Properties

Label 2898.2.a.a.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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 1.1
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} -4.00000 q^{11} -3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{20} +4.00000 q^{22} -1.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} -1.00000 q^{28} -3.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +3.00000 q^{35} -9.00000 q^{37} +3.00000 q^{40} -9.00000 q^{41} -3.00000 q^{43} -4.00000 q^{44} +1.00000 q^{46} +7.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -3.00000 q^{52} +4.00000 q^{53} +12.0000 q^{55} +1.00000 q^{56} +3.00000 q^{58} -6.00000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +9.00000 q^{65} +4.00000 q^{67} +4.00000 q^{68} -3.00000 q^{70} +6.00000 q^{71} -8.00000 q^{73} +9.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} -3.00000 q^{80} +9.00000 q^{82} -4.00000 q^{83} -12.0000 q^{85} +3.00000 q^{86} +4.00000 q^{88} +14.0000 q^{89} +3.00000 q^{91} -1.00000 q^{92} -7.00000 q^{94} -7.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −3.00000 −0.457496 −0.228748 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 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
\(65\) 9.00000 1.11631
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 9.00000 1.04623
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 3.00000 0.323498
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) −12.0000 −1.14416
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −9.00000 −0.789352
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) −9.00000 −0.739795
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 18.0000 1.44579
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) −3.00000 −0.228748
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −19.0000 −1.42013 −0.710063 0.704138i \(-0.751334\pi\)
−0.710063 + 0.704138i \(0.751334\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −3.00000 −0.222375
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 27.0000 1.98508
\(186\) 0 0
\(187\) −16.0000 −1.17004
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −17.0000 −1.22369 −0.611843 0.790979i \(-0.709572\pi\)
−0.611843 + 0.790979i \(0.709572\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 23.0000 1.63868 0.819341 0.573306i \(-0.194340\pi\)
0.819341 + 0.573306i \(0.194340\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 27.0000 1.88576
\(206\) −5.00000 −0.348367
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 9.00000 0.613795
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −3.00000 −0.203186
\(219\) 0 0
\(220\) 12.0000 0.809040
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) −11.0000 −0.730096 −0.365048 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 9.00000 0.558156
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) −9.00000 −0.539784
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −23.0000 −1.37206 −0.686032 0.727571i \(-0.740649\pi\)
−0.686032 + 0.727571i \(0.740649\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −9.00000 −0.528498
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 9.00000 0.523114
\(297\) 0 0
\(298\) 16.0000 0.926855
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) −15.0000 −0.863153
\(303\) 0 0
\(304\) 0 0
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) 15.0000 0.856095 0.428048 0.903756i \(-0.359202\pi\)
0.428048 + 0.903756i \(0.359202\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −18.0000 −1.02233
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −5.00000 −0.280828 −0.140414 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) −7.00000 −0.385922
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 4.00000 0.217571
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.00000 0.161749
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −23.0000 −1.23470 −0.617352 0.786687i \(-0.711795\pi\)
−0.617352 + 0.786687i \(0.711795\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 29.0000 1.54351 0.771757 0.635917i \(-0.219378\pi\)
0.771757 + 0.635917i \(0.219378\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 19.0000 1.00418
\(359\) 1.00000 0.0527780 0.0263890 0.999652i \(-0.491599\pi\)
0.0263890 + 0.999652i \(0.491599\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 3.00000 0.157243
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) 31.0000 1.61819 0.809093 0.587680i \(-0.199959\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −27.0000 −1.40366
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −7.00000 −0.360997
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 17.0000 0.865277
\(387\) 0 0
\(388\) −7.00000 −0.355371
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −23.0000 −1.15872
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 5.00000 0.250627
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 36.0000 1.78445
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) −27.0000 −1.33343
\(411\) 0 0
\(412\) 5.00000 0.246332
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −21.0000 −1.02348 −0.511739 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −9.00000 −0.434019
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) 23.0000 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 3.00000 0.143674
\(437\) 0 0
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −25.0000 −1.18779 −0.593893 0.804544i \(-0.702410\pi\)
−0.593893 + 0.804544i \(0.702410\pi\)
\(444\) 0 0
\(445\) −42.0000 −1.99099
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) 11.0000 0.516256
\(455\) −9.00000 −0.421927
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 21.0000 0.968658
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 27.0000 1.23109
\(482\) 5.00000 0.227744
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 19.0000 0.848012
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) −44.0000 −1.95027 −0.975133 0.221621i \(-0.928865\pi\)
−0.975133 + 0.221621i \(0.928865\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) −15.0000 −0.660979
\(516\) 0 0
\(517\) −28.0000 −1.23144
\(518\) −9.00000 −0.395437
\(519\) 0 0
\(520\) −9.00000 −0.394676
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 42.0000 1.83653 0.918266 0.395964i \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 27.0000 1.16950
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 15.0000 0.640768
\(549\) 0 0
\(550\) 16.0000 0.682242
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −20.0000 −0.849719
\(555\) 0 0
\(556\) 9.00000 0.381685
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 9.00000 0.380659
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 23.0000 0.970196
\(563\) −15.0000 −0.632175 −0.316087 0.948730i \(-0.602369\pi\)
−0.316087 + 0.948730i \(0.602369\pi\)
\(564\) 0 0
\(565\) 27.0000 1.13590
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 13.0000 0.544988 0.272494 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −9.00000 −0.375653
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −16.0000 −0.662652
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −9.00000 −0.369898
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −16.0000 −0.655386
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −3.00000 −0.122271
\(603\) 0 0
\(604\) 15.0000 0.610341
\(605\) −15.0000 −0.609837
\(606\) 0 0
\(607\) 30.0000 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 30.0000 1.21466
\(611\) −21.0000 −0.849569
\(612\) 0 0
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) −15.0000 −0.605351
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 18.0000 0.722897
\(621\) 0 0
\(622\) −28.0000 −1.12270
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 5.00000 0.198575
\(635\) −21.0000 −0.833360
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −31.0000 −1.21312 −0.606562 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 7.00000 0.272888
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −45.0000 −1.71936
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −3.00000 −0.114374
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 23.0000 0.873068
\(695\) −27.0000 −1.02417
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −29.0000 −1.09143
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 18.0000 0.675528
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) −19.0000 −0.710063
\(717\) 0 0
\(718\) −1.00000 −0.0373197
\(719\) 7.00000 0.261056 0.130528 0.991445i \(-0.458333\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −3.00000 −0.111187
\(729\) 0 0
\(730\) −24.0000 −0.888280
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −16.0000 −0.590973 −0.295487 0.955347i \(-0.595482\pi\)
−0.295487 + 0.955347i \(0.595482\pi\)
\(734\) −31.0000 −1.14423
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 27.0000 0.992540
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 48.0000 1.75858
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) −16.0000 −0.585018
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) −9.00000 −0.327761
\(755\) −45.0000 −1.63772
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 25.0000 0.908041
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −3.00000 −0.108607
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 12.0000 0.432450
\(771\) 0 0
\(772\) −17.0000 −0.611843
\(773\) −19.0000 −0.683383 −0.341691 0.939812i \(-0.611000\pi\)
−0.341691 + 0.939812i \(0.611000\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 23.0000 0.819341
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −5.00000 −0.177220
\(797\) 49.0000 1.73567 0.867835 0.496853i \(-0.165511\pi\)
0.867835 + 0.496853i \(0.165511\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) 32.0000 1.12926
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) −18.0000 −0.634023
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) 16.0000 0.562530 0.281265 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) 33.0000 1.15879 0.579393 0.815048i \(-0.303290\pi\)
0.579393 + 0.815048i \(0.303290\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) −36.0000 −1.26180
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) 0 0
\(818\) −16.0000 −0.559427
\(819\) 0 0
\(820\) 27.0000 0.942881
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 7.00000 0.244005 0.122002 0.992530i \(-0.461068\pi\)
0.122002 + 0.992530i \(0.461068\pi\)
\(824\) −5.00000 −0.174183
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −3.00000 −0.104006
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −60.0000 −2.07639
\(836\) 0 0
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 21.0000 0.723708
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) 9.00000 0.308516
\(852\) 0 0
\(853\) 7.00000 0.239675 0.119838 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 9.00000 0.306897
\(861\) 0 0
\(862\) 21.0000 0.715263
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −23.0000 −0.781572
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −3.00000 −0.101593
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −36.0000 −1.21563 −0.607817 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) 10.0000 0.337484
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 54.0000 1.81724 0.908622 0.417619i \(-0.137135\pi\)
0.908622 + 0.417619i \(0.137135\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) 42.0000 1.40784
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 0 0
\(895\) 57.0000 1.90530
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) −11.0000 −0.365048
\(909\) 0 0
\(910\) 9.00000 0.298347
\(911\) 49.0000 1.62344 0.811721 0.584045i \(-0.198531\pi\)
0.811721 + 0.584045i \(0.198531\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −22.0000 −0.724531
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) −36.0000 −1.18367
\(926\) −13.0000 −0.427207
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −9.00000 −0.295280 −0.147640 0.989041i \(-0.547168\pi\)
−0.147640 + 0.989041i \(0.547168\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 13.0000 0.425373
\(935\) 48.0000 1.56977
\(936\) 0 0
\(937\) 25.0000 0.816714 0.408357 0.912822i \(-0.366102\pi\)
0.408357 + 0.912822i \(0.366102\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −21.0000 −0.684944
\(941\) −3.00000 −0.0977972 −0.0488986 0.998804i \(-0.515571\pi\)
−0.0488986 + 0.998804i \(0.515571\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 31.0000 1.00736 0.503682 0.863889i \(-0.331978\pi\)
0.503682 + 0.863889i \(0.331978\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −36.0000 −1.16493
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −27.0000 −0.870515
\(963\) 0 0
\(964\) −5.00000 −0.161039
\(965\) 51.0000 1.64175
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −21.0000 −0.674269
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) −13.0000 −0.416547
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 0 0
\(985\) −69.0000 −2.19852
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 15.0000 0.475532
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −34.0000 −1.07625
\(999\) 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.a.1.1 1
3.2 odd 2 966.2.a.k.1.1 1
12.11 even 2 7728.2.a.j.1.1 1
21.20 even 2 6762.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.k.1.1 1 3.2 odd 2
2898.2.a.a.1.1 1 1.1 even 1 trivial
6762.2.a.y.1.1 1 21.20 even 2
7728.2.a.j.1.1 1 12.11 even 2