375 0.152429 0.0762147 0.997091i \(-0.475717\pi\)
0.0762147 + 0.997091i \(0.475717\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.1047 −0.812668 −0.406334 0.913725i \(-0.633193\pi\)
−0.406334 + 0.913725i \(0.633193\pi\)
\(444\) 0 0
\(445\) 57.0156 2.70280
\(446\) −20.8062 −0.985204
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −27.6125 −1.30311 −0.651557 0.758600i \(-0.725884\pi\)
−0.651557 + 0.758600i \(0.725884\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.29844 0.202182
\(453\) 0 0
\(454\) 26.5078 1.24407
\(455\) −21.1047 −0.989403
\(456\) 0 0
\(457\) −16.2094 −0.758242 −0.379121 0.925347i \(-0.623774\pi\)
−0.379121 + 0.925347i \(0.623774\pi\)
\(458\) 28.2094 1.31814
\(459\) 0 0
\(460\) 3.70156 0.172586
\(461\) −27.4031 −1.27629 −0.638145 0.769916i \(-0.720298\pi\)
−0.638145 + 0.769916i \(0.720298\pi\)
\(462\) 0 0
\(463\) −13.1047 −0.609026 −0.304513 0.952508i \(-0.598494\pi\)
−0.304513 + 0.952508i \(0.598494\pi\)
\(464\) −0.298438 −0.0138546
\(465\) 0 0
\(466\) −28.8062 −1.33442
\(467\) 8.29844 0.384006 0.192003 0.981394i \(-0.438502\pi\)
0.192003 + 0.981394i \(0.438502\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −13.7016 −0.632006
\(471\) 0 0
\(472\) 0.596876 0.0274734
\(473\) 0 0
\(474\) 0 0
\(475\) −47.0156 −2.15722
\(476\) 0 0
\(477\) 0 0
\(478\) 18.8062 0.860178
\(479\) −7.40312 −0.338257 −0.169129 0.985594i \(-0.554095\pi\)
−0.169129 + 0.985594i \(0.554095\pi\)
\(480\) 0 0
\(481\) 21.1047 0.962291
\(482\) 5.10469 0.232512
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −4.08907 −0.185675
\(486\) 0 0
\(487\) −13.1047 −0.593830 −0.296915 0.954904i \(-0.595958\pi\)
−0.296915 + 0.954904i \(0.595958\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 3.70156 0.167220
\(491\) 17.6125 0.794841 0.397420 0.917637i \(-0.369905\pi\)
0.397420 + 0.917637i \(0.369905\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −30.8062 −1.38604
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 7.40312 0.332076
\(498\) 0 0
\(499\) 37.0156 1.65705 0.828523 0.559954i \(-0.189181\pi\)
0.828523 + 0.559954i \(0.189181\pi\)
\(500\) 13.7016 0.612752
\(501\) 0 0
\(502\) −13.9109 −0.620875
\(503\) 11.4031 0.508440 0.254220 0.967146i \(-0.418181\pi\)
0.254220 + 0.967146i \(0.418181\pi\)
\(504\) 0 0
\(505\) −17.0156 −0.757185
\(506\) 0 0
\(507\) 0 0
\(508\) 17.1047 0.758898
\(509\) 14.8062 0.656275 0.328138 0.944630i \(-0.393579\pi\)
0.328138 + 0.944630i \(0.393579\pi\)
\(510\) 0 0
\(511\) 5.40312 0.239020
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −23.6125 −1.04150
\(515\) −63.3141 −2.78995
\(516\) 0 0
\(517\) 0 0
\(518\) −3.70156 −0.162637
\(519\) 0 0
\(520\) 21.1047 0.925502
\(521\) −15.4031 −0.674823 −0.337412 0.941357i \(-0.609551\pi\)
−0.337412 + 0.941357i \(0.609551\pi\)
\(522\) 0 0
\(523\) −38.4187 −1.67993 −0.839967 0.542637i \(-0.817426\pi\)
−0.839967 + 0.542637i \(0.817426\pi\)
\(524\) −15.4031 −0.672889
\(525\) 0 0
\(526\) 16.5078 0.719775
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −34.8062 −1.51189
\(531\) 0 0
\(532\) 5.40312 0.234255
\(533\) −43.9109 −1.90199
\(534\) 0 0
\(535\) −54.8062 −2.36948
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 3.40312 0.146719
\(539\) 0 0
\(540\) 0 0
\(541\) −20.2094 −0.868869 −0.434434 0.900703i \(-0.643052\pi\)
−0.434434 + 0.900703i \(0.643052\pi\)
\(542\) 17.4031 0.747528
\(543\) 0 0
\(544\) 0 0
\(545\) 58.1203 2.48960
\(546\) 0 0
\(547\) −14.2094 −0.607549 −0.303774 0.952744i \(-0.598247\pi\)
−0.303774 + 0.952744i \(0.598247\pi\)
\(548\) −0.895314 −0.0382459
\(549\) 0 0
\(550\) 0 0
\(551\) 1.61250 0.0686947
\(552\) 0 0
\(553\) 0 0
\(554\) −13.4031 −0.569444
\(555\) 0 0
\(556\) −21.1047 −0.895038
\(557\) −21.4031 −0.906879 −0.453440 0.891287i \(-0.649803\pi\)
−0.453440 + 0.891287i \(0.649803\pi\)
\(558\) 0 0
\(559\) 32.5078 1.37493
\(560\) −3.70156 −0.156420
\(561\) 0 0
\(562\) −4.29844 −0.181319
\(563\) −44.7172 −1.88460 −0.942302 0.334763i \(-0.891344\pi\)
−0.942302 + 0.334763i \(0.891344\pi\)
\(564\) 0 0
\(565\) 15.9109 0.669378
\(566\) −10.0000 −0.420331
\(567\) 0 0
\(568\) −7.40312 −0.310628
\(569\) −11.1047 −0.465533 −0.232766 0.972533i \(-0.574778\pi\)
−0.232766 + 0.972533i \(0.574778\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7.70156 0.321457
\(575\) 8.70156 0.362880
\(576\) 0 0
\(577\) −23.6125 −0.983001 −0.491501 0.870877i \(-0.663551\pi\)
−0.491501 + 0.870877i \(0.663551\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) −1.10469 −0.0458696
\(581\) −2.59688 −0.107737
\(582\) 0 0
\(583\) 0 0
\(584\) −5.40312 −0.223583
\(585\) 0 0
\(586\) 3.19375 0.131933
\(587\) 1.19375 0.0492714 0.0246357 0.999696i \(-0.492157\pi\)
0.0246357 + 0.999696i \(0.492157\pi\)
\(588\) 0 0
\(589\) −32.4187 −1.33579
\(590\) 2.20937 0.0909584
\(591\) 0 0
\(592\) 3.70156 0.152133
\(593\) 41.9109 1.72108 0.860538 0.509386i \(-0.170128\pi\)
0.860538 + 0.509386i \(0.170128\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.4031 0.712860
\(597\) 0 0
\(598\) 5.70156 0.233154
\(599\) 29.6125 1.20993 0.604967 0.796251i \(-0.293186\pi\)
0.604967 + 0.796251i \(0.293186\pi\)
\(600\) 0 0
\(601\) 25.4031 1.03622 0.518108 0.855315i \(-0.326637\pi\)
0.518108 + 0.855315i \(0.326637\pi\)
\(602\) −5.70156 −0.232378
\(603\) 0 0
\(604\) 9.10469 0.370464
\(605\) −40.7172 −1.65539
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) −5.40312 −0.219126
\(609\) 0 0
\(610\) 37.0156 1.49872
\(611\) −21.1047 −0.853804
\(612\) 0 0
\(613\) −25.3141 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(614\) 5.10469 0.206008
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) 22.2094 0.891950
\(621\) 0 0
\(622\) 16.2094 0.649937
\(623\) −15.4031 −0.617113
\(624\) 0 0
\(625\) 7.20937 0.288375
\(626\) −4.59688 −0.183728
\(627\) 0 0
\(628\) −13.4031 −0.534843
\(629\) 0 0
\(630\) 0 0
\(631\) 9.19375 0.365997 0.182999 0.983113i \(-0.441420\pi\)
0.182999 + 0.983113i \(0.441420\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −7.70156 −0.305868
\(635\) 63.3141 2.51254
\(636\) 0 0
\(637\) 5.70156 0.225904
\(638\) 0 0
\(639\) 0 0
\(640\) 3.70156 0.146317
\(641\) 11.7016 0.462184 0.231092 0.972932i \(-0.425770\pi\)
0.231092 + 0.972932i \(0.425770\pi\)
\(642\) 0 0
\(643\) 3.19375 0.125949 0.0629746 0.998015i \(-0.479941\pi\)
0.0629746 + 0.998015i \(0.479941\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) 4.20937 0.165488 0.0827438 0.996571i \(-0.473632\pi\)
0.0827438 + 0.996571i \(0.473632\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 49.6125 1.94596
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −47.1047 −1.84335 −0.921674 0.387964i \(-0.873178\pi\)
−0.921674 + 0.387964i \(0.873178\pi\)
\(654\) 0 0
\(655\) −57.0156 −2.22778
\(656\) −7.70156 −0.300695
\(657\) 0 0
\(658\) 3.70156 0.144302
\(659\) −25.6125 −0.997721 −0.498861 0.866682i \(-0.666248\pi\)
−0.498861 + 0.866682i \(0.666248\pi\)
\(660\) 0 0
\(661\) −20.8062 −0.809269 −0.404635 0.914478i \(-0.632601\pi\)
−0.404635 + 0.914478i \(0.632601\pi\)
\(662\) −17.6125 −0.684529
\(663\) 0 0
\(664\) 2.59688 0.100778
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) −0.298438 −0.0115556
\(668\) 21.4031 0.828112
\(669\) 0 0
\(670\) 14.8062 0.572015
\(671\) 0 0
\(672\) 0 0
\(673\) −8.29844 −0.319881 −0.159941 0.987127i \(-0.551130\pi\)
−0.159941 + 0.987127i \(0.551130\pi\)
\(674\) 17.4031 0.670343
\(675\) 0 0
\(676\) 19.5078 0.750300
\(677\) 3.19375 0.122746 0.0613729 0.998115i \(-0.480452\pi\)
0.0613729 + 0.998115i \(0.480452\pi\)
\(678\) 0 0
\(679\) 1.10469 0.0423940
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −3.31406 −0.126624
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 5.70156 0.217370
\(689\) −53.6125 −2.04247
\(690\) 0 0
\(691\) −41.7016 −1.58640 −0.793201 0.608960i \(-0.791587\pi\)
−0.793201 + 0.608960i \(0.791587\pi\)
\(692\) 3.40312 0.129367
\(693\) 0 0
\(694\) −20.5078 −0.778466
\(695\) −78.1203 −2.96327
\(696\) 0 0
\(697\) 0 0
\(698\) 34.2094 1.29484
\(699\) 0 0
\(700\) −8.70156 −0.328888
\(701\) −14.5969 −0.551316 −0.275658 0.961256i \(-0.588896\pi\)
−0.275658 + 0.961256i \(0.588896\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 33.3141 1.25379
\(707\) 4.59688 0.172883
\(708\) 0 0
\(709\) −15.1938 −0.570613 −0.285307 0.958436i \(-0.592095\pi\)
−0.285307 + 0.958436i \(0.592095\pi\)
\(710\) −27.4031 −1.02842
\(711\) 0 0
\(712\) 15.4031 0.577256
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) −5.70156 −0.213077
\(717\) 0 0
\(718\) −9.70156 −0.362059
\(719\) −14.5078 −0.541050 −0.270525 0.962713i \(-0.587197\pi\)
−0.270525 + 0.962713i \(0.587197\pi\)
\(720\) 0 0
\(721\) 17.1047 0.637012
\(722\) 10.1938 0.379372
\(723\) 0 0
\(724\) 16.8062 0.624599
\(725\) −2.59688 −0.0964455
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −5.70156 −0.211314
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) 31.0156 1.14559 0.572794 0.819699i \(-0.305859\pi\)
0.572794 + 0.819699i \(0.305859\pi\)
\(734\) 15.9109 0.587283
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) 38.2094 1.40555 0.702777 0.711410i \(-0.251943\pi\)
0.702777 + 0.711410i \(0.251943\pi\)
\(740\) 13.7016 0.503679
\(741\) 0 0
\(742\) 9.40312 0.345200
\(743\) 21.6125 0.792886 0.396443 0.918059i \(-0.370245\pi\)
0.396443 + 0.918059i \(0.370245\pi\)
\(744\) 0 0
\(745\) 64.4187 2.36012
\(746\) −4.80625 −0.175969
\(747\) 0 0
\(748\) 0 0
\(749\) 14.8062 0.541009
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −3.70156 −0.134982
\(753\) 0 0
\(754\) −1.70156 −0.0619672
\(755\) 33.7016 1.22653
\(756\) 0 0
\(757\) 7.19375 0.261461 0.130731 0.991418i \(-0.458268\pi\)
0.130731 + 0.991418i \(0.458268\pi\)
\(758\) −17.7016 −0.642950
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) −15.6125 −0.565953 −0.282976 0.959127i \(-0.591322\pi\)
−0.282976 + 0.959127i \(0.591322\pi\)
\(762\) 0 0
\(763\) −15.7016 −0.568435
\(764\) −22.8062 −0.825101
\(765\) 0 0
\(766\) −6.80625 −0.245920
\(767\) 3.40312 0.122880
\(768\) 0 0
\(769\) −29.1047 −1.04954 −0.524771 0.851243i \(-0.675849\pi\)
−0.524771 + 0.851243i \(0.675849\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.10469 −0.255703
\(773\) −44.7172 −1.60837 −0.804183 0.594382i \(-0.797397\pi\)
−0.804183 + 0.594382i \(0.797397\pi\)
\(774\) 0 0
\(775\) 52.2094 1.87542
\(776\) −1.10469 −0.0396559
\(777\) 0 0
\(778\) −36.8062 −1.31957
\(779\) 41.6125 1.49092
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −49.6125 −1.77075
\(786\) 0 0
\(787\) 49.8219 1.77596 0.887979 0.459884i \(-0.152109\pi\)
0.887979 + 0.459884i \(0.152109\pi\)
\(788\) −6.50781 −0.231831
\(789\) 0 0
\(790\) 0 0
\(791\) −4.29844 −0.152835
\(792\) 0 0
\(793\) 57.0156 2.02468
\(794\) 6.20937 0.220362
\(795\) 0 0
\(796\) 1.10469 0.0391546
\(797\) 31.7016 1.12293 0.561463 0.827502i \(-0.310239\pi\)
0.561463 + 0.827502i \(0.310239\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 8.70156 0.307647
\(801\) 0 0
\(802\) −28.8062 −1.01718
\(803\) 0 0
\(804\) 0 0
\(805\) −3.70156 −0.130463
\(806\) 34.2094 1.20497
\(807\) 0 0
\(808\) −4.59688 −0.161718
\(809\) 21.4031 0.752494 0.376247 0.926519i \(-0.377214\pi\)
0.376247 + 0.926519i \(0.377214\pi\)
\(810\) 0 0
\(811\) 11.3141 0.397290 0.198645 0.980071i \(-0.436346\pi\)
0.198645 + 0.980071i \(0.436346\pi\)
\(812\) 0.298438 0.0104731
\(813\) 0 0
\(814\) 0 0
\(815\) −44.4187 −1.55592
\(816\) 0 0
\(817\) −30.8062 −1.07777
\(818\) −32.2094 −1.12617
\(819\) 0 0
\(820\) −28.5078 −0.995536
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 46.7172 1.62846 0.814229 0.580543i \(-0.197160\pi\)
0.814229 + 0.580543i \(0.197160\pi\)
\(824\) −17.1047 −0.595870
\(825\) 0 0
\(826\) −0.596876 −0.0207680
\(827\) 33.0156 1.14807 0.574033 0.818832i \(-0.305378\pi\)
0.574033 + 0.818832i \(0.305378\pi\)
\(828\) 0 0
\(829\) −30.2094 −1.04921 −0.524607 0.851344i \(-0.675788\pi\)
−0.524607 + 0.851344i \(0.675788\pi\)
\(830\) 9.61250 0.333655
\(831\) 0 0
\(832\) 5.70156 0.197666
\(833\) 0 0
\(834\) 0 0
\(835\) 79.2250 2.74169
\(836\) 0 0
\(837\) 0 0
\(838\) 2.59688 0.0897076
\(839\) 19.4031 0.669870 0.334935 0.942241i \(-0.391286\pi\)
0.334935 + 0.942241i \(0.391286\pi\)
\(840\) 0 0
\(841\) −28.9109 −0.996929
\(842\) 15.7016 0.541112
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 72.2094 2.48408
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) −9.40312 −0.322905
\(849\) 0 0
\(850\) 0 0
\(851\) 3.70156 0.126888
\(852\) 0 0
\(853\) 41.1047 1.40740 0.703699 0.710498i \(-0.251530\pi\)
0.703699 + 0.710498i \(0.251530\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −14.8062 −0.506067
\(857\) −2.08907 −0.0713611 −0.0356806 0.999363i \(-0.511360\pi\)
−0.0356806 + 0.999363i \(0.511360\pi\)
\(858\) 0 0
\(859\) −3.91093 −0.133439 −0.0667197 0.997772i \(-0.521253\pi\)
−0.0667197 + 0.997772i \(0.521253\pi\)
\(860\) 21.1047 0.719664
\(861\) 0 0
\(862\) −5.10469 −0.173866
\(863\) 10.8062 0.367849 0.183924 0.982940i \(-0.441120\pi\)
0.183924 + 0.982940i \(0.441120\pi\)
\(864\) 0 0
\(865\) 12.5969 0.428307
\(866\) 17.1047 0.581241
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) 22.8062 0.772760
\(872\) 15.7016 0.531722
\(873\) 0 0
\(874\) −5.40312 −0.182763
\(875\) −13.7016 −0.463197
\(876\) 0 0
\(877\) −48.2094 −1.62791 −0.813957 0.580925i \(-0.802691\pi\)
−0.813957 + 0.580925i \(0.802691\pi\)
\(878\) 3.19375 0.107784
\(879\) 0 0
\(880\) 0 0
\(881\) 10.2094 0.343963 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(882\) 0 0
\(883\) 16.5969 0.558529 0.279265 0.960214i \(-0.409909\pi\)
0.279265 + 0.960214i \(0.409909\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −17.1047 −0.574643
\(887\) 52.2094 1.75302 0.876510 0.481384i \(-0.159866\pi\)
0.876510 + 0.481384i \(0.159866\pi\)
\(888\) 0 0
\(889\) −17.1047 −0.573673
\(890\) 57.0156 1.91117
\(891\) 0 0
\(892\) −20.8062 −0.696645
\(893\) 20.0000 0.669274
\(894\) 0 0
\(895\) −21.1047 −0.705452
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −27.6125 −0.921441
\(899\) −1.79063 −0.0597208
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 4.29844 0.142964
\(905\) 62.2094 2.06791
\(906\) 0 0
\(907\) −2.89531 −0.0961373 −0.0480687 0.998844i \(-0.515307\pi\)
−0.0480687 + 0.998844i \(0.515307\pi\)
\(908\) 26.5078 0.879693
\(909\) 0 0
\(910\) −21.1047 −0.699614
\(911\) −1.70156 −0.0563753 −0.0281876 0.999603i \(-0.508974\pi\)
−0.0281876 + 0.999603i \(0.508974\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −16.2094 −0.536158
\(915\) 0 0
\(916\) 28.2094 0.932064
\(917\) 15.4031 0.508656
\(918\) 0 0
\(919\) 1.19375 0.0393782 0.0196891 0.999806i \(-0.493732\pi\)
0.0196891 + 0.999806i \(0.493732\pi\)
\(920\) 3.70156 0.122037
\(921\) 0 0
\(922\) −27.4031 −0.902474
\(923\) −42.2094 −1.38934
\(924\) 0 0
\(925\) 32.2094 1.05904
\(926\) −13.1047 −0.430647
\(927\) 0 0
\(928\) −0.298438 −0.00979670
\(929\) −23.7016 −0.777623 −0.388812 0.921317i \(-0.627114\pi\)
−0.388812 + 0.921317i \(0.627114\pi\)
\(930\) 0 0
\(931\) −5.40312 −0.177080
\(932\) −28.8062 −0.943580
\(933\) 0 0
\(934\) 8.29844 0.271533
\(935\) 0 0
\(936\) 0 0
\(937\) 47.3141 1.54568 0.772841 0.634599i \(-0.218835\pi\)
0.772841 + 0.634599i \(0.218835\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −13.7016 −0.446896
\(941\) 9.31406 0.303630 0.151815 0.988409i \(-0.451488\pi\)
0.151815 + 0.988409i \(0.451488\pi\)
\(942\) 0 0
\(943\) −7.70156 −0.250797
\(944\) 0.596876 0.0194267
\(945\) 0 0
\(946\) 0 0
\(947\) −3.49219 −0.113481 −0.0567405 0.998389i \(-0.518071\pi\)
−0.0567405 + 0.998389i \(0.518071\pi\)
\(948\) 0 0
\(949\) −30.8062 −1.00001
\(950\) −47.0156 −1.52539
\(951\) 0 0
\(952\) 0 0
\(953\) 20.8062 0.673980 0.336990 0.941508i \(-0.390591\pi\)
0.336990 + 0.941508i \(0.390591\pi\)
\(954\) 0 0
\(955\) −84.4187 −2.73173
\(956\) 18.8062 0.608238
\(957\) 0 0
\(958\) −7.40312 −0.239184
\(959\) 0.895314 0.0289112
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 21.1047 0.680442
\(963\) 0 0
\(964\) 5.10469 0.164411
\(965\) −26.2984 −0.846577
\(966\) 0 0
\(967\) 20.4187 0.656623 0.328311 0.944570i \(-0.393521\pi\)
0.328311 + 0.944570i \(0.393521\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) −4.08907 −0.131292
\(971\) 33.4031 1.07196 0.535979 0.844232i \(-0.319943\pi\)
0.535979 + 0.844232i \(0.319943\pi\)
\(972\) 0 0
\(973\) 21.1047 0.676585
\(974\) −13.1047 −0.419901
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 40.2984 1.28926 0.644631 0.764494i \(-0.277011\pi\)
0.644631 + 0.764494i \(0.277011\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.70156 0.118242
\(981\) 0 0
\(982\) 17.6125 0.562037
\(983\) 55.8219 1.78044 0.890221 0.455530i \(-0.150550\pi\)
0.890221 + 0.455530i \(0.150550\pi\)
\(984\) 0 0
\(985\) −24.0891 −0.767541
\(986\) 0 0
\(987\) 0 0
\(988\) −30.8062 −0.980077
\(989\) 5.70156 0.181299
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 7.40312 0.234813
\(995\) 4.08907 0.129632
\(996\) 0 0
\(997\) 48.5969 1.53908 0.769539 0.638600i \(-0.220486\pi\)
0.769539 + 0.638600i \(0.220486\pi\)
\(998\) 37.0156 1.17171
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2898.2.a.bc.1.2 2
3.2 odd 2 966.2.a.m.1.1 2
12.11 even 2 7728.2.a.z.1.1 2
21.20 even 2 6762.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.m.1.1 2 3.2 odd 2
2898.2.a.bc.1.2 2 1.1 even 1 trivial
6762.2.a.bq.1.2 2 21.20 even 2
7728.2.a.z.1.1 2 12.11 even 2